introduction to option pricing - baruch...

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......Introduction to Option Pricing

Liuren Wu

Zicklin School of Business, Baruch College

Options Markets

Liuren Wu (Baruch) Option Pricing Introduction Options Markets 1 / 78

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Outline

...1 General principles and applications

...2 Illustration of hedging/pricing via binomial trees

...3 The Black-Merton-Scholes model

...4 Introduction to Ito’s lemma and PDEs

...5 Real (P) v. risk-neutral (Q) dynamics

...6 implied volatility surface


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Review: Valuation and investment in primary securities

The securities have direct claims to future cash flows.

Valuation is based on forecasts of future cash flows and risk:

DCF (Discounted Cash Flow Method): Discount time-series forecastedfuture cash flow with a discount rate that is commensurate with theforecasted risk.We diversify as much as we can. There are remaining risks afterdiversification — market risk. We assign a premium to the remainingrisk we bear to discount the cash flows.

Investment: Buy if market price is lower than model value; sell otherwise.

Both valuation and investment depend crucially on forecasts of future cashflows (growth rates) and risks (beta, credit risk).


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Compare: Derivative securities

Payoffs are linked directly to the price of an “underlying” security.

Valuation is mostly based on replication/hedging arguments.

Find a portfolio that includes the underlying security, and possiblyother related derivatives, to replicate the payoff of the target derivativesecurity, or to hedge away the risk in the derivative payoff.

Since the hedged portfolio is riskfree, the payoff of the portfolio can bediscounted by the riskfree rate.

No need to worry about the risk premium of a “zero-beta” portfolio.

Models of this type are called “no-arbitrage” models.

Key: No (little) forecasts are involved. Valuation is based, mostly, oncross-sectional comparison.

It is less about whether the underlying security price will go up or down(forecasts of growth rates /risks), but about the relative pricing relationbetween the underlying and the derivatives under all possible scenarios.


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Readings behind the technical jargons: P v. Q

P: Actual probabilities that earnings will be high or low, estimated based onhistorical data and other insights about the company.

Valuation is all about getting the forecasts right and assigning theappropriate price for the forecasted risk — fair wrt future cashflows(and your risk preference).

Q: “Risk-neutral” probabilities that one can use to aggregate expectedfuture payoffs and discount them back with riskfree rate.

It is related to real-time scenarios, but it is not real-time probability.

Since the intention is to hedge away risk under all scenarios anddiscount back with riskfree rate, we do not really care about the actualprobability of each scenario happening.We just care about what all the possible scenarios are and whether ourhedging works under all scenarios.

Q is not about getting close to the actual probability, but about beingfair/consistent wrt the prices of securities that you use for hedging.

Associate P with time-series forecasts and Q with cross-sectionalcomparison.Liuren Wu (Baruch) Option Pricing Introduction Options Markets 5 / 78

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A Micky Mouse example

Consider a non-dividend paying stock in a world with zero riskfree interest rate.Currently, the market price for the stock is $100. What should be the forwardprice for the stock with one year maturity?

The forward price is $100.

Standard forward pricing argument says that the forward price shouldbe equal to the cost of buying the stock and carrying it over tomaturity.The buying cost is $100, with no storage or interest cost or dividendbenefit.

How should you value the forward differently if you have inside informationthat the company will be bought tomorrow and the stock price is going todouble?

Shorting a forward at $100 is still safe for you if you can buy the stockat $100 to hedge.


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Investing in derivative securities without insights

If you can really forecast (or have inside information on) future cashflows,you probably do not care much about hedging or no-arbitrage pricing.

What is risk to the market is an opportunity for you.

You just lift the market and try not getting caught for insider trading.

If you do not have insights on cash flows and still want to invest inderivatives, the focus does not need to be on forecasting, but oncross-sectional consistency.

No-arbitrage pricing models can be useful.

Identifying the right hedge can be as important (if not more so) asgenerating the right valuation.


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What are the roles of an option pricing model?

...1 Interpolation and extrapolation:

Broker-dealers: Calibrate the model to actively traded option contracts,use the calibrated model to generate option values for contractswithout reliable quotes (for quoting or book marking).

Criterion for a good model: The model value needs to match observedprices (small pricing errors) because market makers cannot take bigviews and have to passively take order flows.

Sometimes one can get away with interpolating (linear or spline)without a model.

The need for a cross-sectionally consistent model becomes stronger forextrapolation from short-dated options to long-dated contracts, or forinterpolation of markets with sparse quotes.

The purpose is less about finding a better price or getting a betterprediction of future movements, but more about making sure theprovided quotes on different contracts are “consistent” in the sensethat no one can buy and sell with you to lock in a profit on you.


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...2 Alpha generation

Investors: Hope that market price deviations from the model “fairvalue” are mean reverting.

Criterion for a good model is not to generate small pricing errors, butto generate (hopefully) large, transient errors.

When the market price deviates from the model value, one hopes thatthe market price reverts back to the model value quickly.

One can use an error-correction type specification to test theperformance of different models.

However, alpha is only a small part of the equation, especially forinvestments in derivatives.


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...3 Risk hedge and management:

Hedging and managing risk plays an important role in derivatives.

Market makers make money by receiving bid-ask spreads, but optionsorder flow is so sparse that they cannot get in and out of contractseasily and often have to hold their positions to expiration.

Without proper hedge, market value fluctuation can easily dominatethe spread they make.

Investors (hedge funds) can make active derivative investments basedon the model-generated alpha. However, the convergence movementfrom market to model can be a lot smaller than the market movement,if the positions are left unhedged. — Try the forward example earlier.

The estimated model dynamics provide insights on how tomanage/hedge the risk exposures of the derivative positions.

Criterion of a good model: The modeled risks are real risks and are theonly risk sources.

Once one hedges away these risks, no other risk is left.


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...4 Answering economic questions with the help of options observations

Macro/corporate policies interact with investor preferences andexpectations to determine financial security prices.

Reverse engineering: Learn about policies, expectations, andpreferences from security prices.

At a fixed point in time, one can learn a lot more from the largeamount of option prices than from a single price on the underlying.

Long time series are hard to come by. The large cross sections ofderivatives provide a partial replacement.

The Breeden and Litzenberger (1978) result allows one to infer therisk-neutral return distribution from option prices across all strikes at afixed maturity. Obtaining further insights often necessitates morestructures/assumptions/an option pricing model.

Criterion for a good model: The model contains intuitive, easilyinterpretable, economic meanings.


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Forward pricing

Terminal Payoff: (ST − K ), linear in the underlying security price (ST ).

No-arbitrage pricing:

Buying the underlying security and carrying it over to expiry generatesthe same payoff as signing a forward.

The forward price should be equal to the cost of the replicatingportfolio (buy & carry).

This pricing method does not depend on (forecasts) how the underlyingsecurity price moves in the future or whether its current price is fair.

but it does depend on the actual cost of buying and carrying (not allthings can be carried through time...), i.e., the implementability of thereplicating strategy.

Ft,T = Ste(r−q)(T−t), with r and q being continuously compounded rates of

carrying cost (interest, storage) and benefit (dividend, foreign interest,convenience yield).

When arbitrage trading does not work, q can be traded as a “convenienceyield,” or a prediction of future spot movements.Liuren Wu (Baruch) Option Pricing Introduction Options Markets 12 / 78

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Option pricing

Terminal Payoff: Call — (ST − K )+, put — (K − ST )+.

No-arbitrage pricing:

Can we replicate the payoff of an option with the underlying security sothat we can value the option as the cost of the replicating portfolio?

Can we use the underlying security (or something else liquid and withknown prices) to completely hedge away the risk?

If we can hedge away all risks, we do not need to worry about riskpremiums and one can simply set the price of the contract to the valueof the hedge portfolio (maybe plus a premium).

In most practical situations, options cannot be replicated by the underlying— Options are useful and informative about (i) expectations of the future(ii) risk premiums.

The difficulty lies in how to separate the two.


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Another Mickey Mouse example: A one-step binomial tree

Observation: The current stock price (St) is $20.The current continuously compounded interest rate r (at 3 month maturity)is 12%. (crazy number from Hull’s book).

Model/Assumption: In 3 months, the stock price is either $22 or $18 (nodividend for now).

St = 20 ��PPPP

ST = 22

ST = 18Comments:

Once we make the binomial dynamics assumption, the actual probability ofreaching either node (going up or down) no longer matters.

We have enough information (we have made enough assumption) to priceoptions that expire in 3 months.

Remember: For idealistic derivative no-arbitrage pricing, what matters is thelist of possible scenarios, but not the actual probability of each scenariohappening.


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A 3-month call option

Consider a 3-month call option on the stock with a strike of $21.

Backward induction: Given the terminal stock price (ST ), we can computethe option payoff at each node, (ST − K )+.

The zero-coupon bond price with $1 par value is: 1× e−.12×.25 = $0.9704.

Bt = 0.9704St = 20ct = ?

��

��

QQ

QQ

BT = 1ST = 22cT = 1

BT = 1ST = 18cT = 0

Two angles:

Replicating: See if you can replicate the call’s payoff using stock and bond.⇒ If the payoffs equal, so does the value.

Hedging: See if you can hedge away the risk of the call using stock. ⇒ Ifthe hedged payoff is riskfree, we can compute the present value usingriskfree rate.


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Replicating

Bt = 0.9704St = 20ct = ?

��

��

QQ

QQ

BT = 1ST = 22cT = 1

BT = 1ST = 18cT = 0

Assume that we can replicate the payoff of 1 call using ∆ share of stock andD par value of bond, we have

1 = ∆22 + D1, 0 = ∆18 + D1.

Solve for ∆: ∆ = (1− 0)/(22− 18) = 1/4. ∆ = Change in C/Change in S,a sensitivity measure.

Solve for D: D = − 1418 = −4.5 (borrow).

Value of option = value of replicating portfolio= 1

420− 4.5× 0.9704 = 0.633.


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Hedging

Bt = 0.9704St = 20ct = ?

��

��

QQ

QQ

BT = 1ST = 22cT = 1

BT = 1ST = 18cT = 0

Assume that we can hedge away the risk in 1 call by shorting ∆ share ofstock, such as the hedged portfolio has the same payoff at both nodes:

1−∆22 = 0−∆18.

Solve for ∆: ∆ = (1− 0)/(22− 18) = 1/4 (the same):∆ = Change in C/Change in S, a sensitivity measure.

The hedged portfolio has riskfree payoff: 1− 1422 = 0− 1

418 = −4.5.

Its present value is: −4.5× 0.9704 = −4.3668 = 1ct −∆St .

ct = −4.3668 + ∆St = −4.3668 + 1420 = 0.633.


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One principle underlying two angles

If you can replicate, you can hedge:Long the option contract, short the replicating portfolio.

The replication portfolio is composed of stock and bond.

Since bond only generates parallel shifts in payoff and does not play any rolein offsetting/hedging risks, it is the stock that really plays the hedging role.

The optimal hedge ratio when hedging options using stocks is defined as theratio of option payoff change over the stock payoff change — Delta.

Hedging derivative risks using the underlying security (stock, currency) iscalled delta hedging.

To hedge a long position in an option, you need to short delta of theunderlying.To replicate the payoff of a long position in option, you need to longdelta of the underlying.


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The limit of delta hedging

Delta hedging completely erases risk under the binomial model assumption:The underlying stock can only take on two possible values.

Using two (different) securities can span the two possible realizations.

We can use stock and bond to replicate the payoff of an option.We can also use stock and option to replicate the payoff of a bond.One of the 3 securities is redundant and can hence be priced by theother two.

What will happen for the hedging/replicating exercise if the stock can takeon 3 possible values three months later, e.g., (22, 20, 18)?

It is impossible to find a ∆ that perfectly hedge the option position orreplicate the option payoff.We need another instrument (say another option) to do perfecthedging or replication.


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Introduction to risk-neutral valuation

Bt = 0.9704St = 20

��

��p

QQ

QQ1− p

BT = 1ST = 22

BT = 1ST = 18

Compute a set of artificial probabilities for the two nodes (p, 1− p) suchthat the stock price equals the expected value of the terminal payment,discounted by the riskfree rate.

20 = 0.9704 (22p + 18(1− p)) ⇒ p = 20/0.9704−1822−18 = 0.6523

Since we are discounting risky payoffs using riskfree rate, it is as if we live inan artificial world where people are risk-neutral. Hence, (p, 1− p) are therisk-neutral probabilities.

These are not really probabilities, but more like unit prices:0.9704p is theprice of a security that pays $1 at the up state and zero otherwise.0.9704(1− p) is the unit price for the down state.

To exclude arbitrage, the same set of unit prices (risk-neutral probabilities)should be applicable to all securities, including options.Liuren Wu (Baruch) Option Pricing Introduction Options Markets 20 / 78

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Risk-neutral probabilities

Under the binomial model assumption, we can identify a unique set ofrisk-neutral probabilities (RNP) from the current stock price.

The risk-neutral probabilities have to be probability-like (positive, lessthan 1) for the stock price to be arbitrage free.What would happen if p < 0 or p > 1?

If the prices of market securities do not allow arbitrage, we can identify atleast one set of risk-neutral probabilities that match with all these prices.

When the market is, in addition, complete (all risks can be hedged awaycompletely by the assets), there exist one unique set of RNP.

Under the binomial model, the market is complete with the stock alone. Wecan uniquely determine one set of RNP using the stock price alone.

Under a trinomial assumption, the market is not complete with the stockalone: There are many feasible RNPs that match the stock price.

In this case, we add an option to fully determine a unique set of RNP.That is, we use this option (and the trinomial) to complete the market.


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Market completeness

Market completeness (or incompleteness) is a relative concept: Market maynot be complete with one asset, but can be complete with two...

To me, the real-world market is severely incomplete with primary marketsecurities alone (stocks and bonds).

To complete the market, we need to include all available options, plus modelstructures.

Model and instruments can play similar rules to complete a market.

RNP are not learned from the stock price alone, but from the option prices.

Risk-neutral dynamics need to be estimated using option prices.

Options are not redundant.

Caveats: Be aware of the reverse flow of information and supply/demandshocks.


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Muti-step binomial trees

A one-step binomial model looks very Mickey Mouse: In reality, the stockprice can take on many possible values than just two.

Extend the one-step to multiple steps. The number of possible values(nodes) at expiry increases with the number of steps.

With many nodes, using stock alone can no longer achieve a perfect hedgein one shot.

But locally, within each step, we still have only two possible nodes. Thus,we can achieve perfect hedge within each step.

We just need to adjust the hedge at each step — dynamic hedging.

By doing as many hedge rebalancing at there are time steps, we can stillachieve a perfect hedge globally.

The market can still be completed with the stock alone, in a dynamic sense.

A multi-step binomial tree can still still very useful in practice in, forexample, handling American options, discrete dividends, interest rate termstructures ...


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Constructing the binomial tree

Stft��

QQQQ

Stufu

Stdfd

One way to set up the tree is to use (u, d) to match the stock returnvolatility (Cox, Ross, Rubinstein):

u = eσ√∆t , d = 1/u.

σ— the annualized return volatility (standard deviation).

The risk-neutral probability becomes: p = a−du−d where

a = er∆t for non-dividend paying stock.a = e(r−q)∆t for dividend paying stock.a = e(r−rf )∆t for currency.a = 1 for futures.


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Estimating the return volatility σ

To determine the tree (St , u, d), we set

u = eσ√∆t , d = 1/u.

The only un-observable or “free” parameter is the return volatility σ.

Mean expected return (risk premium) does not matter for derivativepricing under the binomial model because we can completely hedgeaway the risk.St can be observed from the stock market.

We can estimate the return volatility using the time series data:

σ =

√1∆t

[1

N−1

∑Nt=1(Rt − R)2

]. ⇐ 1

∆t is for annualization.

Implied volatility: We can estimate the volatility (σ) by matching observedoption prices.


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The Black-Merton-Scholes (BMS) model

Black and Scholes (1973) and Merton (1973) derive option prices under thefollowing assumption on the stock price dynamics,

dSt = µStdt + σStdWt

The binomial model: Discrete states and discrete time (The number ofpossible stock prices and time steps are both finite).

The BMS model: Continuous states (stock price can be anything between 0and ∞) and continuous time (time goes continuously).

BMS proposed the model for stock option pricing. Later, the model hasbeen extended/twisted to price currency options (Garman&Kohlhagen) andoptions on futures (Black).

I treat all these variations as the same concept and call themindiscriminately the BMS model.


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Primer on continuous time process


The driver of the process is Wt , a Brownian motion, or a Wiener process.

The sample paths of a Brownian motion are continuous over time, butnowhere differentiable.

It is the idealization of the trajectory of a single particle beingconstantly bombarded by an infinite number of infinitessimally smallrandom forces.

Like a shark, a Brownian motion must always be moving, or else it dies.

If you sum the absolute values of price changes over a day (or any timehorizon) implied by the model, you get an infinite number.

If you tried to accurately draw a Brownian motion sample path, yourpen would run out of ink before one second had elapsed.


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Properties of a Brownian motion


The process Wt generates a random variable that is normallydistributed with mean 0 and variance t, ϕ(0, t). (Also referred to as Gaussian.)

Everybody believes in the normal approximation, theexperimenters because they believe it is a mathematical theorem,the mathematicians because they believe it is an experimentalfact!

The process is made of independent normal increments dWt ∼ ϕ(0, dt).

“d” is the continuous time limit of the discrete time difference (∆).∆t denotes a finite time step (say, 3 months), dt denotes an extremelythin slice of time (smaller than 1 milisecond).It is so thin that it is often referred to as instantaneous.Similarly, dWt = Wt+dt −Wt denotes the instantaneous increment(change) of a Brownian motion.

By extension, increments over non-overlapping time periods areindependent: For (t1 > t2 > t3), (Wt3 −Wt2) ∼ ϕ(0, t3 − t2) is independentof (Wt2 −Wt1) ∼ ϕ(0, t2 − t1).


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Properties of a normally distributed random variable


If X ∼ ϕ(0, 1), then a+ bX ∼ ϕ(a, b2).

If y ∼ ϕ(m,V ), then a+ by ∼ ϕ(a+ bm, b2V ).

Since dWt ∼ ϕ(0, dt), the instantaneous price changedSt = µStdt + σStdWt ∼ ϕ(µStdt, σ

2S2t dt).

The instantaneous return dSS = µdt + σdWt ∼ ϕ(µdt, σ2dt).

Under the BMS model, µ is the annualized mean of the instantaneousreturn — instantaneous mean return.σ2 is the annualized variance of the instantaneous return —instantaneous return variance.σ is the annualized standard deviation of the instantaneous return —instantaneous return volatility.


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Geometric Brownian motion

dSt/St = µdt + σdWt

The stock price is said to follow a geometric Brownian motion.

µ is often referred to as the drift, and σ the diffusion of the process.

Instantaneously, the stock price change is normally distributed,ϕ(µStdt, σ

2S2t dt).

Over longer horizons, the price change is lognormally distributed.

The log return (continuous compounded return) is normally distributed overall horizons:d ln St =

(µ− 1

2σ2)dt + σdWt . (By Ito’s lemma)

d ln St ∼ ϕ(µdt − 12σ

2dt, σ2dt).lnSt ∼ ϕ(lnS0 + µt − 1

2σ2t, σ2t).

lnST/St ∼ ϕ((µ− 1

2σ2)(T − t), σ2(T − t)

).

Integral form: St = S0eµt− 1

2σ2t+σWt , ln St = ln S0 + µt − 1

2σ2t + σWt


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Normal versus lognormal distribution

dSt = µStdt + σStdWt , µ = 10%, σ = 20%,S0 = 100, t = 1.

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

PD

F of

ln(S

t/S0)

ln(St/S

0)

50 100 150 200 2500

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

PD

F o

f St

St

S0=100

The earliest application of Brownian motion to finance is by Louis Bachelier in hisdissertation (1900) “Theory of Speculation.” He specified the stock price asfollowing a Brownian motion with drift:

dSt = µdt + σdWt


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Simulate 100 stock price sample paths

dSt = µStdt + σStdWt , µ = 10%, σ = 20%,S0 = 100, t = 1.

0 50 100 150 200 250 300−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

Dai

ly re

turn

s

Days0 50 100 150 200 250 300

60

80

100

120

140

160

180

200

Stoc

k pr

ice

days

Stock with the return process: d ln St = (µ− 12σ

2)dt + σdWt .

Discretize to daily intervals dt ≈ ∆t = 1/252.

Draw standard normal random variables ε(100× 252) ∼ ϕ(0, 1).

Convert them into daily log returns: Rd = (µ− 12σ

2)∆t + σ√∆tε.

Convert returns into stock price sample paths: St = S0e∑252

d=1 Rd .


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Ito’s lemma on continuous processes

Given a dynamics specification for xt , Ito’s lemma determines the dynamics ofany functions of x and time, yt = f (xt , t). Example: Given dynamics on St ,derive dynamics on ln St/S0.For a generic continuous process xt ,

dxt = µxdt + σxdWt ,

the transformed variable y = f (x , t) follows,

dyt =

(ft + fxµx +

1

2fxxσ

2x

)dt + fxσxdWt

You can think of it as aTaylor expansion up to dt term, with (dW )2 = dt

Example 1: dSt = µStdt + σStdWt and y = ln St . We haved ln St =

(µ− 1

2σ2)dt + σdWt

Example 2: dvt = κ (θ − vt) dt + ω√vtdWt , and σt =

√vt .

dσt =

(1

2σ−1t κθ − 1

2κσt −

1

8ω2σ−1

t

)dt +

1

2ωdWt

Example 3: dxt = −κxtdt + σdWt and vt = a+ bx2t . ⇒ dvt =?.


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Ito’s lemma on jumps

For a generic process xt with jumps,

dxt = µxdt + σxdWt +∫g(z) (µ(dz , dt)− ν(z , t)dzdt) ,

= (µx − Et [g ]) dt + σxdWt +∫g(z)µ(dz , dt),

where the random counting measure µ(dz , dt) realizes to a nonzero value for agiven z if and only if x jumps from xt− to xt = xt− + g(z) at time t. The processν(z , t)dzdt compensates the jump process so that the last term is the incrementof a pure jump martingale. The transformed variable y = f (x , t) follows,

dyt =(ft + fxµx +

12 fxxσ

2x

)dt + fxσxdWt

+∫(f (xt− + g(z), t)− f (xt−, t))µ(dz , dt)−

∫fxg(z)ν(z , t)dzdt,

=(ft + fx (µx − Et [g ]) +

12 fxxσ

2x

)dt + fxσxdWt

+∫(f (xt− + g(z), t)− f (xt−, t))µ(dz , dt)


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Ito’s lemma on jumps: Merton (1976) example

dyt =(ft + fxµx +

12 fxxσ

2x

)dt + fxσxdWt

+∫(f (xt− + g(z), t)− f (xt−, t))µ(dz , dt)−

∫fxg(z)ν(z , t)dzdt,

=(ft + fx (µx − Et [g(z)]) +

12 fxxσ

2x

)dt + fxσxdWt

+∫(f (xt− + g(z), t)− f (xt−, t))µ(dz , dt)

Merton (1976)’s jump-diffusion process:

dSt = µSt−dt + σSt−dWt +∫St− (ez − 1) (µ(z , t)− ν(z , t)dzdt) , and

y = ln St . We have

d lnSt =

(µ− 1

2σ2

)dt + σdWt +

∫zµ(dz , dt)−

∫(ez − 1) ν(z , t)dzdt

f (xt− + g(z), t)− f (xt−, t) = lnSt − ln St− = ln(St−ez)− ln St− = z .

The specification (ez − 1) on price guarantees that the maximum downsidejump is no larger than the pre-jump price level St−.

In Merton, ν(z , t) = λ 1σJ

√2πe− (z−µJ )

2

2σ2J .


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The key idea behind BMS option pricing

Under the binomial model, if we cut time step ∆t small enough, thebinomial tree converges to the distribution behavior of the geometricBrownian motion.

Reversely, the Brownian motion dynamics implies that if we slice the timethin enough (dt), it behaves like a binomial tree.

Under this thin slice of time interval, we can combine the option withthe stock to form a riskfree portfolio, like in the binomial case.

Recall our hedging argument: Choose ∆ such that f −∆S is riskfree.

The portfolio is riskless (under this thin slice of time interval) and mustearn the riskfree rate.

Since we can hedge perfectly, we do not need to worry about riskpremium and expected return. Thus, µ does not matter for thisportfolio and hence does not matter for the option valuation.

Only σ matters, as it controls the width of the binomial branching.


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The hedging proof

The stock price dynamics: dS = µSdt + σSdWt .

Let f (S , t) denote the value of a derivative on the stock, by Ito’s lemma:dft =

(ft + fSµS + 1

2 fSSσ2S2)dt + fSσSdWt .

At time t, form a portfolio that contains 1 derivative contract and −∆ = fSof the stock. The value of the portfolio is P = f − fSS .

The instantaneous uncertainty of the portfolio is fSσSdWt − fSσSdWt = 0.Hence, instantaneously the delta-hedged portfolio is riskfree.

Thus, the portfolio must earn the riskfree rate: dP = rPdt.

dP = df − fSdS =(ft + fSµS + 1

2 fSSσ2S2 − fSµS

)dt = r(f − fSS)dt

This lead to the fundamental partial differential equation (PDE):ft + rSfS + 1

2 fSSσ2S2 = rf .

If the stock pays a continuous dividend yield q, the portfolio P&L needs tobe modified as dP = df − fS(dS + qSdt), where the stock investment P&Lincludes both the capital gain (dS) and the dividend yield (qSdt).

The PDE then becomes: ft + (r − q)SfS + 12 fSSσ

2S2 = rf .

No where do we see the drift (expected return) of the price dynamics (µ) inthe PDE.Liuren Wu (Baruch) Option Pricing Introduction Options Markets 37 / 78

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Partial differential equation

The hedging argument leads to the following partial differential equation:

∂f

∂t+ (r − q)S

∂f

∂S+

1

2σ2S2 ∂

2f

∂S2= rf

The only free parameter is σ (as in the binomial model).

Solving this PDE, subject to the terminal payoff condition of the derivative(e.g., fT = (ST − K )+ for a European call option), BMS derive analyticalformulas for call and put option value.

Similar formula had been derived before based on distributional(normal return) argument, but µ was still in.More importantly, the derivation of the PDE provides a way to hedgethe option position.

The PDE is generic for any derivative securities, as long as S followsgeometric Brownian motion.

Given boundary conditions, derivative values can be solved numericallyfrom the PDE.


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How effective is the BMS delta hedge in practice?

If you sell an option, the BMS model says that you can completely removethe risk of the call by continuously rebalancing your stock holding toneutralize the delta of the option.

We perform the following experiment using history S&P options data from1996 - 2011:

Sell a call option

Record the P&L from three strategies

H Hold: Just hold the option.S Static: Perform delta hedge at the very beginning, but with no further

rebalancing.D Dynamic: Actively rebalance delta at the end of each day.

“D” represents a daily discretization of the BMS suggestion.

Compared to “H” (no hedge), how much do you think the daily discretizedBMS suggestion (“D”) can reduce the risk of the call?

[A] 0, [B] 5%, [C ] 50%, [D] 95%


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Example: Selling a 30-day at-the-money call option

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Days

Mean S

quare

d H

edgin

g E

rror

HoldStaticDynamics

0 5 10 15 20 25 3050

55

60

65

70

75

80

85

90

95

100

Days

Perc

enta

ge V

ariance

Reduct

ion

StaticDynamic

Unhedged, the PL variance can be over 100% of the option value.

Un-rebalanced, the delta hedge deteriorates over time as the delta is nolonger neutral after a whole.

Rebalanced daily, 95% of the risk can be removed over the whole sampleperiod.

The answer is “D.”


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Hedging options at different maturities and time periods

Different maturities Different years

0 5 10 15 20 25 3050

55

60

65

70

75

80

85

90

95

100

Days

Perc

enta

ge V

ariance

Reduct

ion

1 month2 month5 month12 month

1996 1998 2000 2002 2004 2006 2008 201050

55

60

65

70

75

80

85

90

95

100

YearP

erc

enta

ge V

ariance

Reduct

ion

1 month2 month5 month12 month

Dynamic delta hedge works equally well on both short and long-term options.

Hedging performance deteriorates in the presence of large moves.


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The BMS formula for European options

ct = Ste−q(T−t)N(d1)− Ke−r(T−t)N(d2),

pt = −Ste−q(T−t)N(−d1) + Ke−r(T−t)N(−d2),

where

d1 =ln(St/K)+(r−q)(T−t)+ 1

2σ2(T−t)

σ√T−t

,

d2 =ln(St/K)+(r−q)(T−t)− 1

2σ2(T−t)

σ√T−t

= d1 − σ√T − t.

Black derived a variant of the formula for futures (which I like better):

ct = e−r(T−t) [FtN(d1)− KN(d2)],

with d1,2 =ln(Ft/K)± 1

2σ2(T−t)

σ√T−t

.

Recall: Ft = Ste(r−q)(T−t).

Once I know call value, I can obtain put value via put-call parity:ct − pt = e−r(T−t) [Ft − Kt ].


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Implied volatility

ct = e−r(T−t) [FtN(d1)− KN(d2)] , d1,2 =ln(Ft/K )± 1

2σ2(T − t)

σ√T − t

Since Ft (or St) is observable from the underlying stock or futures market,(K , t,T ) are specified in the contract. The only unknown (and hence free)parameter is σ.

We can estimate σ from time series return (standard deviation calculation).

Alternatively, we can choose σ to match the observed option price —implied volatility (IV).

There is a one-to-one correspondence between prices and implied volatilities.

Traders and brokers often quote implied volatilities rather than dollar prices.

The BMS model says that IV = σ. In reality, the implied volatilitycalculated from different option contracts (across different strikes,maturities, dates) are usually different.


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Real versus risk-neutral dynamics

Recall: Under the binomial model, we derive a set of risk-neutralprobabilities such that we can calculate the expected payoff from the optionand discount them using the riskfree rate.

Risk premiums are hidden in the risk-neutral probabilities.If in the real world, people are indeed risk-neutral, the risk-neutralprobabilities are the same as the real-world probabilities. Otherwise,they are different.

Under the BMS model, we can also assume that there exists such anartificial risk-neutral world, in which the expected returns on all assets earnrisk-free rate.

The stock price dynamics under the risk-neutral world becomes,dSt/St = (r − q)dt + σdWt .

Simply replace the actual expected return (µ) with the return from arisk-neutral world (r − q) [ex-dividend return].

We label the true probability measure by P andthe risk-neutral measure by Q.


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The risk-neutral return on spots

dSt/St = (r − q)dt + σdWt , under risk-neutral probabilities.

In the risk-neutral world, investing in all securities make the riskfree rate asthe total return.

If a stock pays a dividend yield of q, then the risk-neutral expected returnfrom stock price appreciation is (r − q), such as the total expected return is:dividend yield+ price appreciation =r .

Investing in a currency earns the foreign interest rate rf similar to dividendyield. Hence, the risk-neutral expected currency appreciation is (r − rf ) sothat the total expected return is still r .

Regard q as rf and value options as if they are the same.


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The risk-neutral return on forwards/futures

If we sign a forward contract, we do not pay anything upfront and we do notreceive anything in the middle (no dividends or foreign interest rates). AnyP&L at expiry is excess return.

Under the risk-neutral world, we do not make any excess return. Hence, theforward price dynamics has zero mean (driftless) under the risk-neutralprobabilities: dFt/Ft = σdWt .

The carrying costs are all hidden under the forward price, making the pricingequations simpler.


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Return predictability and option pricing

Consider the following stock price dynamics,

dSt/St = (a+ bXt) dt + σdWt ,dXt = κ (θ − Xt) + σxdW2t , ρdt = E[dWtdW2t ].

Stock returns are predictable by a vector of mean-reverting predictors Xt . Howshould we price options on this predictable stock?

Option pricing is based on replication/hedging — a cross-sectional focus,not based on prediction — a time series behavior.

Whether we can predict the stock price is irrelevant. The key is whether wecan replicate or hedge the option risk using the underlying stock.

Consider a derivative f (S , t), whose terminal payoff depends on S but not onX , then it remains true that dft =

(ft + fSµS + 1

2 fSSσ2S2)dt + fSσSdWt .

The delta-hedged option portfolio (f − fSS) remains riskfree.

The same PDE holds for f : ft + rSfS + 12 fSSσ

2S2 = rf , which has nodependence on X . Hence, the assumption f (S , t) (instead of f (S ,X , t)) iscorrect.


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Return predictability and risk-neutral valuation

Consider the following stock price dynamics (under P),

dSt/St = (a+ bXt) dt + σdWt ,dXt = κ (θ − Xt) + σxdW2t , ρdt = E[dWtdW2t ].

Under the risk-neutral measure (Q), stocks earn the riskfree rate, regardlessof the predictability: dSt/St = (r − q) dt + σdWt

Analogously, the futures price is a martingale under Q, regardless of whetheryou can predict the futures movements or not: dFt/Ft = σdWt

Measure change from P to Q does not change the diffusion componentσdW .

The BMS formula still applies — Option pricing is dictated by the volatility(σ) of the uncertainty, not by the return predictability.


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Stochastic volatility and option pricing

Consider the following stock price dynamics,

dSt/St = µdt +√vtdWt ,

dvt = κ (θ − vt) + ω√vtdW2t , ρdt = E[dWtdW2t ].

How should we price options on this stock with stochastic volatility?

Consider a call option on S . Even though the terminal value does not dependon vt , the value of the option at other periods have to depend on vt .

Assume f (S , t) does not depend on vt , and thus P = f − fSS is ariskfree portfolio, we havedPt = dft − fSdSt =

(ft + fSµS + 1

2 fSSvS2 − fSµS

)dt = r(f − fSS)dt.

ft + rfSS + 12 fSSvS

2 = rf . The PDE is a function of v .Hence, it must be f (S , v , t) instead of f (S , t).

For f (S , v , t), we have (bivariate version of Ito):dft =

(ft + fSµS + 1

2 fSSvtS2 + fvκ (θ − vt) +

12 fvvω

2vt + fSvωvtρ)dt +

fSσS√vtdWt + fvω

√vtdW2t .

Since there are two sources of risk (Wt ,W2t), delta-hedging is not going tolead to a riskfree portfolio.


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Pricing kernel

The real and risk-neutral dynamics can be linked by a pricing kernel.

In the absence of arbitrage, in an economy, there exists at least one strictlypositive process, Mt , the state-price deflator, such that the deflated gainsprocess associated with any admissible strategy is a martingale. The ratio ofMt at two horizons, Mt,T , is called a stochastic discount factor, orinformally, a pricing kernel.

For an asset that has a terminal payoff ΠT , its time-t value ispt = EP

t [Mt,TΠT ].The time-t price of a $1 par riskfree zero-coupon bond that expires atT is pt = EP

t [Mt,T ].In a discrete-time representative agent economy with an additive CRRAutility, the pricing kernel is equal to the ratio of the marginal utilities ofconsumption,

Mt,t+1 = βu′(ct+1)

u′(ct)= β

(ct+1

ct

)−γ


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From pricing kernel to exchange rates

Let Mht,T denote the pricing kernel in economy h that prices all securities in

that economy with its currency denomination.

The h-currency price of currency-f (h is home currency) is linked to thepricing kernels of the two economies by,

S fhT

S fht

=M f

t,T

Mht,T

The log currency return over period [t,T ], ln S fhT /S fh

t equals the differencebetween the log pricing kernels of the two economies.

Let S denote the dollar price of pound (e.g. St = $2.06), then

ln ST/St = lnMpoundt,T − lnMdollar

t,T .

If markets are completed by primary securities (e.g., bonds and stocks),there is one unique pricing kernel per economy. The exchange ratemovement is uniquely determined by the ratio of the pricing kernels.

If markets are not completed by primary securities, exchange rates (and theiroptions) help complete the markets by requiring that the ratio of any twoviable pricing kernels go through the exchange rate and their options.Liuren Wu (Baruch) Option Pricing Introduction Options Markets 51 / 78

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From pricing kernel to measure change

In continuous time, it is convenient to define the pricing kernel via thefollowing multiplicative decomposition:

Mt,T = exp

(−∫ T

t

rsds

)E

(−∫ T

t

γ⊤s dXs

)

r is the instantaneous riskfree rate, E is the stochastic exponentialmartingale operator, X denotes the risk sources in the economy, and γ is themarket price of the risk X .

If Xt = Wt , E(−∫ T

tγsdWs

)= exp

(−∫ T

tγsdWs − 1

2

∫ T

tγ2s ds).

In a continuous time version of the representative agent example,dXs = d ln ct and γ is relative risk aversion.r is usually a function of X .

The exponential martingale E(·) defines the measure change from P to Q.


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Measure change defined by exponential martingales

Consider the following exponential martingale that defines the measure changefrom P to Q:dQdP∣∣t= E

(−∫ t

0γsdWs

),

If the P dynamics is: dS it = µiS i

tdt + σiS itdW

it , with ρidt = E[dWtdW

it ],

then the dynamics of S it under Q is: dS i

t =µiS i

tdt + σiS itdW

it + E[−γtdWt , σ

iS itdW

it ] =

(µit − γσiρi

)S itdt + σiS i

tdWit

If S it is the price of a traded security, we need r = µi

t − γσiρi . The riskpremium on the security is µi

t − r = γσiρi .

How do things change if the pricing kernel is given by:

Mt,T = exp(−∫ T

trsds

)E(−∫ T

tγ⊤s dWs

)E(−∫ T

tγ⊤s dZs

),

where Zt is another Brownian motion independent of Wt or Wit .


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Measure change defined by exponential martingales: Jumps

Let X denote a pure jump process with its compensator being ν(x , t) underP.Consider a measure change defined by the exponential martingale:dQdP∣∣t= E (−γXt),

The compensator of the jump process under Q becomes:ν(x , t)Q = e−γxν(x , t).

Example: Merton (176)’s compound Poisson jump process,

ν(x , t) = λ 1σJ

√2πe− (x−µJ )

2

2σ2J . Under Q, it becomes

ν(x , t)Q = λ 1σJ

√2πe−γx− (x−µJ )

2

2σ2J = λQ 1

σJ

√2πe−

(x−µQJ)2

2σ2J with µQ

J = µJ − γσ2J

and λQ = λe12γ(γσ

2J−2µJ ).

If we start with a symmetric distribution (µJ = 0) under P, positive riskaversion (γ > 0) leads to a negatively skewed distribution under Q.Vassilis Polimenis, Skewness Correction for Asset Pricing, wp.

Reference: Kuchler & Sørensen, Exponential Families of Stochastic Processes, Springer, 1997.


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Implied volatility

Recall the BMS formula:

c(S , t,K ,T ) = e−r(T−t) [Ft,TN(d1)− KN(d2)] , d1,2 =ln

Ft,T

K ± 12σ

2(T − t)

σ√T − t

The BMS model has only one free parameter, the asset return volatility σ.

Call and put option values increase monotonically with increasing σ underBMS.

Given the contract specifications (K ,T ) and the current marketobservations (St ,Ft , r), the mapping between the option price and σ is aunique one-to-one mapping.

The σ input into the BMS formula that generates the market observedoption price (or sometimes a model-generated price) is referred to as theimplied volatility (IV).

Practitioners often quote/monitor implied volatility for each option contractinstead of the option invoice price.


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The relation between option price and σ under BMS

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

45C

all o

ptio

n va

lue,

ct

Volatility, σ

K=80K=100K=120

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

45

50

Put o

ptio

n va

lue,

pt

Volatility, σ

K=80K=100K=120

An option value has two components:

Intrinsic value: the value of the option if the underlying price does notmove (or if the future price = the current forward).Time value: the value generated from the underlying price movement.

Since options give the holder only rights but no obligation, larger movesgenerate bigger opportunities but no extra risk — Higher volatility increasesthe option’s time value.

At-the-money option price is approximately linear in implied volatility.


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Implied volatility versus σ

If the real world behaved just like BMS, σ would be a constant.

In this BMS world, we could use one σ input to match market quoteson options at all days, all strikes, and all maturities.Implied volatility is the same as the security’s return volatility (standarddeviation).

In reality, the BMS assumptions are violated. With one σ input, the BMSmodel can only match one market quote at a specific date, strike, andmaturity.

The IVs at different (t,K ,T ) are usually different — direct evidencethat the BMS assumptions do not match reality.IV no longer has the meaning of return volatility.IV still reflects the time value of the option.The intrinsic value of the option is model independent (e.g.,e−r(T−t)(F −K )+ for call), modelers should only pay attention to timevalue.


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Implied volatility at (t,K ,T )

At each date t, strike K , and expiry date T , there can be two Europeanoptions: one is a call and the other is a put.

The two options should generate the same implied volatility value to excludearbitrage.

Recall put-call parity: c − p = er(T−t)(F − K ).The difference between the call and the put at the same (t,K ,T ) isthe forward value.The forward value does not depend on (i) model assumptions, (ii) timevalue, or (iii) implied volatility.

At each (t,K ,T ), we can write the in-the-money option as the sum of theintrinsic value and the value of the out-of-the-money option:

If F > K , call is ITM with intrinsic value er(T−t)(F − K ), put is OTM.Hence, c = er(T−t)(F − K ) + p.If F < K , put is ITM with intrinsic value er(T−t)(K − F ), call is OTM.Hence, p = c + er(T−t)(K − F ).If F = K , both are ATM (forward), intrinsic value is zero for bothoptions. Hence, c = p.


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Implied volatility quotes on OTC currency options

At each date t and each fixed time-to-maturity τ = (T − t), OTC currencyoptions are quoted in terms of

Delta-neutral straddle implied volatility (ATMV):A straddle is a portfolio of a call & a put at the same strike. The strikehere is set to make the portfolio delta-neutral:eq(T−t)N(d1)− eq(T−t)N(−d1) = 0 ⇒ N(d1) =

12 ⇒ d1 = 0.

25-delta risk reversal: RR25 = IV (∆c = 25)− IV (∆p = 25).

25-delta butterfly spreads:BF25 = (IV (∆c = 25) + IV (∆p = 25))/2− ATMV .

When trading currency options OTC, options and the corresponding BMSdelta of the underlying are exchanged at the same time.


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From delta to strikes

Given these quotes, we can compute the IV at the three moneyness (strike)levels:

IV = ATMV at d1 = 0IV = BF25 + ATMV + RR25/2 at ∆c = 25%IV = BF25 + ATMV − RR25/2 at ∆p = 25%

The three strikes at the three deltas can be inverted as follows:

K = F exp(12 IV

2τ)

at d1 = 0K = F exp

(12 IV

2τ − N−1(∆ceqτ )IV

√τ)

at ∆c = 25%K = F exp

(12 IV

2τ + N−1(|∆p|eqτ )IV√τ)

at ∆p = 25%

Put-call parity is guaranteed in the OTC quotes: one implied volatility ateach delta/strike.

These are not exactly right... They are simplified versions of the muchmessier real industry convention.


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Violations of put-call parities

On the exchange, calls and puts at the same maturity and strike arequoted/traded separately. It is possible to observe put-call parity beingviolated at some times.

Violations can happen when there are market frictions such asshort-sale constraints.For American options (on single name stocks), there only exists aput-call inequality.The “effective” maturities of the put and call American options withsame strike and expiry dates can be different.

The option with the higher chance of being exercised early haseffectively a shorter maturity.

Put-call violations can predict future spot price movements.

One can use the call and put option prices at the same strike tocompute an “option implied spot price:” So

t = (ct − pt + e−rτK ) eqτ .The difference between the implied spot price and the price from thestock market can contain predictive information: St − So

t .

Compute implied forward for option pricing model estimation.


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The information content of the implied volatility surface

At each time t, we observe options across many strikes K and maturitiesτ = T − t.

When we plot the implied volatility against strike and maturity, we obtain animplied volatility surface.

If the BMS model assumptions hold in reality, the BMS model should beable to match all options with one σ input.

The implied volatilities are the same across all K and τ .The surface is flat.

We can use the shape of the implied volatility surface to determine whatBMS assumptions are violated and how to build new models to account forthese violations.

For the plots, do not use K , K − F , K/F or even lnK/F as themoneyness measure. Instead, use a standardized measure, such aslnK/F

ATMV√τ, d2, d1, or delta.

Using standardized measure makes it easy to compare the figuresacross maturities and assets.


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Implied volatility smiles & skews on a stock

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 20.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75AMD: 17−Jan−2006

Moneyness=ln(K/F )

σ√

τ

Impl

ied

Vola

tility Short−term smile

Long−term skew

Maturities: 32 95 186 368 732


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Implied volatility skews on a stock index (SPX)

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 20.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22SPX: 17−Jan−2006

Moneyness=ln(K/F )

σ√

τ

Impl

ied

Vola

tility

More skews than smiles

Maturities: 32 60 151 242 333 704


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Average implied volatility smiles on currencies

10 20 30 40 50 60 70 80 9011

11.5

12

12.5

13

13.5

14

Put delta

Ave

rage im

plie

d v

ola

tility

JPYUSD

10 20 30 40 50 60 70 80 908.2

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

Put delta

Ave

rage

impl

ied

vola

tility

GBPUSD

Maturities: 1m (solid), 3m (dashed), 1y (dash-dotted)


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Return non-normalities and implied volatility smiles/skews

BMS assumes that the security returns (continuously compounding) arenormally distributed. ln ST/St ∼ N

((µ− 1

2σ2)τ, σ2τ

).

µ = r − q under risk-neutral probabilities.

A smile implies that actual OTM option prices are more expensive thanBMS model values.

⇒ The probability of reaching the tails of the distribution is higherthan that from a normal distribution.⇒ Fat tails, or (formally) leptokurtosis.

A negative skew implies that option values at low strikes are more expensivethan BMS model values.

⇒ The probability of downward movements is higher than that from anormal distribution.⇒ Negative skewness in the distribution.

Implied volatility smiles and skews indicate that the underlying securityreturn distribution is not normally distributed (under the risk-neutralmeasure —We are talking about cross-sectional behaviors, not time series).


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Quantifying the linkage

IV (d) ≈ ATMV

(1 +

Skew.

6d +

Kurt.

24d2

), d =

lnK/F

σ√τ

If we fit a quadratic function to the smile, the slope reflects the skewness ofthe underlying return distribution.

The curvature measures the excess kurtosis of the distribution.

A normal distribution has zero skewness (it is symmetric) and zero excesskurtosis.

This equation is just an approximation, based on expansions of the normaldensity (Read “Accounting for Biases in Black-Scholes.”)

The currency option quotes: Risk reversals measure slope/skewness,butterfly spreads measure curvature/kurtosis.Check the VOLC function on Bloomberg.


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Revisit the implied volatility smile graphics

For single name stocks (AMD), the short-term return distribution is highlyfat-tailed. The long-term distribution is highly negatively skewed.

For stock indexes (SPX), the distributions are negatively skewed at bothshort and long horizons.

For currency options, the average distribution has positive butterfly spreads(fat tails).

Normal distribution assumption does not work well.

Another assumption of BMS is that the return volatility (σ) is constant —Check the evidence in the next few pages.


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Stochastic volatility on stock indexes

96 97 98 99 00 01 02 030.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Impl

ied

Vol

atili

ty

SPX: Implied Volatility Level

96 97 98 99 00 01 02 030.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Impl

ied

Vol

atili

ty

FTS: Implied Volatility Level

At-the-money implied volatilities at fixed time-to-maturities from 1 month to 5years.


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Stochastic volatility on currencies

1997 1998 1999 2000 2001 2002 2003 2004

8

10

12

14

16

18

20

22

24

26

28

Impl

ied

vola

tility

JPYUSD

1997 1998 1999 2000 2001 2002 2003 2004

5

6

7

8

9

10

11

12

Impl

ied

vola

tility

GBPUSD

Three-month delta-neutral straddle implied volatility.


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Stochastic skewness on stock indexes

96 97 98 99 00 01 02 030.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Impl

ied

Vol

atili

ty D

iffer

ence

, 80%

−12

0%

SPX: Implied Volatility Skew

96 97 98 99 00 01 02 030

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Impl

ied

Vol

atili

ty D

iffer

ence

, 80%

−12

0%

FTS: Implied Volatility Skew

Implied volatility spread between 80% and 120% strikes at fixedtime-to-maturities from 1 month to 5 years.


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Stochastic skewness on currencies

1997 1998 1999 2000 2001 2002 2003 2004

−20

−10

0

10

20

30

40

50

RR

10 a

nd B

F10

JPYUSD

1997 1998 1999 2000 2001 2002 2003 2004

−15

−10

−5

0

5

10

RR

10 a

nd B

F10

GBPUSD

Three-month 10-delta risk reversal (blue lines) and butterfly spread (red lines).


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What do the implied volatility plots tell us?

Returns on financial securities (stocks, indexes, currencies) are not normallydistributed.

They all have fatter tails than normal (most of the time).The distribution is also skewed, mostly negative for stock indexes (andsometimes single name stocks), but can be of either direction (positiveor negative) for currencies.

The return distribution is not constant over time, but varies strongly.

The volatility of the distribution is not constant.Even higher moments (skewness, kurtosis) of the distribution are notconstant, either.

A good option pricing model should account for return non-normality and itsstochastic (time-varying) feature.


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Abnormal implied volatility shapes

Sometimes, the implied volatility plot against moneyness can show weird shapes:

Implied volatility frown — This does not happen often, but it does happen.

It is more difficult to model than a smile.

The implied volatility shape around corporate actions such as mergers,takeovers, etc.

Although most of our models are time homogeneous, calendar day effectsare important considerations in practice. How to reconcile the two?

Build a business calendar and modify the option time to maturity basedon business activities:

Treat non-trading days as zero (or a small fraction) of one day.

Treat each earnings announcement date as two days.

Pre-process the data as much as possible before applying to a model.

Most deterministic components should be pre-processed.


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Institutional applications of options:

⇒ gain volatility exposures

Buy/sell an out-of-the-money option and delta hedge.

Daily delta hedge is a requirement for most institutional volatilityinvestors and options market makers.

Call or put is irrelevant. What matter is whether one is long/short vega.

Delta hedged P&L is equal to dollar-gamma weighted variance

difference: PLT =∫ T

0er(T−t)(σ2

t − IV 20 )

S2t

2n(d1(St ,t,IV0))

St IV0

√T−t

dt.

Views/quotes are expressed not in terms of dollar option prices, but rather interms of implied volatilities (IV).

Implied volatilities are calculated from the Black-Merton-Scholes(BMS) model.

The fact that practitioners use the BMS model to quote options doesnot mean they agree with the BMS assumptions.

Rather, they use the BMS model as a way to transform/standardizethe option price, for several practical benefits.


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Why BMS implied volatility?

There are several practical benefits in transforming option prices into BMSimplied volatilities.

...1 Information: It is much easier to gauge/express views in terms of impliedvolatilities than in terms of option prices.

Option price behaviors all look alike under different dynamics: Optionprices are monotone and convex in strike...

By contrast, how implied volatilities behavior against strikes reveals theshape of the underlying risk-neutral return distribution.

A flat implied volatility plot against strike serves as a benchmark for anormal return distribution.

Deviation from a flat line reveals deviation from return normality.⇒ Implied volatility smile — leptokurtotic return distribution⇒ Implied volatility smirk — asymmetric return distribution


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Why BMS implied volatility?

...2 No arbitrage constraints:

Merton (1973): model-free bounds based on no-arb. arguments:Type I: No-arbitrage between European options of a fixed strike and maturity

vs. the underlying and cash:call/put prices ≥ intrinsic;call prices ≤ (dividend discounted) stock price;put prices ≤ (present value of the) strike price;put-call parity.

Type II: No-arbitrage between options of different strikes and maturities:bull, bear, calendar, and butterfly spreads ≥ 0.

Hodges (1996): These bounds can be expressed in implied volatilities.Type I: Implied volatility must be positive.

⇒If market makers quote options in terms of a positive implied volatilitysurface, all Type I no-arbitrage conditions are automatically guaranteed.

...3 Delta hedge: The standard industry practice is to use the BMS model tocalculate delta with the implied volatility as the input.No delta modification consistently outperforms this simple practice in allpractical situations.


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When to get away from implied volatility?

At very short maturities for out-of-the-money options.

The implied volatility is computed from a pure diffusion model, under whichout-of-money option value approaches zero with an exponential rate (veryfast) as the time to maturity nears zero.

The chance of getting far out there declines quickly in a diffusive world.

In the presence of jumps, out-of-money option value approaches zero at amuch lower rate (linear).

The chance of getting far out there can still be significant if theprocess can jump over.

The two difference convergence rates imply that implied volatilities forout-of-money options will blow up (become infinity) as the option maturityshrinks to zero, if the underlying process can jump.

One can use the convergence rate difference to test whether the underlyingprocess is purely continuous or contains jumps (Carr & Wu (2003, JF).)

Bid-ask spread can also make the implied volatility calculationdifficult/meaningless at very short maturities.


introduction to option pricing - baruch...

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