FinanceandEconomicsDiscussionSeries

FederalReserveBoard,Washington,D.C.

ISSN1936-2854(Print)

ISSN2767-3898(Online)

ThePricingKernelinOptions

StevenHeston,KrisJacobs,HyungJooKim

2023-053

Pleasecitethispaperas:

Heston,Steven,KrisJacobs,andHyungJooKim(2023).“ThePricingKernelinOptions,”FinanceandEconomicsDiscussionSeries2023-053.Washington:BoardofGovernorsoftheFederalReserveSystem,

/10.17016/FEDS.2023.053

.

NOTE:StafworkingpapersintheFinanceandEconomicsDiscussionSeries(FEDS)arepreliminarymaterialscirculatedtostimulatediscussionandcriticalcomment.TheanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebyothermembersoftheresearchstafortheBoardofGovernors.ReferencesinpublicationstotheFinanceandEconomicsDiscussionSeries(otherthanacknowledgement)shouldbeclearedwiththeauthor(s)toprotectthetentativecharacterofthesepapers.

ThePricingKernelinOptions*

StevenHeston

UniversityofMaryland

KrisJacobs

UniversityofHouston

HyungJooKim

FederalReserveBoard

April7,2023

Abstract

Theempiricaloptionvaluationliteraturespecifiesthepricingkernelthroughthepriceofrisk,ordefinesitimplicitlyastheratioofrisk-neutralandphysicalprobabilities.Instead,weextendtheeconomicallyappealingRubinstein-Brennankernelstoadynamicframeworkthatallowspath-andvolatility-dependence.Becauseoflowstatisticalpower,kernelswithdifferenteconomicpropertiescanproducesimilaroveralloptionfit,evenwhentheyimplycross-sectionalpricinganomaliesandimplausibleriskpremiums.Imposingparsimoniouseconomicrestrictionssuchasmonotonicityandpath-independence(recoverytheory)achievesgoodoptionfitandreasonableestimatesofequityandvarianceriskpremiums,whileresolvingpricingkernelanomalies.

*Heston:sheston@;Jacobs:kjacobs@;Kim:hyungjoo.kim@.WewouldliketothankCaioAlmeida,DavidBates,HiteshDoshi,Bj?rnEraker,XiaohuiGao,StefanoGiglio,MassimoGuidolin,AlexKostakis,PaolaPederzoli,Jean-PaulRenne,seminarparticipantsatthe2023AFAConference,the2022FinanceDownUnderConference,the2022SoFiEConference,the2022FMAConferenceonDerivativesandVolatility,K.U.Leuven,SyracuseUniversity,theUniversitiesofHoustonandLiverpool,andespeciallyourdiscussantsGurdipBakshi,MikhailChernov,andJeroenDalderopforhelpfulcomments.TheanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebytheFederalReserveBoardorothermembersofitsstaff.Co-authorHyungJooKimworkedonthisprojectpriortoemploymentattheFederalReserveBoard,whileaPh.D.candidateattheUniversityofHouston.DatasourceswereobtainedunderpurviewofUniversityofHoustonlicenses.

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1Introduction

Thepricingkernelisacriticalconceptinassetpricing,becauseitdeterminesriskpremiaonallsecurities.Oneapproachtostudythepropertiesofthepricingkernelspecifiesitsrelationtoaggregateconsumptionandestimatestheresultingmodelusingconsumptiondataandreturnsonvariousassets.1Alternatively,buildingontheinsightsofBreedenandLitzenberger(1978),anextensiveliteratureestimatesthepricingkernelusingindexreturnsandindexoptionprices.2Indexoptionsareinterestingfromanempiricalperspectivebecausetheyidentifythepricingkernelundertheassumptionthattheequityindexlevelisequaltoaggregatewealth.However,thisliteraturehasgivenrisetopuzzlingnon-monotonic(U-shaped)estimatesofthepricingkernel.3

Thispapercomplementsthesetwoapproaches.Wespecifyeconomicallyintuitivepricingkernelsinparametricoptionpricingmodelsandestimatethemusingindexreturnsandindexoptionprices.Ourapproachdiffersfromtheexistingparametricliterature,whichtypicallyspecifiespricesofrisk,andtherebydefinesthepricingkernelimplicitlyastheratioofrisk-neutralandphysicalprobabilities.4Instead,weproposeaclassofeconomicallymotivatedpricingkernelsthatextendthepowerkernelsinRubinstein(1976)andBrennan(1979)tobefunctionsofthepathsoflatentvariancev(t)andtheindexlevelS(t).Thesekernelsalsonestpath-independentkernels(Ross,2015)asspecialcases,andareconsistentwiththeconventionalassumptionofaffinedynamicsunderthephysicalandrisk-neutralmeasureinthesquarerootstochasticvolatilitymodel(Heston,1993).5Byincludingseparatecomponentsforvolatilityandstockrisk,thesekernelsallowusto

1See,amongothers,HansenandSingleton(1982),MehraandPrescott(1985),HansenandJagannathan(1991),CampbellandCochrane(1999),BansalandYaron(2004),Gabaix(2012)andWachter(2013)forimportantcontri-butionstothisliterature.

2SeeforinstanceA¨?t-SahaliaandLo(1998),A¨?t-SahaliaandLo(2000),JackwerthandRubinstein(1996),Jack-werth(2000)andRosenbergandEngle(2002).

3ForevidenceonU-shapedpricingkernels,seeforinstanceJackwerth(2000),A¨?t-SahaliaandLo(2000),RosenbergandEngle(2002),Bakshi,Madan,andPanayotov(2010),Chabi-Yo(2012),Christoffersen,Heston,andJacobs(2013),SongandXiu(2016),andCuesdeanuandJackwerth(2018).Linn,Shive,andShumway(2018)andBarone-Adesi,Fusari,Mira,andSala(2020)ontheotherhandarguethatthepricingkerneliswell-behaved.

4Forexamplesofthisapproach,seetheseminalpapersinthisliteraturebyChernovandGhysels(2000),Pan(2002),andEraker(2004).

5Forsimplicity,weusethesimplestpossibleoptionpricingmodelwithastochasticvolatilityfactor,butourapproachcanbeeasilygeneralizedtomorecomplexmodels.

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examinedistinctoriginsoftheequityandvarianceriskpremiums.

Westartouranalysiswiththeoften-usedspecificationofanequity(market)riskpremiumμv(t)andavarianceriskpremiumλv(t).Werefertothisspecificationas“completelyaffine”,adoptingtheterminologyinSingleton(2006,p.392)andthetermstructureliterature.Wecharacterizeapricingkernelthatisconsistentwiththeseriskpremia,andderivetheparameterrestrictionsconsistentwiththemartingaleconditionsandabsenceofarbitrage.Wethenexploreaspecificationwithanequityriskpremiumμ0+μ1v(t)andavarianceriskpremiumλ0+λ1v(t).Werefertothisspecificationas“affine”.Singleton(2006)pointsoutthatthisspecificationisproblematicbecauseitmayviolateno-arbitrageconditions,andnotesthatitwouldbeinterestingtocharacterizetheparameterrestrictionsthatpreventarbitrageforthisspecification.Becauseourkernelisformulatedasafunctionofthestatevariables,itisstraightforwardtospecifysuchrestrictions.

Ourempiricalanalysisusesajointlikelihoodbasedonindexreturnsandarichoptiondataset,usingdatafortheJanuary1996toJune2019period.6Weestimatethisjointlikelihoodforthepricingkernelscorrespondingtothecompletelyaffineandaffinepricesofrisk,andwealsoestimateitsubjecttovariousrestrictionsonparametervaluesandriskpremia.Becausethekernelsareformulatedasafunctionofthestatevariables,itisrelativelystraightforwardtoderivetheimplicationsofeachkernelforthe“marginal”kernelwhichspecifiesstatepriceasafunctionofS(t).Themarginalkernelplaysacentralroleinempiricalapplicationssuchasthepricingkernelpuzzle.

Afirstempiricalresultaddressesthefitandempiricalcontentofthekernelsthatsupportcompletelyaffineandaffinepricesofrisk.Unsurprisingly,wefindthattheaffinepriceofriskspec-ificationprovidessignificantlybetterfitthanthenestedcompletelyaffinespecification.However,theimprovedfitduetotheinterceptsintheaffinespecificationcomesatthecostofimplausibleSharperatiosand/orsignsoftheriskpremiums.Moreover,whilethemarginalpricingkernelforthecompletelyaffinespecificationiswellbehavedandeconomicallyplausible,thekernelinthe

6Muchofthemodernoptionpricingliteraturejointlyconsidersthetime-seriesofobservablereturnsandoptionprices.See,forinstance,Pan(2002),Eraker(2004),Bates(2006),A¨?t-SahaliaandKimmel(2007),Hurn,Lindsay,andMcClelland(2015),andAndersen,Fusari,andTodorov(2017).

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affinecaseimpliesstatepricesthatareS-shapedasafunctionofwealth.Whenweimposeaddi-tionalparameterrestrictionsthatprecludearbitrage,themarginalpricingkerneliswell-behaved,butempiricalfitworsens.Weconcludethattheaffinespecificationcorrespondstoimplausibleeconomicassumptions,andthatsmallandseeminglyinnocuousmodificationstothepriceofriskspecificationsusedintheliteraturecorrespondtodifferentpricingkernelswithradicallydifferenteconomicimplications.Consequently,weadvocatetheuseofthecompletelyaffinepriceofriskspecification.

Asecondsetofresultscomparestheunrestrictedkernelthatsupportsthecompletelyaffinepriceofriskwithrestrictedversions.Werejectrestrictionsthattheequityorvarianceriskpremiumsarezero,butwecannotrejecttheindependenceofthepricingkernelfromeithervarianceormarketreturnshocks.7Weareunabletostatisticallypindowntheoriginsoftheseriskpremiumsbecauseinnovationstomarketreturnsandmarketvariancearehighly(negatively)correlated.Theequityandvarianceriskpremiumeachhavetwocomponents,oneduetovarianceaversionandanotherduetoindexlevelriskaversion.Ifwerestrictoneoftheriskaversionparameterstobezero,theotherparameterabsorbsmostofthateffect.Inotherwords,theseriskslargelyspaneachother.However,restrictingvarianceorindexlevelriskaversiontozeroimpliesradicallydifferentestimatesoftheequityandvarianceriskpremia,aswellaslargedifferencesinthestatepricesembodiedinthemarginalpricingkernelasafunctionofwealth.Therealizedtimeseriespathofthekernelwithoutvolatilityriskisalsosubstantiallylessvariablecomparedtotheunrestrictedkernel,especiallyincrisisperiods.Afinalobservationisthatwhilewecannotstatisticallyrejectthepath-independencerestrictionusedinRoss(2015),itimpliesanimplausibleestimateofthevarianceaversionparameter.

Ourthirdfindingshedslightonthepricingkernelpuzzle–thefindingthatthemarginalpricingkernelisU-shaped–intheexistingliterature.WeshowthatU-shapedmarginalkernelscanresultfromanunderlyingpricingkernelthatisamonotonicfunctionofvolatilityandthestockprice.Therefore,U-shapedpricingkernelsarenotanomalousnordotheyconstituteanassetpricing

7Notethatthehypothesisthatthevariance-aversionparameterequalszeroamountstotheabsenceofaninde-pendentvarianceriskpremium,whichamountstologarithmicutilityintheMerton(1973)ICAPM.

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puzzle.

Ourfourthfindingaddressesestimationofriskpremiums.FollowingBreedenandLitzenberger’s(1978)insightthattherisk-neutraldensitycanbeinferredfromoptionprices,financialeconomistshaveemphasizedfittingoptionsandreturnsjointlytoidentifyriskpremia.Wefindthatthesedatahavelowpowertodistinguishdifferentpricingkernels,becauseidentifyingpricingkernelsisequivalenttotheestimationofconditionalriskpremia,anditisdifficulttoestimateaveragereturnsovershortperiods.8DifferentparameterrestrictionsleadtowidelydifferentSharperatiosandequityandvarianceriskpremia,butdonottranslateintolargedecreasesinthelikelihood.Merton(1980)convincinglyarguesthatverylongtimeseriesofreturnsarerequiredtoobtainreliableestimatesoftheequitypremium.OurfindingsextendMerton’sobservationtojointestimationofequityandvarianceriskpremia.WealsoreinforceMerton’sconclusionthateconomicrestrictionsincreasepowertoidentifymarketriskpremia.WhileMerton(1980)advocatesimposingapositivityrestrictiononthepathoftheconditionalequityriskpremium,wefindthatimposinganegativityrestrictiononthemarketvarianceriskpremiumleadstomoreplausibleandreliableestimates.OurfindingsalsoconfirmtheresultsinBakshi,Crosby,andGao(2022)thatsomeoptionmodelparametersarehardtoidentifybecauseof(darkmatter)unspannedrisksthataffectriskpremiums. Ourpaperisrelatedtoseveralotherstrandsofliteraturebesidesthoseontheestimationofparametricoptionpricingmodelsandthepricingkernelpuzzle.Severalstudiesuseconsumption-basedmodelstoanalyzehowpreferencesandpricingkernelsimpactindexoptionprices.9SomeofthesestudiesusetherecursivepreferencesofKrepsandPorteus(1978),EpsteinandZin(1989)andDuffieandEpstein(1992),whichresultinstochasticvolatilityofindexreturns.OurproposedpricingkernelsareextensionsofthepowerutilityofRubinstein(1976).Whileconsumptionisnotastatevariableinoursetup,ourapproachprovidesadirectrelationwithexistingempiri-

8Thisstatementisspecifictoplainvanillaoptionprices,whicharesensitivetotheprobabilitiesatexpirationbutnotveryinformativeaboutthepath-dependentpropertiesofthepricingkernel.PricingkernelswithwidelydifferenteconomicimplicationscanthereforeproducesimilarvaluesforEuropeanoptions.

9See,forinstance,Garcia,Luger,andRenault(2003),ErakerandShaliastovich(2008),Drechsler(2013),Shalias-tovich(2015),ErakerandYang(2019),andSeoandWachter(2019).Liu,Pan,andWang(2005)andErakerandWu(2017)userelatedmodelswiththedividendpayoutrateandcashflowrespectivelyasthestatevariable.

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calimplementationsof(reduced-form)parametricdynamicoptionpricingmodels.Itisthereforestraightforwardtoimplementusingoptiondata,whichallowsustoexploretheimpactofstockindexvolatilityonthepricingkernel.

Fromanempiricalperspective,arelatedpaperisChernov(2003),whoreverseengineersthepricingkernelbasedonoptionsonvarioussecurities.Chernov(2003)alsostudiesthetimepathoftherealizedpricingkerneltolearnaboutstatevariablesandtherelationbetweenthepricingkernelandeconomicconditions.Ghosh,Julliard,andTaylor(2017)alsoexploretherelationbetweenthepricingkernelandbusinesscyclefluctuations,butdonotuseoptionstoestimatethekernel.Brennan,Liu,andXia(2006)specifyandestimatepricingkernelswithmultiplestatevariables.BeasonandSchreindorfer(2022)analyzetheimplicationsofoptiondataformacro-financemodels.Dew-BeckerandGiglio(2022)studytheimplicationsofsyntheticputsforthepropertiesofthemarginalpricingkernel.

Thepaperproceedsasfollows.Section2discussesthedata.Section3reviewstheHeston(1993)stochasticvolatilitymodelanddiscussesourestimationapproachbasedonreturnsandoptionsdata.Section4specifiestheclassofpricingkernelsthatconnecttherisk-neutralandphysicaldynamics.Section5presentstheestimationresultsandSection6discussestheireconomicimplications.Section7concludes.

2Data

Ourempiricalanalysisusesout-of-the-money(OTM)S&P500callandputoptionswithmaturitiesbetween14and365daysfortheJanuary1996toJune2019period.WeobtaintheoptiondatafromOptionMetrics.Weapplythefollowingfilters:

1.Discardoptionswithimpliedvolatilitysmallerthan5%orgreaterthan150%.

2.Discardoptionswithvolumeoropeninterestlessthantencontracts.

3.Discardoptionswithmidpricelessthan$0.50orbidpricelessthan$0.375toavoidlow-valuedoptions.

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4.Discardoptionswithdataerrors–wherebidpriceexceedsofferprice,oranegativepriceisimpliedthroughput-callparity.

5.Discardoptionswithmoneyness<0.75or>1.25.

Thenwekeepthesixmostactivelytradedstrikepricesforeachavailablematurity.Itisimportanttouseaslongatimeperiodaspossibletoidentifykeyaspectsofthemodel,includingvolatilitypersistence.10Ontheotherhand,estimationusinglargeoptionpanelsandlongtimeseriesisverytime-intensive.Ratherthanusingashorttimeseriesofdailyoptiondata,weuseanextendedtimeperiod,butweselectoptioncontractsforonedayperweekonly.Followingseveralexistingstudies(see,e.g.,HestonandNandi,2000;Christoffersen,Heston,andJacobs,2013),weuseWednesdaydatabecauseitisthedayoftheweekleastlikelytobeaholiday.Itisalsolesslikelythanotherdaystobeaffectedbyday-of-the-weekeffects.Thesestepsresultinadatasetwith62,483optioncontracts.Table1presentsdescriptivestatistics.

WeobtainS&P500indexreturnsfromCRSP.WeusedatafortheJanuary1990toJune2019period.Thissamplestartsbeforetheoptionsampletohelpwiththeidentificationofthereturnparametersunderthephysicalmeasure,asinChristoffersen,Heston,andJacobs(2013).WealsousedataontheVIXfromJanuary1990toJune2019,whichweobtainfromtheFederalReserveBankofSt.LouisEconomicDatabase.Thetimeseriesfortherisk-freerateisproxiedbytheone-monthTreasuryBillrateobtainedfromCRSP.Followingexistingwork,optionsarevaluedusingamaturity-specificrisk-freerate.WeapplyacubicsplineinterpolationtothedataobtainedfromOptionMetrics.

3Return-BasedandOption-BasedParameterEstimates

WeestimatethestylizedaffineHeston(1993)stochasticvolatilitymodel.Weobtainparameterestimatesforthismodelunderthephysicalmeasure,exclusivelybasedonreturns,andundertherisk-neutralmeasure,exclusivelybasedonoptions.Thenwecomparetheresultingestimates.

10See,forinstance,Broadie,Chernov,andJohannes(2007)foradiscussion.

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3.1TheModel

Wefocusonthesimplestpossiblestochasticvolatilitymodelwithasinglediffusivevolatilityfactor.Werecognizethattheexistingliteraturehasclearlyestablishedthatadditionalvolatilityfactors,jumpsinreturnsandvarianceand/ortailfactorsarerequiredtoimproveoptionfitandpricingperformance.However,wedeliberatelyfocusonthesimplestpossiblemodelbecauseitsufficestoillustrateourmainargumentandwewanttoavoidcomparisonsbetweenmodelsandfactors.Ouranalysiscanberepeatedusingmoregeneralmodels,butatthecostofmuchgreatercomplexity.Webelievethatmostoftheissueswehighlighthereusingasimplemodelareevenmorerelevantinmorecomplexmodels,butweleavethisanalysisforfuturework.

WeemploytheHeston(1993)continuous-timestochasticsquarerootvolatilitymodeltospecifystockpricedynamicsaswellasoptionprices.Foroptionvaluation,therisk-neutralstockpricedynamicissufficient.Thesquarerootstochasticvolatilitymodelspecifiestherisk-neutraldynamicsofthespotindexS(t)anditsstochasticvariancev(t)asfollows:

(1)

dS(t)/S(t)=rdt+^v(t)dz(t),

dv(t)=κ?(θ??v(t))dt+σ^v(t)dz(t),

wheredzanddzareWienerprocesseswithcorrelationcoefficientρ.Therisk-freeratercanbeeitherconstantortime-varying;thishasnegligibleimplicationsforourresults.Itisalsostraight-forwardtospecifyastochasticmodelfortherisk-freerate,butitiswell-knownfromtheexistingliteraturethatthisdoesnothaveamajorimpactonoptionvaluation(Bakshi,Cao,andChen,1997).Wethereforedeliberatelyfocusonthesimplestpossiblemodel.Consistentwithmostoftheexistingliterature,wefocusonaphysicaldynamicthathasthesamefunctionalformastherisk-neutraldynamic:

(2)

dS(t)/S(t)=[r+μ(v(t))]dt+^v(t)dz1(t),

dv(t)=κ(θ?v(t))dt+σ^v(t)dz2(t),

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whereμ(v(t))denotestheequitypremiumasafunctionofv(t),anddz1anddz2areWienerprocessesunderthephysicalmeasure.Notethatσ,thevarianceofvarianceparameter,andρ,thecorrelationbetweenz1andz2,areassumedtobeidenticaltothecorrespondingparametersintherisk-neutraldynamics.However,thelong-runphysicalvarianceθandmeanreversionκdifferfromthelong-runrisk-neutralvarianceθ?andmeanreversionκ?.Thisspecificationisconsistentwiththeexistingliterature.Itrepresentsthemostgeneralcombinationofphysicalandrisk-neutraldynamicsthatareconsistentwiththeaffinespecificationandGirsanov’stheorem.Weanalyzethismappinginmoredetailbelowinourdiscussionof(the)pricingkernel(s).

3.2TheInstantaneousStochasticVarianceandtheVIX

IntheHeston(1993)model,aswellasinitsmanygeneralizationsstudiedintheliterature,thestochasticvarianceisunknown.Thislatencyistypicallyaddressedinestimationbyusingfiltering-orsimulation-basedtechniques(see,e.g.,Eraker,Johannes,andPolson,2003;Eraker,2004;Bates,2006;Christoffersen,Jacobs,andMimouni,2010).Itiswell-knownthattheimplementationofsuchtechniquesiscomputationallyverydemanding,especiallywhenusinglongtimeseriesand

largecross-sectionsofoptionpricesinestimation.

Toalleviatethiscomputationalburden,wefollowadifferentapproach.11Weusethefactthatthestochasticvariancev(t)canberepresentedasalinearfunctionofVIX2(t).Thisdirectlyfollowsfromthemodelspecification:Whenv(t)followsaCIRprocess,VIX2(t)isalinearfunctionofv(t).Specifically,themodel-impliedVIX2(t)isgivenby:

(3)

VIX2(t)=E[\tt+?1mv(u)du]

=θ?+e11(v(t)?θ?),

11SeeBates(2000)andAndersen,Fusari,andTodorov(2015)foralternativeapproaches.

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w

where?1m≈30/365.Rearrangingequation(3)yields

VIX2(t)?θ?(1?w)

,

v(t)=(4)

wherew=(1?exp(?κ??1m))/(κ??1m).Inimplementation,wecanaddameasurementerror

becauseequations(3)and(4)usethemodel-impliedVIX2(t).Equation(3)inconjunctionwiththemeasurementerroryieldsameasurementequationwhichcanbeusedtofilterthelatentstatevariable.Jones(2003),Cheung(2008),andChernov,Graveline,andZviadadze(2018)usethismeasurementequationandaBayesianframeworkwithMarkovchainMonteCarlomethodstoestimateoptionpricingmodels.Wefurthersimplifythesetup:Wedonotusethemeasurementequation,butrelaxtherestrictionsonthecoefficientsinequation(4)andomitthemeasurement

error.Specifically,weassume:

v(t)=η0+η1VIX2(t).(5)

Wethenuseequation(5)inthevaluationformulaforalloptionsinthesample.Asaresult,optionsareafunctionnotonlyofthestochasticv(t),butalsooftheobservableVIX.ThisimplementationfollowsA¨?t-SahaliaandKimmel(2007),whouseitinasamplewhichcontainsasingleshort-maturityat-the-moneyoptionateachtimet.

Wenextdiscussthedetailsofthisestimationapproachwhenusingreturnsandwhenusingoptions.OuruseoftheVIXasaproxyforthestochasticvariancehasimplicationsforbothestimationexercises.

3.3Return-BasedEstimation

ThemainpurposeoftheassumptionthatthestochasticvarianceisanaffinefunctionofVIXistoalleviatethecomputationalburdenwhenestimatingthemodelusingoptiondata.However,thisassumptionalsohasimplicationsforthereturn-basedestimation.SinceweobservethetotalreturnofthestockindexandVIXateachtimet,wecanformulatethejointlikelihoodfunctionofthereturnandVIX2toestimatethephysicalparameters.Inmostexistingestimations,thevarianceis

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insteadfilteredfromtheunderlyingreturns,andtheVIXisnotusedinestimation.

Tocharacterizethelikelihoodfunction,wefirstapplyIto’slemmaandtheEulerdiscretizationtoequation(2),whichresultsin:

(6)

logR(t+?)=[r+μ(v(t))?v(t)]?+?R(t+?),

v(t+?)?v(t)=κ(θ?v(t))?+?v(t+?),

whereR(t+?)=S(t+?)/S(t)representsthegrossreturnand?=1/252.12Theerrors?(t+?)=(?R(t+?),?v(t+?))\followajointnormaldistribution,andtheirmeanandvariance-covariance

matrixaregivenby

0=Σ(t)=?.

Thejointlog-likelihoodfunctionisgivenby:

logLR

=

=

=

T?1

之logf(logR(t+?),VIX2(t+?)|VIX2(t))

t=1

之logf(logR(t+?),v(t+?)|v(t))×J(t+?)

T?1

t=1

之?log(2π)?log|Σ(t)|??(t+?)Σ\?1(t)?(t+?)+logη1,

T?1

t=1

wheref(logR(t+?),v(t+?)|v(t))istheconditionaldensityofthediscretizedlogR(t+?)andv(t+?),J(t+?)istheJacobianbetweenVIX2(t+?)andv(t+?),whichisgivenbyη1fromequation(5),andtrepresentstimemeasuredindays.LetΘ={μ,κ,θ,σ,ρ,η0,η1}bethesetofphysicalparameters.ToestimateΘ,wesolvethefollowingoptimizationproblem:

maxlogLR.

Θ

(7)

12NotethatlogR(t+?)isthedailylogreturnbetweentandt+?whilev(t)istheannualizedvarianceattimet.

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3.4Option-BasedEstimation

Therisk-neutralparametersforthedynamicinequation(1)canbeestimatedinvariousways,buteachimplementationrequiresanoptionvaluationtechnique.WefollowthefastFourierimplemen-tationofCarrandMadan(1999).ThepriceofacalloptionwithitsstrikepriceKandmaturityτisexpressedbyaquasiclosedformuptoanumericalintegration,anditisgivenby

C(S(t),v(t),t)=\0∞Re[e?iukψ(u)]du,(8)

wherekisthenaturallogofK.Thefunctionψ(u)istheFouriertransformofamodifiedcallprice,whichisthecallpricemultipliedbyeαkforα>0.Wefoundthatα=4workswell.Thefunctionψ(u)iscalculatedasfollows:

(α+iu)(α+1+iu),

ψ(u)=e?rτf(u?i(α+1)|S(t),v(t))

whereiistheimaginaryunit,andf(?|S(t),v(t))=E[ei?logS(t+τ)]istherisk-neutralcon-ditionalcharacteristicfunctionoflogS(t+τ).Theclosed-formexpressionoff(?|S(t),v(t))followsHeston(1993).13Thepriceofaputoptionwiththesamestrikepriceandmaturitycanbeobtainedthroughput-callparity.Notethattheoptionpricingformulainequation(8)doesnotaccountfordividends.Wefollowtheexistingliteratureanduseafuture-dividend-adjustedindexprice.Specifically,weuseS(t)e?qτ,whereqisthedividendyieldattimet.

13WhenlogS(t)andv(t)arecharacterizedby

dlogS(t)=[r+uv(t)]dt+^v(t)dz1(t),

dv(t)=(a?bv(t))dt+σ^v(t)dz2(t),

thecharacteristicfunctionsolutionisgivenby

f(?|S(t),v(t))=eC+Dv(t)+i?logS(t),

where

C=r?iτ+{(b?ρσ?i+d)τ?2log[]},D=[],

g=,andd=^(ρσ?i?b)2?σ2(2u?i??2).

(9)

(10)

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Weusevega-weightedoptionpricingerrors.LetOandOd