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The Heston Model and Its Extensions in Matlab and C# Foundedin1807,JohnWiley&Sonsistheoldestindependentpublishingcompany in the United States. With offices in North America, Europe, Australia, and Asia, Wiley is globally committed to developing and marketing print and electronic products and services for our customers’ professional and personal knowledge and understanding. The Wiley Finance series contains books written specifically for finance and investment professionals as well as sophisticated individual investors and their financial advisors. Book topics range from portfolio management to e-commerce, riskmanagement,financialengineering,valuation,andfinancialinstrumentanalysis, aswellasmuchmore. Foralistofavailabletitles,visitourwebsiteatwww.WileyFinance.com The Heston Model and Its Extensions in Matlab and C# FABRICE DOUGLAS ROUAH Coverillustration:GillesGheerbrant,‘‘1234auhasard’’(1976); CGillesGheerbrant Coverdesign:GillesGheerbrant CopyrightC2013byFabriceDouglasRouah.Allrightsreserved. PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey. PublishedsimultaneouslyinCanada. 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HG6024.A3R67772013 332.64(cid:2)53028553–dc23 2013019475 PrintedintheUnitedStatesofAmerica. 10 9 8 7 6 5 4 3 2 1 Contents Foreword ix Preface xi Acknowledgments xiii CHAPTER1 TheHestonModelforEuropeanOptions 1 ModelDynamics 1 TheEuropeanCallPrice 4 TheHestonPDE 5 ObtainingtheHestonCharacteristicFunctions 10 SolvingtheHestonRiccatiEquation 12 DividendYieldandthePutPrice 17 ConsolidatingtheIntegrals 18 Black-ScholesasaSpecialCase 19 SummaryoftheCallPrice 22 Conclusion 23 CHAPTER2 IntegrationIssues,ParameterEffects,andVarianceModeling 25 RemarksontheCharacteristicFunctions 25 ProblemsWiththeIntegrand 29 TheLittleHestonTrap 31 EffectoftheHestonParameters 34 VarianceModelingintheHestonModel 43 MomentExplosions 56 BoundsonImpliedVolatilitySlope 57 Conclusion 61 CHAPTER3 DerivationsUsingtheFourierTransform 63 TheFourierTransform 63 RecoveryofProbabilitiesWithGil-PelaezFourierInversion 65 DerivationofGatheral(2006) 67 Attari(2004)Representation 69 CarrandMadan(1999)Representation 73 BoundsontheCarr-MadanDampingFactorandOptimalValue 76 TheCarr-MadanRepresentationforPuts 82 TheRepresentationforOTMOptions 84 Conclusion 89 v vi CONTENTS CHAPTER4 TheFundamentalTransformforPricingOptions 91 ThePayoffTransform 91 TheFundamentalTransformandtheOptionPrice 92 TheFundamentalTransformfortheHestonModel 95 OptionPricesUsingParseval’sIdentity 100 VolatilityofVolatilitySeriesExpansion 108 Conclusion 113 CHAPTER5 NumericalIntegrationSchemes 115 TheIntegrandinNumericalIntegration 116 Newton-CotesFormulas 116 GaussianQuadrature 121 IntegrationLimitsandKahlandJa¨ckelTransformation 130 IllustrationofNumericalIntegration 136 FastFourierTransform 137 FractionalFastFourierTransform 141 Conclusion 145 CHAPTER6 ParameterEstimation 147 EstimationUsingLossFunctions 147 SpeedinguptheEstimation 158 DifferentialEvolution 162 MaximumLikelihoodEstimation 166 Risk-NeutralDensityandArbitrage-FreeVolatilitySurface 170 Conclusion 175 CHAPTER7 SimulationintheHestonModel 177 GeneralSetup 177 EulerScheme 179 MilsteinScheme 181 MilsteinSchemefortheHestonModel 183 ImplicitMilsteinScheme 185 TransformedVolatilityScheme 188 Balanced,Pathwise,andIJKSchemes 191 Quadratic-ExponentialScheme 193 AlfonsiSchemefortheVariance 198 MomentMatchingScheme 201 Conclusion 202 Contents vii CHAPTER8 AmericanOptions 205 Least-SquaresMonteCarlo 205 TheExplicitMethod 213 Beliaeva-NawalkhaBivariateTree 217 Medvedev-ScailletExpansion 228 ChiarellaandZiogasAmericanCall 253 Conclusion 261 CHAPTER9 Time-DependentHestonModels 263 GeneralizationoftheRiccatiEquation 263 BivariateCharacteristicFunction 264 LinkingtheBivariateCFandtheGeneralRiccatiEquation 269 MikhailovandNo¨gelModel 271 ElicesModel 278 Benhamou-Miri-GobetModel 285 Black-ScholesDerivatives 299 Conclusion 300 CHAPTER10 MethodsforFiniteDifferences 301 ThePDEinTermsofanOperator 301 BuildingGrids 302 FiniteDifferenceApproximationofDerivatives 303 TheWeightedMethod 306 BoundaryConditionsforthePDE 315 ExplicitScheme 316 ADISchemes 321 Conclusion 325 CHAPTER11 TheHestonGreeks 327 AnalyticExpressionsforEuropeanGreeks 327 FiniteDifferencesfortheGreeks 332 NumericalImplementationoftheGreeks 333 GreeksUndertheAttariandCarr-MadanFormulations 339 GreeksUndertheLewisFormulations 343 GreeksUsingtheFFTandFRFT 345 AmericanGreeksUsingSimulation 346 AmericanGreeksUsingtheExplicitMethod 349 AmericanGreeksfromMedvedevandScaillet 352 Conclusion 354 viii CONTENTS CHAPTER12 TheDoubleHestonModel 357 Multi-DimensionalFeynman-KACTheorem 357 DoubleHestonCallPrice 358 DoubleHestonGreeks 363 ParameterEstimation 368 SimulationintheDoubleHestonModel 373 AmericanOptionsintheDoubleHestonModel 380 Conclusion 382 Bibliography 383 AbouttheWebsite 391 Index 397 Foreword I am pleased to introduce The Heston Model and Its Extensions in Matlab and C# byFabriceRouah.AlthoughIwasalreadyfamiliarwithhispreviousbookentitled Option Pricing Models and Volatility Using Excel/VBA, I was pleasantly surprised todiscoverhehadwrittenabookdevotedexclusivelytothemodelthatIdeveloped in1993andtothemanyenhancementsthathavebeenbroughttotheoriginalmodel in the twenty years since its introduction. Obviously, this focus makes the book more specialized than his previous work. Indeed, it contains detailed analyses and extensive computer implementations that will appeal to careful, interested readers. Thisbookshouldinterestabroadaudienceofpractitionersandacademics,including graduatestudents,quantsontradingdesksandinriskmanagement,andresearchers inoptionpricingandfinancialengineering. There are existing computer programs for calculating option prices, such as those in Rouah’s prior book or those available on Bloomberg systems. But this bookoffersmore.Inparticular,itcontains detailedtheoreticalanalyses inaddition to practical Matlab and C# code for implementing not only the original model, but also the many extensions that academics and practitioners have developed specificallyforthemodel.Thebookanalyzesnumericalintegration,thecalculation of Greeks, American options, many simulation-based methods for pricing, finite difference numerical schemes, and recent developments such as the introduction of time-dependent parameters and the double version of the model. The breadth of methods covered in this book provides comprehensive support for implementation bypractitionersandempiricalresearcherswhoneedfastandreliablecomputations. The methods covered in this book are not limited to the specific application of option pricing. The techniques apply to many option and financial engineering models.Thebookalsoillustrateshowimplementationofseeminglystraightforward mathematical models can raise many questions. For example, one colleague noted that a common question on the Wilmott forums was how to calculate a complex logarithm while still guaranteeing that the option model produces real values. Obviously, an imaginary option value will cause problems in practice! This book resolves many similar difficulties and will reward the dedicated reader with clear answersandpracticalsolutions.IhopeyouenjoyreadingitasmuchasIdid. ProfessorStevenL.Heston RobertH.SmithSchoolofBusiness UniversityofMaryland January3,2013 ix Preface I n the twenty years since its introduction in 1993, the Heston model has become one of the most important models, if not the single most important model, in a then-revolutionary approach to pricing options known as stochastic volatility modeling.Tounderstandwhythismodelhasbecomesoimportant,wemustrevisit an event that shook financial markets around the world: the stock market crash of October1987anditssubsequentimpactonmathematicalmodelstopriceoptions. The exacerbation of smiles and skews in the implied volatility surface that resultedfromthecrashbroughtintoquestiontheabilityoftheBlack-Scholesmodel to provide adequate prices in a new regime of volatility skews, and served to highlight the restrictive assumptions underlying the model. The most tenuous of theseassumptionsisthatofcontinuouslycompoundedstockreturnsbeingnormally distributed with constant volatility. An abundance of empirical studies since the 1987 crash have shown that this assumption does not hold in equities markets. It is now a stylized fact in these markets that returns distributions are not normal. Returns exhibit skewness, and kurtosis—fat tails—that normality cannot account for.Volatilityisnotconstantintime,buttendstobeinverselyrelatedtoprice,with high stock prices usually showing lower volatility than low stock prices. A number ofresearchershavesoughttoeliminatethisassumptionintheirmodels,byallowing volatilitytobetime-varying. One popular approach for allowing time-varying volatility is to specify that volatilitybedrivenbyitsownstochasticprocess.Themodelsthatusethisapproach, including the Heston (1993) model, are known as stochastic volatility models. The models of Hull and White (1987), Scott (1987), Wiggins (1987), Chensey and Scott (1989), and Stein and Stein (1991) are among the most significant stochastic volatilitymodelsthatpre-dateSteveHeston’smodel.TheHestonmodelwasnotthe first stochastic volatility model to be introduced to the problem of pricing options, but it has emerged as the most important and now serves as a benchmark against whichmanyotherstochasticvolatilitymodelsarecompared. Allowing for non-normality can be done by introducing skewness and kurtosis in the option price directly, as done, for example, by Jarrow and Rudd (1982), Corrado and Su (1997), and Backus, Foresi, and Wu (2004). In these models, skewness and kurtosis are specified in Edgeworth expansions or Gram-Charlier expansions. In stochastic volatility models, skewness can be induced by allowing correlationbetweentheprocessesdrivingthestockpriceandtheprocessdrivingits volatility.Alternatively,skewnesscanarisebyintroducingjumpsintothestochastic processdrivingtheunderlyingassetprice. The parameters of the Heston model are able to induce skewness and kurtosis, and produce a smile or skew in implied volatilities extracted from option prices generatedbythemodel.Themodeleasilyallowsfortheinverserelationshipbetween price level and volatility in a manner that is intuitive and easy to understand. Moreover, the call price in the Heston model is available in closed form, up to an xi