21
Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch * Case Studies in Financial Modelling Course Notes, Courant Institute of Mathematical Sciences, Fall Term, 2004 Abstract In the course of the following lectures, we will investigate why equity options are priced as they are. In so doing, we will apply many of the techniques students will have learned in previous semesters and develop some intuition for the pricing of both vanilla and exotic equity options. By considering specific examples, we will see that in pricing options, it is often as important to take into account the dynamics of underlying variables as it is to match known market prices of other claims. My hope is that these lectures will prove particularly useful to those who end up specializing in the structuring, pricing, trading and risk management of equity derivatives. * I am indebted to Peter Friz, our former Teaching Assistant, for carefully reading these lecture notes, providing corrections and suggesting useful improvements.

Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

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Page 1: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

Lecture 1: Stochastic Volatility andLocal Volatility

Jim Gatheral, Merrill Lynch∗

Case Studies in Financial Modelling Course Notes,Courant Institute of Mathematical Sciences,

Fall Term, 2004

Abstract

In the course of the following lectures, we will investigate whyequity options are priced as they are. In so doing, we will apply manyof the techniques students will have learned in previous semesters anddevelop some intuition for the pricing of both vanilla and exotic equityoptions. By considering specific examples, we will see that in pricingoptions, it is often as important to take into account the dynamics ofunderlying variables as it is to match known market prices of otherclaims. My hope is that these lectures will prove particularly usefulto those who end up specializing in the structuring, pricing, tradingand risk management of equity derivatives.

∗I am indebted to Peter Friz, our former Teaching Assistant, for carefully reading theselecture notes, providing corrections and suggesting useful improvements.

Page 2: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

1 Stochastic Volatility

1.1 Motivation

That it might make sense to model volatility as a random variable should beclear to the most casual observer of equity markets. To be convinced, oneneed only recall the stock market crash of October 1987. Nevertheless, giventhe success of the Black-Scholes model in parsimoniously describing marketoptions prices, it’s not immediately obvious what the benefits of makingsuch a modeling choice might be.

Stochastic volatility models are useful because they explain in a self-consistent way why it is that options with different strikes and expirationshave different Black-Scholes implied volatilities (“implied volatilities” fromnow on) – the “volatility smile”. In particular, traders who use the Black-Scholes model to hedge must continuously change the volatility assumptionin order to match market prices. Their hedge ratios change accordingly inan uncontrolled way. More interestingly for us, the prices of exotic optionsgiven by models based on Black-Scholes assumptions can be wildly wrongand dealers in such options are motivated to find models which can take thevolatility smile into account when pricing these.

From Figure 1, we see that large moves follow large moves and smallmoves follow small moves (so called “volatility clustering”). From Figures 2and 3 (which shows details of the tails of the distribution), we see that thedistribution of stock price returns is highly peaked and fat-tailed relative tothe Normal distribution. Fat tails and the high central peak are character-istics of mixtures of distributions with different variances. This motivatesus to model variance as a random variable. The volatility clustering featureimplies that volatility (or variance) is auto-correlated. In the model, this isa consequence of the mean reversion of volatility 1.

There is a simple economic argument which justifies the mean reversionof volatility (the same argument that is used to justify the mean reversionof interest rates). Consider the distribution of the volatility of IBM in onehundred years time say. If volatility were not mean-reverting ( i.e. if thedistribution of volatility were not stable), the probability of the volatilityof IBM being between 1% and 100% would be rather low. Since we believethat it is overwhelmingly likely that the volatility of IBM would in fact lie

1Note that simple jump-diffusion models do not have this property. After a jump, thestock price volatility does not change.

2

Page 3: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

Figure 1: SPX daily log returns from 1/1/1990 to 31/12/1999

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Figure 2: Frequency distribution of SPX daily log returns from 1/1/1990 to31/12/1999 compared with the Normal distribution

-0.06 -0.04 -0.02 0.02 0.04 0.06

10

20

30

40

50

60

in that range, we deduce that volatility must be mean-reverting.Having motivated the description of variance as a mean-reverting random

variable, we are now ready to derive the valuation equation.

3

Page 4: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

Figure 3: Tails of SPX frequency distribution

-0.06 -0.04 -0.02 0.02 0.04 0.06

0.25

0.5

0.75

1

1.25

1.5

1.75

2

1.2 Derivation of the Valuation Equation

In this section, we follow Wilmott (1998) closely. We suppose that the stockprice S and its variance v satisfy the following SDEs:

dS(t) = µ(t)S(t)dt +√

v(t)S(t)dZ1 (1)

dv(t) = α(S, v, t)dt + η β(S, v, t)√

v(t)dZ2 (2)

with〈dZ1 dZ2〉 = ρ dt

where µ(t) is the (deterministic) instantaneous drift of stock price returns,η is the volatility of volatility and ρ is the correlation between random stockprice returns and changes in v(t). dZ1 and dZ2 are Wiener processes.

The stochastic process (1) followed by the stock price is equivalent to theone assumed in the derivation of Black and Scholes (1973). This ensures thatthe standard time-dependent volatility version of the Black-Scholes formula(as derived in section 8.6 of Wilmott (1998) for example) may be retrievedin the limit η → 0. In practical applications, this is a key requirement of astochastic volatility option pricing model as practitioners’ intuition for thebehavior of option prices is invariably expressed within the framework of theBlack-Scholes formula.

4

Page 5: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

In the Black-Scholes case, there is only one source of randomness – thestock price, which can be hedged with stock. In the present case, randomchanges in volatility also need to be hedged in order to form a riskless port-folio. So we set up a portfolio Π containing the option being priced whosevalue we denote by V (S, v, t), a quantity −∆ of the stock and a quantity−∆1 of another asset whose value V1 depends on volatility. We have

Π = V −∆ S −∆1 V1

The change in this portfolio in a time dt is given by

dΠ =

{∂V

∂t+

1

2v S2(t)

∂2V

∂S2+ ρη vβ S(t)

∂2V

∂v∂S+

1

2η2vβ2∂2V

∂v2

}dt

−∆1

{∂V1

∂t+

1

2v S2(t)

∂2V1

∂S2+ ρη vβ S(t)

∂2V1

∂v∂S+

1

2η2vβ2∂2V1

∂v2

}dt

+

{∂V

∂S−∆1

∂V1

∂S−∆

}dS

+

{∂V

∂v−∆1

∂V1

∂v

}dv

To make the portfolio instantaneously risk-free, we must choose

∂V

∂S−∆1

∂V1

∂S−∆ = 0

to eliminate dS terms, and

∂V

∂v−∆1

∂V1

∂v= 0

to eliminate dv terms. This leaves us with

dΠ =

{∂V

∂t+

1

2v S2∂2V

∂S2+ ρηvβ S

∂2V

∂v∂S+

1

2η2 vβ2∂2V

∂v2

}dt

−∆1

{∂V1

∂t+

1

2v S2∂2V1

∂S2+ ρηvβ S

∂2V1

∂v∂S+

1

2η2 vβ2∂2V1

∂v2

}dt

= r Π dt

= r(V −∆S −∆1V1) dt

where we have used the fact that the return on a risk-free portfolio mustequal the risk-free rate r which we will assume to be deterministic for our

5

Page 6: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

purposes. Collecting all V terms on the left-hand side and all V1 terms onthe right-hand side, we get

∂V∂t

+ 12v S2 ∂2V

∂S2 + ρη v β S ∂2V∂v∂S

+ 12η2vβ2 ∂2V

∂v2 + rS ∂V∂S− rV

∂V∂v

=∂V1

∂t+ 1

2v S2 ∂2V1

∂S2 + ρη vβ S ∂2V1

∂v∂S+ 1

2η2vβ2 ∂2V1

∂v2 + rS ∂V1

∂S− rV1

∂V1

∂v

The left-hand side is a function of V only and the right-hand side is a functionof V1 only. The only way that this can be is for both sides to be equal tosome function f of the independent variables S, v and t. We deduce that

∂V

∂t+

1

2v S2∂2V

∂S2+ρη v β S

∂2V

∂v∂S+

1

2η2vβ2∂2V

∂v2+rS

∂V

∂S− rV = − (α− ϕβ)

∂V

∂v(3)

where, without loss of generality, we have written the arbitrary function fof S, v and t as (α− ϕβ). Conventionally, ϕ(S, v, t) is called the marketprice of volatility risk because it tells us how much of the expected returnof V is explained by the risk (i.e. standard deviation) of v in the CapitalAsset Pricing Model framework.

2 Local Volatility

2.1 History

Given the computational complexity of stochastic volatility models and theextreme difficulty of fitting parameters to the current prices of vanilla op-tions, practitioners sought a simpler way of pricing exotic options consis-tently with the volatility skew. Since before Breeden and Litzenberger(1978), it was understood that the risk-neutral pdf could be derived from themarket prices of European options. The breakthrough came when Dupire(1994) and Derman and Kani (1994) noted that under risk-neutrality, therewas a unique diffusion process consistent with these distributions. The cor-responding unique state-dependent diffusion coefficient σL(S, t) consistentwith current European option prices is known as the local volatility func-tion.

It is unlikely that Dupire, Derman and Kani ever thought of local volatil-ity as representing a model of how volatilities actually evolve. Rather, it is

6

Page 7: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

likely that they thought of local volatilities as representing some kind ofaverage over all possible instantaneous volatilities in a stochastic volatilityworld (an “effective theory”). Local volatility models do not therefore reallyrepresent a separate class of models; the idea is more to make a simplify-ing assumption that allows practitioners to price exotic options consistentlywith the known prices of vanilla options.

As if any proof had been needed, Dumas, Fleming, and Whaley (1998)performed an empirical analysis which confirmed that the dynamics of theimplied volatility surface were not consistent with the assumption of constantlocal volatilities.

In section 2.5, we will show that local volatility is indeed an averageover instantaneous volatilities, formalizing the intuition of those practition-ers who first introduced the concept.

2.2 A Brief Review of Dupire’s Work

For a given expiration T and current stock price S0 , the collection{C (S0, K, T ) ; K ∈ (0,∞)} of undiscounted option prices of different strikesyields the risk neutral density function ϕ of the final spot ST through therelationship

C (S0, K, T ) =∫ ∞

KdST ϕ (ST , T ; S0) (ST −K)

Differentiate this twice with respect to K to obtain

ϕ (K, T ; S0) =∂2C

∂K2

so the Arrow-Debreu prices for each expiration may be recovered by twicedifferentiating the undiscounted option price with respect to K. This processwill be familiar to any option trader as the construction of an (infinite size)infinitesimally tight butterfly around the strike whose maximum payoff isone.

Given the distribution of final spot prices ST for each time T conditionalon some starting spot price S0, Dupire shows that there is a unique risk neu-tral diffusion process which generates these distributions. That is, given theset of all European option prices, we may determine the functional form ofthe diffusion parameter (local volatility) of the unique risk neutral diffusion

7

Page 8: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

process which generates these prices. Noting that the local volatility will ingeneral be a function of the current stock price S0, we write this process as

dS

S= µ (t) dt + σ (S, t; S0) dZ

Application of Ito’s Lemma together with risk neutrality, gives rise to a par-tial differential equation for functions of the stock price which is a straightfor-ward generalization of Black-Scholes. In particular, the pseudo probabilitydensities ϕ (K, T ; S0) = ∂2C

∂K2 must satisfy the Fokker-Planck equation. Thisleads to the following equation for the undiscounted option price C in termsof the strike price K:

∂C

∂T=

σ2K2

2

∂2C

∂K2+ (r(T )−D(T ))

(C −K

∂C

∂K

)(4)

where r(t) is the risk-free rate, D(t) is the dividend yield and C is short forC (S0, K, T ). See the Appendix for a derivation of this equation.

Were we to express the option price as a function of the forward priceFT = S0 exp

{∫ T0 µ(t)dt

}2, we would get the same expression minus the drift

term. That is∂C

∂T=

1

2σ2 K2 ∂2C

∂K2

where C now represents C (FT , K, T ). Inverting this gives

σ2(K,T, S0) =∂C∂T

12

K2 ∂2C∂K2

(5)

The right hand side of equation (5) can be computed from known Europeanoption prices. So, given a complete set of European option prices for allstrikes and expirations, local volatilities are given uniquely by equation (5).

We can view equation (5) as a definition of the local volatility functionregardless of what kind of process (stochastic volatility for example) actuallygoverns the evolution of volatility.

2From now on, µ(T ) represents the risk-neutral drift of the stock price process whichis the risk-free rate r(T ) minus the dividend yield D(T )

8

Page 9: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

2.3 Transforming to Black-Scholes Implied VolatilitySpace

Market prices of options are quoted in terms of Black-Scholes implied volatil-ity σBS (K, T ; S0). In other words, we may write

C (S0, K, T ) = CBS (S0, K, σBS (S0, K, T ) , T )

It will be more convenient for us to work in terms of two dimensionlessvariables: the Black-Scholes implied total variance w defined by

w (S0, K, T ) ≡ σ2BS (S0, K, T ) T

and the log-strike y defined by

y = ln(

K

FT

)

where FT = S0 exp{∫ T

0 dt µ(t)}

gives the forward price of the stock at time0. In terms of these variables, the Black-Scholes formula for the future valueof the option price becomes

CBS (FT , y, w) = FT {N (d1)− eyN (d2)}= FT

{N

(− y√

w+

√w

2

)− eyN

(− y√

w−√

w

2

)}(6)

and the Dupire equation (4) becomes

∂C

∂T=

vL

2

{∂2C

∂y2− ∂C

∂y

}+ µ (T ) C (7)

with vL = σ2 (S0, K, T ) representing the local variance. Now, by takingderivatives of the Black-Scholes formula, we obtain

∂2CBS

∂w2=

(−1

8− 1

2 w+

y2

2 w2

)∂CBS

∂w

∂2CBS

∂y∂w=

(1

2− y

w

)∂CBS

∂w

∂2CBS

∂y2− ∂CBS

∂y= 2

∂CBS

∂w(8)

9

Page 10: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

We may transform equation (7) into an equation in terms of implied varianceby making the substitutions

∂C

∂y=

∂CBS

∂y+

∂CBS

∂w

∂w

∂y

∂2C

∂y2=

∂2CBS

∂y2+ 2

∂2CBS

∂y∂w

∂w

∂y+

∂2CBS

∂w2

(∂w

∂y

)2

+∂CBS

∂w

∂2w

∂y2

∂C

∂T=

∂CBS

∂T+

∂CBS

∂w

∂w

∂T=

∂CBS

∂w

∂w

∂T+ µ (T ) CBS

where the last equality follows from the fact that the only explicit depen-dence of the option price on T in equation (6) is through the forward price

FT = S0 exp{∫ T

0 dt µ (t)}

. Equation (4) now becomes (cancelling µ (T ) C

terms on each side)

∂CBS

∂w

∂w

∂T

=vL

2

∂CBS

∂y+

∂2CBS

∂y2− ∂CBS

∂w

∂w

∂y+ 2

∂2CBS

∂y∂w

∂w

∂y+

∂2CBS

∂w2

(∂w

∂y

)2

+∂CBS

∂w

∂2w

∂y2

=vL

2

∂CBS

∂w

2− ∂w

∂y+ 2

(1

2− y

w

)∂w

∂y+

(−1

8− 1

2w+

y2

2w2

) (∂w

∂y

)2

+∂2w

∂y2

Then, taking out a factor of ∂CBS

∂w and simplifying, we get

∂w

∂T= vL

1− y

w

∂w

∂y+

1

4

(−1

4− 1

w+

y2

w2

) (∂w

∂y

)2

+1

2

∂2w

∂y2

Inverting this gives our final result:

vL =∂w∂T

1− yw

∂w∂y

+ 14

(−1

4− 1

w + y2

w2

) (∂w∂y

)2+ 1

2∂2w∂y2

2.4 Special Case: No Skew

If the skew ∂w∂y

is zero3 , we must have

vL =∂w

∂T3Note that this implies that ∂

∂K σBS (S0,K, T ) is zero

10

Page 11: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

So the local variance in this case reduces to the forward Black-Scholes impliedvariance. The solution to this is of course

w (T ) =∫ T

0vL (t) dt

2.5 Local Variance as a Conditional Expectation ofInstantaneous Variance

In this section, we review the elegant derivation of Derman and Kani (1998).We assume the same stochastic process for the stock price as in equation (1)

but write it in terms of the forward price Ft,T = St exp{∫ T

t ds µs

}.

dFt,T =√

vtFt,T dZ (9)

Note that dFT,T = dST . The undiscounted value of a European option withstrike K expiring at time T is given by

C (S0, K, T ) = E[(ST −K)+

]

Differentiating once with respect to K gives

∂C

∂K= −E [θ (ST −K)]

where θ(·) is the Heaviside function. Differentiating again with respect toK gives

∂2C

∂K2= E [δ (ST −K)]

where δ(·) is the Dirac δ function.Now, a formal application of Ito’s Lemma to the terminal payoff of the

option (and using dFT,T = dST ) gives the identity

d (ST −K)+ = θ (ST −K) dST +1

2vT S2

T δ (ST −K) dT

Taking conditional expectations of each side, and using the fact that Ft,T isa Martingale, we get

dC = dE[(ST −K)+

]=

1

2E

[vT S2

T δ (ST −K)]

dT

11

Page 12: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

Also, we can write

E[vT S2

T δ (ST −K)]

= E [vT |ST = K ]1

2K2 E [ δ (ST −K)]

= E [vT |ST = K ]1

2K2 ∂2C

∂K2

Putting this together, we get

∂C

∂T= E [vT |ST = K ]

1

2K2 ∂2C

∂K2

Comparing this with the definition of local volatility (equation (5)), we seethat

σ2(K, T, S0) = E [vT |ST = K ]

That is, local variance is the risk-neutral expectation of the instantaneousvariance conditional on the final stock price ST being equal to the strikeprice K.

3 The Heston Model

3.1 The Model

The Heston model (Heston (1993)) corresponds to choosing α(S, v(t), t) =−λ(v(t) − v) and β(S, v, t) = 1 in equations (1) and (2). These equationsthen become

dS(t) = µ(t)S(t)dt +√

v(t)S(t)dZ1 (10)

anddv(t) = −λ(v(t)− v)dt + η

√v(t)dZ2 (11)

with〈dZ1 dZ2〉 = ρ dt

where λ is the speed of reversion of v(t) to its long term mean v.The process followed by v(t) may be recognized as a version of the square

root process described by Cox, Ingersoll, and Ross (1985). It is a (jump-free) special case of a so-called affine jump diffusion (AJD) which is roughlyspeaking a jump-diffusion process for which the drifts and covariances andjump intensities are linear in the state vector (which is {x, v} in this case with

12

Page 13: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

x = log(S)). Duffie, Pan, and Singleton (2000) show that AJD processes areanalytically tractable in general. The solution technique involves computingan “extended transform” which in the Heston case is a conventional Fouriertransform.

We now substitute the above values for α(S, v, t) and β(S, v, t) into thegeneral valuation equation (equation (3)). We obtain

∂V

∂t+

1

2v S2∂2V

∂S2+ ρη v S

∂2V

∂v∂S+

1

2η2v

∂2V

∂v2+ rS

∂V

∂S− rV

= (λ(v − v)− ϕ )∂V

∂v(12)

Now, to be able to use the AJD results, the market price of volatility riskalso needs to be affine. Various economic arguments can be made (see forexample Wiggins (1987)) that the market price of volatility risk ϕ should beproportional to the variance v. Then, let ϕ = θv for some constant θ.

Now define the risk-adjusted parameters λ′ and v′ through λ′ = λ −θ, λ′v′ = λv. Substituting this into equation (12) gives

∂V

∂t+

1

2v S2∂2V

∂S2+ ρη v S

∂2V

∂v∂S+

1

2η2v

∂2V

∂v2+ rS

∂V

∂S− rV

= (λ′(v − v′))∂V

∂v(13)

Note that equation (13) is now identical to equation (12) with no explicitrisk preference related parameters except that the parameters λ′ and v′ arenow risk adjusted. From now on we will drop the primes on λ′ and v′ andassume that we are dealing with the risk-adjusted parameters.

3.2 The Heston Solution for European Options

This section repeats the derivation of the Heston formula for the value of aEuropean-style option first presented in Heston (1993) but with rather moredetail than is provided in that paper.

Before solving equation (13) with the appropriate boundary conditions,we can simplify it by making some suitable changes of variable. Let K bethe strike price of the option, T be its expiry date and F (t, T ) the forwardprice of the stock index to expiry. Then let

x = ln

(F (t, T )

K

)

13

Page 14: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

Further, suppose that we consider only the future value to expiration Cof the European option price rather than its value today and define τ = T−t.Then equation (13) simplifies to

−∂C

∂τ+

1

2v C11 − 1

2v C1 +

1

2η2v C22 + ρη v C12 − λ(v − v) C2 = 0 (14)

where the subscripts 1 and 2 refer to differentiation with respect to x and vrespectively.

According to Duffie, Pan, and Singleton (2000), the solution of equation(14) has the form

C(x, v, τ) = K {ex P1(x,v,τ)−P0(x,v,τ)} (15)

where the first term in the brackets represents the pseudo-expectation of thefinal index level given that the option is in-the-money and the second termrepresents the pseudo-probability of exercise.

Substituting the proposed solution (15) into equation (14) shows that P0

and P1 must satisfy the equation

−∂Pj

∂τ+

1

2v∂2Pj

∂x2−

(12− j

)v∂Pj

∂x+

1

2η2v

∂2Pj

∂v2+ρηv

∂2Pj

∂x∂v+(a− bjv)

∂Pj

∂v= 0

(16)for j = 0, 1 where

a=λ v, bj=λ−jρη

subject to the terminal condition

limτ→0

Pj(x, v, τ) =

{1 if x > 00 if x ≤ 0

≡ θ(x) (17)

We solve equation (16) subject to the condition (17) using a Fourier trans-form technique. To this end define the Fourier transform of Pj through

P (k, v, τ) =∫ ∞

−∞dx e−ikx P (x, v, τ)

Then

P (k, v, 0) =∫ ∞

−∞dx e−ikx θ(x) =

1

ik

14

Page 15: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

The inverse transform is given by

P (x, v, τ) =∫ ∞

−∞dk

2πeikx P (k, v, τ) (18)

Substituting this into equation (16) gives

− ∂Pj

∂τ− 1

2k2 v Pj − (1

2− j) ik v Pj

+1

2η2 v

∂2Pj

∂v2+ ρη ikv

∂Pj

∂v+ (a− bjv)

∂Pj

∂v= 0 (19)

Now define

α = −k2

2− ik

2+ ijk

β = λ− ρηj − ρηik

γ =η2

2

Then equation (19) becomes

v

{α Pj − β

∂Pj

∂v+ γ

∂2Pj

∂v2

}+a

∂Pj

∂v−∂Pj

∂τ= 0 (20)

Now substitute

Pj(k, v, τ) = exp {C(k, τ) v + D(k, τ) v} Pj(k, v, 0)

=1

ikexp {C(k, τ) v + D(k, τ) v}

It follows that

∂Pj

∂τ=

{v∂C

∂τ+ v

∂D

∂τ

}Pj

∂Pj

∂v= D Pj

∂2Pj

∂v2= D2 Pj

15

Page 16: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

Then equation (20) is satisfied if

∂C

∂τ= λD

∂D

∂τ= α− β D + γ D2

= γ(D − r+)(D − r−) (21)

where we define

r± =β ±√β2 − 4αγ

2γ≡ β ± d

η2

Integrating (21) with the terminal conditions C(k, 0) = 0 and D(k, 0) = 0gives

D(k, τ) = r−1− e−dτ

1− ge−dτ

C(k, τ) = λ

{r−τ − 2

η2ln

(1− ge−dτ

1− g

)}

where we defineg ≡ r−

r+

Taking the inverse transform using equation (18) and performing the com-plex integration carefully gives the final form of the pseudo-probabilities Pj

in the form of an integral of a real-valued function.

Pj(x, v, τ) =1

2+

1

π

∫ ∞

0dk Re

{exp{Cj(k, τ) v + Dj(k, τ) v + ikx}

ik

}

This integration may be performed using standard numerical methods.It is worth noting that taking derivatives of the Heston formula with

respect to x or v in order to derive risk parameters is extremely straightfor-ward because the functions C(k, τ) and D(k, τ) are independent of x andv.

In Appendix B, we show that the Heston characteristic function is givenby

φT (u) = exp {C(u, τ) v + D(u, τ) v} (22)

as might be guessed from the form of the Heston formula.

16

Page 17: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

3.3 Simulating the Heston process

Recall the Heston process

dS = µSdt +√

v S dZ1

dv = −λ(v − v) dt + η√

v dZ2 (23)

with〈dZ1 dZ2〉 = ρ dt

A simple Euler discretization of the variance process may give rise to anegative variance. To deal with this problem, practitioners generally adoptone of two approaches: either the absorbing assumption: if v < 0 thenv = 0 or the reflecting assumption: if v < 0 then v = −v. In practice,with the kinds of parameter values that are required to fit equity index optionprices, a very high number of time-steps is required to achieve convergence.

As Paul Glasserman points out in his excellent book (Glasserman 2004),the problem of negative variances may be avoided by sampling from theexact transition law of the process. Broadie and Kaya (2003) show in detailhow this may be done for the Heston process but their method turns outalso to be very time consuming as it involves integration of a characteristicfunction expressed in terms of Bessel functions.

It is nevertheless instructive to follow their argument. The exact solutionof (23) may be written as

St = S0 exp{−1

2

∫ t

0vs ds + ρ

∫ t

0

√vs dZs +

√1− ρ2

∫ t

0

√vs dZ⊥

s

}

vt = v0 + λ v t− λ∫ t

0vs ds + η

∫ t

0

√vs dZs

with ⟨dZs dZ⊥

s

⟩= 0

The Broadie-Kaya simulation procedure is as follows:

• Generate a sample from the distribution of vt given v0.

• Generate a sample from the distribution of∫ t0 vs ds given vt and v0.

• Recover∫ t0

√vs dZs given

∫ t0 vs ds, vt and v0.

• Generate a sample from the distribution of St given∫ t0

√vs dZs and∫ t

0 vs ds.

17

Page 18: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

Note that in the final step, the distribution of∫ t0

√vs dZ⊥

s is normal withvariance

∫ t0 vs ds because dZ⊥

s and vs are independent by construction.Andersen and Brotherton-Ratcliffe (2001) suggest that processes like the

square-root variance process should be simulated by sampling from a distri-bution that is similar to the true distribution but not necessarily the same;this approximate distribution should have the same mean and variance asthe true distribution.

Applying their suggested approach to simulating the Heston process, wewould have to find the means and variances of

∫ t0

√vs dZs,

∫ t0 vs ds, vt and

v0.

References

Andersen, Leif B.G., and Rupert Brotherton-Ratcliffe, 2001, Extended LI-BOR market models with stochastic volatility, Discussion paper Gen ReSecurities.

Black, F., and M. Scholes, 1973, The pricing of options and corporate lia-bilities, Journal of Political Economy 81, 631–659.

Breeden, D., and R. Litzenberger, 1978, Prices of state-contingent claimsimplicit in option prices, Journal of Business 51, 621–651.

Broadie, Mark, and Ozgur Kaya, 2003, Exact simulation of stochastic volatil-ity and other affine jump diffusion processes, Discussion paper ColumbiaUniversity, Graduate School of Business.

Cox, John C., Jonathan E. Ingersoll, and Steven A. Ross, 1985, A theory ofthe term structure of interest rates, Econometrica 53, 385–407.

Derman, Emanuel, and Iraj Kani, 1994, Riding on a smile, Risk 7, 32–39.

, 1998, Stochastic implied trees: Arbitrage pricing with stochasticterm and strike structure of volatility, International Journal of Theoreticaland Applied Finance 1, 61–110.

Duffie, Darrell, Jun Pan, and Kenneth Singleton, 2000, Transform analysisand asset pricing for affine jump diffusions, Econometrica 68, 1343–1376.

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Page 19: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

Dumas, B., J. Fleming, and R. E. Whaley, 1998, Implied volatility functions:Empirical tests, The Journal of Finance 53.

Dupire, Bruno, 1994, Pricing with a smile, Risk 7, 18–20.

Glasserman, Paul, 2004, Monte Carlo Methods in Financial Engineering .chap. 3.4, pp. 120–134 (Springer Verlag: New York).

Heston, Steven L., 1993, A closed-form solution for options with stochas-tic volatility, with application to bond and currency options, Review ofFinancial Studies 6, 327–343.

Kohn, Robert V., 2000, PDE for finance class notes, .

Wiggins, J.B., 1987, Option values under stochastic volatility, Journal ofFinancial Economics 15, 351–372.

Wilmott, Paul, 1998, Derivatives. The Theory and Practice of FinancialEngineering . chap. 23, pp. 299–304 (John Wiley & Sons: Chichester).

19

Page 20: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

A Derivation of the Dupire Equation

Suppose the stock price diffuses with risk-neutral drift µ (t) and local volatil-ity σ (S, t) according to the equation:

dS

S= µ (t) dt + σ (S, t) dZ

The undiscounted risk-neutral value C (S0, K, T ) of a European option withstrike K and expiration T is given by

C (S0, K, T ) =∫ ∞

KdST ϕ (ST , T ; S0) (ST −K) (A-1)

Here ϕ (ST , T ; S0) is the pseudo probability density of the final spot at timeT . It evolves according to the Fokker-Planck equation4 :

1

2

∂2

∂S2T

(σ2S2

T ϕ)− S

∂ST

(µST ϕ) =∂ϕ

∂T

Differentiating with respect to K gives

∂C

∂K= −

∫ ∞

KdST ϕ (ST , T ; S0)

∂2C

∂K2= ϕ (K, T ; S0)

Now, differentiating (A-1) with respect to time gives

∂C

∂T=

∫ ∞

KdST

{∂

∂Tϕ (ST , T ; S0)

}(ST −K)

=∫ ∞

KdST

{1

2

∂2

∂S2T

(σ2S2

T ϕ)− ∂

∂ST

(µST ϕ)

}(ST −K)

Integrating by parts twice gives:

∂C

∂T=

σ2K2

2ϕ +

∫ ∞

KdST µST ϕ

=σ2K2

2

∂2C

∂K2+ µ (T )

(C −K

∂C

∂K

)

4See Section 5 of Robert Kohn’s PDE for Finance Class Notes for a very readableaccount of this topic

20

Page 21: Lecture 1: Stochastic Volatility and Local Volatility · Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch⁄ Case Studies in Financial Modelling Course

which is the Dupire equation when the underlying stock has risk-neutraldrift µ. That is, the forward price of the stock at time T is given by

F (T ) = S0 exp

{∫ T

0dt µ (t)

}

B Derivation of the Heston Characteristic Func-

tion

By definition, the characteristic function is given by

φT (u) ≡ E[eiuxT |xt = 0

]

The probability of the final log-stock price xT being greater than the strikeprice is given by

Pr(xT > x) = P0(x, v, τ) =1

2+

1

π

∫ ∞

0dk Re

{exp{C(k, τ) v + D(k, τ) v + i k x}

ik

}

with x = ln(St/K) and τ = T − t. Let the log-strike y be defined byy = ln(K/St) = −x. Then, the probability density function p(y) must begiven by

p(y) = −∂P0

∂y

=1

∫ ∞

−∞dk exp{C(k, τ) v + D(k, τ) v − i k y}

Then

φT (u) =∫ ∞

−∞dy p(y) eiuy

=1

∫ ∞

−∞dk exp{C(k, τ) v + D(k, τ) v}

∫ ∞

−∞dy ei(u−k)y

=∫ ∞

−∞dk exp{C(k, τ) v + D(k, τ) v} δ(u− k)

= exp{C(u, τ) v + D(u, τ) v}

21