modern methods of option pricing -...

Modern Methods of Option Pricing

Denis Belomestny

Weierstraß Institute Berlin

Motzen, 14 June 2007

Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 1 / 30

Overview1 Introduction

Contingent claimsOptions and Risks

2 Bermudan pricing problemBermudan derivativesSnell Envelope ProcessBackward Dynamic ProgrammingExercise policies and lower bounds

3 Upper BoundsDual Upper BoundsRiesz upper bounds

4 Fast upper boundsDoob-Meyer MartingaleMartingale RepresentationProjection Estimator

5 ApplicationsDenis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 2 / 30

Introduction Contingent claims

Contingent claims

DefinitionA derivative security or a contingent claim is a financial assetwhose payment depends on the value of some underlying variable(stock, interest rate and so on).

DefinitionAn option is derivative security that gives the right to buy or sell theunderlying asset, at (European option) or not later than (American orBermudan option) some maturity date T , for a prespecified price K ,called strike.

A call (put) option is a right to buy (sell). These plain vanilla optionshave been introduced in 1973.


Introduction Options and Risks

Options and Risks

Risk reductionAny option is a kind of insurance policy that put floor on the lossesrelated to the specific behavior of the underlyings.

ExampleThe seller of a call exchanges his upside risk (gains above the strikeprice) for the certainty of cash in hand (the premium). The buyer of aput limits his downside risk for a price - like buying fire insurance fora house. The buyer of a straddle limits both risks.


Introduction Options and Risks

Options and Risks

0 1 2 3 4

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Call Option

x

Pay

off

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

Put Option

xP

ayof

f

0 1 2 3 4

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Straddle Option

x

Pay

off

0.0 0.2 0.4 0.6 0.8 1.0

01

23

4

Time

Sto

ck

0.0 0.2 0.4 0.6 0.8 1.0

01

23

4

Time

Sto

ck

0.0 0.2 0.4 0.6 0.8 1.00

12

34

Time

Sto

ck


Bermudan pricing problem Bermudan derivatives

Bermudan derivatives

Let Lt ∈ RD be an underlying and T := {T0,T1, . . . ,TJ } be a set ofexercise dates.

Bermudan derivative: An option to exercise a cashflow C(Tτ ,L(Tτ ))at a future time Tτ ∈ T, to be decided by the option holder.

ExampleThe callable snowball swap pays semi-annually a constant rate Iover the first year and in the forthcoming years

(Previous coupon + A− Libor)+,

semi-annually, where A is given in the contract.



Bermudan derivatives



Valuation

If N, with N(0) = 1, is a numeraire and P is the associated pricingmeasure, then with

Zτ := C(Tτ ,L(Tτ ))/N(Tτ ),

the t = 0 price of the derivative is given by the solution of an optimalstopping problem

V0 = supτ∈{0,...,J}

EF0Zτ ,

where the supremum runs over all stopping indexes τ with respect to{FTj , 0 ≤ j ≤ J}, where (Ft)t≥0 is the usual filtration generated by L.



Optimal stopping

Mathematical problem:

Optimal stopping (calling) of a reward (cash-flow) process Zdepending on an underlying (e.g. interest rate) process L

Typical difficulties:

L is usually high dimensional, for Libor interest rate models,D = 10 and up, so PDE methods do not work in general

Z may only be virtually known, e.g. Zi = EFi∑

j≥i C(Lj) for somepay-off function C, rather than simply Zi = C(Li)

Z may be path-dependent


Bermudan pricing problem Snell Envelope Process

Snell Envelope Process

At a future time point t , when the option is not exercised before t , theBermudan option value is given by

Vt = N(t) supτ∈{κ(t),...,J}

EFt Zτ

with κ(t) := min{m : Tm ≥ t}.The process

Y ∗t :=

Vt

N(t)

is called the Snell-envelope process and is a supermartingale, i.e.

EFsY ∗t ≤ Y ∗

s .


Bermudan pricing problem Backward Dynamic Programming

Backward Dynamic Programming

Set Y ∗j := Y ∗(Tj), Lj = L(Tj) Fj := FTj . At the last exercise date TJ

we have

Y ∗J = ZJ

and for 0 ≤ j < J ,

Y ∗j = max

(Zj ,EFj Y ∗

j+1

).

Observation

Nested Monte Carlo simulation of the price Y ∗0 would require Nk

samples when conditional expectations are computed with N samples


Bermudan pricing problem Backward Dynamic Programming

Backward Dynamic Programming


Bermudan pricing problem Exercise policies and lower bounds

Construction of exercise policies

Any stopping family (policy) (τj) satisfies

j ≤ τj ≤ J , τJ = J , τj > j ⇒ τj = τj+1, 0 ≤ j < J .

A lower bound Y for the Snell envelope Y ∗,

Yi := EFi Zτi ≤ Y ∗i

ExampleThe policy

τi := inf{j ≥ i : L(Tj) ∈ G ⊂ RD}

exercises when the underlying process L enters a certain region G.


Bermudan pricing problem Exercise policies and lower bounds

Regression Methods

An exercise policy τ can be constructed via

τJ = J ,τ j = jχ{Cj (Lj )≤Zj} + τ j+1χ{Cj (Lj )>Zj}, j < J ,

where a continuation value Cj(Lj) := EFj Y ∗j+1 can be approximated as

Cj(x) ≈R∑

r=1

βjrψr (x), j = 0,1, . . . ,J − 1,

using a sample from (Lj ,Zτ j ) and a set of basis functions {ψr}.

QuestionIs the policy τ a good one ?


Upper Bounds Dual Upper Bounds

Dual upper bounds

Consider a discrete martingale(Mj

)j=0,...,J with M0 = 0 with respect to

the filtration(Fj

)j=0,...,J . Following Rogers, Haugh and Kogan, we

observe that

Y0 = supτ∈{0,,...,J}

EF0 [Zτ −Mτ ] ≤ EF0 max0≤j≤J

[Zj −Mj

].

Hence the r.h.s. with an arbitrary martingale gives an upper bound forthe Bermudan price Y0.

QuestionWhat martingale does lead to equality ?


Upper Bounds Dual Upper Bounds

Dual upper bounds

Theorem (Rogers (2001), Haugh & Kogan (2001))

Let M∗ be the (unique) Doob-Meyer martingale part of(

Y ∗j

)0≤j≤J

, i.e.

M∗j is an

(Fj

)-martingale which satisfies

Y ∗j = Y ∗

0 + M∗j − A∗j , j = 0, ...,J ,

with M∗0 := A∗0 := 0 and A∗ being such that A∗j is Fj−1 measurable for

j = 1, ...,J . Then we have

Y ∗0 = EF0 max

0≤j≤J

[Zj −M∗

j

].


Upper Bounds Riesz upper bounds

Riesz upper bounds

Doob-Meyer decomposition

Y ∗j = Y ∗

0 + M∗j − A∗j , j = 0, ...,J ,

and Y ∗J = ZJ imply Riesz decomposition

Y ∗j = EFj ZJ + EFj (A∗J − A∗j )

Since A∗i+1 − A∗i = Y ∗i − EFi Y ∗

i+1 = [Zi − EFi Y ∗i+1]

+,

Y ∗j = EFj ZJ + EFj

J−1∑i=j

[Zi − EFi Y ∗i+1]

+.



Riesz upper bounds

Theorem (Belomestny & Milstein (2005))If Yi is a lower approximation for Y ∗

i , then

Y upj = EFj ZJ + EFj

J−1∑i=j

[Zi − EFi Yi+1]+

is an upper approximation for Y ∗j , that is

Yj ≤ Y ∗j ≤ Y up

j , j = 0, . . . ,J .



Riesz upper bounds

PropertiesMonotonicity

Yi ≥ Yi −→ Y upi ≤ Y up

i

LocalityLet {Y α

i , α ∈ Ii} be a family of local lower bounds at i, then

Y α,upj = EFj ZJ + EFj

J−1∑i=j

[Zi − maxα∈Ii+1

EFi Y αi+1]

+

is an upper bound.


Fast upper bounds Doob-Meyer Martingale

Doob-Meyer Martingale

For any martingale MTj with respect to the filtration (FTj ; 0 ≤ j ≤ J )starting at M0 = 0

Y up0 (M) := EF0

[max

0≤j≤J(ZTj −MTj )

]is an upper bound for the price of the Bermudan option with thediscounted cash-flow ZTj .

Exact Bermudan price is attained at the martingale part M∗ of theSnell envelope,

Y ∗Tj

= Y ∗T0

+ M∗Tj− A∗Tj

,

M∗T0

= A∗T0= 0 and A∗Tj

is FTj−1 measurable.


Fast upper bounds Doob-Meyer Martingale

Doob-Meyer Martingale

Assume that YTj = u(Tj ,L(Tj)) is an approximation of the Snellenvelope Y ∗

Tj, 0 ≤ j ≤ J , with Doob decomposition

YTj = YT0 + MTj − ATj ,

where MT0 = AT0 = 0 and ATj is FTj−1 measurable.

It then holds:

MTj+1 −MTj = YTj+1 − ETj [YTj+1 ]

Observation

The computation of MTj by MC leads to quadratic Monte Carlo for Y up0


Fast upper bounds Martingale Representation

Martingale Representation

If L is Markovian and F is generated by a D-dimensional Brownianmotion W , then due to the martingale representation theorem

MTj =:

∫ Tj

0HtdWt

=:

∫ Tj

0h(t ,L(t))dWt , j = 0, . . . ,J ,

where Ht is a square integrable and previsible process.

ObservationFor any square integrable function h(·, ·) we get a martingale


Fast upper bounds Projection Estimator

Projection Estimator

We are going to estimate Ht on partition π = {t0, . . . , tI} with t0 = 0,tI = T , and {T0, . . . ,TJ } ⊂ π.

Write formally,

YTj+1 − YTj ≈∑

tl∈π;Tj≤tl<Tj+1

Htl · (Wtl+1 −Wtl ) + ATj+1 − ATj .

By multiplying both sides with (W dti+1

−W dti ), Tj ≤ ti < Tj+1, and taking

Fti -conditional expectations, we get by the FTj -measurability of ATj+1 ,

Hdti ≈

1ti+1 − ti

EFti

[(W d

ti+1−W d

ti ) · YTj+1

].




In fact for Tj ≤ ti < Tj+1

Hπti = hπ(ti ,L(ti)) =

1∆π

iE i

[(∆πWi)

>u(Tj+1,L(Tj+1))]

and this expectation can be computed by regression.

1 Take basis functions

ψ(ti , ·) = (ψr (ti , ·), r = 1, . . . ,R)

2 Simulate N independent samples

(ti , nL(ti)), n = 1, . . . ,N

from L(ti) constructed using the Brownian increments ∆πnWi .



Regression

3 Construct the matrix A⊕ti := (A>ti Ati )−1A>ti , where

Ati = {(ψr (ti , nL(ti))) , n = 1, . . . ,N, r = 1, . . . ,R} .

4 Define

hπ(ti , x) = ψ(ti , x) A⊕ti

(∆π· Wi

∆πi

·YTj+1

)=: ψ(ti , x)βti ,

where βti is the R × D matrix of estimated regression coefficientsat time ti .



Fast MC Upper Bound

Finally we construct

Y up0 =

1

N

N∑n=1

max0≤j≤J

[z(Tj , nL(Tj))− Mj

],

withMj =

∑ti∈π;0≤ti<Tj

hπ(ti , L(Tj)) · (∆πWi)

by simulating new paths (nL(Tj),∆πnWi), n = 1, . . . , N.

Observationis always a martingale, so the upper bound is true!


Applications

Max Call on D assets

Black-Scholes model:

dX dt = (r − δ)X d

t dt + σX dt dW d

t , d = 1, ...,D,

Pay-off:Zt := z(Xt) := (max(X 1

t , ...,XDt )− κ)+.

TJ = 3yr, J = 9 (ex. dates), κ = 100, r = 0.05, σ = 0.2, δ = 0.1,D = 2 and different x0

D x0 Lower Bound Upper Bound A&B PriceY0 Y up

0 (Mπ) Interval90 8.0242±0.075 8.0891±0.068 [8.053, 8.082]

2 100 13.859±0.094 13.958±0.085 [13.892, 13.934]110 21.330±0.109 21.459±0.097 [21.316, 21.359]


Applications


Basis functions at (t , x) with Tj−1 ≤ t < Tj

SI{

1, X k ∂ EP(t ,X ; Tj )

∂xk , X k ∂ EP(t ,X ; TJ )∂xk , k = 1, . . . ,D

}SII

{1, X k ∂ EP(t ,X ; Tj )

∂xk , k = 1, . . . ,D}

SIII Pol3(X ,EP(t , x ; Tj))SIV Pol3(X )SV 1


Applications



Literatur

Belomestny, D. and Milstein, G.Monte Carlo evaluation of American options using consumptionprocesses.International Journal of Theoretical and Applied Finance,02(1):65–69, 2000.

Belomestny, D., Milstein, G. and Spokoiny, V.Regression methods in pricing American and Bermudan optionsusing consumption processes.Journal of Quantitative Finance, tentatively accepted.

Belomestny, D., Bender, Ch. and Schoenmakers, J.True upper bounds for Bermudan products via non-nested MonteCarlo.Mathematical Finance, tentatively accepted.


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