The Heston Model and its Extensions in Matlab and C#, + Website

416 Pages·2013·17.967 MB·English

by  Fabrice D. Rouah, Steven L. Heston (Foreword by)

The Heston Model
and Its Extensions in
Matlab and C#
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The Heston Model
and Its Extensions in
Matlab and C#
FABRICE DOUGLAS ROUAH
Coverillustration:GillesGheerbrant,‘‘1234auhasard’’(1976);
CGillesGheerbrant
Coverdesign:GillesGheerbrant
CopyrightC2013byFabriceDouglasRouah.Allrightsreserved.
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LibraryofCongressCataloging-in-PublicationData:
Rouah,Fabrice,1964-
TheHestonmodelanditsextensionsinMatlabandC#/FabriceDouglasRouah.
pagescm.–(Wileyfinanceseries)
Includesbibliographicalreferencesandindex.
ISBN978-1-118-54825-7(paper);ISBN978-1-118-69518-0(ebk);ISBN978-1-118-69517-3(ebk)
1. Options(Finance)–Mathematicalmodels.2. Options(Finance)–Prices.3. Finance–Mathematical
models.4. MATLAB.5. C#(Computerprogramlanguage)I. Title.
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