Inside Volatility Filtering: Secrets of the Skew

321 Pages·2015·6.44 MB·English

by  Alireza Javaheri

Inside Volatility
Filtering
Wiley Finance series contains books written specifically for finance and
investment professionals as well as sophisticated individual investors and
their financial advisers. Book topics range from portfolio management
to e-commerce, risk management, financial engineering, valuation and
financial instrument analysis, as well as much more. For a list of available
titles,visitourwebsiteatwww.WileyFinance.com.
Foundedin1807,JohnWiley&Sonsistheoldestindependentpublish-
ing company in the United States. With offices in North America, Europe,
Australia,andAsia,Wileyisgloballycommittedtodevelopingandmarket-
ingprintandelectronicproductsandservicesforourcustomers’professional
andpersonalknowledgeandunderstanding.
Inside Volatility
Filtering
Secrets of the Skew
Second Edition
ALIREZA JAVAHERI
Copyright©2015byAlirezaJavaheri.Allrightsreserved.
PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey.
TheFirstEditionofthisbookwaspublishedbyJohnWileyandSonsin2005underthetitleInside
VolatilityArbitrage:TheSecretsofSkewness.
PublishedsimultaneouslyinCanada.
Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformor
byanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptaspermitted
underSection107or108ofthe1976UnitedStatesCopyrightAct,withouteitherthepriorwritten
permissionofthePublisher,orauthorizationthroughpaymentoftheappropriateper-copyfeetothe
CopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,[phone],fax(978)
646-8600,orontheWebatwww.copyright.com.RequeststothePublisherforpermissionshouldbe
addressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ07030,
[phone],fax[phone],oronlineathttp://www.wiley.com/go/permissions.
LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbesteffortsin
preparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyor
completenessofthecontentsofthisbookandspecificallydisclaimanyimpliedwarrantiesof
merchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedorextendedbysales
representativesorwrittensalesmaterials.Theadviceandstrategiescontainedhereinmaynotbesuitable
foryoursituation.Youshouldconsultwithaprofessionalwhereappropriate.Neitherthepublishernor
authorshallbeliableforanylossofprofitoranyothercommercialdamages,includingbutnotlimitedto
special,incidental,consequential,orotherdamages.
Forgeneralinformationonourotherproductsandservicesorfortechnicalsupport,pleasecontactour
CustomerCareDepartmentwithintheUnitedStatesat(800)762-2974,outsidetheUnitedStatesat(317)
572-3993,orfax(317)572-4002.
Wileypublishesinavarietyofprintandelectronicformatsandbyprint-on-demand.Somematerial
includedwithstandardprintversionsofthisbookmaynotbeincludedine-booksorinprint-on-demand.
IfthisbookreferstomediasuchasaCDorDVDthatisnotincludedintheversionyoupurchased,you
maydownloadthismaterialathttp://booksupport.wiley.com.FormoreinformationaboutWileyproducts,
visitwww.wiley.com.
LibraryofCongressCataloging-in-PublicationData:
ISBN978-1-118-94397-7(cloth)
ISBN978-1-118-94399-1(epdf)
ISBN978-1-118-94398-4(epub)
CoverDesign:Wiley
CoverImage:AlirezaJavaheri
PrintedintheUnitedStatesofAmerica
10987654321
Contents
Foreword ix
Acknowledgments(SecondEdition) xi
Acknowledgments(FirstEdition) xiii
Introduction(SecondEdition) xv
Introduction(FirstEdition) xvii
Summary xvii
ContributionsandFurtherResearch xxiii
DataandPrograms xxiv
CHAPTER1
TheVolatilityProblem 1
Introduction 1
TheStockMarket 2
TheStockPriceProcess 2
HistoricVolatility 3
TheDerivativesMarket 5
TheBlack-ScholesApproach 5
TheCoxRossRubinsteinApproach 7
JumpDiffusionandLevel-DependentVolatility 8
JumpDiffusion 8
Level-DependentVolatility 11
LocalVolatility 14
TheDupireApproach 14
TheDermanKaniApproach 17
StabilityIssues 18
CalibrationFrequency 19
StochasticVolatility 21
StochasticVolatilityProcesses 21
GARCHandDiffusionLimits 22
v
vi CONTENTS
ThePricingPDEunderStochasticVolatility 26
TheMarketPriceofVolatilityRisk 26
TheTwo-FactorPDE 27
TheGeneralizedFourierTransform 28
TheTransformTechnique 28
SpecialCases 30
TheMixingSolution 32
TheRomanoTouziApproach 32
AOne-FactorMonte-CarloTechnique 34
TheLong-TermAsymptoticCase 35
TheDeterministicCase 35
TheStochasticCase 37
ASeriesExpansiononVolatility-of-Volatility 39
LocalVolatilityStochasticVolatilityModels 42
StochasticImpliedVolatility 43
JointSPXandVIXDynamics 45
Pure-JumpModels 47
VarianceGamma 47
VarianceGammawithStochasticArrival 51
VarianceGammawithGammaArrivalRate 53
CHAPTER2
TheInferenceProblem 55
Introduction 55
UsingOptionPrices 58
ConjugateGradient(Fletcher-Reeves-Polak-Ribiere)
Method 59
Levenberg-Marquardt(LM)Method 59
DirectionSet(Powell)Method 61
NumericTests 62
TheDistributionoftheErrors 65
UsingStockPrices 65
TheLikelihoodFunction 65
Filtering 69
TheSimpleandExtendedKalmanFilters 72
TheUnscentedKalmanFilter 74
Kushner’sNonlinearFilter 77
ParameterLearning 80
ParameterEstimationviaMLE 95
Diagnostics 108
ParticleFiltering 111
ComparingHestonwithOtherModels 133
Contents vii
ThePerformanceoftheInferenceTools 141
TheBayesianApproach 158
UsingtheCharacteristicFunction 172
IntroducingJumps 174
Pure-JumpModels 184
Recapitulation 201
ModelIdentification 201
ConvergenceIssuesandSolutions 202
CHAPTER3
TheConsistencyProblem 203
Introduction 203
TheConsistencyTest 206
TheSetting 206
TheCross-SectionalResults 206
Time-SeriesResults 209
FinancialInterpretation 210
The“Peso”Theory 214
Background 214
NumericResults 215
TradingStrategies 216
SkewnessTrades 216
KurtosisTrades 217
DirectionalRisks 217
AnExactReplication 219
TheMirrorTrades 220
AnExampleoftheSkewnessTrade 220
MultipleTrades 225
HighVolatility-of-VolatilityandHighCorrelation 225
Non-GaussianCase 230
VGSA 232
AWordofCaution 236
ForeignExchange,FixedIncome,andOtherMarkets 237
ForeignExchange 237
FixedIncome 238
CHAPTER4
TheQualityProblem 241
Introduction 241
AnExactSolution? 241
NonlinearFiltering 242
StochasticPDE 243
viii CONTENTS
WienerChaosExpansion 244
First-OrderWCE 247
Simulations 248
Second-OrderWCE 251
QualityofObservations 251
HistoricSpotPrices 252
HistoricOptionPrices 252
Conclusion 262
Bibliography 263
Index 279
Foreword
A
revolution has been quietly brewing in quantitative finance for the last
quarter century. Inthe 1980s and 1990s, quant groups were dominated
byphysicistsandmathematicianswellversedinsophisticatedmathematical
techniques related to physical phenomena such as heat flow and diffusion.
RiversofdataflowedthroughthecanyonsofWallStreet,butlittlethought
wasgiventocapturingthedataforthepurposeofextractingthesignalfrom
thenoise.Thedominantworldviewatthetimeviewedfinanceasprimarilya
socialenterprise,renderingitfundamentallydistinctfromthephysicalworld
thatthequantshadleftbehind.Ithinkthatmostquantsofthetimewould
haveagreedwithGeorgeBernardShaw,whosaidthat:
Hegel was right when he said that the only thing we learn from
historyisthatmanneverlearnsanythingfromhistory.
WhenaneconometriciannamedRobertLittermaninterviewedforaposition
atGoldmanSachsin1986,thelegendaryFischerBlackbegantheinterview
byasking,“Whatmakesyouthinkthataneconometricianhasanythingto
contributetoWallStreet?”
I suspect we will never know how the candidate answered that ques-
tion,buthadthecandidatebeenMarkTwain,theanswerwouldsurelyhave
beenthat:
Historydoesn’trepeatitself,butitdoesrhyme.
Six years after Black interviewed Litterman, they co-authored the Black–
Litterman model that blends theoretical notions such as economic equilib-
rium with investor views guided by econometric time series models. Since
then,econometriciansandstatisticianshaveenjoyedasteadilygrowingrole
inbothpractitionerandacademiccircles.IntheMay2003issueofWilmott
magazine, an article profiled past winners of the Nobel Memorial Prize in
Economic Sciences. In a side note titled “A Winner You’d like to See,” the
Theviewsrepresentedhereinaretheauthor’sownviewsanddonotnecessarilyrep-
resenttheviewsofMorganStanleyoritsaffliates,andarenotaproductofMorgan
Stanleyresearch.
ix