An Option Pricing Formula for the GARCH diffusion

AnOptionPricingFormulafortheGARCHdiffusionModelGiovanniBarone-Adesi¤HenrikObbekaerRasmussenyClaudiaRavanellizMay2003AbstractWederiveanalyticallythefirstfourconditionalmomentsoftheintegratedvarianceimpliedbytheGARCHdiffusionprocess.AnanalyticalclosedformapproximationforEuropeanoptionsundertheGARCHdiffusionmodelisobtainedfromthesemoments.UsingMonteCarlosimulationsweshowthatthisapproximationisaccurateforalargesetofreasonableparameters.Theclosed-formoptionpricingsolutionallowstostudyeasilyimpliedvolatilitysurfacesinducedbytheGARCHdiffusionmodel.Correspondingauthor:[e-mail].¤InstituteofFinance,UniversityofSouthernSwitzerlandandCityUniversityBusinessSchool.yMathematicalInstitute,UniversityofOxfordzInstituteofFinance,UniversityofSouthernSwitzerland.11IntroductionThispaperanalysesEuropeanoptionpricesinstochasticvolatilitymodelswheretheunder-lyingassetfollowsageometricBrownianmotionwithinstantaneousvariancedrivenbytheGARCHdiffusionprocess.StochasticvolatilitymodelswerefirstintroducedbyHullandWhite(1987),Scott(1987)andWiggins(1987)toovercomethedrawbacksoftheBlackandScholesmodel(1973).Volatilities,stochasticallychangingovertime,accountforrandombe-havioursofimpliedandhistoricalvariancesandgeneratesomeofthelog-returnfeaturesobservedinempiricalstudies.1Unfortunately,itisdifficulttoderiveclosedoranalyticallytractableop-tionpricingformulasinstochasticvolatilitymodels,evenforvanillaoptions.OnlytheHullandWhite(1987)andtheHeston(1993)modelshave,respectively,ananalyticapproximationandaquasi-analyticformulatopriceEuropeanoptions.Forothermodelsnumericalmethodsareavailablebuttheseproceduresarecomputationallyintensive2.Inthispaper,wederiveananalyticalclosed-formapproximationforEuropeanoptionpriceswherethevarianceisdrivenbyaGARCHdiffusionprocess3.Thisapproximationisveryaccurateandeasytoimplement.TheGARCHdiffusionprocesshasseveraldesirableproperties.Itispositive,meanrevertingandwithastationaryinverseGammadistribution.Hence,itsatisfiestherestrictionthatbothhistoricalandimpliedvariancesbepositive.Italsofitstheobservationthatvariancesappeartobestationaryandmeanreverting(Scott(1987),Taylor(1992),Jorion(1995)andGuo1996).Moreover,theGARCHdiffusionisaflexiblemodelandallowsforrichpatternsofbehaviourofvolatilitiesandassetprices.Forinstance,itproduceslargecorrelationsinthesquaredlog-returns,arbitrarylargekurtosisand,asobservedinempiricalstudies,finiteunconditionalmo-mentsoflog-returndistributionsuptoagivenorder.Bycontrast,thesquarerootprocess(cf.Heston1993)islessappropriatetomodelthepreviouslog-returncharacteristics.Specifi-cally,itsstationaryGammadistributionimplieslog-returnswithexcess-kurtosisupto3,finiteunconditionalmomentsofanyorderandcorrelationofsquaredlog-returns1=5atmost(cf.Sorensen2000).Furthermore,thepresentmodelisthe‘meanreverting’extensionofHullandWhite(1987)optionpricingmodel4wherethevarianceprocessfollowsanindependentlog-normalprocesswithoutdrift.TheGARCHdiffusionprocessmakesamarkedimprovementovertheHulland1SeeforinstanceMandelbrot(1963)andFama(1965).2Whenlargetradingbookshavetobequicklyandfrequentlyevaluatedmanyproceduresarepracticallynotfeasible.3ThismodelwasfirstintroducedbyWong(1964)andpopularisedbyNelson(1990).Formorerecentstudiessee,forinstance,Lewis(2000),Andersen,BollerslevandMeddahi(2002)andreferencestherein.4ThismodelisalsostudiedbyGuo(1996),GesserandPoncet(1997)andLewis(2000).2Whitemodel.Precisely,themean-revertingdriftgivesstationaryvarianceandlog-returnpro-cesses(cf.Genon-Catalot,JeantheauandLaredo,1999)anditpermitstheinclusionofthevolatilityriskpremiuminthevarianceprocess.Bycontrast,fortheHullandWhitemodeltheoptionpricingapproximationisavailableonlywhenthedriftisequaltozero.5Furthermore,themeanreversionofthevarianceallowstoapproximatelongmaturityoptionprices,whileintheHullandWhitemodeltheoptionpricingapproximationholdsonlywhenthetimetomaturityissmall(cf.HullandWhite(1987)andGesserandPoncet1997).Finally,forthepresentmodel,thenastyproblemofmakinginferenceoncontinuous-timeparameterscanbereducedtoaGARCHmodelestimation(cf.forinstanceEngleandLee(1996)andLewis2000).Indeed,Nel-son(1990)showsthatthismodelisacontinuoustimelimitofthesimplediscreteGARCH(1,1)modelandtheparameterscanbeconsistentlyinferredfromthediscreteGARCH(1,1)param-eters6.ThisisanimportantadvantageoverothermodelssuchastheHeston(1993)oneforwhichtheparameterestimationismoreinvolved.InordertoderiveoptionpricesundertheGARCHdiffusionmodelwestartfromtheHullandWhite(1987)optionpricingapproximationthatholdswhentheassetpriceanditsinstantaneousvarianceareindependent.Thisindependenceimpliesvolatility‘smiles’,i.e.asymmetricshapepatternoftheimpliedvolatilityasfunctionofstrikeprices(HullandWhite(1987),RenaultandTouzi,1996).Suchasymmetricshapetipicallycharacterizesimpliedvolatilitiesofcurrencyoptionmarkets(cf.Bates(1996),ChesneyScott(1989),MelinoandTurbull(1990),TaylorandXu(1994)andBollerslevandZhou2002).Aswewillshow,alsothedependenceoftheimpliedvolatilityontimetomaturitiesinducedbythemodelisinqualitativeagreementwiththe‘termstructures’offoreignexchangeoptionmarkets.Forthe