Delta-Hedged Gains and the Negative Market Volatil

Delta-HedgedGainsandtheNegativeMarketVolatilityRiskPremiumGurdipBakshiandNikunjKapadiaApril9,2001BakshiisatDepartmentofFinance,SmithSchoolofBusiness,UniversityofMaryland,CollegePark,MD20742,andKapadiaisatDepartmentofFinance,SchoolofManagement,UniversityofMassachusetts-Amhert,MA01003.BakshicanbereachedatTel:301-405-2261,Email:[e-mail],Website:nance/gbakshi/;andKapadiaatTel:413-545-5643,Email:[e-mail],wethankDoronAvramov,CharlesCao,BentChristensen,SanjivDas,StephenFiglewski,ChristopherJones,HosseinKazemi,LeonidKogan,DilipMadan,GeorgeMartin,VasantNaik,JunPan,JayPatel,AllenPoteshman,N.R.Prabhala,RangarajanSundaram,BobWhaley,andGregoryWillette.PartsofthearticlebuildonKapadia’sthesiswrittenatNewYorkUniversity.EarlierversionsofthepaperwerepresentedatBostonUniversity,UniversityofMassachusettsandVirginiaPolytechnicInstitute.Conferenceparticipantsatthe1998WFA(Monterey)and2001AFA(NewOrleans)meetingsprovidedmanyusefulsuggestions.NickBollen(AFAdiscussant)andJeFleming(WFAdiscussant)providedextremelyconstructivecomments.ThereportsofBernardDumas(theEditor)andtwoanonymousrefereeshavesubstantiallyimprovedthispaper.KristapsLicishaspro-videdexcellentresearchassistance.BentChristensengraciouslysharedhisoptiondataset.KapadiaacknowledgesnancialsupportfromtheCenterofInternationalDerivativesandSecuritiesMarkets.The1998versionofthepaperwascirculatedunderthetitleDoEquityOptionsPriceVolatilityRisk?0Delta-HedgedGainsandtheNegativeMarketVolatilityRiskPremiumAbstractWeinvestigatewhetherthevolatilityriskpremiumisnegativebyexaminingthestatisticalpropertiesofdelta-hedgedoptionportfolios(buytheoptionandhedgewithstock).Withinastochasticvolatilityframework,wedemonstrateacorrespondencebetweenthesignandmag-nitudeofthevolatilityriskpremiumandthemeandelta-hedgedportfolioreturns.UsingasampleofS&P500indexoptions,weprovideempiricalteststhathavethefollowinggeneralresults.First,thedelta-hedgedstrategyunderperformszero.Second,thedocumentedunder-performanceislessforoptionsawayfromthemoney.Third,theunderperformanceisgreaterattimesofhighervolatility.Fourth,thevolatilityriskpremiumsignicantlyaectsdelta-hedgedgainsevenafteraccountingforjump-fears.Ourevidenceissupportiveofanegativemarketvolatilityriskpremium.1Thenotionthatvolatilityofequityreturnsisstochastichasarmfootinginnancialeco-nomics.However,alessthanunderstoodphenomenoniswhethervolatilityriskiscompensated,andwhetherthiscompensationishigherorlowerthantherisk-freerate.Istheriskfromchangesinmarketvolatilitypositivelycorrelatedwiththeeconomy-widepricingkernelprocess?Ifso,howdoesitaecttheequityandtheoptionmarkets?Evidencethatmarketvolatilityriskpremiummaybenon-zerocanbemotivatedbythreeempiricalndings:1.Purchasedoptionsarehedgesagainstsignicantmarketdeclines.Thisisbecauseincreasedrealizedvolatilitycoincideswithdownwardmarketmoves(French,SchwertandStambaugh(1987)andNelson(1991)).Oneeconomicinterpretationisthatbuyersofmarketvolatilityarewillingtopayapremiumfordownsideprotection.Thehedgingmotiveisindicativeofanegativevolatilityriskpremium;2.At-the-moneyBlack-Scholesimpliedvolatilitiesaresystematicallyandconsistentlyhigherthanrealizedvolatilities(JackwerthandRubinstein(1996)).Apotentialexplanationforthispuzzlingempiricalregularityisthatthevolatilityriskpremiumisnegative.Ceterisparibus,anegativevolatilityriskpremiumincreasestherisk-neutraldriftofthevolatilityprocessand,thus,raisesequityoptionprices;3.Equityindexoptionsarenon-redundantsecurities(Bakshi,CaoandChen(2000)andBuraschiandJackwerth(2001)).Indexoptionmodelsomittingtheeconomicimpactofamarketvolatilityriskpremiummaybeinconsistentwithobservedoptionpricingdynamics.Thisarticleinvestigates,boththeoreticallyandempirically,whetherthevolatilityriskpremiumisnegativeinindexoptionmarkets.Thisisdonewithoutimposinganypriorstructureonthepricingkernel,andwithoutparameterizingtheevolutionofthevolatilityprocess.Thesetupisaportfolioofalongcallposition,hedgedbyashortpositioninthestock,suchthatthenetinvestmentearnstherisk-freeinterestrate.Thecentralideaunderlyingouranalysisisthatifoptionpricesincorporateanon-zerovolatilityriskpremium,thenwecaninferitsexistencefromthereturnsofanoptionportfoliothathasdynamicallyhedgedallrisksexceptvolatilityrisk.Ifvolatilityisconstant,orthepriceprocessfollowsaone-dimensionalMarkovdiusion,thenourtheoreticalanalysisimpliesthatthenetgain,henceforth\delta-hedgedgains,onthedelta-hedgedportfolioispreciselyzero.Asimilarconclusionobtainswhenvolatilityisstochastic,butvolatilityriskisunpriced.Inthisparticularcase,weshowthatthedistributionofthedelta-hedgedgainshasanexpectedvalueofzero.However,ifvolatilityriskispriced,thenthesignandmagnitudeofaveragedelta-hedgedgainsaredeterminedbythevolatilityriskpremium.1Ourtheoreticalcharacterizationspointtoempiricalimplicationsthatcanbetestedusingdiscretedelta-hedgedgains.First,inthetime-series,theat-the-moneydelta-hedgedgainsmustberelatedtovolatilityriskpremium.Second,cross-sectionalvariationsindelta-hedgedgains(inthestrikedimension)arer