Hes Ton

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The Society for Financial Studies

A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond andCurrency OptionsAuthor(s): Steven L. HestonSource: The Review of Financial Studies, Vol. 6, No. 2 (1993), pp. 327-343

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A Closed-Form Solution forOptions with StochasticVolatility with Applicationsto Bond and CurrencyOptionsSteven L. HestonYale University

I use a new technique to derive a closed-form solu-tionfor theprice of a European call option on anasset with stochastic volatility. The model allowsarbitrary correlation between volatility and spot-asset returns. I introduce stochastic interest ratesand show how to apply the model to bond optionsand foreign currency options. Simulations showthat correlation between volatility and the spotasset's price is important for explaining returnskewness and strike-price biases in the Black-Scholes (1973) model. The solution technique isbased on characteristic functions and can beapplied to other problems.

Manyplaudits have been aptlyused to describe Blackand Scholes' (1973) contribution to option pricingtheory. Despite subsequent development of optiontheory,the originalBlack-Scholes formulafora Euro-pean call option remains the most successful andwidely used application. This formula is particularlyuseful because it relates the distributionof spotreturns

I thank Hans Knoch for computational assistance. I am grateful for thesuggestions of Hyeng Keun (the referee) and for comments by participantsat a 1992 National Bureau of Economic Research seminar and the Queen'sUniversity 1992 Derivative Securities Symposium. Any remaining errors aremy responsibility. Address correspondence to Steven L.Heston, Yale Schoolof Organization and Management, 135 Prospect Street, New Haven, CT06511.The Review of Financial Studies 1993 Volume 6, number 2, pp. 327-343? 1993 The Review of Financial Studies 0893-9454/93/$1.50

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The Review of Financial Studies / v 6 n 2 1993

to the cross-sectional properties of option prices. In this article, Igeneralize the model while retaining this feature.Although the Black-Scholes formula is often quite successful inexplaining stock option prices [Black and Scholes (1972)], it doeshave known biases [Rubinstein (1985)]. Its performance also is sub-stantially worse on foreign currency options [Melino and Turnbull(1990, 1991), Knoch (1992)]. This is not surprising, since the Black-Scholes model makes the strong assumptionthat (continuously com-pounded) stock returnsare normally distributed with known meanand variance. Since the Black-Scholes formula does not depend onthe mean spot return,it cannot be generalized by allowing the meanto vary.But the varianceassumptionis somewhat dubious. Motivatedby this theoretical consideration, Scott (1987), Hull and White (1987),and Wiggins (1987) have generalized the model to allow stochasticvolatility. Melino and Turnbull (1990, 1991) report that this approachis successful in explaining the prices of currency options. Thesepapers have the disadvantage that their models do not have closed-form solutions and require extensive use of numerical techniques tosolve two-dimensional partial differential equations. Jarrow and

Eisenberg (1991) and Stein and Stein (1991) assume that volatilityis uncorrelated with the spot asset and use an average of Black-Scholes formulavalues overdifferentpathsof volatility.But since thisapproach assumes that volatility is uncorrelated with spot returns, itcannot capture importantskewness effects that arise from such cor-relation. I offer a model of stochastic volatility that is not based onthe Black-Scholes formula. It provides a closed-form solution fortheprice of a European call option when the spot asset is correlatedwithvolatility, and it adapts the model to incorporate stochastic interestrates.Thus, the model can be applied to bond options and currencyoptions.

1. Stochastic Volatility ModelWebegin by assumingthat the spot asset at time tfollows the diffusion

dS(t) = tSdt + VSv tiSdzl (t), (1)where z1 t) is a Wiener process. If the volatilityfollows an Ornstein-Uhlenbeck process [e.g., used by Stein and Stein (1991)],dVv(tY = -fVvtdt + 6 dz2(t), (2)

then Ito's lemma shows that the variance v(t) follows the process328

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Closed-Form Solution for Options with Stochastic Volatility

dv(t) = [62 - 21v(t)]dt + 26V\ dz2(t). (3)This can be written as the familiarsquare-rootprocess [used by Cox,Ingersoll, and Ross (1985)]

dv(t) = K[O - v(t)]dt + o/vYt5~dz2(t), (4)where z2(t) has correlation p with z, (t). For simplicity at this stage,we assume a constant interest rate r. Therefore,the price at time t ofa unit discount bond that matures at time t + r is

P(t, t + r) = e--. (5)These assumptions are still insufficient to price contingent claimsbecause we have not yet made an assumption that gives the "priceof volatility risk." Standardarbitragearguments [Black and Scholes(1973), Merton (1973)] demonstratethat the value of any asset U(S,v, t) (including accruedpayments) must satisfythe partialdifferentialequation (PDE)

1 02U C2U 1 02U aU-vS2-~ + pcUVS ~+ -o2v-~ + rS-2 OS2 O9S-v 2 0 v22 Sa

+ {K[O - v(t)] - X(S, v, t)} - rU+ a =0. (6)ov atThe unspecified termX(S, v, t) represents the price of volatility risk,and must be independent of the particular asset. Lamoureux andLastrapes 1993) present evidence thatthis term is nonzero forequityoptions. To motivatethe choice of X(S,v, t), we note thatin Breeden's(1979) consumption-based model,

X(S,v, t) dt = yCov[dv, dC/C], (7)where C(t) is the consumption rateand y is the relative-riskaversionof an investor. Consider the consumption process that emerges inthe (general equilibrium) Cox, Ingersoll, and Ross (1985) model

dC(t) = i, v(t) Cdt + a c-/tCYdz3(t), (8)where consumption growth has constant correlation with the spot-asset return. This generates a riskpremium proportional to v,X(S, v,t) = Xv. Although we will use this form of the risk premium, thepricing results are obtained by arbitrageand do not depend on theother assumptionsof the Breeden (1979) or Cox, Ingersoll, and Ross(1985) models. However,we note that the model is consistent withconditional heteroskedasticity in consumption growth as well as inasset returns.In theory,the parameterXcould be determined by one

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The Review of Financial Studies/ v 6 n 2 1993

volatility-dependent asset and then used to price all other volatility-dependent assets.'A European call option with strikeprice K and maturingat time Tsatisfies the PDE (6) subject to the following boundaryconditions:

U(S, v, T) = Max(O, S-K),U(O, v, t) = O,OUau (o,v, t) = 1,(rSa (S, 0, t) + KO-a (S, 0, t)

- rU(S, 0, t) + Ut(S, 0, t) = 0,U(S, 0o, t) = S.

By analogy with the Black-Scholes formula,we guess a solution ofthe formC(S, v, t) = SP, - KP(t, T)P2, (10)

where the first erm is the present value of the spot assetupon optimalexercise, and the second term is the present value of the strike-pricepayment. Both of these terms must satisfythe original PDE (6). It isconvenient to write them in terms of the logarithm of the spot price

x = ln[S]. (11)Substituting the proposed solution (10) into the original PDE (6)shows that P1and P2must satisfythe PDEs1 92P. a2pi 1 02P. Pj

2 O)x2 Ox v + 2 2v O9v2+ (r + ujv)OP. OP.+ (a )V) At=0 (12)

forj= 1,2, whereul = ?1, u2 =-?V2, a= KO, b, = K + X-pp, b2 = K + X.

Forthe option price to satisfythe terminalcondition in Equation (9),these PDEs (12) are subject to the terminal condition

Pj(x, v, T; ln[K]) = 1(x2:n[q. (13)Thus, they may be interpreted as "adjusted" or "risk-neutralized"probabilities (See Coxand Ross (1976)). The Appendix explains thatwhen x follows the stochastic process

I This is analogous to extractingan impliedvolatilityparameter n the Black-Scholesmodel.

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dx(t) = [r + uJv]dt + V 7(t)5dz1(t), (14)dv = (aj - bjv)dt + uV(tGdz2(t),where the parameters uJ, a1, and bjare defined as before, then Pjisthe conditional probabilitythat the option expires in-the-money:

Pj(X, V, T; ln[K]) = Pr[x(T) - ln[K] I x(t) = x, v(t) = v]. (15)The probabilities arenot immediatelyavailablein closed form.How-ever, the Appendix shows that their characteristicfunctions, tf(x, v,T;0) andf2(x, v, T;c) respectively,satisfythe same PDEs (12), subjectto the terminalcondition

f1(x, v, T; 4) = eikx. (16)The characteristicfunction solution is

fj(x, v, t; 0) = ec(1- ; ) + D(T- t;)v+ iOx (17)where

C(r; r)=rqii + a (bj - pa0i + d)-r -2 ln[ gedT]b. paoid Li g Jj'b1-pohi+ dFledTD(&; ) a2- gedTjand

b- paoi + dg bj-ppai- d'd V\(pai- bj)2 - u2(2uji- 02)

One can invert the characteristicfunctions to get the desired prob-abilities:P(x, v, T; ln[K]) = + - Re e1n xv, T; )]d. (18)

The integrand in Equation (18) is a smooth function that decaysrapidlyand presents no difficulties.2Equations (10), (17), and (18) give the solution for Europeancalloptions. In general, one cannot eliminate the integrals in Equation(18), even in the Black-Scholes case. However, they canbe evaluatedin a fraction of a second on a microcomputerby using approximations

2 Note that characteristic functions always exist; Kendall and Stuart (1977) establish that the integralconverges.

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similar to the standardones used to evaluatecumulativenormalprob-abilities.32. Bond Options, Currency Options, and Other Extensions

One can incorporatestochastic interest rates into the option pricingmodel, following Merton(1973) and Ingersoll (1990). In this manner,one can apply the model to options on bonds or on foreign currency.This section outlines these generalizations to show the broad appli-cability of the stochastic volatility model. These generalizations areequivalent to the model of the previous section, except that certainparametersbecome time-dependent to reflect the changing charac-teristics of bonds as they approachmaturity.To incorporatestochastic interest rates,we modify Equation (1) toallow time dependence in the volatility of the spot asset:

dS(t) = AsSdt + aS(t)VjSdz1(t). (19)This equation is satisfied by discount bond prices in the Cox, Inger-soll, and Ross (1985) model and multiple-factormodels of Heston(1990). Although the results of this section do not depend on thespecific form of as, if the spot asset is a discount bond then Usmustvanish at maturity n order for the bond price to reachpar with prob-ability 1. The specificationof the drift term.uss unimportantbecauseit will not affect option prices. We specify analogous dynamics forthe bond price:

dP(t; T) = ,ApP(t;T)dt + cp(t)VvjtYP(t; T)dz2(t). (20)Note that, for parsimony,we assume that the variances of both thespot asset and the bond are determined by the same variable v(t). Inthis model, the valuation equation is

1 02U 1 02U 1 C02U2 ?5(t)2VS2 s2 + -02 (t)VP20 + a2 2+ pSP5(t)Oap(t)vSP aP + Psv5s th)aVS dv

asU au a+ Ppvop(t)cravPZ + rS - + rP-+P Av as OP+ (K[O - V(t)] - XV)-a - rU + -u = 0 (149v 49 ~ ~ (21

I Note thatwhen evaluatingmultiple optionswith different trike options, one need not recomputethe characteristic unctionswhen evaluating he integralin Equation(18).

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where px denotes the correlationbetween stochasticprocesses x andy. Proceeding with the substitution (10) exactly as in the previoussection shows that the probabilities P1 and P2must satisfy the PDE:1 a2Op1 1 (92p.x(t)2 v-2 + PXV(t)0cX(t)-v J + - U2V(V22 Oxx OxOv 2 Ov

+ uj(t)va + (aj- bj(t)v) + -- 0, (22)ox Ov O9tforj= 1,2, where

x= ln[P(t T)]aX(t) 2= ?2S(t)2 - p5PSu(t)Up(t) + ?2o2 (t),Pxv(t) = P axs(t)a -ap(t)ul (t) = ?a2X(t), U2(t) =-2ax(t) 2,

a= K@,bl(t) = K + X - p&,u5(t)U, b2(t) = K + X-pp,,p(t) U.

Note that Equation (22) is equivalent to Equation (12) with sometime-dependent coefficients.The availabilityof closed-form solutionsto Equation (22) will depend on the particular erm structuremodel[e.g., the specification of ax(t)]. In any case, the method used in theAppendix shows that the characteristic function takes the form ofEquation (17), where the functions C(r) and D(r) satisfy certainordinarydifferentialequations. The option price is then determinedby Equation (18). While the functions C(r) and D(r) may not haveclosed-formsolutions forsome termstructuremodels, this representsan enormous reduction compared to solving Equation (21) numeri-cally.One can also apply the model when the spot asset S(t) is the dollarprice of foreigncurrency.We assume that the foreign price of aforeigndiscount bond, F(t; T), follows dynamicsanalogous to the domesticbond in Equation (20):

dF(t; T) = ,upF(t;T)dt + cp(t)V-JGYF(t;T)dz2(t). (23)Forclarity,we denote the domestic interest rateby rDand the foreigninterest rateby rF.Following the argumentsin Ingersoll (1990), thevaluation equation is

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1 12U 1 O2U 1 O2U 1 CO2U-oS(t)2VS2 + -202(t)vP2 8 + - 2 (t)VF2 8 + -u2v2 OS2 2~ OP2 2 OF2 2 O9v2a2u O2U+ Pspas(t)p(t) vsPd + PSPoS(t)F(t) VSF SOFO2U ___a2+ PSPP(t)cF(t) VPFd + pSvUS(t)vS Ov

_2U a2U aU aU+l(pap v + P aF(t)VFF + rDS - + rJ'-P~~~cAt)cTVP0Ov OFov OS OP+ rFF- + (K[O - v(t)] - Xv)- - rU + - = 0.clF OV Ot (24)

Solvingthis five-variablePDEnumericallywould be completely infea-sible. But one can use Garmenand Kohlhagen's (1983) substitutionanalogous to Equation (10):C(S, v, t) = SF(t, T)P1 - KP(t, T)P2. (25)

ProbabilitiesP1and P2 must satisfythe PDE1 C2P. 2p. 1 02P. OP-U"(t)2V X2 + p 'JUt)av()r Jd + - 2Va2Vd2 + uj(t) v ",IL2x Ox V xO4v 2 0v29 x

OP. O9P.+ (aj- b(t) v) - + - =O, (26)O9v O9tforj= 1,2, where

X=InSF(t; T)ltP(t; T)].(t)2= ?2S(t)2 + ?U2 p(t) + ?24(t) - pspoS(t)op(t)

+ PSF4S(t)rF(t) - PPFPW(t)?F(t),

~()=Psvos (t)c -f pPV? ( t)cTf + PFv/1F(t) fPxv(t) PS= S6 - PPvUPff +PaW

ul(t) = ?ox(t)2, u2(t) = -V20rX(t)2,a = KO,

b1(t)= K + X - psvas(t)a - PFv0F(t), b2(t) = K + X PPvUPW(t)c.Once again,the characteristic unction hasthe formof Equation(17),where C(r) and D(T) depend on the specificationof cx(t), pxv(t),andbj(t) (see the Appendix).334

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Although the stochastic interest rate models of this section aretractable, they would be more complicated to estimate than the sim-pler model of the previous section. For short-maturityoptions onequities, anyincrease in accuracywould likely be outweighed by theestimation error introduced by implementing a more complicatedmodel. As option maturitiesextend beyond one year, however, theinterest rate effects can become more important[Koch (1992)]. Themore complicatedmodels illustratehow the stochasticvolatilitymodelcan be adapted to a variety of applications. For example, one couldvalue U.S. options by adding on the early exercise approximationofBarone-Adesiand Whalley (1987). The solution technique has otherapplications,too. See the Appendixforapplicationto Steinand Stein's(1991) model (with correlated volatility) and see Bates (1992) forapplication to jump-diffusionprocesses.

3. Effects of the Stochastic Volatility Model Options PricesIn this section, I examine the effects of stochasticvolatilityon optionsprices andcontrastresultswith the Black-Scholes model. Manyeffectsare related to the time-series dynamics of volatility. For example, ahigher variance v(t) raises the prices of all options, justas it does inthe Black-Scholes model. In the risk-neutralizedpricing probabili-ties, the variance follows a square-rootprocess

dv(t) = K*[0* -v(t)]dt + aV~v(t)Yz2(t), (27)where

K* = K + X and 0* K/ (K + X)We analyze the model in terms of this risk-neutralizedvolatilitypro-cess instead of the "true"process of Equation (4), because the risk-neutralizedprocess exclusively determinesprices.4The variancedriftstowarda long-runmean of 0*,with mean-reversion peed determinedby K*. Hence, an increase in the average variance 0 * increases theprices of options. The mean reversion then determines the relativeweights of the currentvarianceand the long-runvarianceon optionprices. When mean reversion is positive, the variance has a steady-state distribution [Cox, Ingersoll, and Ross (1985)] with mean 0*.Therefore, spot returns over long periods will have asymptoticallynormal distributions, with variance per unit of time given by 0*.Consequently, the Black-Scholes model should tend to work wellfor long-term options. However, it is importantto realize that the

I This occurs for exactly the same reason that the Black-Scholes formula does not depend on themean stock return. See Heston (1992) for a theoretical analysis that explains when parameters dropout of option prices.

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Table 1Default parameters for simulation of option pricesdS(t) = AS dt + Vv3(t)S dz,(t), (10)dv(t) = X*[6* - v(t)]dt + ar%/ji{dz22(t). (30)

Parameter ValueMean reversion K* = 2Long-run variance 0* = .01Current variance v(t) = .01Correlation of zl(t) and z2(t) p = 0.Volatility of volatility parameter a= .1Option maturity .5 yearInterest rate r = 0Strike price K= 100

implied variance 0* from option prices may not equal the varianceof spot returns given by the "true"process (4). This difference iscaused by the risk premium associated with exposure to volatilitychanges. AsEquation (27) shows, whether0* is largeror smallerthanthe true average variance0 depends on the sign of the risk-premiumparameterX.One could estimate 0* and other parametersby usingvalues implied by option prices. Alternatively,one could estimate 0and K from the true spot-price process. One could then estimate therisk-premiumparameterXby using average returns on option posi-tions that are hedged against the risk of changes in the spot asset.The stochastic volatility model can conveniently explain propertiesof option prices in termsof the underlyingdistributionof spot returns.Indeed, this is the intuitive interpretationof the solution (10), sinceP2 corresponds to the risk-neutralized probability that the optionexpires in-the-money.To illustrateeffects on options prices, we shalluse the default parameters n Table 1.5 For comparison, we shall usethe Black-Scholes model with avolatility parameter hat matches the(square root of the) variance of the spot return over the life of theoption.6 This normalization focuses attention on the effects of sto-chasticvolatilityon one option relativeto anotherby equalizing "aver-age" option model prices across differentspot prices. The correlationparameterp positively affectsthe skewness of spot returns.Intuitively,a positive correlation results in high variance when the spot assetrises, and this "spreads" the right tail of the probability density.Conversely, the left tail is associated with low variance and is notspread out. Figure 1 shows how a positive correlation of volatilitywith the spot return creates a fat right tail and a thin left tail in the

5These parameters roughly correspond to Knoch's (1992) estimates with yen and deutsche markcurrency options, assuming no risk premium associated with volatility. However, the mean-reversionparameter is chosen to be more reasonable.6 This variance can be determined by using the characteristic function.

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ProbabilityDensity 00p =.5.3-

0. 2-0. ?L

-0. 3 -0. 2 -0. 1 0.1 0.2 0.3Spot Return

Figure 1Conditional probability density of the continuously compounded spot return over a six-month horizonSpot-asset ynamics redS(t) = juSdt + \RjtjS dz,t), wheredv(t) = *[O v(t)]dt+ ev-{tYdz2(t).Exceptfor the correlationp between z, and z2 shown, parameter alues are shown in Table 1. Forcomparison, he probabilitydensities are normalizedto have zero mean and unit variance.

PriceDifference($)p = .5

0. :.0. 05

80 90. I0v 110. . L30.-0. 05 V \ A Spot Price($)

-0. 1LFigure 2Option prices from the stochastic volatility model minus Black-Scholes values with equalvolatility to option maturityExceptfor the correlationpbetweenz, andz2shown, parameter aluesareshownin Table1.Whenp = -.5 and p = .5, respectively,the Black-Scholes volatilities are7.10 percent and 7.04 percent,and at-the-moneyoption values are $2.83and $2.81.

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Probability Density = .40.6/ .20.0

.3,/3 \},a=O0.2-/0. I

-0. 3 -0. 2 -0. 1 0.1 0.2 0.3Spot Return

Figure 3Conditional probability density of the continuously compounded spot return over a six-month horizonSpot-asset dynamics are dS(t) =,uSdt+ \/~JtjSdz,(t), where dv(t) = -*[* v(t)]dt+ aV/it~dz2(t).Exceptfor the volatilityof volatility parametera shown, parameter alues are shown in Table 1.Forcomparison, he probabilitydensities arenormalized o have zero mean and unit variance.

distributionof continuouslycompoundedspotreturns.7 igure2 showsthatthis increasesthe pricesof out-of-the-moneyoptionsand decreasesthe pricesof in-the-moneyoptions relativeto the Black-Scholesmodelwith comparablevolatility. Intuitively,out-of-the-moneycall optionsbenefit substantiallyfrom a fat right tail and pay little penalty for anincreased probability of an average or slightly below average spotreturn. A negative correlation has completely opposite effects. Itdecreases the prices of out-of-the-money options relative to in-the-money options.The parametera controls the volatilityof volatility.When a is zero,the volatility is deterministic, and continuously compounded spotreturnshaveanormal distribution.Otherwise,a increases the kurtosisof spot returns.Figure 3 shows how this creates two fat tails in thedistribution of spot returns.As Figure 4 shows, this has the effect ofraising far-in-the-moneyand far-out-of-the-moneyoption prices andlowering near-the-money prices. Note, however, that there is littleeffect on skewness or on the overallpricing of in-the-moneyoptionsrelative to out-of-the-moneyoptions.These simulations show that the stochastic volatility model can

7This illustration s motivatedby Jarrow nd Rudd (1982) and Hull (1989).

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PriceDifference(S)0. 050. 025-

_ -4-0. 025- Spot Price ($)

-0. 05- zs.2 o-O . 075-\

Figure 4Option prices from the stochastic volatility model minus Black-Scholes values with equalvolatility to option maturityExceptfor the volatilityof volatilityparameter shown,parameter aluesare shownin Table 1. Inbothcurves,the Black-Scholesvolatility s 7.07percentand the at-the-money ptionvalueis $2.82.

produce a rich variety of pricing effects compared with the Black-Scholes model. The effects just illustrated assumed that variance wasat its long-run mean, 0*. In practice, the stochastic variance will driftabove and below this level, but the basic conclusions should notchange. An important insight from the analysis is the distinctionbetween the effects of stochastic volatility per se and the effects ofcorrelation of volatility with the spot return. If volatility is uncorre-lated with the spot return, then increasing the volatility of volatility(a) increases the kurtosis of spot returns, not the skewness. In thiscase, random volatility is associated with increases in the prices offar-from-the-money options relative to near-the-money options. Incontrast, the correlation of volatility with the spot return producesskewness. And positive skewness is associated with increases in theprices of out-of-the-money options relative to in-the-money options.Therefore, it is essential to choose properly the correlation of volatilitywith spot returns as well as the volatility of volatility.

4. ConclusionsI present a closed-form solution for options on assets with stochasticvolatility. The model is versatile enough to describe stock options,bond options, and currency options. As the figures illustrate, themodel can impart almost any type of bias to option prices. In partic-ular, it links these biases to the dynamics of the spot price and thedistribution of spot returns. Conceptually, one can characterize the

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option models in terms of the firstfour moments of the spot return(under the risk-neutral probabilities). The Black-Scholes (1973)model shows that the mean spot return does not affect option pricesat all, while variance has a substantial effect. Therefore, the pricinganalysisof this article controls for the variancewhen comparingoptionmodels with different skewness and kurtosis. The Black-Scholes for-mula produces option prices virtually identical to the stochastic vol-atility models for at-the-money options. One could interpret this assaying that the Black-Scholes model performs quite well. Alterna-tively, all option models with the same volatility are equivalent forat-the-moneyoptions. Sinceoptionsareusuallytradednear-the-money,this explains some of the empirical support for the Black-Scholesmodel. Correlationbetween volatility and the spot price is necessaryto generate skewness. Skewness in the distribution of spot returnsaffects he pricingof in-the-moneyoptions relativeto.out-of-themoneyoptions. Without this correlation, stochastic volatility only changesthe kurtosis. Kurtosisaffects the pricing of near-the-moneyversus far-from-the-money options.With proper choice of parameters,the stochastic volatility modelappears to be a very flexible and promising description of optionprices. It presents a number of testable restrictions, since it relatesoption pricing biases to the dynamics of spot prices and the distri-bution of spot returns. Knoch (1992) has successfully used the modelto explain currency option prices. The model mayeventually explainother option phenomena. For example, Rubinstein (1985) foundoption biases thatchanged throughtime. There is also some evidencethat implied volatilities from options prices do not seem properlyrelated to future volatility. The model makes it feasible to examinethese puzzles and to investigate other features of option pricing.Finally,the solution technique itself canbe applied to other problemsand is not limited to stochastic volatility or diffusion problems.

Appendix: Derivation of the Characteristic FunctionsThis appendix derives the characteristicfunctions in Equation (17)and shows how to apply the solution technique to other valuationproblems. Suppose that x(t) and v(t) follow the (risk-neutral) pro-cesses in Equation (15). Consider any twice-differentiable functionf(x, v, t) that is a conditional expectation of some function of x andv at a later date, T, g(x(T), v(T)):

f(x, v, t) = E[g(x(T), v(T)) I x(t) = x, v(t) = v]. (Al)340

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Ito's lemma shows thatdfj! i+ Of+! 2vL+ (r + Ov fJ\(2 OX2 O Ov 2 Cv2 ( )x

+ (a- bjv)Lf+ L) dt+ (r+ uv)- f dz, + (a- b,v) f dz2. (A2)

By iterated expectations, we know thatf must be a martingale:E[df] = 0. (A3)

Applying this to Equation (A2) yields the Fokker-Planck forwardequation: 1 0l2f O2f 1 Ofvd_ + pav + - va2 Ox" OlxOcv 2 0v+ (r + ujv) Of + (a Ob.0'f tf (A)Olx Olv at

[see KarlinandTaylor(1975) for moredetails]. Equation(Al) imposesthe terminal conditionf(x, v, T) = g(x, v). (A5)

This equation has manyuses. Ifg(x, v) = 6(x - x0), then the solutionis the conditional probabilitydensity at time t thatx( T) = x,. And ifg(x, v) = 1jx2In[K]j) then the solution is the conditional probabilityattime t that x(T) is greaterthan ln[K]. Finally, if g(x, v) = ex, thenthe solution is the characteristicfunction. For properties of charac-teristic functions, see Feller (1966) orJohnson and Kotz (1970).To solve for the characteristicfunction explicitly, we guess thefunctional formf(x, v, t) = exp[C(T- t) + D(T- t)v+ iox]. (A6)

This "guess"exploits the linearityof the coefficientsin the PDE(A2).Following Ingersoll (1989, p. 397), one can substitute this functionalforminto the PDE (A2) to reduce it to two ordinarydifferentialequa-tions,

12q62+ pa1iD + 1D2 + uODi-biD + d = ?'nfi+ aD+ - = 0? (A7)Oi3t

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subject toC(O) = 0, D(O) = 0.

These equations can be solved to produce the solution in the text.One can apply the solution technique of this article to other prob-lems in which the characteristicfunctions are known. Forexample,Stein and Stein (1991) specify a stochasticvolatilitymodel of the formdVv7t = [a - fA/ t] dt + 6 dz2(t), (A8)

From Ito's lemma, the process for the varianceisdv(t) = [62 + 2a\v - 23v ]dt + 26\/vYtYdz2(t) (A9)

Although Stein and Stein (1991) assume that the volatilityprocess isuncorrelated with the spot asset, one can generalize this to allowz1(t) and z2(t) to have constantcorrelation.The solution method ofthis article applies directly, except that the characteristicfunctionstake the formffx, v, t; 0) = exp[C(T- t) + D(T- t)v + E(T- t)\yv + X6x].(A10)

Bates (1992) provides additional applications of the solution tech-nique to mixed jump-diffusionprocesses.

ReferencesBarone-Adesi, G., and R. E. Whalley, 1987, "Efficient Analytic Approximation of American OptionValues," Journal of Finance, 42, 301-320.Bates, D. S., 1992, "Jumps and Stochastic Processes Implicit in PHLX Foreign Currency Options,"working paper, Wharton School, University of Pennsylvania.Black, F., and M. Scholes, 1972, "The Valuation of Option Contracts and a Test of Market Efficiency,"Journal of Finance, 27, 399-417.Black, F., and M. Scholes, 1973, "The Valuation of Options and Corporate Liabilities," Journal ofPolitical Economy, 81, 637-654.Breeden, D. T., 1979, "An Intertemporal Asset Pricing Model with Stochastic Consumption andInvestment Opportunities," Journal of Financial Economics, 7, 265-296.Cox, J. C., J. E. Ingersoll, and S. A. Ross, 1985, "A Theory of the Term Structure of Interest Rates,"Econometrica, 53, 385-408.Cox, J. C., and S. A. Ross, 1976, "The Valuation of Options for Alternative Stochastic Processes,"Journal of Financial Economics, 3, 145-166.Eisenberg, L. K., and R. A. Jarrow, 1991, "Option Pricing with Random Volatilities in CompleteMarkets," Federal Reserve Bank of Atlanta Working Paper 91-16.Feller, W., 1966, An Introduction to Probability Theory and Its Applications (Vol. 2), Wiley, NewYork.

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Garman, M. B., and S. W. Kohlhagen, 1983, "Foreign Currency Option Values," Journal of Inter-national Money and Finance, 2, 231-237.Heston, S. L., 1990, "Testing Continuous Time Models of the Term Structure of Interest Rates,"Ph.D. Dissertation, Carnegie Mellon University Graduate School of Industrial Administration.Heston, S. L., 1992, "Invisible Parameters in Option Prices," working paper, Yale School of Orga-nization and Management.Hull, J. C., 1989, Options, Futures, and Other Derivative Instruments, Prentice-Hall, EnglewoodCliffs, NJ.Hull, J. C., and A. White, 1987, "The Pricing of Options on Assets with Stochastic Volatilities,"Journal of Finance, 42, 281-300.Ingersoll, J. E., 1989, Theory of Financial Decision Making Rowman and Littlefield, Totowa, NJ.Ingersoll, J. E., 1990, "Contingent Foreign Exchange Contracts with Stochastic Interest Rates,"working paper, Yale School of Organization and Management.Jarrow, R., and A. Rudd, 1982, "Approximate Option Valuation for ArbitraryStochastic Processes,"Journal of Financial Economics, 10, 347-369.Johnson, N. L., and S. Kotz, 1970, Continuous Univariate Distributions, Houghton Mifflin, Boston.Karlin, S., and H. M. Taylor, 1975, A First Course in Stochastic Processes, Academic, New York.Kendall, M., and A. Stuart, 1977, The Advanced Theory of Statistics (Vol. 1), Macmillan, New York.Knoch, H. J., 1992, "The Pricing of Foreign Currency Options with Stochastic Volatility," Ph.D.Dissertation, Yale School of Organization and Management.Lamoureux, C. G., and W. D. Lastrapes, 1993, "Forecasting Stock-Return Variance: Toward anUnderstanding of Stochastic Implied Volatilities," Review of Financial Studies, 6, 293-326.Melino, A., and S. Turnbull, 1990, "The Pricing of Foreign Currency Options with StochasticVolatility," Journal of Econometrics, 45, 239-265.Melino, A., and S. Turnbull, 1991, "The Pricing of Foreign Currency Options," Canadian Journalof Economics, 24, 251-281.Merton, R. C., 1973, "Theory of Rational Option Pricing," BellJournal of Economics and Man-agement Science, 4, 141-183.Rubinstein, M., 1985, "Nonparametric Tests of Alternative Option Pricing Models UsingAll ReportedTrades and Quotes on the 30 Most Active CBOE Option Classes from August 23, 1976 throughAugust 31, 1978," Journal of Finance, 40, 455-480.Scott, L.O., 1987, "Option Pricing When the Variance Changes Randomly: Theory, Estimation, andan Application," Journal of Financial and Quantitative Analysis, 22, 419-438.Stein, E. M., and J. C. Stein, 1991, "Stock Price Distributions with Stochastic Volatility: An AnalyticApproach," Review of Financial Studies, 4, 727-752.Wiggins,J. B., 1987, "Option Values under Stochastic Volatilities,"Journal ofFinancialEconomics,19, 351-372.

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