The Heston Model and Its Extensions in VBA

349 Pages·2015·42.87 MB·English

by  Fabrice D. Rouah

The Heston Model
and Its Extensions
in VBA
Founded in 1807, John Wiley & Sons is the oldest independent publishing com-
pany in the United States. With offices in North America, Europe, Australia, and
Asia,Wileyisgloballycommittedtodevelopingandmarketingprintandelectronic
products and services for our customers’ professional and personal knowledge and
understanding.
The Wiley Finance series contains books written specifically for finance and
investmentprofessionalsaswellassophisticatedindividualinvestorsandtheirfinan-
cial advisors. Book topics range from portfolio management to e-commerce, risk
management, financial engineering, valuation and financial instrument analysis, as
wellasmuchmore.
Foralistofavailabletitles,visitourwebsiteatwww.WileyFinance.com.
The Heston Model
and Its Extensions
in VBA
FABRICE DOUGLAS ROUAH
Coverimage:©GillesGheerbrant(1234auhasard)
Coverdesign:©GillesGheerbrant
Copyright©2015byFabriceDouglasRouah.Allrightsreserved.
PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey.
PublishedsimultaneouslyinCanada.
Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyform
orbyanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptas
permittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,withouteithertheprior
writtenpermissionofthePublisher,orauthorizationthroughpaymentoftheappropriateper-copyfeeto
theCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,[phone],fax
[phone],orontheWebatwww.copyright.com.RequeststothePublisherforpermissionshould
beaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ
07030,[phone],fax[phone],oronlineatwww.wiley.com/go/permissions.
LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbesteffortsin
preparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyor
completenessofthecontentsofthisbookandspecificallydisclaimanyimpliedwarrantiesof
merchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedorextendedbysales
representativesorwrittensalesmaterials.Theadviceandstrategiescontainedhereinmaynotbesuitable
foryoursituation.Youshouldconsultwithaprofessionalwhereappropriate.Neitherthepublishernor
authorshallbeliableforanylossofprofitoranyothercommercialdamages,includingbutnotlimited
tospecial,incidental,consequential,orotherdamages.
Forgeneralinformationonourotherproductsandservicesorfortechnicalsupport,pleasecontactour
CustomerCareDepartmentwithintheUnitedStatesat(800)762-2974,outsidetheUnitedStatesat
[phone]orfax(317)572-4002.
Wileypublishesinavarietyofprintandelectronicformatsandbyprint-on-demand.Somematerial
includedwithstandardprintversionsofthisbookmaynotbeincludedine-booksorin
print-on-demand.IfthisbookreferstomediasuchasaCDorDVDthatisnotincludedintheversion
youpurchased,youmaydownloadthismaterialathttp://booksupport.wiley.com.Formoreinformation
aboutWileyproducts,visitwww.wiley.com.
LibraryofCongressCataloging-in-PublicationData:
Rouah,Fabrice,1964–
TheHestonmodelanditsextensionsinVBA+website/FabriceD.Rouah.
pagescm.–(Wileyfinance)
Includesbibliographicalreferencesandindex.
ISBN978-1-119-00330-4(paperback)–ISBN978-1-119-00332-8(ePDF)–
ISBN978-1-119-00331-1(ePub) 1.Options(Finance)–Mathematicalmodels.
2.Options(Finance)–Prices. 3.Finance–Mathematicalmodels. 4.VisualBasic
forApplications(Computerprogramlanguage) I.Title.
HG6024.A3R67782015
332.64′5302855133–dc23
2014043249
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
TO GIOVANNA

Contents
Foreword xi
Preface xiii
Acknowledgments xv
AboutThisBook xvii
VBALibraryforComplexNumbers xix
Chapter1
TheHestonModelforEuropeanOptions 1
ModelDynamics 1
TheHestonEuropeanCallPrice 2
DividendYieldandthePutPrice 8
ConsolidatingtheIntegrals 9
Black-ScholesasaSpecialCase 10
Conclusion 12
Chapter2
IntegrationIssues,ParameterEffects,andVarianceModeling 13
RemarksontheCharacteristicFunctions 14
ProblemswiththeIntegrand 16
TheLittleHestonTrap 18
EffectoftheHestonParameters 20
VarianceModelingintheHestonModel 26
MomentExplosions 38
BoundsonImpliedVolatilitySlope 40
Conclusion 42
Chapter3
DerivationsUsingtheFourierTransform 45
DerivationofGatheral(2006) 46
Attari(2004)Representation 47
CarrandMadan(1999)Representation 49
Conclusion 61
vii
viii CONTENTS
Chapter4
TheFundamentalTransformforPricingOptions 63
ThePayoffTransform 64
OptionPricesUsingParseval’sIdentity 70
VolatilityofVolatilitySeriesExpansion 75
Conclusion 81
Chapter5
NumericalIntegrationSchemes 83
TheIntegrandinNumericalIntegration 84
Newton-CotesFormulas 85
GaussianQuadrature 90
IntegrationLimits,MultidomainIntegration,and
KahlandJa¨ckelTransformation 98
IllustrationofNumericalIntegration 103
FastFourierTransform 106
FractionalFastFourierTransform 108
Conclusion 114
Chapter6
ParameterEstimation 115
EstimationUsingLossFunctions 116
SpeedingUptheEstimation 126
DifferentialEvolution 128
MaximumLikelihoodEstimation 132
Risk-NeutralDensityandArbitrage-FreeVolatilitySurface 135
Conclusion 140
Chapter7
SimulationintheHestonModel 143
GeneralSetup 144
EulerScheme 146
MilsteinScheme 147
ImplicitMilsteinScheme 149
TransformedVolatilityScheme 152
Balanced,Pathwise,andIJKSchemes 155
Quadratic-ExponentialScheme 157
AlfonsiSchemefortheVariance 161
Moment-MatchingScheme 165
Conclusion 167