IntroductionMotivationApproachHeston ModelValidationReduce to Black-ScholesResultsHeston ModelDiscussionConvergenceSummarySummaryIntroduction Approach Validation Results Discussion SummaryApplication of Moment ExpansionMethod to Option Square Root ModelYun ZhouAdvisor: Professor Steve HestonUniversity of MarylandMay 5, 20091 / 19Introduction Approach Validation Results Discussion SummaryMotivationBlack-Scholes Model successfully explain stock option priceEquity price follows a Geometric Brownian MotionAssumption: Log return is normal distribution with constantvolatilityReality: Log return is NOT normal distribution, volatility isNOT constant2 / 19Introduction Approach Validation Results Discussion SummaryComparison Between Heston Model and Black-ScholesVolatility Log Return DistributionBlack-Scholes Constant NormalHeston Model Stochastic Not Normal3 / 19Introduction Approach Validation Results Discussion SummaryMethods to Solve Heston ModelClosed Form Exact Solution (Heston, 1993)Fast Fourier Transform (Carr and Madan, 1999):Characteristic Function NeededMoment Expansion (This Project, 2009)can work for Stochastic Volatility Models (no exact solution,Characteristic Function hard to get)Other methods4 / 19Introduction Approach Validation Results Discussion SummaryWhat I did in this project?Get Moments of Log Return in Heston Model.Apply Gram-Charlier Expansion ApproximationCompare the Approximation with Exact SolutionDiscuss Convergence of this Method5 / 19Introduction Approach Validation Results Discussion SummaryHeston ModeldSt= rStdt +√νtStdWstdνt= κ(θ − νt)dt + σ√νtdWνtdWst, dWνtBrownian Motion with Correlation ρStStock Price at Time tνtVariance at Time tr Rate of Returnθ Average Varianceκ Mean Reversion Rateσ Volatility of Volatility6 / 19Introduction Approach Validation Results Discussion SummaryMoment Expansion MethodUse Backward Equation to get any order of momentsUse Gram-Charlier to seek an approximate distributionNormal distribution + series approximation related to moments andHermite PolynomialsReplace normal distribution by the approximate distribution inoption price formula7 / 19Introduction Approach Validation Results Discussion SummaryGram-Charlier Expansiong(z) = n(z)(1 +Pi =3µi−normii !Hi(z))z =ln(St/S0)−(r −σ2/2)tσ√tg(z) Approximate Distribution of Log Returnn(z) Probability Density Function of Standard NormalµiMoments of Desired DistributionnormiMoments of Standard Normal DistributionHi(z) Hermite Polynomial8 / 19Introduction Approach Validation Results Discussion SummaryOption PriceC = e−rTE (ST− K)+=e−rTR∞−∞(elnS0+(r −σ22)t+σ√T z− K)+n(z)dzReplace n(z) by g(z)Call(GC ) = Call(BS) +Pi =3Qi(µi− normi)QiCoefficient part involving integral of Hermite Polynomial9 / 19Introduction Approach Validation Results Discussion SummaryMoments ComputingAnalytical : Up to 4th order (Mathematica By Heston )Numerical : Matrix Exponential MethodThey are the same10 / 19Introduction Approach Validation Results Discussion SummaryValidationdSt= rStdt +√νtStdWstdνt= κ(θ − νt)dt + σ√νtdWνtMake Volatility as a constantσ = 0 and θ = νtMoments from Heston Model = Moments of Standard NormalCall Option Price by Gram Charlie = Call Option Price byBlack-ScholesNumerical Results make an agreement with above conditions11 / 19Introduction Approach Validation Results Discussion SummaryResults12 / 19Introduction Approach Validation Results Discussion SummaryRMSEFor Gram-Charlier, 4th order might be good13 / 19Introduction Approach Validation Results Discussion SummaryConvergence of Gram-Charlier ExpansionPoor Convergence Properties (Cramer 1957)Souce of Divergence: g(x) must fall to 0 faster than e−x24Cramer’s Condition for Convergence:R∞−∞ex24g(x)dx < ∞14 / 19Introduction Approach Validation Results Discussion SummaryExamples−5 0 5−0.100.10.20.30.40.50.60.70.8xprobability density Hermite n=4Hermite n=6Hermite n=8Hermite n=10Normal(0,0.5)−5 0 5−20−15−10−5051015xprobability density Hermite n=4Hermite n=6Hermite n=8Hermite n=10Normal(0,2)σ = 0.5 σ = 2Convergence Divergence15 / 19Introduction Approach Validation Results Discussion SummaryConvergence of g(x)PDF of Log Return in Heston Model (Dragulescu andYakovenko, 2002)Properties of PDFFall to Zero Slower than e−x24Cramers Condition can not hold16 / 19Introduction Approach Validation Results Discussion SummarySummaryMoment Expansion Method is applied to Stochastic VolatilityModel (Heston Model)Up to certain order of moments, adding higher moments cannot increase accuracy of the approximationConvergence condition is disscussed17 / 19Introduction Approach Validation Results Discussion SummaryAcknowledgementDr. Zimin and Dr. Balan for suggestions and feedbackDr. Heston advises me on the project18 / 19Introduction Approach Validation Results Discussion SummaryRefenercesCorrado and Su, 1996, Skewness and Kurtosis in SP 500 IndexReturns Implied by Option Prices, The Journal of FinancialResearch, Vol. XIX, No.2 175-192Corrado and Su, 1997, Implied Volatility Skews and Stock IndexSkewness and Kurtosis Implied by SP 500 Index Option Prices,Journal of Derivatives, 4, 8-19Heston, S.L., 1993, A Closed Form Solutions with StochasticVolatility with Applications to Bond and Currency Options, Reviewof Financial Studies, 6, 327-4419 /