Heston Jim Gatheral

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Lecture 1: Stochastic Volatility andLocal VolatilityJim Gatheral, Merrill LynchCase Studies in Financial Modelling Course Notes,Courant Institute of Mathematical Sciences,Fall Term, 2004AbstractIn 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 specic 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.1 Stochastic Volatility1.1 MotivationThat 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, its not immediately obvious what the benets 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 dierent strikes and expirationshave dierent 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 nd 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 dierent 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 justies 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 lie1Note that simple jump-diusion models do not have this property. After a jump, thestock price volatility does not change.2Figure 1: SPX daily log returns from 1/1/1990 to 31/12/1999-0.06-0.04-0.0200. 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 0.06102030405060in that range, we deduce that volatility must be mean-reverting.Having motivated the description of variance as a mean-reverting randomvariable, we are now ready to derive the valuation equation.3Figure 3: Tails of SPX frequency distribution-0.06 -0.04 -0.02 0.02 0.04 Derivation of the Valuation EquationIn this section, we follow Wilmott (1998) closely. We suppose that the stockprice S and its variance v satisfy the following SDEs:dS(t) = (t)S(t)dt +_v(t)S(t)dZ1 (1)dv(t) = (S, v, t)dt + (S, v, t)_v(t)dZ2 (2)withdZ1 dZ2 = dtwhere (t) is the (deterministic) instantaneous drift of stock price returns, is the volatility of volatility and 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 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.4In 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 containing the option being priced whosevalue we denote by V (S, v, t), a quantity of the stock and a quantity1 of another asset whose value V1 depends on volatility. We have = V S 1 V1The change in this portfolio in a time dt is given byd =_Vt + 12v S2(t)2VS2 + v S(t) 2VvS + 122v22Vv2_dt1_V1t + 12v S2(t)2V1S2 + v S(t) 2V1vS + 122v22V1v2_dt+_VS 1V1S _dS+_Vv 1V1v_dvTo make the portfolio instantaneously risk-free, we must chooseVS 1V1S = 0to eliminate dS terms, andVv 1V1v = 0to eliminate dv terms. This leaves us withd =_Vt + 12v S22VS2 + v S 2VvS + 122v22Vv2_dt1_V1t + 12v S22V1S2 + v S 2V1vS + 122v22V1v2_dt= r dt= r(V S 1V1) dtwhere 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 our5purposes. Collecting all V terms on the left-hand side and all V1 terms onthe right-hand side, we getVt + 12v S2 2VS2 + v S 2VvS + 122v2 2Vv2 + rSVS rVVv=V1t + 12v S2 2V1S2 + v S 2V1vS + 122v2 2V1v2 + rSV1S rV1V1vThe 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 thatVt +12v S22VS2+ v S 2VvS+122v22Vv2 +rSVS rV = ( ) Vv(3)where, without loss of generality, we have written the arbitrary function fof S, v and t as ( ). Conventionally, (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 Volatility2.1 HistoryGiven the computational complexity of stochastic volatility models and theextreme diculty of tting 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 diusion process consistent with these distributions. The cor-responding unique state-dependent diusion coecient L(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 is6likely that they thought of local volatilities as representing some kind ofaverage over all possible instantaneous volatilities in a stochastic volatilityworld (an eective 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 conrmed 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 rst introduced the concept.2.2 A Brief Review of Dupires WorkFor a given expiration T and current stock price S0 , the collection{C (S0, K, T) ; K (0, )} of undiscounted option prices of dierent strikesyields the risk neutral density function of the nal spot ST through therelationshipC (S0, K, T