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Nikos SKANTZOS 2010 Stochastic methods in Finance

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Nikos SKANTZOS 2010. Stochastic methods in Finance. What is the fair price of an option? Consider a call option on an asset. Delta hedging: For every time interval: Buy/Sell the asset to make the position: Call – Spot * nbr Assets insensitive to variations of the Spot - PowerPoint PPT Presentation

TRANSCRIPT

Page 1: Nikos SKANTZOS 2010

Nikos SKANTZOS

2010

Stochastic methods in Finance

Fair price What is the fair price of an

option Consider a call option on an

asset

Delta hedging For every time interval BuySell the asset to

make the positionCall ndash Spot nbr Assetsinsensitive to variations of the Spot

Fair price is the amount spent during delta-hedgingOption Price ==Δ1+ Δ2+ Δ3 + Δ4+ Δ5

It is fair because that is how much we spent

Δ1 Δ2 Δ3 Δ4 Δ5

Black-Scholes the mother model Black-Scholes based option-pricing on

no-arbitrage amp delta hedging

Previously pricing was based mainly on intuition and risk-based calculations

Fair value of securities was unknown

Black-Scholes main ideasAssume a Spot Dynamics

The rule for updating the spot has two terms

Drift the spot follows a main trend

Vol the spot fluctuates around the main trend

Black-Scholes assume as update rule

This process is a ldquolognormalrdquo process

ldquolognormalrdquo means that drift and fluctuations are proportional to S t

nsfluctuatiomain trend

tttttt WStSSS

tttt dWSdtSdS

Tttt SSSSS 10

σ size of fluctuations

μ steepness of main trend

ΔWt random variable (posneg)

Black-Scholes assumptions No-arbitrage drift = risk-free rate

Impose no-arbitrage by requiring thatexpected spot = market forward

Calculations simplify if fluctuations are normal

is Gaussian normal of zero mean variance ~ T

Volatility (size of fluctuations) is assumed constant

Risk-free rate is assumed constant

No-transaction costs underlying is liquid etc

tW

Black-Scholes formula Call option = endashrT ∙ E[ max(S(T)-K0)]

= discounted average of the call-payoff over various realizations of final spot

Solution

)()( 2121 dNKedNSeC TrTr

T

TrrKS

d

212

1

21

ln

T

TrrKS

d

212

2

21

ln

Interpretation of BS formula

money in the finishes

spot y that probabilit

2

long go toshares ofnumber option theof Delta

1 )()( 21 dNKedNeSC TrTr

Price = value of position at maturity ndash value of cash-flow at maturity

How much does the portfolio value change when spot changes

Δ=0 Delta-neutral value

if S S+dS then portfolio value does not change

Vega=0 Vega-neutral value

if σ σ+dσ then portfolio value does not change

ldquoGreeksrdquo measure sensitivity of portfolio value

portfolio value

S

Delta neutral position S

S+dSpartPortfoliopartS=0

Black-Scholes vs market Comparison with market

BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility

smile Inverse calculation ldquoimplied volrdquo Call on EURUSD

0

10000

20000

30000

40000

50000

60000

70000

80000

12000 12500 13000 13500 14000 14500 15000 15500

strike

USD

cas

h

Black-Scholes

Market

Smile

1300

1350

1400

1450

1500

12000 12500 13000 13500 14000 14500 15000 15500 16000

Strike

Vola

tility

Black-Scholes

Market

Spot probability density Distribution of terminal spot

(given initial spot) obtained from

Fat tails

Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes

Main causes

bullSpot dynamics is not lognormal

bullSpot fluctuations (vol) are not constant

2

mkt2

0

Call2

KeSSP Tr

T

Market observable

What information does the smile give

It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price

It is not the volatility of the spot dynamics

It does not give any information about the spot dynamics even if we combine smiles of various tenors

Therefore it cannot be used (directly) to price path-dependent options

The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)

If there was an instantaneous volatility σ(t) the BS could be interpreted as

dtT

T

t

22BS

1 the accumulated vol

Types of smile quotes The smile is a static representation of the

implied volatilities at a given moment of time

What if the spot changes

Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change

Sticky strike if spot changes implied vol of a given strike doesnrsquot change

Moneyness Δ=DF1N(d1)

Spotladders price delta amp gamma

Vanilla

Knock-out spot=128 strike=125 barrier=15

0051

152

253

35

08 1 12 14 16 18

spot

gamma

06y

1y

0

01

02

03

04

05

06

08 1 12 14 16 18

spot

price

06y

1y

0

02

04

06

08

1

12

08 1 12 14 16 18

spot

delta

06y

1y

Linear regime S-K

1 underlying is needed to hedge

Sensitivity of Delta to spot is maximum

0000050001

000150002

000250003

000350004

00045

08 09 1 11 12 13 14

spot

price

06y

1y

-006

-005

-004

-003

-002

-001

0

001

002

003

08 09 1 11 12 13spot

delta

06y

1y

-06

-04

-02

0

02

04

08 1 12spot

gamma

06y

1y

Spot is far from barrier and far from OTM risk is minimum price is maximum

Δlt0 price gets smaller if spot increases

Spotladders vega vanna amp volga Vanilla

Knock-out spot=128 strike=125 barrier=135

0

000001

000002

000003

000004

000005

000006

08 1 12 14 16 18

spot

vega

06y

1y

-2

-1

0

1

2

3

08 1 12 14 16 18

spotvanna

06y

1y

0051

152

253

35

08 1 12 14 16 18

spot

volga

06y

1y

-0000008

-0000006

-0000004

-0000002

0

0000002

0000004

08 1 12

spot

vega

06y

1y

-1

-05

0

05

1

15

08 09 1 11 12 13

spot

vanna

06y

1y

-1

-05

0

05

1

15

2

08 1 12spot

volga

06y

1y

Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 2: Nikos SKANTZOS 2010

Fair price What is the fair price of an

option Consider a call option on an

asset

Delta hedging For every time interval BuySell the asset to

make the positionCall ndash Spot nbr Assetsinsensitive to variations of the Spot

Fair price is the amount spent during delta-hedgingOption Price ==Δ1+ Δ2+ Δ3 + Δ4+ Δ5

It is fair because that is how much we spent

Δ1 Δ2 Δ3 Δ4 Δ5

Black-Scholes the mother model Black-Scholes based option-pricing on

no-arbitrage amp delta hedging

Previously pricing was based mainly on intuition and risk-based calculations

Fair value of securities was unknown

Black-Scholes main ideasAssume a Spot Dynamics

The rule for updating the spot has two terms

Drift the spot follows a main trend

Vol the spot fluctuates around the main trend

Black-Scholes assume as update rule

This process is a ldquolognormalrdquo process

ldquolognormalrdquo means that drift and fluctuations are proportional to S t

nsfluctuatiomain trend

tttttt WStSSS

tttt dWSdtSdS

Tttt SSSSS 10

σ size of fluctuations

μ steepness of main trend

ΔWt random variable (posneg)

Black-Scholes assumptions No-arbitrage drift = risk-free rate

Impose no-arbitrage by requiring thatexpected spot = market forward

Calculations simplify if fluctuations are normal

is Gaussian normal of zero mean variance ~ T

Volatility (size of fluctuations) is assumed constant

Risk-free rate is assumed constant

No-transaction costs underlying is liquid etc

tW

Black-Scholes formula Call option = endashrT ∙ E[ max(S(T)-K0)]

= discounted average of the call-payoff over various realizations of final spot

Solution

)()( 2121 dNKedNSeC TrTr

T

TrrKS

d

212

1

21

ln

T

TrrKS

d

212

2

21

ln

Interpretation of BS formula

money in the finishes

spot y that probabilit

2

long go toshares ofnumber option theof Delta

1 )()( 21 dNKedNeSC TrTr

Price = value of position at maturity ndash value of cash-flow at maturity

How much does the portfolio value change when spot changes

Δ=0 Delta-neutral value

if S S+dS then portfolio value does not change

Vega=0 Vega-neutral value

if σ σ+dσ then portfolio value does not change

ldquoGreeksrdquo measure sensitivity of portfolio value

portfolio value

S

Delta neutral position S

S+dSpartPortfoliopartS=0

Black-Scholes vs market Comparison with market

BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility

smile Inverse calculation ldquoimplied volrdquo Call on EURUSD

0

10000

20000

30000

40000

50000

60000

70000

80000

12000 12500 13000 13500 14000 14500 15000 15500

strike

USD

cas

h

Black-Scholes

Market

Smile

1300

1350

1400

1450

1500

12000 12500 13000 13500 14000 14500 15000 15500 16000

Strike

Vola

tility

Black-Scholes

Market

Spot probability density Distribution of terminal spot

(given initial spot) obtained from

Fat tails

Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes

Main causes

bullSpot dynamics is not lognormal

bullSpot fluctuations (vol) are not constant

2

mkt2

0

Call2

KeSSP Tr

T

Market observable

What information does the smile give

It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price

It is not the volatility of the spot dynamics

It does not give any information about the spot dynamics even if we combine smiles of various tenors

Therefore it cannot be used (directly) to price path-dependent options

The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)

If there was an instantaneous volatility σ(t) the BS could be interpreted as

dtT

T

t

22BS

1 the accumulated vol

Types of smile quotes The smile is a static representation of the

implied volatilities at a given moment of time

What if the spot changes

Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change

Sticky strike if spot changes implied vol of a given strike doesnrsquot change

Moneyness Δ=DF1N(d1)

Spotladders price delta amp gamma

Vanilla

Knock-out spot=128 strike=125 barrier=15

0051

152

253

35

08 1 12 14 16 18

spot

gamma

06y

1y

0

01

02

03

04

05

06

08 1 12 14 16 18

spot

price

06y

1y

0

02

04

06

08

1

12

08 1 12 14 16 18

spot

delta

06y

1y

Linear regime S-K

1 underlying is needed to hedge

Sensitivity of Delta to spot is maximum

0000050001

000150002

000250003

000350004

00045

08 09 1 11 12 13 14

spot

price

06y

1y

-006

-005

-004

-003

-002

-001

0

001

002

003

08 09 1 11 12 13spot

delta

06y

1y

-06

-04

-02

0

02

04

08 1 12spot

gamma

06y

1y

Spot is far from barrier and far from OTM risk is minimum price is maximum

Δlt0 price gets smaller if spot increases

Spotladders vega vanna amp volga Vanilla

Knock-out spot=128 strike=125 barrier=135

0

000001

000002

000003

000004

000005

000006

08 1 12 14 16 18

spot

vega

06y

1y

-2

-1

0

1

2

3

08 1 12 14 16 18

spotvanna

06y

1y

0051

152

253

35

08 1 12 14 16 18

spot

volga

06y

1y

-0000008

-0000006

-0000004

-0000002

0

0000002

0000004

08 1 12

spot

vega

06y

1y

-1

-05

0

05

1

15

08 09 1 11 12 13

spot

vanna

06y

1y

-1

-05

0

05

1

15

2

08 1 12spot

volga

06y

1y

Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 3: Nikos SKANTZOS 2010

Black-Scholes the mother model Black-Scholes based option-pricing on

no-arbitrage amp delta hedging

Previously pricing was based mainly on intuition and risk-based calculations

Fair value of securities was unknown

Black-Scholes main ideasAssume a Spot Dynamics

The rule for updating the spot has two terms

Drift the spot follows a main trend

Vol the spot fluctuates around the main trend

Black-Scholes assume as update rule

This process is a ldquolognormalrdquo process

ldquolognormalrdquo means that drift and fluctuations are proportional to S t

nsfluctuatiomain trend

tttttt WStSSS

tttt dWSdtSdS

Tttt SSSSS 10

σ size of fluctuations

μ steepness of main trend

ΔWt random variable (posneg)

Black-Scholes assumptions No-arbitrage drift = risk-free rate

Impose no-arbitrage by requiring thatexpected spot = market forward

Calculations simplify if fluctuations are normal

is Gaussian normal of zero mean variance ~ T

Volatility (size of fluctuations) is assumed constant

Risk-free rate is assumed constant

No-transaction costs underlying is liquid etc

tW

Black-Scholes formula Call option = endashrT ∙ E[ max(S(T)-K0)]

= discounted average of the call-payoff over various realizations of final spot

Solution

)()( 2121 dNKedNSeC TrTr

T

TrrKS

d

212

1

21

ln

T

TrrKS

d

212

2

21

ln

Interpretation of BS formula

money in the finishes

spot y that probabilit

2

long go toshares ofnumber option theof Delta

1 )()( 21 dNKedNeSC TrTr

Price = value of position at maturity ndash value of cash-flow at maturity

How much does the portfolio value change when spot changes

Δ=0 Delta-neutral value

if S S+dS then portfolio value does not change

Vega=0 Vega-neutral value

if σ σ+dσ then portfolio value does not change

ldquoGreeksrdquo measure sensitivity of portfolio value

portfolio value

S

Delta neutral position S

S+dSpartPortfoliopartS=0

Black-Scholes vs market Comparison with market

BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility

smile Inverse calculation ldquoimplied volrdquo Call on EURUSD

0

10000

20000

30000

40000

50000

60000

70000

80000

12000 12500 13000 13500 14000 14500 15000 15500

strike

USD

cas

h

Black-Scholes

Market

Smile

1300

1350

1400

1450

1500

12000 12500 13000 13500 14000 14500 15000 15500 16000

Strike

Vola

tility

Black-Scholes

Market

Spot probability density Distribution of terminal spot

(given initial spot) obtained from

Fat tails

Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes

Main causes

bullSpot dynamics is not lognormal

bullSpot fluctuations (vol) are not constant

2

mkt2

0

Call2

KeSSP Tr

T

Market observable

What information does the smile give

It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price

It is not the volatility of the spot dynamics

It does not give any information about the spot dynamics even if we combine smiles of various tenors

Therefore it cannot be used (directly) to price path-dependent options

The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)

If there was an instantaneous volatility σ(t) the BS could be interpreted as

dtT

T

t

22BS

1 the accumulated vol

Types of smile quotes The smile is a static representation of the

implied volatilities at a given moment of time

What if the spot changes

Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change

Sticky strike if spot changes implied vol of a given strike doesnrsquot change

Moneyness Δ=DF1N(d1)

Spotladders price delta amp gamma

Vanilla

Knock-out spot=128 strike=125 barrier=15

0051

152

253

35

08 1 12 14 16 18

spot

gamma

06y

1y

0

01

02

03

04

05

06

08 1 12 14 16 18

spot

price

06y

1y

0

02

04

06

08

1

12

08 1 12 14 16 18

spot

delta

06y

1y

Linear regime S-K

1 underlying is needed to hedge

Sensitivity of Delta to spot is maximum

0000050001

000150002

000250003

000350004

00045

08 09 1 11 12 13 14

spot

price

06y

1y

-006

-005

-004

-003

-002

-001

0

001

002

003

08 09 1 11 12 13spot

delta

06y

1y

-06

-04

-02

0

02

04

08 1 12spot

gamma

06y

1y

Spot is far from barrier and far from OTM risk is minimum price is maximum

Δlt0 price gets smaller if spot increases

Spotladders vega vanna amp volga Vanilla

Knock-out spot=128 strike=125 barrier=135

0

000001

000002

000003

000004

000005

000006

08 1 12 14 16 18

spot

vega

06y

1y

-2

-1

0

1

2

3

08 1 12 14 16 18

spotvanna

06y

1y

0051

152

253

35

08 1 12 14 16 18

spot

volga

06y

1y

-0000008

-0000006

-0000004

-0000002

0

0000002

0000004

08 1 12

spot

vega

06y

1y

-1

-05

0

05

1

15

08 09 1 11 12 13

spot

vanna

06y

1y

-1

-05

0

05

1

15

2

08 1 12spot

volga

06y

1y

Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 4: Nikos SKANTZOS 2010

Black-Scholes main ideasAssume a Spot Dynamics

The rule for updating the spot has two terms

Drift the spot follows a main trend

Vol the spot fluctuates around the main trend

Black-Scholes assume as update rule

This process is a ldquolognormalrdquo process

ldquolognormalrdquo means that drift and fluctuations are proportional to S t

nsfluctuatiomain trend

tttttt WStSSS

tttt dWSdtSdS

Tttt SSSSS 10

σ size of fluctuations

μ steepness of main trend

ΔWt random variable (posneg)

Black-Scholes assumptions No-arbitrage drift = risk-free rate

Impose no-arbitrage by requiring thatexpected spot = market forward

Calculations simplify if fluctuations are normal

is Gaussian normal of zero mean variance ~ T

Volatility (size of fluctuations) is assumed constant

Risk-free rate is assumed constant

No-transaction costs underlying is liquid etc

tW

Black-Scholes formula Call option = endashrT ∙ E[ max(S(T)-K0)]

= discounted average of the call-payoff over various realizations of final spot

Solution

)()( 2121 dNKedNSeC TrTr

T

TrrKS

d

212

1

21

ln

T

TrrKS

d

212

2

21

ln

Interpretation of BS formula

money in the finishes

spot y that probabilit

2

long go toshares ofnumber option theof Delta

1 )()( 21 dNKedNeSC TrTr

Price = value of position at maturity ndash value of cash-flow at maturity

How much does the portfolio value change when spot changes

Δ=0 Delta-neutral value

if S S+dS then portfolio value does not change

Vega=0 Vega-neutral value

if σ σ+dσ then portfolio value does not change

ldquoGreeksrdquo measure sensitivity of portfolio value

portfolio value

S

Delta neutral position S

S+dSpartPortfoliopartS=0

Black-Scholes vs market Comparison with market

BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility

smile Inverse calculation ldquoimplied volrdquo Call on EURUSD

0

10000

20000

30000

40000

50000

60000

70000

80000

12000 12500 13000 13500 14000 14500 15000 15500

strike

USD

cas

h

Black-Scholes

Market

Smile

1300

1350

1400

1450

1500

12000 12500 13000 13500 14000 14500 15000 15500 16000

Strike

Vola

tility

Black-Scholes

Market

Spot probability density Distribution of terminal spot

(given initial spot) obtained from

Fat tails

Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes

Main causes

bullSpot dynamics is not lognormal

bullSpot fluctuations (vol) are not constant

2

mkt2

0

Call2

KeSSP Tr

T

Market observable

What information does the smile give

It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price

It is not the volatility of the spot dynamics

It does not give any information about the spot dynamics even if we combine smiles of various tenors

Therefore it cannot be used (directly) to price path-dependent options

The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)

If there was an instantaneous volatility σ(t) the BS could be interpreted as

dtT

T

t

22BS

1 the accumulated vol

Types of smile quotes The smile is a static representation of the

implied volatilities at a given moment of time

What if the spot changes

Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change

Sticky strike if spot changes implied vol of a given strike doesnrsquot change

Moneyness Δ=DF1N(d1)

Spotladders price delta amp gamma

Vanilla

Knock-out spot=128 strike=125 barrier=15

0051

152

253

35

08 1 12 14 16 18

spot

gamma

06y

1y

0

01

02

03

04

05

06

08 1 12 14 16 18

spot

price

06y

1y

0

02

04

06

08

1

12

08 1 12 14 16 18

spot

delta

06y

1y

Linear regime S-K

1 underlying is needed to hedge

Sensitivity of Delta to spot is maximum

0000050001

000150002

000250003

000350004

00045

08 09 1 11 12 13 14

spot

price

06y

1y

-006

-005

-004

-003

-002

-001

0

001

002

003

08 09 1 11 12 13spot

delta

06y

1y

-06

-04

-02

0

02

04

08 1 12spot

gamma

06y

1y

Spot is far from barrier and far from OTM risk is minimum price is maximum

Δlt0 price gets smaller if spot increases

Spotladders vega vanna amp volga Vanilla

Knock-out spot=128 strike=125 barrier=135

0

000001

000002

000003

000004

000005

000006

08 1 12 14 16 18

spot

vega

06y

1y

-2

-1

0

1

2

3

08 1 12 14 16 18

spotvanna

06y

1y

0051

152

253

35

08 1 12 14 16 18

spot

volga

06y

1y

-0000008

-0000006

-0000004

-0000002

0

0000002

0000004

08 1 12

spot

vega

06y

1y

-1

-05

0

05

1

15

08 09 1 11 12 13

spot

vanna

06y

1y

-1

-05

0

05

1

15

2

08 1 12spot

volga

06y

1y

Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 5: Nikos SKANTZOS 2010

Black-Scholes assumptions No-arbitrage drift = risk-free rate

Impose no-arbitrage by requiring thatexpected spot = market forward

Calculations simplify if fluctuations are normal

is Gaussian normal of zero mean variance ~ T

Volatility (size of fluctuations) is assumed constant

Risk-free rate is assumed constant

No-transaction costs underlying is liquid etc

tW

Black-Scholes formula Call option = endashrT ∙ E[ max(S(T)-K0)]

= discounted average of the call-payoff over various realizations of final spot

Solution

)()( 2121 dNKedNSeC TrTr

T

TrrKS

d

212

1

21

ln

T

TrrKS

d

212

2

21

ln

Interpretation of BS formula

money in the finishes

spot y that probabilit

2

long go toshares ofnumber option theof Delta

1 )()( 21 dNKedNeSC TrTr

Price = value of position at maturity ndash value of cash-flow at maturity

How much does the portfolio value change when spot changes

Δ=0 Delta-neutral value

if S S+dS then portfolio value does not change

Vega=0 Vega-neutral value

if σ σ+dσ then portfolio value does not change

ldquoGreeksrdquo measure sensitivity of portfolio value

portfolio value

S

Delta neutral position S

S+dSpartPortfoliopartS=0

Black-Scholes vs market Comparison with market

BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility

smile Inverse calculation ldquoimplied volrdquo Call on EURUSD

0

10000

20000

30000

40000

50000

60000

70000

80000

12000 12500 13000 13500 14000 14500 15000 15500

strike

USD

cas

h

Black-Scholes

Market

Smile

1300

1350

1400

1450

1500

12000 12500 13000 13500 14000 14500 15000 15500 16000

Strike

Vola

tility

Black-Scholes

Market

Spot probability density Distribution of terminal spot

(given initial spot) obtained from

Fat tails

Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes

Main causes

bullSpot dynamics is not lognormal

bullSpot fluctuations (vol) are not constant

2

mkt2

0

Call2

KeSSP Tr

T

Market observable

What information does the smile give

It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price

It is not the volatility of the spot dynamics

It does not give any information about the spot dynamics even if we combine smiles of various tenors

Therefore it cannot be used (directly) to price path-dependent options

The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)

If there was an instantaneous volatility σ(t) the BS could be interpreted as

dtT

T

t

22BS

1 the accumulated vol

Types of smile quotes The smile is a static representation of the

implied volatilities at a given moment of time

What if the spot changes

Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change

Sticky strike if spot changes implied vol of a given strike doesnrsquot change

Moneyness Δ=DF1N(d1)

Spotladders price delta amp gamma

Vanilla

Knock-out spot=128 strike=125 barrier=15

0051

152

253

35

08 1 12 14 16 18

spot

gamma

06y

1y

0

01

02

03

04

05

06

08 1 12 14 16 18

spot

price

06y

1y

0

02

04

06

08

1

12

08 1 12 14 16 18

spot

delta

06y

1y

Linear regime S-K

1 underlying is needed to hedge

Sensitivity of Delta to spot is maximum

0000050001

000150002

000250003

000350004

00045

08 09 1 11 12 13 14

spot

price

06y

1y

-006

-005

-004

-003

-002

-001

0

001

002

003

08 09 1 11 12 13spot

delta

06y

1y

-06

-04

-02

0

02

04

08 1 12spot

gamma

06y

1y

Spot is far from barrier and far from OTM risk is minimum price is maximum

Δlt0 price gets smaller if spot increases

Spotladders vega vanna amp volga Vanilla

Knock-out spot=128 strike=125 barrier=135

0

000001

000002

000003

000004

000005

000006

08 1 12 14 16 18

spot

vega

06y

1y

-2

-1

0

1

2

3

08 1 12 14 16 18

spotvanna

06y

1y

0051

152

253

35

08 1 12 14 16 18

spot

volga

06y

1y

-0000008

-0000006

-0000004

-0000002

0

0000002

0000004

08 1 12

spot

vega

06y

1y

-1

-05

0

05

1

15

08 09 1 11 12 13

spot

vanna

06y

1y

-1

-05

0

05

1

15

2

08 1 12spot

volga

06y

1y

Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 6: Nikos SKANTZOS 2010

Black-Scholes formula Call option = endashrT ∙ E[ max(S(T)-K0)]

= discounted average of the call-payoff over various realizations of final spot

Solution

)()( 2121 dNKedNSeC TrTr

T

TrrKS

d

212

1

21

ln

T

TrrKS

d

212

2

21

ln

Interpretation of BS formula

money in the finishes

spot y that probabilit

2

long go toshares ofnumber option theof Delta

1 )()( 21 dNKedNeSC TrTr

Price = value of position at maturity ndash value of cash-flow at maturity

How much does the portfolio value change when spot changes

Δ=0 Delta-neutral value

if S S+dS then portfolio value does not change

Vega=0 Vega-neutral value

if σ σ+dσ then portfolio value does not change

ldquoGreeksrdquo measure sensitivity of portfolio value

portfolio value

S

Delta neutral position S

S+dSpartPortfoliopartS=0

Black-Scholes vs market Comparison with market

BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility

smile Inverse calculation ldquoimplied volrdquo Call on EURUSD

0

10000

20000

30000

40000

50000

60000

70000

80000

12000 12500 13000 13500 14000 14500 15000 15500

strike

USD

cas

h

Black-Scholes

Market

Smile

1300

1350

1400

1450

1500

12000 12500 13000 13500 14000 14500 15000 15500 16000

Strike

Vola

tility

Black-Scholes

Market

Spot probability density Distribution of terminal spot

(given initial spot) obtained from

Fat tails

Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes

Main causes

bullSpot dynamics is not lognormal

bullSpot fluctuations (vol) are not constant

2

mkt2

0

Call2

KeSSP Tr

T

Market observable

What information does the smile give

It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price

It is not the volatility of the spot dynamics

It does not give any information about the spot dynamics even if we combine smiles of various tenors

Therefore it cannot be used (directly) to price path-dependent options

The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)

If there was an instantaneous volatility σ(t) the BS could be interpreted as

dtT

T

t

22BS

1 the accumulated vol

Types of smile quotes The smile is a static representation of the

implied volatilities at a given moment of time

What if the spot changes

Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change

Sticky strike if spot changes implied vol of a given strike doesnrsquot change

Moneyness Δ=DF1N(d1)

Spotladders price delta amp gamma

Vanilla

Knock-out spot=128 strike=125 barrier=15

0051

152

253

35

08 1 12 14 16 18

spot

gamma

06y

1y

0

01

02

03

04

05

06

08 1 12 14 16 18

spot

price

06y

1y

0

02

04

06

08

1

12

08 1 12 14 16 18

spot

delta

06y

1y

Linear regime S-K

1 underlying is needed to hedge

Sensitivity of Delta to spot is maximum

0000050001

000150002

000250003

000350004

00045

08 09 1 11 12 13 14

spot

price

06y

1y

-006

-005

-004

-003

-002

-001

0

001

002

003

08 09 1 11 12 13spot

delta

06y

1y

-06

-04

-02

0

02

04

08 1 12spot

gamma

06y

1y

Spot is far from barrier and far from OTM risk is minimum price is maximum

Δlt0 price gets smaller if spot increases

Spotladders vega vanna amp volga Vanilla

Knock-out spot=128 strike=125 barrier=135

0

000001

000002

000003

000004

000005

000006

08 1 12 14 16 18

spot

vega

06y

1y

-2

-1

0

1

2

3

08 1 12 14 16 18

spotvanna

06y

1y

0051

152

253

35

08 1 12 14 16 18

spot

volga

06y

1y

-0000008

-0000006

-0000004

-0000002

0

0000002

0000004

08 1 12

spot

vega

06y

1y

-1

-05

0

05

1

15

08 09 1 11 12 13

spot

vanna

06y

1y

-1

-05

0

05

1

15

2

08 1 12spot

volga

06y

1y

Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 7: Nikos SKANTZOS 2010

Interpretation of BS formula

money in the finishes

spot y that probabilit

2

long go toshares ofnumber option theof Delta

1 )()( 21 dNKedNeSC TrTr

Price = value of position at maturity ndash value of cash-flow at maturity

How much does the portfolio value change when spot changes

Δ=0 Delta-neutral value

if S S+dS then portfolio value does not change

Vega=0 Vega-neutral value

if σ σ+dσ then portfolio value does not change

ldquoGreeksrdquo measure sensitivity of portfolio value

portfolio value

S

Delta neutral position S

S+dSpartPortfoliopartS=0

Black-Scholes vs market Comparison with market

BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility

smile Inverse calculation ldquoimplied volrdquo Call on EURUSD

0

10000

20000

30000

40000

50000

60000

70000

80000

12000 12500 13000 13500 14000 14500 15000 15500

strike

USD

cas

h

Black-Scholes

Market

Smile

1300

1350

1400

1450

1500

12000 12500 13000 13500 14000 14500 15000 15500 16000

Strike

Vola

tility

Black-Scholes

Market

Spot probability density Distribution of terminal spot

(given initial spot) obtained from

Fat tails

Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes

Main causes

bullSpot dynamics is not lognormal

bullSpot fluctuations (vol) are not constant

2

mkt2

0

Call2

KeSSP Tr

T

Market observable

What information does the smile give

It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price

It is not the volatility of the spot dynamics

It does not give any information about the spot dynamics even if we combine smiles of various tenors

Therefore it cannot be used (directly) to price path-dependent options

The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)

If there was an instantaneous volatility σ(t) the BS could be interpreted as

dtT

T

t

22BS

1 the accumulated vol

Types of smile quotes The smile is a static representation of the

implied volatilities at a given moment of time

What if the spot changes

Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change

Sticky strike if spot changes implied vol of a given strike doesnrsquot change

Moneyness Δ=DF1N(d1)

Spotladders price delta amp gamma

Vanilla

Knock-out spot=128 strike=125 barrier=15

0051

152

253

35

08 1 12 14 16 18

spot

gamma

06y

1y

0

01

02

03

04

05

06

08 1 12 14 16 18

spot

price

06y

1y

0

02

04

06

08

1

12

08 1 12 14 16 18

spot

delta

06y

1y

Linear regime S-K

1 underlying is needed to hedge

Sensitivity of Delta to spot is maximum

0000050001

000150002

000250003

000350004

00045

08 09 1 11 12 13 14

spot

price

06y

1y

-006

-005

-004

-003

-002

-001

0

001

002

003

08 09 1 11 12 13spot

delta

06y

1y

-06

-04

-02

0

02

04

08 1 12spot

gamma

06y

1y

Spot is far from barrier and far from OTM risk is minimum price is maximum

Δlt0 price gets smaller if spot increases

Spotladders vega vanna amp volga Vanilla

Knock-out spot=128 strike=125 barrier=135

0

000001

000002

000003

000004

000005

000006

08 1 12 14 16 18

spot

vega

06y

1y

-2

-1

0

1

2

3

08 1 12 14 16 18

spotvanna

06y

1y

0051

152

253

35

08 1 12 14 16 18

spot

volga

06y

1y

-0000008

-0000006

-0000004

-0000002

0

0000002

0000004

08 1 12

spot

vega

06y

1y

-1

-05

0

05

1

15

08 09 1 11 12 13

spot

vanna

06y

1y

-1

-05

0

05

1

15

2

08 1 12spot

volga

06y

1y

Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 8: Nikos SKANTZOS 2010

How much does the portfolio value change when spot changes

Δ=0 Delta-neutral value

if S S+dS then portfolio value does not change

Vega=0 Vega-neutral value

if σ σ+dσ then portfolio value does not change

ldquoGreeksrdquo measure sensitivity of portfolio value

portfolio value

S

Delta neutral position S

S+dSpartPortfoliopartS=0

Black-Scholes vs market Comparison with market

BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility

smile Inverse calculation ldquoimplied volrdquo Call on EURUSD

0

10000

20000

30000

40000

50000

60000

70000

80000

12000 12500 13000 13500 14000 14500 15000 15500

strike

USD

cas

h

Black-Scholes

Market

Smile

1300

1350

1400

1450

1500

12000 12500 13000 13500 14000 14500 15000 15500 16000

Strike

Vola

tility

Black-Scholes

Market

Spot probability density Distribution of terminal spot

(given initial spot) obtained from

Fat tails

Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes

Main causes

bullSpot dynamics is not lognormal

bullSpot fluctuations (vol) are not constant

2

mkt2

0

Call2

KeSSP Tr

T

Market observable

What information does the smile give

It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price

It is not the volatility of the spot dynamics

It does not give any information about the spot dynamics even if we combine smiles of various tenors

Therefore it cannot be used (directly) to price path-dependent options

The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)

If there was an instantaneous volatility σ(t) the BS could be interpreted as

dtT

T

t

22BS

1 the accumulated vol

Types of smile quotes The smile is a static representation of the

implied volatilities at a given moment of time

What if the spot changes

Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change

Sticky strike if spot changes implied vol of a given strike doesnrsquot change

Moneyness Δ=DF1N(d1)

Spotladders price delta amp gamma

Vanilla

Knock-out spot=128 strike=125 barrier=15

0051

152

253

35

08 1 12 14 16 18

spot

gamma

06y

1y

0

01

02

03

04

05

06

08 1 12 14 16 18

spot

price

06y

1y

0

02

04

06

08

1

12

08 1 12 14 16 18

spot

delta

06y

1y

Linear regime S-K

1 underlying is needed to hedge

Sensitivity of Delta to spot is maximum

0000050001

000150002

000250003

000350004

00045

08 09 1 11 12 13 14

spot

price

06y

1y

-006

-005

-004

-003

-002

-001

0

001

002

003

08 09 1 11 12 13spot

delta

06y

1y

-06

-04

-02

0

02

04

08 1 12spot

gamma

06y

1y

Spot is far from barrier and far from OTM risk is minimum price is maximum

Δlt0 price gets smaller if spot increases

Spotladders vega vanna amp volga Vanilla

Knock-out spot=128 strike=125 barrier=135

0

000001

000002

000003

000004

000005

000006

08 1 12 14 16 18

spot

vega

06y

1y

-2

-1

0

1

2

3

08 1 12 14 16 18

spotvanna

06y

1y

0051

152

253

35

08 1 12 14 16 18

spot

volga

06y

1y

-0000008

-0000006

-0000004

-0000002

0

0000002

0000004

08 1 12

spot

vega

06y

1y

-1

-05

0

05

1

15

08 09 1 11 12 13

spot

vanna

06y

1y

-1

-05

0

05

1

15

2

08 1 12spot

volga

06y

1y

Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 9: Nikos SKANTZOS 2010

Black-Scholes vs market Comparison with market

BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility

smile Inverse calculation ldquoimplied volrdquo Call on EURUSD

0

10000

20000

30000

40000

50000

60000

70000

80000

12000 12500 13000 13500 14000 14500 15000 15500

strike

USD

cas

h

Black-Scholes

Market

Smile

1300

1350

1400

1450

1500

12000 12500 13000 13500 14000 14500 15000 15500 16000

Strike

Vola

tility

Black-Scholes

Market

Spot probability density Distribution of terminal spot

(given initial spot) obtained from

Fat tails

Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes

Main causes

bullSpot dynamics is not lognormal

bullSpot fluctuations (vol) are not constant

2

mkt2

0

Call2

KeSSP Tr

T

Market observable

What information does the smile give

It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price

It is not the volatility of the spot dynamics

It does not give any information about the spot dynamics even if we combine smiles of various tenors

Therefore it cannot be used (directly) to price path-dependent options

The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)

If there was an instantaneous volatility σ(t) the BS could be interpreted as

dtT

T

t

22BS

1 the accumulated vol

Types of smile quotes The smile is a static representation of the

implied volatilities at a given moment of time

What if the spot changes

Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change

Sticky strike if spot changes implied vol of a given strike doesnrsquot change

Moneyness Δ=DF1N(d1)

Spotladders price delta amp gamma

Vanilla

Knock-out spot=128 strike=125 barrier=15

0051

152

253

35

08 1 12 14 16 18

spot

gamma

06y

1y

0

01

02

03

04

05

06

08 1 12 14 16 18

spot

price

06y

1y

0

02

04

06

08

1

12

08 1 12 14 16 18

spot

delta

06y

1y

Linear regime S-K

1 underlying is needed to hedge

Sensitivity of Delta to spot is maximum

0000050001

000150002

000250003

000350004

00045

08 09 1 11 12 13 14

spot

price

06y

1y

-006

-005

-004

-003

-002

-001

0

001

002

003

08 09 1 11 12 13spot

delta

06y

1y

-06

-04

-02

0

02

04

08 1 12spot

gamma

06y

1y

Spot is far from barrier and far from OTM risk is minimum price is maximum

Δlt0 price gets smaller if spot increases

Spotladders vega vanna amp volga Vanilla

Knock-out spot=128 strike=125 barrier=135

0

000001

000002

000003

000004

000005

000006

08 1 12 14 16 18

spot

vega

06y

1y

-2

-1

0

1

2

3

08 1 12 14 16 18

spotvanna

06y

1y

0051

152

253

35

08 1 12 14 16 18

spot

volga

06y

1y

-0000008

-0000006

-0000004

-0000002

0

0000002

0000004

08 1 12

spot

vega

06y

1y

-1

-05

0

05

1

15

08 09 1 11 12 13

spot

vanna

06y

1y

-1

-05

0

05

1

15

2

08 1 12spot

volga

06y

1y

Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 10: Nikos SKANTZOS 2010

Spot probability density Distribution of terminal spot

(given initial spot) obtained from

Fat tails

Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes

Main causes

bullSpot dynamics is not lognormal

bullSpot fluctuations (vol) are not constant

2

mkt2

0

Call2

KeSSP Tr

T

Market observable

What information does the smile give

It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price

It is not the volatility of the spot dynamics

It does not give any information about the spot dynamics even if we combine smiles of various tenors

Therefore it cannot be used (directly) to price path-dependent options

The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)

If there was an instantaneous volatility σ(t) the BS could be interpreted as

dtT

T

t

22BS

1 the accumulated vol

Types of smile quotes The smile is a static representation of the

implied volatilities at a given moment of time

What if the spot changes

Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change

Sticky strike if spot changes implied vol of a given strike doesnrsquot change

Moneyness Δ=DF1N(d1)

Spotladders price delta amp gamma

Vanilla

Knock-out spot=128 strike=125 barrier=15

0051

152

253

35

08 1 12 14 16 18

spot

gamma

06y

1y

0

01

02

03

04

05

06

08 1 12 14 16 18

spot

price

06y

1y

0

02

04

06

08

1

12

08 1 12 14 16 18

spot

delta

06y

1y

Linear regime S-K

1 underlying is needed to hedge

Sensitivity of Delta to spot is maximum

0000050001

000150002

000250003

000350004

00045

08 09 1 11 12 13 14

spot

price

06y

1y

-006

-005

-004

-003

-002

-001

0

001

002

003

08 09 1 11 12 13spot

delta

06y

1y

-06

-04

-02

0

02

04

08 1 12spot

gamma

06y

1y

Spot is far from barrier and far from OTM risk is minimum price is maximum

Δlt0 price gets smaller if spot increases

Spotladders vega vanna amp volga Vanilla

Knock-out spot=128 strike=125 barrier=135

0

000001

000002

000003

000004

000005

000006

08 1 12 14 16 18

spot

vega

06y

1y

-2

-1

0

1

2

3

08 1 12 14 16 18

spotvanna

06y

1y

0051

152

253

35

08 1 12 14 16 18

spot

volga

06y

1y

-0000008

-0000006

-0000004

-0000002

0

0000002

0000004

08 1 12

spot

vega

06y

1y

-1

-05

0

05

1

15

08 09 1 11 12 13

spot

vanna

06y

1y

-1

-05

0

05

1

15

2

08 1 12spot

volga

06y

1y

Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 11: Nikos SKANTZOS 2010

What information does the smile give

It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price

It is not the volatility of the spot dynamics

It does not give any information about the spot dynamics even if we combine smiles of various tenors

Therefore it cannot be used (directly) to price path-dependent options

The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)

If there was an instantaneous volatility σ(t) the BS could be interpreted as

dtT

T

t

22BS

1 the accumulated vol

Types of smile quotes The smile is a static representation of the

implied volatilities at a given moment of time

What if the spot changes

Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change

Sticky strike if spot changes implied vol of a given strike doesnrsquot change

Moneyness Δ=DF1N(d1)

Spotladders price delta amp gamma

Vanilla

Knock-out spot=128 strike=125 barrier=15

0051

152

253

35

08 1 12 14 16 18

spot

gamma

06y

1y

0

01

02

03

04

05

06

08 1 12 14 16 18

spot

price

06y

1y

0

02

04

06

08

1

12

08 1 12 14 16 18

spot

delta

06y

1y

Linear regime S-K

1 underlying is needed to hedge

Sensitivity of Delta to spot is maximum

0000050001

000150002

000250003

000350004

00045

08 09 1 11 12 13 14

spot

price

06y

1y

-006

-005

-004

-003

-002

-001

0

001

002

003

08 09 1 11 12 13spot

delta

06y

1y

-06

-04

-02

0

02

04

08 1 12spot

gamma

06y

1y

Spot is far from barrier and far from OTM risk is minimum price is maximum

Δlt0 price gets smaller if spot increases

Spotladders vega vanna amp volga Vanilla

Knock-out spot=128 strike=125 barrier=135

0

000001

000002

000003

000004

000005

000006

08 1 12 14 16 18

spot

vega

06y

1y

-2

-1

0

1

2

3

08 1 12 14 16 18

spotvanna

06y

1y

0051

152

253

35

08 1 12 14 16 18

spot

volga

06y

1y

-0000008

-0000006

-0000004

-0000002

0

0000002

0000004

08 1 12

spot

vega

06y

1y

-1

-05

0

05

1

15

08 09 1 11 12 13

spot

vanna

06y

1y

-1

-05

0

05

1

15

2

08 1 12spot

volga

06y

1y

Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 12: Nikos SKANTZOS 2010

Types of smile quotes The smile is a static representation of the

implied volatilities at a given moment of time

What if the spot changes

Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change

Sticky strike if spot changes implied vol of a given strike doesnrsquot change

Moneyness Δ=DF1N(d1)

Spotladders price delta amp gamma

Vanilla

Knock-out spot=128 strike=125 barrier=15

0051

152

253

35

08 1 12 14 16 18

spot

gamma

06y

1y

0

01

02

03

04

05

06

08 1 12 14 16 18

spot

price

06y

1y

0

02

04

06

08

1

12

08 1 12 14 16 18

spot

delta

06y

1y

Linear regime S-K

1 underlying is needed to hedge

Sensitivity of Delta to spot is maximum

0000050001

000150002

000250003

000350004

00045

08 09 1 11 12 13 14

spot

price

06y

1y

-006

-005

-004

-003

-002

-001

0

001

002

003

08 09 1 11 12 13spot

delta

06y

1y

-06

-04

-02

0

02

04

08 1 12spot

gamma

06y

1y

Spot is far from barrier and far from OTM risk is minimum price is maximum

Δlt0 price gets smaller if spot increases

Spotladders vega vanna amp volga Vanilla

Knock-out spot=128 strike=125 barrier=135

0

000001

000002

000003

000004

000005

000006

08 1 12 14 16 18

spot

vega

06y

1y

-2

-1

0

1

2

3

08 1 12 14 16 18

spotvanna

06y

1y

0051

152

253

35

08 1 12 14 16 18

spot

volga

06y

1y

-0000008

-0000006

-0000004

-0000002

0

0000002

0000004

08 1 12

spot

vega

06y

1y

-1

-05

0

05

1

15

08 09 1 11 12 13

spot

vanna

06y

1y

-1

-05

0

05

1

15

2

08 1 12spot

volga

06y

1y

Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 13: Nikos SKANTZOS 2010

Spotladders price delta amp gamma

Vanilla

Knock-out spot=128 strike=125 barrier=15

0051

152

253

35

08 1 12 14 16 18

spot

gamma

06y

1y

0

01

02

03

04

05

06

08 1 12 14 16 18

spot

price

06y

1y

0

02

04

06

08

1

12

08 1 12 14 16 18

spot

delta

06y

1y

Linear regime S-K

1 underlying is needed to hedge

Sensitivity of Delta to spot is maximum

0000050001

000150002

000250003

000350004

00045

08 09 1 11 12 13 14

spot

price

06y

1y

-006

-005

-004

-003

-002

-001

0

001

002

003

08 09 1 11 12 13spot

delta

06y

1y

-06

-04

-02

0

02

04

08 1 12spot

gamma

06y

1y

Spot is far from barrier and far from OTM risk is minimum price is maximum

Δlt0 price gets smaller if spot increases

Spotladders vega vanna amp volga Vanilla

Knock-out spot=128 strike=125 barrier=135

0

000001

000002

000003

000004

000005

000006

08 1 12 14 16 18

spot

vega

06y

1y

-2

-1

0

1

2

3

08 1 12 14 16 18

spotvanna

06y

1y

0051

152

253

35

08 1 12 14 16 18

spot

volga

06y

1y

-0000008

-0000006

-0000004

-0000002

0

0000002

0000004

08 1 12

spot

vega

06y

1y

-1

-05

0

05

1

15

08 09 1 11 12 13

spot

vanna

06y

1y

-1

-05

0

05

1

15

2

08 1 12spot

volga

06y

1y

Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 14: Nikos SKANTZOS 2010

Spotladders vega vanna amp volga Vanilla

Knock-out spot=128 strike=125 barrier=135

0

000001

000002

000003

000004

000005

000006

08 1 12 14 16 18

spot

vega

06y

1y

-2

-1

0

1

2

3

08 1 12 14 16 18

spotvanna

06y

1y

0051

152

253

35

08 1 12 14 16 18

spot

volga

06y

1y

-0000008

-0000006

-0000004

-0000002

0

0000002

0000004

08 1 12

spot

vega

06y

1y

-1

-05

0

05

1

15

08 09 1 11 12 13

spot

vanna

06y

1y

-1

-05

0

05

1

15

2

08 1 12spot

volga

06y

1y

Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 15: Nikos SKANTZOS 2010

Simple analytic techniques ldquomoment matchingrdquo

Average-rate option payoff with N fixing dates

Basket option with two underlyings

TV pricing can be achieved quickly via ldquomoment matchingrdquo

Mark-to-market requires correlated stochastic processes for spotsvols (more complex)

01

max Asian 1

KSN

N

ii

0max Basket

2

22

1

11 K

tS

TSa

tS

TSa

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 16: Nikos SKANTZOS 2010

ldquoMoment matchingrdquo To price Asian (average option) in TV we consider

that The spot process is lognormal The sum of all spots is lognormal also

Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)

Central idea of moment matching Find first and second moment of sum of lognormals

E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known

moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 17: Nikos SKANTZOS 2010

Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T

(time to maturity)

E[Wt]=0 E[Wt2]=t

If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1

2] = t1

(because Wt1 is independent of Wt2-Wt1)

More generally E[Wt1∙Wt2] = min(t1t2)

From this and with some algebra it follows that E[St1 ∙ St2] = S0

2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 18: Nikos SKANTZOS 2010

Asian options analytics (2) Asian payoff contains sum of spots

What are its mean (first moment) and variance

Looks complex but on the right-hand side all quantities are known and can be easily calculated

Therefore the first and second moment of the sum of spots can be calculated

N

iiSN

X1

1

N

ji

ttttrN

iji

NN

jj

N

ii

N

i

trN

i

NttrN

ii

N

ii

jiji

iii

eSN

SSN

SSN

X

eSN

eSN

SN

SN

X

1

)(min202

1 1j2

112

2

10

1

)10(0

11

2

221

E1

E11

EE

1E

1E

11EE

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 19: Nikos SKANTZOS 2010

Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ

the drift

Has solution (as in standard Black-Scholes)

Take averages in above and obtain first and second moment in terms of μλ

Solving for drift and vol produces

tttt dWXdtXdX

TWTT eSX 2

21

0

TT

WTT

TT

eXeeSX

eSXT

2221 2222

02

0

EEE

E

0

Elog

1

S

X

TT

TT

X

X

T 2

2

E

Elog

1

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 20: Nikos SKANTZOS 2010

Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula

With

And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates

The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla

Basket is based on similar ideas

)()(DFAsian 210 dNKdNSe T

T

TK

S

d

20

1

2

1ln

T

TK

S

d

20

2

2

1ln

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 21: Nikos SKANTZOS 2010

Smile-dynamics models Large number of alternative models

Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip

A successful model must allow quick and exact pricing of vanillas to reproduce smile

Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 22: Nikos SKANTZOS 2010

Dupire Local Vol Comes from a need to price path-dependent

options while reproducing the vanilla mkt prices

Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic

Local vol is a time- and spot-dependent vol(something the BS implied vol is not)

No-arbitrage fixes drift μ to risk-free rate

ttttt dWtSSdtSdS

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 23: Nikos SKANTZOS 2010

Local Vol

tTSK

KK

KTt tCK

CrrKCrCtS

2

21

1212

Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36

They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives

Or equivalently in terms of BS implied-vols

tTSKt t

dd

KKtTK

d

KtTKK

KrrK

TtTtS

BS

21

2

BS2BS

2

0BS

1BS

02

BS

221

BS12

BS

0

BS21

2

21

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 24: Nikos SKANTZOS 2010

Dupire Local Vol

Contains derivatives of mkt quotes with respect to

Maturity Strike

The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary

The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step

Can be used to price path-dependent options

T

tStStS SSS TT 112211 2

1

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 25: Nikos SKANTZOS 2010

Local Vol rule of thumb Rule of thumb

Local vol varies with index level twice as fast as implied vol varies with strike

(Derman amp Kani)

Sinitial

Sfinal

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 26: Nikos SKANTZOS 2010

Local-Vol and vanillas

Example Take smile quotes Build local-vol Use them in simulation

and price vanillas Compare resulting price

of vanillas vs market quotes(in smile terms)

By design the local-vol model reproduces automatically vanillas

No further calibration necessary only market quotes needed

EURUSD market

Lines market quotes

Markers LV pricer

Blue 3 years maturity

Green 5 years maturity

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 27: Nikos SKANTZOS 2010

Analytic Local-Vol (2)

Alternative assume a form for the local-vol σ(Stt)

Do that for example by

From historical market data calculate log-returns

These equal to the volatility

Make a scatter plot of all these Pass a regression The regression will give an idea of

the historically realised local-vol function

tSS

St

t

tt log

Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 28: Nikos SKANTZOS 2010

Analytic Local-Vol (2) A popular choice is

Ft the forward at time t Three calibration parameters

σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)

Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)

+

SF

F

F

FtS tt

2

000 111

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 29: Nikos SKANTZOS 2010

Stochastic models Stochastic models introduce one extra source of

randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model

Main problem Calibration minimize

(model output ndash market observable)2

Example (model ATM vol ndash market ATM vol)2

Parameter space should not be too small model cannot reproduce all market-quotes

across tenors too large more than one solution exists to calibration

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 30: Nikos SKANTZOS 2010

Heston model Coupled dynamics of underlying and volatility

Interpretation of model parameters

μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance

Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic

volatility Rev Fin Stud (1993) v6 pp327-343

1dWSvdtSdS tttt

2dWvdtvvdv ttt

dtdWdWE 21

Processes Lognormal for spot Mean-reverting for

variance Correlated Brownian

motions

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 31: Nikos SKANTZOS 2010

Effect of Heston parameters on smile

Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin

Affecting skew Correlation ρ Vol of variance ε

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 32: Nikos SKANTZOS 2010

Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics

Rule of thumb skewed smiles use Local Vol convex smiles use Heston

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 33: Nikos SKANTZOS 2010

Hull-White model It models mean-reverting underlyings such as

Interest rates Electricity oil gas etc

3 parameters to calibrate obtained from historical data

rmean (describes long-term mean) obtained from calibration

a speed of mean reversion σ volatility

Has analytic solution for the bond price P = E[ e-

intr(t)dt ]

ttt dWdtrardr mean

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 34: Nikos SKANTZOS 2010

Three-factor model in FOREX

Three factor model in FOREX spot + domesticforeign rates

To replicate FX volatilities match

FXmkt with FXmodel

Θ(s) is a function of all model parameters FXdfadaf

ffff

meanff

ddddmean

dd

FXfd

dWdtrardr

dWdtrardr

dWSdtSrrdS

T

t

dsstT

22modelFX

1

Hull-White is often coupled to another underlying

Common calibration issue Variance squeezeldquo

FX vol + IR vols up to a certain date have exceeded the FX-model vol

Solution (among other possibilities)

Time-dependent parameters (piecewise constant)

parameter

time

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 35: Nikos SKANTZOS 2010

Two-factor model in commodities

Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)

δ = benefit of direct access ndash cost of carry

Not observable but related to physical ownership of asset

No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]

δt is taken as a correction to the drift of the spot price process

What is the process for St rt δt

Problem δt is unobserved Spot is not easy to observe

for electricity it does not exist For oil the future is taken as a proxy

Commodity models based on assumptions on δ

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 36: Nikos SKANTZOS 2010

Gibson-Scwartz model Classic commodities model

Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting

Very similar to interest rate modeling (although δt can be posneg)

Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates

Analysis based on combining techniques Calculate implied convenience yield from observed

future prices

2

1

ttt

ttttttt

dWdtd

dWSdtrSdS

Miltersen extension

Time-dependent parameters

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 37: Nikos SKANTZOS 2010

Merton jump model This model adds a new element to the

stochastic models jumps in spot Motivated by real historic data

Disadvantages Risk cannot be

eliminated by delta-hedging as in BS

Hedging strategy is not clear

Advantages Can produce smile Adds a realistic

element to dynamics Has exact solution

for vanillas

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 38: Nikos SKANTZOS 2010

Merton jump modelExtra term to the Black-Scholes process

If jump does not occur

If jump occurs Then

Therefore Y size of the jump

Model has two extra parameters size of the jump Y frequency of the jump λ

tt

t dWdtS

dS

1 YdWdtS

dSt

t

t

YSS

YSSSS

tt

tttt

jump beforejumpafter

jump beforejump beforejumpafter 1

Jump size amp jump times

Random variables

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 39: Nikos SKANTZOS 2010

Merton model solution Merton assumed that

The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real

Jump times Poisson-distributed with mean λ Prob(n jumps)=e-

λT(λT)n n Jump times independent from jump sizes

The model has solution a weighted sum of Black-Scholes formulas

σn rn λrsquo are functions of σr and the jump-statistics given by η γ

nn

nT rTKS

n

TBS

e price Call 0

0n

-

T

TrK

S

KeT

TrK

S

SerTKSn

nnTrr

n

nnTr

nnn

22102

210

0

loglogBS 11

21 e

T

nn

222 2

21

12 12

21

T

nerrrn

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 40: Nikos SKANTZOS 2010

Merton model properties The model is able to produce a smile effect

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 41: Nikos SKANTZOS 2010

Vanna-Volga method Which model can reproduce market dynamics

Market psychology is not subject to rigorous math modelshellip

Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc

Buthellip Difficult to implement Hard to calibrate Computationally inefficient

Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient

Buthellip It is not a rigorous model Has no dynamics

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 42: Nikos SKANTZOS 2010

Vanna-Volga main idea The vol-sensitivities

Vega Vanna Volga

are responsible the smile impact

Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which

zero out the VegaVannaVolga of exotic option at hand

Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)

Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of

vanillas

Price

S

Price2

2

2Price

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 43: Nikos SKANTZOS 2010

Vanna-Volga hedging portfolio Select three liquid instruments

At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM

KATM

KATM

K25ΔP K25ΔC

KATM

K25ΔP K25ΔC

ATM Straddle 25Δ Risk-Reversal

25Δ Butterfly

RR carries mainly Vanna BF carries mainly Volga

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 44: Nikos SKANTZOS 2010

Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF

∙ BF

What are the appropriate weights wATM wRR wBF

Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes

vol-sensitivities of portfolio P = vol-sensitivities of exotic X

solve for the weights

volga

vanna

vega

volgavolgavolga

vannavannavanna

vegavegavega

volga

vanna

vega

w

w

w

BFRRATM

BFRRATM

BFRRATM

X

X

X

XAw -1

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 45: Nikos SKANTZOS 2010

Vanna-Volga price Vanna-Volga market price

is

XVV = XBS + wATM ∙ (ATMmkt-ATMBS)

+ wRR ∙ (RRmkt-RRBS)

+ wBF ∙ (BFmkt-BFBS)

Other market practices exist

Further weighting to correct price when spot is near barrier

It reproduces vanilla smile accurately

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 46: Nikos SKANTZOS 2010

Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in

F Bossens G Rayee N Skantzos and G Delstra

Vanna-Volga methods in FX derivatives from theory to market practiseldquo

Int J Theor Appl Fin (to appear)

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 47: Nikos SKANTZOS 2010

Models that go the extra mile

Local Stochastic Vol model Jump-vol model Bates model

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 48: Nikos SKANTZOS 2010

Local stochastic vol model Model that results in both a skew (local vol) and a convexity

(stochastic vol)

For σ(Stt) = 1 the model degenerates to a purely stochastic model

For ξ=0 the model degenerates to a local-volatility model

Calibration hard

Several calibration approaches exist for example

Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option

market

2

1

tttt

tttttt

dWVdtVdV

dWVtSdtSdS

222LV Dupire ttt VtStS

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 49: Nikos SKANTZOS 2010

Jump vol model Consider two implied volatility surfaces

Bumped up from the original Bumped down from the original

These generate two local vol surfaces σ1(Stt) and σ2(Stt)

Spot dynamics

Calibrate to vanilla prices using the bumping parameter and the probability p

ptS

ptStS

dWtSSdtSdS

t

tt

ttttt

-1 prob with

prob with

2

1

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 50: Nikos SKANTZOS 2010

Bates model Stochastic vol model with jumps

Has exact solution for vanillas

Analysis similar to Heston based on deriving the Fourier characteristic function

More info D S Bates ldquoJumps and Stochastic Volatility

Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107

2

1

tttt

ttttt

dWVdtd

dZdWdtSdS

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 51: Nikos SKANTZOS 2010

Which model is better

Good for Skew smiles

Good for simple exotics

Good for convex smiles

Allows fat-tails

Good for barrier options lt1y

Fast + accurate for simple exoticsOTKODKOhellip

Good for maturitiesgt1y

Good if product has spot amp rates as underlying

Can price most types of products (in theory)

Not good for convex smiles

Approximates numerical derivatives outside mkt quotes

Not good for Skew smiles

Often needs time-dependent params to fit term structure

Cannot be used for path-dependent optionsTARFLKBhellip

Not useful if rates are approx constant

Often unstable

Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol

Pros

Cons

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 52: Nikos SKANTZOS 2010

Choice of model Model should fit vanilla market (smile)

and a liquid exotic market (OT)

Model must reproduce market quotes across various tenors (term structure)

No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 53: Nikos SKANTZOS 2010

One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range

0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0

OT table

-700

-600

-500

-400

-300

-200

-100

000

100

200

300

0 02 04 06 08 1

TV price

mkt

- m

od

el

VannaVolga

LocalVol

Heston

OT tables depend on

nbr barriers

Type of underlying

Maturity

mkt conditions

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 54: Nikos SKANTZOS 2010

Numerical MethodsMonte Carlo Advantages

Easy to implement Easy for multi-factor

processes Easy for complex payoffs

Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of

random number generator

PDE Disadvantages

Hard to implement Hard for multi-factor

processes Hard for complex payoffs

Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random

numbers

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 55: Nikos SKANTZOS 2010

Monte Carlo vs PDE

Monte CarloBased on discounted average payoff over realizations of

spot

Outline of Monte Carlo simulation For each path

At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot

Calculate payoff for this path Calculate average payoff across all paths

Pathsnbr

1

)(payoffPathsnbr

1

payoffE PriceOption

i

iT

Tr

TTr

Se

Se

number random

tttttt WStSSS

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 56: Nikos SKANTZOS 2010

Monte Carlo vs PDE

Partial Differential Equation (PDE)Based on alternative formulation of option price problem

Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS

Apply payoff at maturity and solve PDE backwards till today

PrS

P

S

PS

t

P

2

22

2

1

PrS

SPSPSP

S

SPSPS

t

tPtP

22 )()(2)(

2

1

2

)()()()(

time

Spot

today maturity

S0

K

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 57: Nikos SKANTZOS 2010

Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise

options Likelihood ratio method

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 58: Nikos SKANTZOS 2010

Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)

mean=0 variance=1 This means that if we sum all random numbers we should get 0 and

stdev=1 In practise we draw uniform random numbers in [01] and convert them

to Normal-Gaussian random numbers using the normal inverse cumulative function

A typical simulation requires 105 paths amp 102 steps 107 random numbers

Deviations away from the required statistics produce unwanted bias in option price

Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of

steps number of paths) increases

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 59: Nikos SKANTZOS 2010

Pseudo-random number generators RNG generate numbers in the interval [01]

With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)

Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock

After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition

occurs ldquoMersennerdquo random numbers have a period that is a

Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)

Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 60: Nikos SKANTZOS 2010

ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous

LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the

probability density will produce the correct density of points

0

1

hom

og

enous

nu

mbers

form

[0

1]

Gaussian cumulative function

Non-homogenous numbers in (-infin infin)

Gaussian probability

function

Higher density of points here

ldquoPeakrdquo implies that more points should be sampled from here

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 61: Nikos SKANTZOS 2010

Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr

Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random

Calculating the Greeks with finite difference requires the same sequence of random numbers

The calculation of the Greeks should differ only in the ldquobumpedrdquo param

S

SSSS

2

PricePrice

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 62: Nikos SKANTZOS 2010

Random number quality

1 2 3 4 5 6 70 0 0 0 0 0 0

05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075

0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875

06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375

059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375

Draw (n x m) table of Sobolrsquo numbers

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

2 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 10 20 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 13 40 )

0000

0100

0200

0300

0400

0500

0600

0700

0800

0900

1000

0000 0200 0400 0600 0800 1000

Pair( 20 881 )

Plot pairs of columns(12) (1020)

Non-uniform filling for large dimensions

(1340) (20881)

Nbr Steps Nbr Paths

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 63: Nikos SKANTZOS 2010

Barrier options

Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit

Consider a (slightly) complex barrier pattern

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 64: Nikos SKANTZOS 2010

Barrier options There is analytic expression for ldquosurvival probabilityrdquo

=probability of not hitting

We rewrite the pattern in terms of ldquonot-hittingrdquo events

This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB

Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)

hitnot isA ANDhit not is BProbhitnot isA Prob

hitnot isA Probhitnot isA GIVENhit not is BProb1

hitnot isA Probhitnot isA GIVENhit is BProb

hitnot is A ANDhit is BProb rule Bayes

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 65: Nikos SKANTZOS 2010

Barrier option replication

Prob(A is hit) = Prob(A is hit in [t1t2])∙

Prob(A is hit in [t2t3])

Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 66: Nikos SKANTZOS 2010

Barrier options formula

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 67: Nikos SKANTZOS 2010

Barrier option formula

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 68: Nikos SKANTZOS 2010

American exercise in Monte Carlo

When is it optimal to exercise the option

Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then

start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise

now if (on average) final spot finishes less in-the-money exercise now

today

K

S0

today t maturity

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 69: Nikos SKANTZOS 2010

Least-squares Monte Carlo Since this has to be done for every time step t

Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by

Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea

Work backwards starting from maturity At each step compare immediate exercise value with expected

cashflow from continuing Exercise if immediate exercise is more valuable

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 70: Nikos SKANTZOS 2010

Least-squares Monte Carlo (1)

Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)

Out-of-the-money

In-the-money

Npaths

Nsteps

Spot Paths

0000

0000

0000

0000

0000

0000

0000

Cashflows

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 71: Nikos SKANTZOS 2010

Least-squares Monte Carlo (2)

If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0

Out-of-the-money

In-the-money

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

CF=(Sthis path(T)-K)+

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 72: Nikos SKANTZOS 2010

Least-squares Monte Carlo (3)

Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue

Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0

Spot Paths

000

000

0000

000

0000

000

000

Cashflows

Y1(T-Δt)

Y3(T-Δt)=0

Y5(T-Δt)=0

Y6(T-Δt)

Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 73: Nikos SKANTZOS 2010

Least-squares Monte Carlo (4)

On the pairs Spath iYpath i pass a regression of the form

E(S) = a0+a1∙S +a2∙S2

This function is an approximation to the expected payoff from continuing to hold the option from this time point on

If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0

If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps

Y

S

E(S)

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 74: Nikos SKANTZOS 2010

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 75: Nikos SKANTZOS 2010

Least-squares Monte Carlo (5)

Proceed similarly till the first time step and populate the matrix of cashflows

There should be one non-zero cashflow per path(the option can be exercised only once)

Callables are priced with the same idea

000000

000000

000000

000000

000000

000000

000000

exer

1

exer

paths

tCFttodayDF 1

Amerpaths

ii

N

ii S

N

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 76: Nikos SKANTZOS 2010

Greeks in Monte Carlo To calculate Greeks with Monte Carlo

Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile

curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example

This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders

S

SSSS

2

PricePrice 2

PricePrice2Price

S

SSSSS

PricePriceVega

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 77: Nikos SKANTZOS 2010

Likelihood ratio method (1)

This method allows us to calculate all Greeks within a single Monte Carlo

Main idea

Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation

Note The analytics of the method simplify if spot is assumed to follow

lognormal process (as in BS) The LR greeks will not be in general the same as the finite

difference greeks This is because of the modification of the market data when using the

finite difference method

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 78: Nikos SKANTZOS 2010

Likelihood ratio method (2)

Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path

PDF probability density function of the spot

zi the Gaussian random number used to make the jump S i-1 Si

Probsurv the total survival probability for the spot path (given some barrier levels)

For explicit expressions for the survprob of KO or DKO see previous slides

PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS

m

i

z

iii

m

iim

ieSt

SSS1

21

1

2211

2

1PDFPDF

ii

iii

i

it

trS

S

z

2

1 21

log

m

i

tt ii

1survsurv

1ProbProb

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 79: Nikos SKANTZOS 2010

Likelihood ratio method (3)

Sensitivity with respect to a parameter α (=spot vol etc)

This is simple derivatives over analytic functions (see previous slide)

For example Delta becomes the new payoff

To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable

m

m

m

mm

mmm

S

S

S

SdS

SSdS

1surv

1surv

1

11

1surv11

Prob

Prob

1PDF

PDF

1PayoffDF

ProbPDF PayoffDF

Exotic

0

surv

surv0

1

1

10

10

Prob

Prob

1PDF

PDF

1PayoffDF

SS

S

S

tt

ttm

m

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer

Page 80: Nikos SKANTZOS 2010

References Options

ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall

ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley

Numerical methods PDE Pricing Financial Instruments The Finite

Difference Method D Tavella and C Randall (2000) Wiley

Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley

Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer