TheImpactofJumpsinVolatilityandReturnsBJ
RNERAKER,MICHAELJOHANNES,andN - PDF document

TheImpactofJumpsinVolatilityandReturnsBJ
RNERAKER,MICHAELJOHANNES,andNICHOLASPOLSONAbstractThispaperexaminescontinuous-timestochasticvolatilitymodelsincorporat-ingjumpsinreturnsandvolatility.Wedevelopalikelihood-basedestimationstrategyandprovideestimatesofparameters,spotvolatility,jumptimes,and creasegraduallyviaasequenceofsmallnormallydistributedincrements.Jumpsinvolatility¢llthegapbetweenjumpsinreturnsanddi¡usivevolatilitybypro-vidingarapidlymovingbutpersistentfactorthatdrivestheconditionalvolati-lityofreturns.WefocusontheroleofjumpsinvolatilityandreturnsinS&P500andNasdaq100indexreturns,twoprominentindiceswithactivelytradedfuturesandEur-opeanoptioncontracts.Ourempiricalapproachdepartsfromtheusualroutineofestimatingparametersandperformingspeci¢cationtestsinthatwealsoesti-matetheunobservedjumptimes,jumpsizes,andspotvolatilities.Theseesti-matesprovideadynamicpictureoftherolesthesefactorsplayandareusefulforanalyzingperiodsofmarketstress.Itisespeciallyimportanttodeterminethecontributionofjumpstoperiodsofmarketstressbecausejumprisk,eitherinreturnsorinvolatility,cannottypicallybehedgedaway,andinvestorsmaydemandalargepremiatocarrytheserisks.Weconsidertwomodelswithjumpsinvolatilityandreturns,onewithcontem-poraneousarrivalsandcorrelatedjumpsizesandanotherwithindependentarrivalsandsizes,bothintroducedbyDu‡e,Pan,andSingleton(2000).Inthesemodels,we¢ndstrongevidenceforthepresenceofbothjumpsinvolatilityandreturns.First,addingjumpsinreturnstothesquare-rootstochasticvolatilitymodeldramaticallychangesthebehaviorofstochasticvolatility.Atcertainpointsintime,theadditionofjumpsreducesannualizedspotvolatilitybyasmuchas20percent,from50percentto30percent.Whilejumpsinreturnsareinfrequentevents(onetotwoperyear),theyaretypicallylargeandexplain8to15percentofthetotalvarianceofreturns.Jumpsinvolatilityarealsoimportantastheyallowvolatilitytorapidlyin-crease.Forexample,inthemarketstressofFall1987,volatilityjumpedupfromroughly20percenttoover50percent.Onceatthishighlevel,volatilitymeanrevertsbacktoitslong-runlevel,whichshowsthepersistente¡ectofjumpsinvolatilityonthedistributionofreturns.We¢ndlittle,ifany,misspeci¢cationinthemodelswithjumpsinvolatility.ThisprovidesformalevidencesupportingBates(2000),Du‡e,Pan,andSingleton(2000),andPan(2002),whosuggestthatjumpsinvolatilitymayremovethemisspeci¢cationdocumentedinmodelswithdi¡usivestochasticvolatility.Itisimportanttonotethatthepresenceofjumpsinvolatilitydoesnotelimi-natetheneedforjumpsinreturns.Withbothtypesofjumps,jumpsinreturnsoccurlessoften,buttheystillplayanimportantrole,astheygeneratethelarge,thoughinfrequentlyobserved,crashlikemovements.Forexample,inbothofthemodelswithjumpsinvolatilityandreturns,jumpsinreturnsgeneratemorethanhalfofthecrashin1987whilehighvolatilityexplainstherest.Ananalysisofthethreeperiodsofmarketstressinoursample,1987,1997,and1998,indicatesthatjumpsinvolatilityandreturnsplayagreaterrolethandi¡usivestochasticvola-tilityingeneratingtheseepisodes.Thissuggeststhatjumpcomponentsshouldcommandrelativelylargerriskpremiathandi¡usiveones,astheircontributiontoperiodsofmarketstressisgreater.Pan(2002)¢ndsevidenceforlargejumpriskpremia.TheJournalofFinance1270 Speci¢cationdiagnosticsprovideinsightintoexactlywhymodelswithoutjumpsinvolatilityaremisspeci¢ed.Hestons(1993)square-rootstochasticvolati-litymodelrequiresimplausiblylargeshockstogeneratethelargestobservedmovementsinreturns.Forexample,thesquare-rootmodelrequiresalmostaneightstandarddeviationshockinreturnstogeneratethecrash.Thisisnotduetothesquare-rootspeci¢cation:Jacquier,Polson,andRossi(2001)¢ndthesamemisspeci¢cationusingamore£exiblelog-variancemodel.Diagnosticsindicatethatamodelwithdi¡usivestochasticvolatilityandonlyjumpsinreturnsisalsomisspeci¢ed.Estimatesindicatethatjumptimesareclustered,evidenceincontrasttotheconstantarrivalintensityassumption.Forexample,inthemodelwithjumpsinreturnsanddi¡usivestochasticvolatility,weestimatethreejumparrivalsduringtheweekofthecrashin1987.Althoughtheevidenceisslightlydi¡erent,theseresultsreinforcetheconclusionsofBak-shietal.(1997),Bates(2000),andPan(2002),who,usingadditionalinformationinoptionprices,¢ndstrongevidenceformisspeci¢cationinmodelswithdi¡usivevolatilityandjumpsinreturns.Thefactthatwearriveatthesameconclusionusingonlyreturnsdataisnotacoincidence.Duetotheabsolutecontinuityofchangesinprobabilitymeasures,returnsandoptionsdatashouldcontainthesameinformationaboutthesourcesofrisks,althoughtheirimpactmaybealteredduetoriskpremia.Allofthemodelsweconsidergeneratenearclosed-formoptionprices,andwenextexaminehowthedi¡erentfactorsimpactoptionprices.Usingimpliedvolatilityasametric,we¢ndthatthetwotypesofjumpsinduceimportantdi¡er-encesinboththetermstructureandcrosssectionofimpliedvolatilities.Com-paredtothestochasticvolatilitymodel,addingjumpsinreturnssteepenstheslopeoftheimpliedvolatilitycurves.Jumpsinvolatilityfurthersteepenimpliedvolatilitycurvesandincreaseimpliedvolatilityforin-the-moneyoptions.Thislat-tere¡ectfoundbyDu‡e,Pan,andSingleton(2000)andPan(2002)waslabeledasthehookortippingattheende¡ect.Althoughthemotivationforjumpsinvo-latilitywastoimproveonthedynamicsofvolatility,theresultsindicatethatjumpsinvolatilityalsohaveanimportantcross-sectionalimpactonoptionprices.Finally,weevaluatethee¡ectofestimationriskonoptionprices.Wedecom-poseestimationriskintotwocomponents:parameterandspotvolatilityestima-tionrisk.Thesesourcesofestimationriskhaveverydi¡erente¡ects.Forshortmaturityoptions,volatilityuncertaintyisprimarilyanat-the-moneye¡ectandhaslittleimpactonout-of-the-moneyoptions.Parameteruncertainty,ontheotherhand,haslittleat-the-moneye¡ectforshortmaturityoptions,butgener-atesnearlyalloftheout-of-the-moneyuncertainty.Forlongmaturityoptions,parameteruncertaintydominatesastheuncertaintyregardingtheaveragelevelofvolatility,speedofmeanreversioninvolatility,andvolatilityofvolatilityplayamoreimportantrole.Spotvolatilityuncertaintyhaslittleimpactonlong-datedoptions.Liu,Longsta¡,andPan(2003)considerrelatedmodelswithjumpsinreturnsand/orjumpsinvolatilityand¢ndthatthesefactorshaveimportantimplicationsforoptimalportfolioallocation.TheImpactofJumpsinVolatilityandReturns1271 Therestofthepaperproceedsasfollows.SectionIintroducesthemodels,andSectionIIdescribesourestimationapproach.SectionIIIsummarizestheempiri-calresultsfortheS&P500andNasdaq100indices.SectionIVanalyzestheoptionpricingimplications,andSectionVconcludes.I.Jump-Di¡usionModelsofReturnsandVolatilityAnumberofrecentpapersexamineequitypricemodelswithjumpsinreturnsandstochasticvolatility(seeBakshi,Cao,andChen(1997),Bates(2000),Ander-sen,Benzoni,andLund(2002),Chernovetal.(2002),andPan(2002)).Whileitisclearthatbothstochasticvolatilityandjumpsinreturnsareimportantcompo-nents,Bakshi,Cao,andChen(1997),Bates(2000),andPan(2002)¢ndstrongevi-denceformisspeci¢cationinthevolatilityprocess.Speci¢cally,Bakshi,Cao,andChen(1997),usingatestdevelopedinBates(1996),¢ndthattheimpliedstructuralvolatilityparametersareinconsistentwithtime-seriesestimatesusingimpliedvolatilities.Additionally,Bates(2000)andPan(2002)¢ndthatthehighermomentsofvolatilitychangesareinconsistentwiththedi¡usionspeci¢cation.Together,theseresultspointtothepresenceofanaddi-tional,rapidlymovingfactordrivingconditionalvolatility,which,unlikejumpsinreturns,hasapersistentcomponent.Jumpsinvolatilityprovidesuchafactor.Weassumethatthelogarithmofassetsprice,log(S),solveswhereisastandardBrownianmotioninandarePois-sonprocesseswithconstantintensitiesand,andandarethejumpsizesinreturnsandvolatility,respectively.Weassumethattheparametersandinitialconditionshavesu‡cientregularityforthesolutionof(1)tobewellde¢ned.Thisspeci¢cationnestsmanyofthepopularmodelsusedforoptionpricingandportfolioallocationapplications.Withoutjumps,0,(1)reducestoHestons(1993)square-rootstochasticvolatilitymodel,theSVmodel.Bates(1996)SVJmodelhasnormallydistributedjumpsinreturns,,butnojumpsinvolatility,0.Du‡e,Pan,andSingleton(2000)introducedthemodelswithjumpsinvolatility.TheSVIJmodelhasindependentlyarrivingjumpsinvolatility,exp(),andjumpsinreturns,TheSVCJmodelhascontemporaneousarrivals,,andcorrelatedjumpsizes,exp()andAndersenetal.(2001)andAlizadeh,BrandtandDiebold(2002)argue,forU.S.equitiesandanumberofforeignexchangerates,thattherearetwofactorsgeneratingvolatility,onehighlypersistentandslowlymoving,theotherrapidlymoving.Thisbehaviorisnicelycap-turedinthemodelsweconsiderwithvolatilitydrivenbydi¡usiveandjumpcomponents.Weoriginallyincludedavarianceriskpremiaterminthereturndrift,.Itwasin-signi¢cantandwasthereforedroppedfromtheanalysis,consistentwithAndersen,Benzoni,andLund(2002)andPan(2002),whoalsofoundthisparameterinsigni¢cantinmodelswithjumpsinreturns.TheJournalofFinance1272 IntheSVandSVJmodels,controlsthelong-runmeanofHowever,inthepresenceofjumpsinvolatility,isonlythedi¡usive,andnottotal,long-runmeanof.Withjumpsinvolatility,isthelong-runmean.Toseehowjumpsinreturnsandvolatilitya¡ectthesecondmo-mentsofreturnsandvolatility,TableIprovidestheinstantaneousvarianceandcovarianceofandforeachofthemodels.TheexponentiallydistributedjumpsincapturethelargepositiveoutliersinvolatilitydocumentedinBates(2000)andguaranteethatispositive.TheSVCJandSVIJmodelshavethreefactors(di¡usivestochasticvolatility,jumpsinreturns,andjumpsinvolatility),andwenowdiscusshowthesefactorsimpactthedistributionofreturns.Althoughjumpsinreturnsanddi¡usivestochasticvolatilitycanbothgener-aterealisticpatternsofunconditionalnonnormalitiesinreturns,theygenerateverydi¡erentpatternsofconditionalnonnormalities.Jumpsinreturnsresultinadiscretemixtureofnormaldistributionsforreturns,whicheasilygeneratesunconditionalandconditionalnonnormalitiesovershortfrequenciessuchasdailyorweekly.Overlongerintervals,acentrallimite¡ectresultsindecreasingamountsofexcessskewnessandkurtosis(seeDu‡eandPan(1997)orDasandSundaram(1999)).Di¡usivestochasticvolatilitymodelsinduceadi¡erentdistributionalstruc-ture.Conditionalonthecurrentlevelofvolatility,returnsareapproximately TableIConditionalMomentsThistablesummarizestheinstananeousconditionalmomentsforthefourmodelsundercon-sideration.InthecaseoftheSVCJmodel,thesecondmomentofthejumpsizesisSVSVJSVCJSVIJ 1dtvartðdYtÞVtVtþlyðm2yþs2yÞVtþlyEðxyÞ2tþlym2yþs2y dtvartðdVtÞs2vVts2Vts2Vtþlymvs2vVtþlvmv Tocalculatethis,integratetheSDEtogetSincethestochasticintegralagainsttheBrownianmotionismeanzero,FubinistheoremimpliesthatSinceandthejumparrivalsarePoisson,wehavethatwhichimpliesthat Thenotation isaheuristicforlim.Sinceourinitialconditions,drift,di¡usion,jumpintensity,andjumpsizedistributionshavesu‡cientregular-ity,thelimitingoperationisvalid.TheImpactofJumpsinVolatilityandReturns1273 normallydistributedovershorttimehorizonssuchasdailyorevenweekly.DasandSundaram(1999)¢ndthatdi¡usivestochasticvolatilitymodels,withreason-ableparameters,cangeneraterealisticamountsofconditionalnonnormalitiesinreturnsonlyoverlongerhorizons,suchasmonthsorevenyears.Thisexplainswhydi¡usivestochasticvolatilitymodels(square-rootorotherwise)generatevery£atimpliedvolatilitycurvesforshort-datedoptionswithplausiblepara-meters(seeDasandSundaram(1999)orJones(2002)).Modelswithonlyjumpsinreturnsanddi¡usivestochasticvolatilitycangen-eraterealisticpatternsofbothunconditionalandconditionalnonnormalities,buttheyhavedi‡cultycapturingthedynamicsoftheconditionalvolatilityofreturns.IntheSVJmodel,theconditionalvarianceofreturnsis(seeTableI).Whenisadi¡usion,theconditionalvolatilityofreturnsistimevaryingandpersistent,butmovesslowlyasitisdrivenbynormallydistributedshocks.AspointedoutbyBates(2000)andPan(2002),thiscreatesmisspeci¢ca-tion,asthey¢ndthatvolatilityneedstoincreaserapidly.Jumpsinvolatilityprovideafactorthatcombinesfeaturesfrombothjumpsinreturnsanddi¡usivestochasticvolatility.Likejumpsinreturnsandunlikesto-chasticvolatility,jumpsinvolatilityarearapidlymovingfactordrivingreturns.Likedi¡usivestochasticandunlikejumpsinreturnswhoseimpactonreturnsistransient,ajumpinvolatilitypersists.Thus,jumpsinvolatilityprovidearapidlymovingbutpersistentfactordrivingvolatility.Thefactthateachfactorgener-atesverydi¡erentbehaviorishelpfulforeconometricidenti¢cation.Therearealternativeexplanationsforthefailureofmodelswithonlyjumpsinreturnsandsquare-rootstochasticvolatility.Theseincludeadditionalsquare-rootvolatilityfactors,more£exible,parametric,single-factorstochasticvolatilityspe-ci¢cations,orcombinationsofthesetwo.More£exiblesingle-factorspeci¢cationssuchasthelogandCEVmodelsallowforthevolatilityofvolatilitytobestatedependent,apropertyabsentinsquare-rootmodels.Thesemodelsarelimited,be-causetheydonotprovideclosed-formoptionprices.Andersen,Benzoni,andLund(2002)andChernovetal.(2002)¢ndthatthelog-volatilityandsquare-rootmodelsprovideanearidentical¢ttothedataandthatneithermodelcancapturethefattailsinthereturndistribution.ChackoandViceira(2001)andJones(2002)consid-eraCEVstochasticvolatilityspeci¢cation.Whilemore£exiblethanthelogmod-el,returnsintheCEVmodelarestillconditionallynormalovershorttimeintervals.ChackoandViceira¢ndnonlinearitiesinthevarianceof,butthee¡ectdisappearswhenjumpsinreturnsareadded.Consistentwiththeargu-mentsabove,JonesshowsthattheCEVmodelgeneratesrealisticunconditionalnonnormalities,butovershorttimeintervals,themodelo¡ersonlyamodestim-provementoverthesquare-rootmodelingeneratingconditionalnonnormalities.Bates(2000)andChernovetal.(2002)considerandrejecttwo-factorsquare-rootstochasticvolatilitymodels.Infact,Bates(p.218)arguesthatthepostu-latedsquare-rootprocessforimplicitfactorevolutionisfundamentallymisspe-ci¢ed,thatthetwo-factormodelshaveevengreaterdi‡cultiesthantheone-factormodelsingeneratingsamplepathsconsistentwiththepostulatedprocessandtheimplicitparameters(p.214),andadditionallyarguesthatthesemodelsareover¢t.TheJournalofFinance1274 Toinvestigateanyimprovementsgeneratedbyadditionalvolatilityfactors,weestimatedatwo-factorindependentsquare-rootstochasticvolatilitymodelwhereConsistentwiththe¢ndingsofBates(2000)andChernovetal.(2002),wefoundlittleevidencethatthismodelprovidedanysubstantiveimprovementoverthesingle-factorsquare-rootmodel.Infact,itsu¡eredfromthesamemisspeci¢cationasthesquare-rootandlog-volatilitymodels:Itcannotcapturethetailsofthereturndistribution.Thisisnotsurprising,asthemultifactormodelisinstantaneouslyGaussian,condi-tionalontotalvolatility.Forthesereasons,wefocusonasingle-factorsquare-rootmodelofstochasticvolatilityandextensionsincorporatingjumpsinreturnsandvolatility.II.EstimatingStochasticVolatilityJumpDi¡usionsThissectiondevelopsalikelihood-basedestimationapproachforestimatingmultivariatejump-di¡usionmodelsusingMarkovChainMonteCarlo(MCMC)methods.RobertandCasella(1999)provideageneraldiscussionofthesemethods,andJohannesandPolson(2002)provideanoverviewofMCMCestimationofcontinuous-timemodels.Thisapproachhasfouradvantagesoverotherestimationmethods:(1)MCMCprovidesestimatesofthelatentvola-tility,jumptimes,andjumpssizes;(2)MCMCaccountsforestimationrisk;(3)MCMCmethodshavebeenshowninrelatedsettingstohavesuperiorsam-plingpropertiestocompetingmethods;and(4)MCMCmethodsarecomputa-tionallye‡cientsothatwecanchecktheaccuracyofthemethodusingsimulations.Ourapproachusesonlyreturnsdatatoestimateandtestthemodels,althoughitcanbeextendedinastraightforwardmannertoincludeoptionpricedata(seeEraker(2002)).Duetotheabsolutecontinuityofthechangeinmeasurefromob-jectivetoriskneutral,thepresenceofjumpsinreturnsorvolatilityunderoneWethanktherefereeforsuggestingthis.Forexample,inamodelwithtwoindependentsquarerootvolatilityfactors,theinstanta-neousdistributionofpricechanges, isstillGaussian: OthermethodsthathavebeenusedtoestimatemodelswithstochasticvolatilityandjumpsincludeEMM,simulatedmaximumlikelihood(BrandtandSanta-Clara(2002),Dur-hamandGallant(2001),Piazessi(2001)),calibration(Bates(1996,2000),Bakshi,Cao,andChen(1997)),andtheimplied-stateGMMmethodofPan(2002).Jacquier,Polson,andRossi(1994)¢ndinsimulationsthatMCMCoutperformsGMMandQMLEinestimationofstochasticvolatilitymodels,andAndersen,Chung,andSorensen(1999)¢ndthatMCMCoutperformsEMM.TheImpactofJumpsinVolatilityandReturns1275 measureimpliestheirpresenceundertheother,althoughduetoriskpremia,theirimpactmaybedistorted.Thus,forspeci¢cationanalysis,returnsdatashouldleadtothesameconclusionasoptionpricedata.Themainadvantageofusingonlyreturnsdataismorepragmatic:Analysesusingoptionpricedatatendtouserelativelyshorttimespans.Forexample,Bakshi,Cao,andChen(1997)usedatafrom1988to1991whilePan(2002)usesdatafrom1989to1996.Thisisespeciallyimportantwhenestimatingmodelswithjumps,whichweexpecttooccurinfrequently.Longersamplesspanningperiodsofmarketstress(theepisodesin1987,1997,and1998)willprovidemoreaccurateparameterestimatesandbetterinsightsintotherelativerolesplayedbyjumpsandstochasticvolatility.ThebasisforourMCMCestimationisatime-discretizationof(1)whereindicatesajumparrival,andarestan-dardnormalrandomvariableswithcorrelationandisthetime-discretizationinterval(oneday).ThejumpsizesretaintheirdistributionalstructureandthejumptimesareBernoullirandomvariableswithconstantintensities,and.Thisprocedurecouldintroduceadiscretizationbias,althoughthebiasistypicallyquitesmallwithdailydata.Weprovidesimulationsbelowtosupportthisclaim.Inthissection,wefocusontheSVCJmodel,asithasthemostcomplicateddistributionalstructure.Theposteriordistributionsummarizesthesamplein-formationregardingtheparameters,,andthelatentvolatility,jumptimes,andjumpsizes:where,andarevectorscontainingthetimeseriesoftherelevantvari-ables.Theposteriorcombinesthelikelihood,),andtheprior,Anadvantageofourapproachistheabilitytoformallyincorporatepriorinfor-mation.Theneedforthisisnotuniquetoourapproach,butiscommoninestimat-ingmodelswithjumps.Honore(1998)showsthatwithoutpriorparameterrestrictions,atimediscretizationofMertons(1976)jump-di¡usionmodelgener-atesanunboundedlikelihoodfunction.Moreover,thepriorcontainsinformationaboutboththeparametersandthestructureofthelatentprocesses:thestochas-ticspeci¢cationsofthejumpsizes,jumptimes,andvolatility.Thisreinforcesthelinkbetweenparametersandmodelspeci¢cationthatisoftenheuristicallyusedtomotivatethepresenceofjumps.Typically,jumpsaredescribedaslarge,butOurframeworkisnotlimitedtothecaseinwhichthediscretizationinterval,equalstheobservedfrequency.Toreduceanybias,wecouldintroduceadditionalunobserveddatapointsbetweendatesand1andtreatthemasmissingdatapointstobeincludedintheMCMCsimulation(seeEraker(2001)).TheJournalofFinance1276 infrequentmovementsinreturns.Thisisaformofpriorinformationasthepara-metersareassumedtoinduceinfrequentbutrelativelylargemovements(lowandlargeand/or),asopposedtofrequentbutsmalljumps.Ourpriorsarealwaysconsistentwiththeintuitionthatjumpsarelargeandinfrequent.Morespeci¢cally,wechooseaprioronthatplaceslowprobabilityonthejumpsizesbeingsmall,saylessthanonepercent.For,ourpriorplaceslowprobabilityonthedailyjumpprobabilitybeinggreaterthan10percentandweplaceanuninformativeprioron.Fortheotherparameters,wespecifyex-tremelyuninformativepriors.For,andweusemeanzeronormalpriorswithlargevariancesandtheprioronisuniformover[1,1].AppendixApro-videsfurtherdetailsonthepriors.Itisimportanttonotethatweimposeverylittleinformationthroughourpriors.Astheposteriordistributionisnotknowninclosedform,ourMCMCalgo-rithmgeneratessamplesbyiterativelydrawingfromthefollowingconditionalposteriors:parametersjumptimesjumpsizes...volatilitywheredenotestheelementsoftheparametervectorexcept.Drawingfromthesedistributionsisstraightforward,withtheexceptionofvolatility,asthedis-tributionisnotofstandardform(AppendixAprovidesdetails).Thealgorithmproducesasetofdrawswhicharesamplesfrom).JohannesandPolson(2002)provideareviewofthetheorybe-hindMCMCalgorithms.A.EstimatingVolatilityandJumpsForspeci¢cationanalysisandtoidentifytherelativeimportanceofjumpsandvolatility,werequireestimatesofthelatentvolatility,jumptimes,andjumpsizes.Withcontinuousrecordobservations,allareobserved,butwithdiscretelysampledobservations,itisnotobvioushowtoseparateoutthee¡ectsofjumpsandtime-varyingvolatility.Forexample,isalargemovementinreturnsgener-atedbyajumpinreturnsorbyhighvolatility?Standardlatentvariableestima-tionmethodssuchastheKalman¢lterdonotapply,asourmodelisneitherlinearnorGaussian.OurMCMCapproachprovidesastraightforwardmethodtoestimatethevola-tilities,jumptimes,andjumpsizesbycomputingtheposteriorexpectationofthesevariables.We,therefore,provideaMonteCarlosolutiontotheclassical,latentvariableestimationproblem.ThekeyisthatourMCMCalgorithmgeneratessamplesofthespotvolatilities,jumptimes,andjumpsizes,drawnfromthejointposteriordistribution.Giventhesesamples,theMonteCarloestimateofthemeanoftheposteriorvolatilitydistribution,forexample,isTheImpactofJumpsinVolatilityandReturns1277 E½VtDjY ,whereisthevarianceattimeintheiterationofthealgorithm.Noadditionalcalculationsarerequired:Latentvariableestima-tionisjustaby-productofouralgorithm.Jumptimeandsizeestimatesaresimi-larlycalculated.Theseestimatestakeintoaccountparameteruncertainty.Toseethis,notethatweestimateteVtDjY]andnot.Theformerdistributionintegratesoutalloftheparameteruncertainty.Thelatterdistributiontreatstheparameteres-timatesasknown,ignoringthefactthattheyarerandomvariables.B.ModelDiagnosticsandSpeci¢cationTestsThespotvolatility,jumptime,andjumpsizeestimatesgenerateanumberofinformaldiagnosticsthatareusefulinassessingtheabilityofthevariousmodelsto¢ttheobserveddata.Forexample,considerthereturnresiduals: Itiseasytocomputetheposterioroftheseresidualsusingtheparameterandlatentvariablesamples.Theseresidualsneednotbeexactlynormallydistributed(sinceweuseatimediscretizationoftheoriginalmodel),buttheyshouldbeapproximatelynormal.Extremelylargeresidualssuggestmisspeci¢cationasthemodelrequiresabnormallylargeshocksto¢ttheobserveddata.Jumptimeandsizeestimatesprovidefurtherdiagnostictools.Forinstance,anyevidenceofclusteredjumptimescontradictsthei.i.d.arrivalassumption.Jumpsonneigh-boringdaysofoppositesigns,thereversale¡ectofSchwert(1990),isalsoevidenceagainstthei.i.d.jumparrivalsandsizes.Wealsocomputeformalspeci¢cationtests.Unlikestandardteststhatleadtoanoverallevaluationofthe¢tofthemodel(e.g.,omnibuschi-squaretests),wecomparethemarginallikelihoodsofthemodels.Theadvantageofthisapproachisthatitdoesnotrelyonlargesampledistributiontheoryandprovidesanintui-tiveapproachtoevaluatingtherelativemeritsofcompetingmodels.OurapproachforcomparingnestedmodelsissimilartothatconsideredinJacquierandPolson(1999).ConsidertestingSVversusSVJ.Ifweassumepositiveprioroddsonthemodels,SVJ0(priorignorancesetsthisto1),Bayestheoremimpliesthattheposterioroddsare pðSVjYÞpðSVJjYÞ¼ SVJ where,SVJisknownastheBayesFactor.Notethatthisanalysisdoesnotassumethatthemodelsarenecessarilyexhaustive,thatis,thatSVJAssumingpriorignorance,theOddsRatio,isinterpretedinthefol-lowingmanner:Ariskneutralbookmakerwouldlayodds, to1,onmodelSVJversusSV.Anadvantageofthisapproachisthatthetestresultsinasinglenumberthatistherelativeoddsofthemodelsgiventhedata,andthereisnoappealtoapproximatelimitingdistributionsorsigni¢cancelevels.AppendixBTheJournalofFinance1278 derivesthemarginallikelihoodsandBayesFactorsforthemodelsunderconsid-erationandprovidesMCMCestimators.Thefactthatthemodelswithjumpsinvolatilityarenotformallynestedpro-videsnoproblemforcomputingtheBayesFactors,forexample,SVCJversusSVIJ.WecomputetheBayesFactorforthenonnestedmodelsusingthefactthat SVCJSVIJ SVCJSVJ SVJOnecaveatisthattheMonteCarlostandarderrorsmaybelargerthaninthecaseofthesimplenestedcomparisons,astheOddsRatioisnowtheproductoftwoOddsRatios.FollowingKassandRaftery(1995),weusethefollowingscaletointerprettheBayesFactors.EvidenceagainstamodelispositiveifthelogOddsRatioisbetween2and6,strongifitisbetween6and10,andverystrongifitisgreaterthan10.ItisimportanttonotethatOddsRatiosdonotnecessarilyfavormorecomplexmodels,astheycontainapenaltyforusingmoreparameters(duetotheirmarginalnature).Becauseofthis,OddsRatiosareoftenreferredtoasanautomaticOccamsrazor(seeSmithandSpiegelhalter(1982)).C.SimulationResultsWeperformedMonteCarlosimulationstocheckthereliabilityofourestima-tionapproach.Thisisimportantfortworeasons.First,sincewetimediscretizethecontinuous-timemodel,itisimportanttocheckthatthisdoesnotintroduceanybiasesinparameterestimates.Second,methodsforestimatingmultivariatejumpdi¡usionmodelsarenotwelldevelopedanditisimportanttoverifythatwecanreliablyestimatetheparametersforthegivensamplesize.AppendixCdescribesoursimulationstudy,andTablesVIIandVIIIinthatap-pendixsummarizetheresults.Theresultsindicatethatourprocedureprovidesaccurateinference.Someparametersareestimatedlesspreciselythanothers,forexample,butallareclosetotheirtruevalues.Theresultsalsoindicatethatourpriorsarenotinformative,asweusethesamepriorsforparameterscommoninboththeSVJandSVCJmodels,eventhoughtheestimationresultsandtruepara-metersdi¡er.III.EmpiricalResultsWeestimatethemodelsusingS&P500andNasdaq100indexreturnsfromJan-uary2,1980,toDecember31,1999,andSeptember24,1985,toDecember31,1999,respectively.Excludingweekendsandholidays,wehave5,054dailyobservationsfortheS&Pand3,594observationsfortheNasdaq.TableIIprovidessummarystatisticsforthecontinuouslycompoundedreturns,scaledby100.Inthissection,wediscusstheestimationresultsfortheS&P500,foreachofthemodels,howtheNasdaqestimatesdi¡erfromtheS&P,andtheroleofjumpsandvolatilityinthethreeperiodsofmarketstressinoursample.TablesIIIandIVsummarizeTheImpactofJumpsinVolatilityandReturns1279 parameterestimationforallofthemodelsfortheS&P500andNasdaq100,respectively.A.S&P500ThesecondcolumnofTableIIIprovidesparameterposteriormeansandstan-darddeviationsfortheSVmodel.Theleft-handpanelofFigure1providesspotvolatilityestimatesovertwotimeperiods,1987to1989and1997to1999.Thepara-meterestimatesareconsistentwithprevious¢ndings.Theaverageannualizedvolatility,252,is15.10percentandisclosetothesamplevolatilityof15.89 TableIISummaryStatisticsThistableprovidessummarystatisticsfordailyreturndataontheS&P500fromJanuary2,1980,toDecember31,1999,andtheNasdaq100fromSeptember24,1985,toDecember31,1999.MeanVolatilitySkewnessKurtosisMinMaxS&P50013.018515.88382.606462.032822.83308.7089Nasdaq10024.584123.21820.728711.942416.34059.7984 TableIIIS&P500ParameterEstimatesParameterestimatesfortheS&P500indexdata,January2,1980,toDecember31,1999.ThemodelsandparameterizationsaregiveninSectionIandtheestimatescorrespondtopercen-tagechangesintheindexvalue.Foreachparameter,wereportthemeanoftheposteriordevia-tionandthestandarddeviationoftheposteriorinparentheses.SVSVJSVCJSVIJ0.0444(0.0110)0.0496(0.0109)0.0554(0.0112)0.0506(0.0111)0.9052(0.1077)0.8136(0.1244)0.5376(0.0539)0.5585(0.0811)0.0231(0.0068)0.0128(0.0039)0.0260(0.0041)0.0250(0.0057)0.1434(0.0128)0.0954(0.0104)0.0790(0.0074)0.0896(0.0115)2.5862(1.3034)1.7533(1.5566)3.0851(3.2485)0.6008(0.9918)4.0720(1.7210)2.8864(0.5679)2.9890(0.7486)1.4832(0.3404)1.7980(0.5737)0.3974(0.0516)0.4668(0.0579)0.4838(0.0623)0.5040(0.0661)0.0060(0.0021)0.0066(0.0020)0.0046(0.0020)0.0055(0.0032)FortheSV,SVJ,andSVCJmodels,theMCMCalgorithmappearstoconvergequickly.Wediscardthe¢rst10,000iterationsasaburn-inperiodandusethelast90,000toformtheMonteCarloestimates.FortheSVIJmodel,thealgorithmappearstoconvergemoreslowlyandthusweranitfor200,000iterations,discardingthe¢rst10,000asaburn-inperiod.FortheNasdaq100,thealgorithmconvergedquicklyforallmodels,andthuswediscardedthe¢rst10,000drawsandusedthelast90,000draws.TheJournalofFinance1280 percent.Ourestimateof0.40,isclosetotheestimateobtainedbyJacquier,Polson,andRossi(2001)inalog-volatilitymodel(0.39)andAndersen,Ben-zoni,andLund(2002)(0.38),butisslightlysmallerthanthoseobtainedfromstudiesusingoptionpricedata(Bakshi,Cao,andChen(1997),Bates(2000),andPan(2002)obtainestimatesaround0.5).TheSVmodelismisspeci¢ed.Thiscanbeseeninanumberofways.First,con-sidertheQQornormalprobabilityplotoftheresidualsintheupperleftpanelinFigure2.Notetheextremenonnormalityoftheresiduals,strongevidenceofmis-speci¢cation.Whydoesthemodelrequiresuchlargeshocks?Considerthecrashin1987.Onthedayofthecrash,wasaboutthreepercent,whichimpliesthatanalmosteightstandarddeviationreturnshockisneededtodeliverthepercentmove.Thistypeofmisspeci¢cationwasalsonotedinstochasticvolatilitymodelsbyJacquier,Polson,andRossi(2001),Andersen,Benzoni,andLund(2002),andChernovetal.(2002).Second,spotvolatilityincreasedfor37consecutivedayspriortothecrashin1987.Thisisextremelyunlikelyasthemeanrevertingdriftwasexertingdownwardpressureonvolatilitythroughoutthisperiod.Final-ly,theBayesFactorsreportedinTableVIbelowprovidesevidenceagainsttheSVmodelinfavoroftheothermodels.ThethirdcolumninTableIIIprovidesparameterestimatesfortheSVJmodel.Addingjumpsinreturnshastheexpectede¡ectofreducingthedemandsonthevolatilityprocess.Forexample,averagevolatilityfallsfrom15.10percentto14.32percentandbothandfalldramatically,indicatingalessvolatile,moreper-sistentvolatilityprocess.EstimatesofspotvolatilityinFigure1showthatjumpsinreturnsreducespotvolatilitydramaticallyduringperiodsofmarketstress,asjumpsgeneratethelargestmovements.Alsonotethatafterthecrashin1987,spotvolatilityintheSVmodelremainedhigherthanintheSVJmodelforalongtime. TableIVNasdaq100ParameterEstimatesParameterestimatesfortheNasdaq100indexdata,September24,1985,toDecember31,1999.ThemodelsandparameterizationsaregiveninSectionIandtheestimatescorrespondtoper-centagechangesintheindexvalue.Foreachparameter,wereportthemeanoftheposteriordeviationandthestandarddeviationoftheposteriorinparentheses.SVSVJSVCJSVIJ0.1070(0.0196)0.1240(0.0205)0.1284(0.0206)0.1164(0.0196)2.0242(0.2309)1.9067(0.2853)0.9249(0.1547)1.0593(0.1506)0.0264(0.0072)0.0176(0.0052)0.0414(0.0106)0.0371(0.0080)0.2155(0.0226)0.1692(0.0190)0.1216(0.0188)0.1395(0.0196)2.4755(1.0276)1.8868(0.72.6231(2.5946)0.0993(0.1692)2.0788(0.4375)1.8452(0.3079)2.0389(0.4485)2.1054(0.3989)2.5227(0.5946)0.2869(0.0553)0.3366(0.0695)0.3427(0.0975)0.3910(0.0801)0.0172(0.0097)0.0202(0.0074)0.0081(0.0044)0.0140(0.0058)TheImpactofJumpsinVolatilityandReturns1281 Thus,thee¡ectofomittingjumpsinreturnslingersafterthejumparrivedthroughitsimpactonestimatedvolatility.IntheSVJmodel,jumpsinreturnsareinfrequentevents(about1.5peryear),tendtobenegative,andarelargerelativetonormalday-to-daymovements.Athreestandarddeviationjumpmoveisabout15percent.TableVdecomposesthetotalvarianceofreturnsintostochasticvolatilityandjumpcomponents.Theproportionofvarianceduetojumpsis andis14.65percentintheSVJmodel.ThenormalityplotsintheSVJmodelareimprovedandtheresidualsnowhaveslightlythintailsasjumpsinreturnscapturenearlyallofthelargemovementsinreturns.Figure3,however,providesevidencethattheSVJmodelismisspeci¢ed. 19871987. 5 19881988. 5 1989 5 15 25 35 45 55 Spot VolatilityS&P 500, SV and SVJ SVJ 19971997. 5 19981998. 5 1999 5 10 15 20 25 30 35 40 Spot VolatilityS&P 500, SV and SVJ SVJ 19871987. 5 19881988. 5 1989 5 15 25 35 45 55 S&P 500, SVCJ and SVIJSpot Volatility 19971997. 5 19981998. 5 1999 5 10 15 20 25 30 35 40 Spot VolatilityS&P 500, SVCJ and SVIJ SVCJ Figure1.EstimatedvolatilitypathsfortheS&P500indexforthefourmodels.TheJournalofFinance1282 Duringtheweekofthecrashin1987,therewerethreedaysonwhichtheesti-matedjumpprobabilitieswereextremelyhigh,indicatingaclusterofjumps.Whydidthisoccur?DuringOctober1987,dailyvolatilitywasalwayslessthantwopercent,implyingthatathreestandarddeviationmoveinreturnsduetovo-latilitywasonlysixpercent.Duetothis,allofthemoveslargerthansixpercentwereattributedtojumpsinreturns.Similarly,inOctober1997,therewerejumpsestimatedonneighboringdayswithoppositesigns,thereversale¡ectofSchwert(1990).Theclusteringofjumparrivalsandsizereversalsareextremelyunlikelygiventhei.i.d.jumptimeandsizespeci¢cationsandtheinfrequentnatureofjumps(1.5peryear).ThissuggeststhattheSVJmodelismisspeci¢ed.TheBayes -4 -2 0 2 0.001 0.003 0.01 0.02 0.05 0.10 0.25 Normal Probability Plot, SV Model 0 2 0.001 0.003 0.01 0.02 0.05 0.10 0.25 Normal Probability Plot, SVCJ Model 0 2 0.001 0.003 0.01 0.02 0.05 0.10 0.25 DataProbabilityNormal Probability Plot, SVJ Model -2 0 2 0.001 0.003 0.01 0.02 0.05 0.10 0.25 Figure2.QQornormalityplotoftheresidualsforeachofthemodels.Johannes,KumarandPolson(1999)documentthispatternanddevelopmodelswithstate-dependentarrivalsandjumpsizestocapturethisphenomenon.TheImpactofJumpsinVolatilityandReturns TableVVarianceDecompositionsThesecondandthirdcolumnsgivetheaverageannualizedspotstochasticvolatilityforeachofthemodelsandfortheS&P500andNasdaq100.Thefourthand¢fthcolumnsgivetheaverageannualizedtotalvolatility,whichisthesumofstochasticvolatilityandjumpsinreturnscom-ponents.Thelasttwocolumnsgivetheproportionofthevarianceduetojumps.SpotVolatilityTotalVolatilityReturnJumpVariance(%ofTotal)0S&PNDXS&PNDXS&PNDXSV15.1022.5915.1022.59^^SVJ14.3221.9215.4924.0614.6517.00SVCJ15.1822.1815.9923.599.9611.63SVIJ15.5122.5116.1823.228.175.99 1985 1990 1995 2000 -20 -15 -10 -5 0 5 10 Jump SizeS&P 500, SVJ 1985 1990 1995 2000 0 0.250. 5 0.75 1 S&P 500, SVJJump Probability 1980 1985 1990 1995 2000 -4 0 4 Jump SizeS&P 500, SVCJ 1985 1990 1995 2000 0 0.25 0.75 1 Figure3.EstimatedjumptimesandsizesfortheSVJandSVCJmodelsfortheS&P500index.TheJournalofFinance1284 factorsinTableVIprovideadditionalevidenceagainsttheSVJmodelandinfavorofthemodelswithjumpsinvolatilityInconclusion,theSVandSVJmodelssu¡erfromsimilarproblems.Periodsofmarketstressarecharacterizedbyashorttimeperiodwithmultiplelargemove-mentsandneithermodelcangeneratethesemovements.IntheSVmodel,theselargemovementsappearasabnormallylargeshocks,whileintheSVJmodel,theyappearasclusteredjumps.Inbothcases,di¡usivevolatilitycannotincreasera-pidlyenoughtogeneratetheseepisodesandthemodelsaremisspeci¢ed.ThefourthcolumnofTableIIIprovidesparameterestimatesfortheSVCJmod-elandtheresultsindicatethatjumpsinvolatilityplayanimportantrole.Whenajumparrives,volatilityincreasesfrom,andtheparameteres-timatespointtowardsigni¢cantincreases.Forexample,whenvolatilityis15per-cent,anaveragesizejumpincreasesvolatilityto24percent.ThesolidlineintherighthandpanelsofFigure1showstheestimatedvolatilitypathfortheSVCJmodel,anditisverydi¡erentfromthoseintheSVandSVJmodels.Asjump-drivenhighvolatilitygeneratesmanyofthelargemovesinthevolati-lityprocess,issmallerthanintheSVJmodel.IntheSVCJmodel,volatilitymeanrevertsfaster,asisalmostdoubleitsvalueintheSVJmodel.Jumpsstillarriveattherateofabout1.5peryear,butjumpsinreturnsaresmaller.JumpsinreturnsplayalesserroleintheSVCJmodelthanintheSVJmodel,asthepropor-tionoftotalvariancecomingfromjumpsinreturnsisnowonly9.96percent.ThejumpsizeestimatesintheupperrighthandpanelofFigure3indicatethatnearlyallofthejumpsarenegative,astheincreasedvolatilityafterajumpgeneratesthereversale¡ect.Theestimateoffallsagain,whichisconsistentwithDu‡e,Pan,andSingleton(2000).ThenormalresidualplotinFigure3givesnoindicationofmisspeci¢cationintheSVCJmodel.Asjumpsinreturnsaresmaller,theyexplainlessofthelargemovementsandtheBrownianincrementsarealmostperfectlynormallydistrib-uted.Althoughitisdi‡culttoseeinFigure2,therewasonlyasinglejumpdur-ingtheweekofthecrashin1987onOctober19,whenreturnshadanegativejump14percentandvolatilityjumpedupwardfrom40percenttojustover50per-cent.TheBayesfactorsinTableVIsupportSVCJovertheSVandSVJmodels.Figure1showsthattheSVIJandSVCJmodelsexhibitverysimilarbehavior,asspotvolatilityinthetwomodelsisalmostidentical.Thejumptimesandsizesarealsosimilar,andarethereforenotreported.Allowingvolatilitytojumpindepen-dentlyofreturnsprovidesadditional£exibilityovertheSVCJmodel,althoughthemodelishardertoestimateasjumpsinvolatilityarenotsignalledbyajump TableVILog-BayesFactorsfortheS&P500andNasdaq100SVvs.SVJSVvs.SVCJSVvs.SVIJSVJvs.SVIJSVCJvs.SVIJS&P50047.6249.3259.4533.8310.12Nasdaq10027.1628.9556.0333.1927.08TheImpactofJumpsinVolatilityandReturns inreturns.Thisadded£exibilityallowsvolatilitytoplayanevenmoreimportantrolethanintheSVCJmodel,astheproportionofvarianceduetojumpsinre-turnsislessthanintheSVJorSVCJmodels.Diagnosticsdonotprovideanyevi-denceofmisspeci¢cationintheSVIJmodel.TheBayesFactorcomparingSVCJandSVIJisover10,evidencethatfavorstheSVIJmodel.Acaveatisinorder:TheSVIJmodelismoredi‡culttoestimate,asthejumpsdonotoccursimulta-neouslyandthisresultsingreaterposteriorparameteruncertaintyforallofthejumpsizeparameters.B.Nasdaq100TableIVprovidestheNasdaqparametersestimates,usingthesamepriorsthatwereusedfortheS&P,andFigures4and5providevolatilityandjumpestimates.Thebiggestdi¡erencebetweentheNasdaqandS&PreturnsisthatNasdaqs 19871987. 5 19881988. 5 1989 10 20 30 40 50 60 70 Nasdaq 100, SV and SVJSpot Volatility SVJ 19971997. 5 19981998. 5 1999 15 20 25 30 35 40 45 50 55 60 Spot VolatilityNasdaq 100, SV and SVJ 19871987. 5 19881988. 5 1989 10 20 30 40 50 60 70 Nasdaq 100, SVCJ and SVIJSpot Volatility 19971997. 5 19981998. 5 1999 15 20 25 30 35 40 45 50 55 60 Spot VolatilityNasdaq 100, SVCJ and SVIJ SVCJ Figure4.EstimatedvolatilitypathsfortheNasdaq100forthefourmodels.TheJournalofFinance1286 volatilityismuchhigherandmorevolatilethanS&Pvolatility.Forexample,intheSVJmodel,theaveragedailyvariancefortheNasdaqis1.91,morethandoubletheaveragefortheS&P,0.91.Foreachofthemodels,isroughly50percenthigh-erfortheNasdaqthanfortheS&PandthespeedofvolatilitymeanreversionisalsohigherfortheNasdaq.Althoughvolatilityissystematicallyhigher,thisdoesnotimplythatjumpsplayalesserrole;infact,thecontraryistrue.Jumpsarriveaboutthreetimesmoreoften,althoughthesizesaretypicallysmaller.SincethelargestsingledaymoveintheNasdaqissmallerthanintheS&P(16percentcomparedto23per-cent),jumpsinreturnsneednotgenerateasmanyextremelylargemoves(seeFigure5).Thecombinationoftheslightlylowerjumpsizevolatilitywiththehigh-erarrivalrateresultsintheproportionoftotalvarianceduetojumpsincreasingslightly,exceptfortheSVIJmodel(seeTableV).AneasierwaytoseehowthevolatilitystructureoftheS&PandNasdaqindicesdi¡ersistocomparespot 1988 1992 1994 1998 -7 -6 -5 -4 -3 -2 -1 0 1 Nasdaq 100, SVJJump Size 1988 1992 1994 1998 0.250. 5 0.75 1 Nasdaq 100, SVJJump Probability 1986 1988 -7 -6 -5 -4 -3 -2 -1 1 Nasdaq 100, SVCJJump Size 1988 0 0.250. 5 0.75 1 Figure5.EstimatedjumptimesandsizesfortheSVJandSVCJmodelsfortheNasdaq100index.TheImpactofJumpsinVolatilityandReturns1287 volatilityestimatesinFigure6.Thevolatilityofthetwoindiceswassimilarpriorto1990,butsincethen,thevolatilityoftheNasdaqhasbeenhigher,oftenthreetimeshigherthanthevolatilityoftheS&P.Theleveragee¡ect,,islesspronouncedfortheNasdaq.Thisisconsistentwiththesamplestatistics,asNasdaqsskewnessissmallerthanthatoftheS&P.AsinthecaseoftheS&P,increases,inabsolutevalue,fromSVtoSVIJ.TheaveragesizeofajumpinvolatilityintheNasdaqislargerthanintheS&P.However,asNasdaqvolatilityishigherthanthevolatilityoftheS&P,theproportionalin-creaseinvolatilityduetoanaverage-sizedjumpissmallerfortheNasdaq.Thespeci¢cationdiagnostics,bothformalandinformal,indicatethatmodelswithjumpsinvolatilityarepreferredoverthosewithoutjumpsinvolatility.C.PeriodsofMarketsStress:JumpsandVolatilityTheestimatesofspotvolatilityandjumpsprovideameanstoevaluatethecontributionofthesefactorstothethreeperiodsofmarketstressinoursample: 1988 1989 1990 1991 1992 1993 0 10 20 40 50 Spot VolatilityNasdaq 100 and S&P 500 S&P 500Nasdaq 100 1993 1994 1995 1996 1997 1998 1999 2000 0 10 20 30 40 50 Spot VolatilityNasdaq 100 and S&P 500 S&P 500Nasdaq 100 Figure6.ImpliedvolatilitypathsfortheS&P500andNasdaq100fortheSVJmodel.TheJournalofFinance1288 October1987,October1997,andlateSummer^Fall1998.Apriori,itisunclearifjumpsinreturnsorexcessvolatilitywereresponsiblefortheseperiods,asbothcangeneratelargemovements.Understandinghowthesefactorscontributetotheseperiodsisimportant,becauseitispreciselytheextrememovementsduringperiodsofmarketstressthatmaycauseinvestorstodemandlargepremiatocar-rytheserisks.WefocusonthebehaviorofvolatilityandjumpsintheSVJandSVCJmodelsfortheS&P500index.InOctober1987,theSVJmodelattributesmostofthelargemovementstojumpsinreturns.Ofthe23percentdeclineonthedayofthecrash,ajumpinreturnsgenerated18.8percentofthemoveandtherewerealsotwootherjumpsduringtheweek.However,whenallowedtojump,volatilityplaysamorepromi-nentroleingeneratingmarketstress.IntheSVCJmodel,therewasajumpabouttwoweeksbeforethecrash,whichincreasedvolatilityfrom21percentto36per-cent,andonOctober16,volatilityjumpedfrom35to42percent.Anotherjumparrivedonthedayofthecrash,increasingvolatilityto50percentanddelivering14percentjumpinreturns.ThisimpliesthatintheSVCJmodel,jumpsinreturnsgeneratedmorethanhalfofthecrash.Thelargemovementsafterthecrash(8percent,4percent,and8percent)areattributedtoBrownianshockscombinedwiththehighvolatility,incontrasttotheSVJmodel.TheSVIJmodeldeliverssimilarresults,althoughtheexacttimingofjumpsinvolatilityandreturnsisslightlydi¡erent.InOctober1997,therewasa7percentmoveonthe27thanda5percentmoveonthe28th.TheSVJmodelattributesbothmovestojumpsinreturnsandvolatilitystaysconstantatabout19percent.SVCJestimatesindicatethattherewasasinglejumponthe27ththatloweredreturnsby6percentandincreasedvolatilityfrom16percentto25percent.Thusthe¢rstmovewasduetoajumpinreturns,whilethesecondwasduetohighvolatility,amoreplausibleexplanationthanback-to-backjumpsinreturns.Theseperiodsindicatethatbothjumpsinvolatilityandreturnsareimportantcomponentsofperiodsofmarketstress.LateSummerandearlyFall1998wereperiodswithanumberofrelativelylargemovesinthethreeto¢vepercentrange.Ourestimatesindicatethatmostofthemovesweregeneratedbyhighvolatility(asopposedtojumpsinreturns),which,inturn,wasgeneratedbyjumpsinvolatility.The¢rst,onJuly21,1998,occurredcoincidentaltoAlanGreenspanscommentsontheeconomyandonedayaftertheWallStreetJournal¢rstreportedthelargelossesatLongTermCapitalManage-ment.AlthoughthemoveintheS&P500wasonly1.6percent,intheSVCJmodel,ouralgorithmestimatedthatajumpinvolatilityoccurredwithrelativelyhighprobability(about50percent),increasingvolatilityfrom12percenttojustover20percent.JumpsinreturnsandvolatilityalsooccurredattheendofAugust,correspondingtothehaltintradingoftheRussianruble.Overtime,volatilitygraduallymean-revertedbacktorelativelynormallevels.Thesethreeperiodsindicatethatjumpsinvolatilityandreturnsarethepri-marycomponentsthatgeneratedtheperiodsofmarketstress,whiledi¡usivestochasticvolatilityplaysasecondaryrole.Thissuggeststhatitmaybemoreap-propriatetoassignriskpremiatojumpsinreturnsandvolatilitycomponents,ratherthandi¡usivestochasticvolatility.TheImpactofJumpsinVolatilityandReturns IV.OptionPricingImplicationsThissectionanalyzestwooptionpricingimplications:howthemodelsinducedi¡erentoptionpricesandthee¡ectofparameterandspotvolatilityestimationriskonoptionprices.Eraker(2002)providesamoredetailedanalysisoftheoptionpricingimplications.A.Di¡erencesacrossModelsThetime-seriesresultsindicatethatjumpsinvolatilityandreturnsplayanimportantroleindeterminingthedynamicsofreturns.Toevaluatetheireconom-icsigni¢cance,weexaminethecrosssectionandtermstructureofimpliedvola-tilitiesforthedi¡erentmodels,conditionalonestimatedparametersandspotvolatility.TheseresultsarecomparabletothoseinDu‡e,Pan,andSingleton(2000)withoneimportantdi¡erence.Du‡e,Pan,andSingletoncalibratetheparametersandspotvolatilitytominimizeoptionpricingerrorsonagivenday,whileoursuseonlytheinformationcontainedinhistoricalreturns.Figure7displaysimpliedvolatility(IV)curvesfortheSV,SVJ,andSVCJmodelsforthreematurities.TheSVCJandSVIJmodelsdeliversimilarcurves,soweomittedtheSVIJcurves.TheIVcurveswerecomputedusingcalloptionpricesconditionalontheposteriormeanofspotvolatilityandparametersforvariousstrikes.Toframetheresults,wealsoplotmarketimpliedvolatilitiesforoptionstradedonOctober31,1994,arandomlyselectedaveragevolatilitydayinoursample.Resultsindicatethatjumpsinreturnsandvolatilityinduceimportantdi¡er-encesintheshapeoftheIVcurves,especiallyforshortmaturities.First,andmostapparent,theadditionofjumpsinreturnsandjumpsinvolatilitysigni¢-cantlyincreasesthecurvatureoftheIVcurves.Forshortmaturityoptions,thedi¡erencebetweentheSV,SVJ,andSVCJIVcurvesforfarin-the-money(ITM)orout-of-themoney(OTM)optionsisquitelarge.Thisshouldnotbeasurprise,asjumpsinreturnsandvolatilityincreasetheconditionalnonnormalitiesoftheunderlyingdistribution.Ourresultsaredi¡erentfromthoseinAndersen,Benzoni,andLund(2002)andPan(2002)whorequiresigni¢cantjumpriskpremiatogeneratedi¡erencesbetweentheIVcurvesgeneratedbytheSVandSVJmodel.NotethesubstantialincreaseinIVforITMoptionsfortheSVCJmodel.Thishookortippingattheende¡ect(Du‡e,Pan,andSingleton(2000)andPan(2002))isnotpresentintheSVJmodel.Itappearstobeanimportantfeatureofmarketimpliedvolatilities,althoughtheSVCJmodel,withoutvolatilityjumpriskpremia,doesnotgenerateasharpenoughhooktomatchobservedIVcurves.Third,theSVmodelgeneratesvery£atIVcurves,asitdoesnotgenerateanysubstantiveconditionaldeparturesfromnormality.Last,we¢ndasigni¢cant£atteningoute¡ectastimetomaturityincreasesforallofthemodels.Thisoc-cursbecauseasmaturityincreases,thefat-tailsandasymmetriesinthecondi-tionaldistributionaredriventoalargerextentbydi¡usivevolatility,ratherthanjumps.Wethanktherefereeforsuggestingthis.TheJournalofFinance1290 Toseetheimpactofaddingameanjumpsizeriskpremium,therighthandpanelofFigure7displaystheIVcurveswithamodesttwopercentriskpremiuminthemeanjumpsizeinreturns.ThepremiumincreasestheOTMslopeoftheIVcurves,butthee¡ectissmall.Casualobservationindicatesthattheriskpre-miumimprovesthemodelsabilityto¢toptionprices,especiallyforlongermatu-rities.ThefactthatthissmallriskpremiumbringsmodelimpliedpricesclosetomarketpricesisincontrasttoPan(2002),whoestimatestheriskpremiumtobe18percent.B.EstimationRiskandOptionPricesOurMCMCapproach,throughtheposteriordistribution,quanti¢esestima-tionrisk:theuncertaintyinherentinestimatingparametersandspotvolatility. 0.850. 9 0.95 1 1.051. 10. 1 0.150. 2 0.250. 3 0.35 3 Weeks  No PremiumImplied Volatility 80. 11. 0. 1 0.150. 2 0.25 3 Months  No PremiumImplied Volatility 80. 11. 0. 1 0.150. 2 0.25 3 Months  2% Jump Risk Premium 0.850. 9 0.95 1 1.051. 10. 1 0.150. 2 0.250. 3 0.35 3 Weeks  2% Jump Risk Premium 80. 11. 0.12 0.160. 2 0.24 1 Year 80. 11. 0.12 0.160. 2 0.24 Figure7.ImpliedvolatilitycurvesfortheSV,SVJ,andSVCJmodelsandmarketimpliedvolatilitycurvesforarandomlyselecteddayinoursample,October31,1994.TheImpactofJumpsinVolatilityandReturns1291 Thissectionexamineshowthisuncertaintyimpactsoptionprices.Ifandwereknown,thepriceofacalloptionstruckat,withdaystomaturity,condi-tionalonthecurrentspotindexlevel,,istheusualoptionpricingfunction,max,whereisthepricingmeasure.Thisignoresthatandareunknownandmustbeestimated.Inoursetting,,whichourMCMCalgorithmcomputes,quanti¢estheuncertaintyinestimatingand.Fromtheeconometriciansperspective,withuncertainspotvolatilityandparameters,thepriceofacalloptionisgivenbywhichintegratesoutallposterioruncertainty.Thisassumesthattheagentspri-cingtheoptionsknowthetruevolatilityandparameters.Wedonotaddressthedi‡cultissueofevaluatingtheimpactofparameteruncertaintyonequilibriumprices(see,e.g.,Anderson,Hansen,andSargent(2000)foroneapproachtothisissue).Toquantifytheimpactofestimationrisk,wecomputetheposteriordistribu-tionofoptionpriceswhenbothandareuncertain(Case1),whenonlyunknown(Case2),andwhenonlyisuncertain(Case3).Wedothisbyintegrat-ingouttheuncertaintyinand/or,assummarizedbytheposteriordistribu-tion.Inthecasewhereareuncertain,weconditionontheposteriormeanfortheotherquantity.Figure8displaystheposteriormeananda(10percent,90percent)posteriorcoverageintervalforeachcase,forthreematuritiesfortheSVCJmodel,whichisrepresentativeoftheotherjumpmodels.AcomparisonofthegraphsintheupperandlowerleftpanelsindicatesthatnearlyalloftheATMuncertaintyisgeneratedbyuncertaintyin.Theoppositee¡ectoccursforOTMoptions.Forthiscase,themiddleleftpanelindicatesthatparameteruncertaintydominates.Whydoesthisoccur?Notethatwhilethepos-teriormeansofandare1.7and2.9,aonestandarddeviationsymmetriccoverageintervalis(0.2percent,3.3percent)forand(2.3percent,3.5percent)for.Sincethetailsintheconditionaldistributionovershortintervalsloadheavilyontheseparameters,anyuncertaintyovertheseparametershasalargeimpactonOTMoptions.AsimilarargumentholdsforandAtmediumtimehorizons,anduncertaintyisstillslightlygreaterforATMandOTMoptions,butthee¡ectissmaller.Atlonghorizons,adi¡erente¡ectoccurs:Parameteruncertaintydominatesandspotvolatilityuncertaintyhasaminimale¡ect.Theintuitionforthisisclear.Theconditionaldistributionoftheindexvalueoverayearislargelydeterminedbythelong-runbehaviorofthemodel,whichinturnisdeterminedbytheparametersdrivingvolatility:andV.ConclusionsThispaperanalyzedmodelswithjumpsinreturnsandinvolatility.ForboththeS&P500andNasdaq100index,resultsindicatethatbothofthesejumpcom-TheJournalofFinance ponentsareimportant,andmodelswithoutjumpsinvolatilityaremisspeci¢ed.Modelswithonlydi¡usivestochasticvolatilityandjumpsinreturnsaremisspe-ci¢ed,becausetheydonothaveacomponentdrivingtheconditionalvolatilityofreturns,whichisrapidlymoving.Theinformationinthetimeseriesofreturnsindicatesthatjumpsinreturnsandjumpsinvolatilityhaveastrongimpactonoptionprices.Comparedtoamod-elwithonlyjumpsinreturns,modelswithjumpsinvolatilityresultinasigni¢-cantincreaseinIVfordeepin-the-moneyorout-of-the-moneyoptions.ThefactthatjumpsinvolatilityhavesuchalargeimpactonIVissomewhatsurprising,astheoriginalmotivationforincludingjumpsinvolatilitywastoimproveonthe 1.1 0.12 0.16 2 Weeks, Case 1Implied Volatility 1.1 0.12 0.16 2 Weeks, Case 2Implied Volatility 1.1 0.12 0.16 2 Weeks, Case 3Implied Volatility 0.14 0.16 0.18 0.2 2 Months, Case 1 0.9 0.14 0.16 0.18 0.2 2 Months, Case 2 0.9 0.14 0.16 0.18 0.2 2 Months, Case 3 0.8 0.14 0.16 0.18 1 Year, Case 1 0.8 0.14 0.16 0.18 1 Year, Case 2 0.8 0.14 0.16 0.18 Figure8.Theimpactofestimationrisk.Thetoppanelsintegrateoutbothvolatilityandparameteruncertainty.Themiddlepanelsintegrateoutonlyparameteruncertainty(conditionalonestimatedvolatility).Thebottompanelsintegrateoutonlyvolatilityun-certainty(conditionalonposteriormeanofparameters).TheImpactofJumpsinVolatilityandReturns dynamicsofspotvolatilityandnotnecessarilytogeneratemorerealisticIVcurves,although,ofcourse,thesetwogoalsareclearlyrelated.Parameterandvolatilityestimationriskalsohasanimportantimpactonop-tionprices.Whileparameteruncertaintymightresultinsomeoptionpriceuncer-taintyforshortmaturityat-the-moneyoptions,theimpactcanbeaslargeasimpliedvolatilitiesforout-of-the-moneyoptions.Volatilityuncertaintyforat-the-moneyshortmaturityoptionsleadstopricesthatdi¡erby2impliedvolatili-ties,fargreaterthanthebid^askspread.Althoughtheresultsonparameterun-certaintyinoptionpricingarenew,theimportanceofvolatilityuncertaintyisnotnew.Merton(1980)recognizedthisasanimportantfeatureandstatesthatthemostimportantdirectionistodevelopaccuratevarianceestimationmodelswhichtakeintoaccountoftheerrorsinvarianceestimates(p.355).OurMCMCalgorithmprovidessuchamethod.Ourresultsindicatethatparameterandspotvolatilityestimationriskissub-stantial.Toobtainmoreaccurateestimates,optionpricedatamaybeextremelyuseful.ChernovandGhysels(2000),Eraker(2002),andPan(2002)allusespotandoptionpricedatatoestimatevariousmodels.However,whiletheadditionofop-tionpriceswillaidintheestimationofthespotvolatility,itisunclearifitwillsigni¢cantlyreducetheparameteruncertainty.Thisoccursbecausetheriskpre-miaembeddedinoptionpricesintroduceadditionalparameters,whicharetypi-callydi‡culttoestimate.UnlesstheriskpremiaarerestrictedasinPan(2002),itisnotcleariftheaggregateparameteruncertaintywillincreaseordecreasewiththeadditionofoptionpricedata.AppendixA.PosteriorDistributionsforJumpsandVolatilityTheconditionalposteriorsforthejumpsizesandjumptimesarenewandarederivedasfollows.WeusetheSVCJmodel,becauseitisthemostcomplex,intermsofdistributionalstructure.RecallthepriorstructureforthejumpsizesinSVCJ,and.Thisspeci¢cationallowsustousetheGibbssamplertoexploittheconditionalindependenceofthejumpsinvolatility,asitiseasytodrawfromtheconditionalposterior.Theconditionalposteriorforthejumpsizestovolatilityiswhich,byBayesruleisproportionaltowherethe¢rstterm,,isabi-variatenormaldistributionandthesecondtermis,byBayesrule,proportionaltotheproductofand.Thisisaproductofanormalandanexponential.To¢ndthefullconditionalposterior,completingthesquareforallthreetermsasafunctionofleadstoatrun-catednormalTheJournalofFinance1294 whereandarestraightforwardtocompute.Theconditionalposter-iorgivenajumpisthereforeatruncatednormaldistribution.Whentheconditionalposterioris,asthedataprovidesnoinformationaboutthejumpsize.Theposteriorforthejumpsinreturnsissimilarlyderived.BayesRuleimpliesthatSincebothofthedensitiesareGaussian,wehavethatwhereandareeasytocompute.Forthejumptimes,whichareassumedtoarrivecontemporaneously,thepos-teriorcombinesinformationfromthereturnsandfromthevolatility.Ascantakeonlytwovalues,itsposteriorisBernoulli,.TocomputetheBernoulliprobability,weusetheconditionalindependenceofincrementstovo-latilityandreturnstogetthatwhich,again,iseasytocalculate,sinceisabivariateGaussiandensity.Computingtheconditionalposteriorfor0proceedssimilarly,whichgivestheBernoulliprobability.Theconditionalposteriorforvolatility,,isnotaknowndistribution.Tosamplefromit,weusearandom-walkMetropolisalgorithm(seeJohannesandPolson(2002)).Thecon-ditionalposteriorsfortheparametersarestandard.Givenourconjugatepriors,theconditionalposteriorsforareallstandarddis-tributions,andweomitthederivationsastheycanbefoundinstandardtexts.ForweuseanindependenceMetropolisalgorithmwithaproposaldensitycen-teredatthesamplecorrelationbetweentheBrownianincrements.Wenowdiscussourchoicesofpriordistributionsandparameters.Whereverpossible,wechoosestandardconjugatepriors,whichallowsustodirectlydrawfromtheconditionalposteriors.Ourpriordistributionsfortheparametersare:(1,25),N(0,1),(0,1),Uð(2,40),(0,100),IGðGð,and(0,4)wherereferstoaGammadistribution,referstotheInverseGammadistribution,andastan-darduniformdistribution.Allofthepriordistributionsareuninformativewiththeexceptionofand.Thesimulationresultsgivenbelowdemonstratethattheinformationimposedbythesepriorsisminorrelativetotheinformationinthelikelihoodfunction.ThisisfurtherdemonstratedbythefactthatweusethesamepriorsfortheNasdaqandS&Pdata,whichdeliverdrasticallydi¡erentparameterestimates.TheImpactofJumpsinVolatilityandReturns AppendixB.BayesFactorCalculationsSupposethatyouwishtocompareSVandSVJ.Letbeamatrixoflatentvariablesandparameters,whichgivesthemarginallikeli-hoodsasSVJSVJNow,astheSVJmodelembedsthesimpleSVmodelwhentheentirevectorofjumptimesiszero(0),wehavetheidentitythatSVJ).Moreover,ifweassumeinthepriorsthatSVJ)(para-metersincommonhavethesamepriors),thenSVJBayesrulealsoimpliesthat SVJandwehavetheimportantidentitythatweusetodevelopanMCMCestimator, SVJThisissimplyaratioofpriorordinatetoposteriorordinate,andthekeyobserva-tionisthatthiscanbedirectlycomputedfromtheMCMCoutput,whereasthemarginallikelihoodsabovearenotavailable(seeJacquierandPolson(1999)).IntheSVJmodel,wecancomputeSVJ la01ð1lÞb010;b0Þdl¼ Fortheotherportion,asimilarargumentgivesSVJwhichastraightforwardcomputationyields TheJournalofFinance1296 sinceSVJ ComputingthisintegralwiththeMonteCarlosamplesgives GXGg¼1 andwethereforehaveanMCMCestimator:svj 0;b0Þ0;Tþb0Þ 1GXGg¼1 ThecomputationoftheotherOddsRatiosissimilar.Infollowingtheliterature,wecomputelogoddsratios.AppendixC.SimulationExperimentsOursimulationsused100arti¢cialdatasetsconsistingof4,000datapoints.ThedataweregeneratedusingtheEulerdiscretizationofthecontinuous-timemodelwith1/20.Hence,ourestimationmethodismeasuredagainstthearti-¢cialdatageneratedbythetruecontinuoustimeprocess.Weusedaposteriorsamplesizeof50,000foreachoftheMCMCruns.Thenumberofdatapointsandthetrueparametersaresuggestiveofwhatonecanexpectfromdailyequitypricedata,althoughwetriedtostackthedeckagainstourmethodologybymakingthejumpsextremelyunlikely.Anintensityof0.006indicatesthatthereareonly1.5jumpsperyear.With4,000dailyobserva-tions,thisimpliesthatweessentiallyonlyhave26jumpspersample.TablesVIIandVIIIreportsimulationsfortheSVJandSVCJmodels.Thetablesreportthemeansandtherootmeansquarederror(RMSE)andtheresultsindi-catethatthealgorithmdeliversextremelyaccurateestimatesformostoftheparameters,withtheexceptionof,whichappearstobeslightlybiased.Thisisnotnecessarilysurprising:Thisparametergovernsthespeedofmeanreversionofstochasticvolatilityandisdi‡culttoestimate.OurresultsaresimilartothosefoundreportedinJacquieretal.(1994)forthestochasticvolatilitypara-meters.Toseehowmuchoftheerrorinestimatingjumpcomponentswasduetotheextremelyrarejumpsinreturns,TableVIIIreportstheresultswhenweincreasedthearrivalintensityto0.015(3.6jumpsperyear).Asexpected,increasingthear-rivalrateofjumpsimprovestheabilityofthealgorithmtoestimatethepara-metersofthejumpdistribution,althoughtheincreasednoisemakesestimationofthestochasticvolatilitycomponentsslightlylessaccurate.TheresultsfortheTheImpactofJumpsinVolatilityandReturns SVCJmodelaresimilar,althoughwearenotabletoreliablyestimatethepara-meter.Thevariableisessentiallyaregressionparameterbetweenthejumpsinreturnsandvolatility,whereNð.Sincejumpsarelatentandjumpsarerareevents,thisisnotsurprising.ReferencesAlizadeh,Sassan,MichaelBrandt,andFrancisX.Diebold,2002,Range-basedestimationofstochas-ticvolatilitymodels,JournalofFinance57,1047^1091.Andersen,Torben,LucaBenzoni,andJesperLund,2002,Towardsanempiricalfoundationforcontin-uous-timeequityreturnmodels,JournalofFinance57,1239^1284.Andersen,Torben,TimBollerslev,FrancisDiebold,andHeikoEbens,2001,Thedistributionofrealizedstockreturnvolatility,JournalofFinancialEconomics61,43^76.Andersen,Torben,Hyung-JinChung,andBentSorensen,1999,E‡cientmethodofmomentsestima-tionofastochasticvolatilitymodel:AMonteCarlostudy,JournalofEconometrics91,61^87. 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TableVIIISimulationResultsfortheSVCJModelThistableprovidesasummaryofthesimulationresultsfortheSVCJmodel.Onehundredsam-plepathsweresimulatedo¡thetrueparametersandwereporttheaverageacrossthesimula-tionsaswellastherootmeansquarederror(RMSE).ayksTrue0.05000.50000.03000.10000.5000Mean0.05070.47550.03300.08170.4921RMSE0.01140.06010.00620.00760.0711True2.00003.50000.40001.00000.0080Mean2.31402.91160.05991.42720.0093RMSE1.10870.42590.43690.36220.0027TheJournalofFinance1298 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