Analytical approximations for the GJR-GARCH and EG

ApproximatingtheGJR-GARCHandEGARCHOptionPricingModelsAnalyticallyJin-ChuanDuan,GenevieveGauthier,CarolineSasseville,Jean-GuySimonato(February3,2004)AbstractInDuan,GauthierandSimonato(1999),ananalyticalapproximateformulaforEuropeanoptionsintheGARCHframeworkwasdeveloped.TheformulaishoweverrestrictedtothenonlinearasymmetricGARCHmodel.ThispaperextendsthesameapproachtotwootherimportantGARCHspecications-GJR-GARCHandEGARCH.Weprovidethecorrespondingformulasandstudytheirnumericalperformance.keywords:Optionpricing,EGARCH,GJR-GARCH,analyticalapproximationDuaniswithRotmanSchoolofManagement,UniversityofToronto;GauthierandSimonatoarewithHECMontreal;SassevilleisaPh.D.candidateattheKellogGraduateBusinessSchool.Duan,GauthierandSimonatoacknowledgethenancialsupportfromtheNaturalSciencesandEngineeringResearchCouncilofCanada(NSERC),LesFondspourlaFormationdeChercheursetl'AidealaRechercheduQuebec(FCAR)andfromtheSocialSciencesandHumanitiesResearchCouncilofCanada(SSHRC).DuanalsoacknowledgessupportreceivedastheManulifeChairinFinancialServices.11IntroductionDuringthepastdecade,researchershavebeguntostudygeneralizedautoregressiveconditionalheteroskedasticitic(GARCH)modelsforoptionpricingduetothesuperiorabilityofthisclassofmodelstodescribeassetreturndynamics.Duan(1995)developedatheorywithrespecttowhichoptionscanbepricedwhentheevolutionoftheassetreturnfollowstheGARCHprocess.Empiri-cally,Duan(1996),HestonandNandi(2000),HsiehandRitchken(2000),HardleandHafner(2000),DuanandZhang(2001)andChristoersenandJacobs(2002)haveshowedthattheGARCHmodelcanbeusedtocapturethepricingbehaviorofexchange-tradedEuropeanoptions.AnalyticallypricingEuropeanoptionsrequirestheknowledgeoftherisk-neutraldistributionofthecumulativereturnwithrespecttoagivenmodel.However,theanalyticalformofthedistributionforthetime-aggregatedreturn(ie.,cumulativereturn)isunknownforallGARCHspecications,andthuscomputingoptionpricesmustrelyononsometime-consumingnumericalprocedures.Inrecentyears,researchershavetriedtospeedupthevaluationofEuropeanoptionsunderGARCHbydevelopinganalyticalsolutionsandanalyticalapproximationsforspecicformsoftheGARCHmodel.HestonandNandi(2000)developedananalyticalformulatopriceEuropeanop-tionswhenthedynamicoftheconditionalvarianceisgivenbyaspecicGARCHprocess.1Incontrast,Duan,GauthierandSimonato(1999)(DGShereafter)developedananalyticalapprox-imationtotheEuropeanoptionpriceunderGARCH.TheirapproachutilizestheideaofJarrowandRudd(1982)tondanapproximateoptionpriceundergeneralstochasticprocesses.Specically,DGS(1999)usedanEdgeworthexpansiontoobtainanapproximatepricingformulaforthenonlinearasymmetricGARCHspecicationofEngleandNg(1993)(NGARCH).TheNGARCHoptionpricingmodelistheoptionpricingmodelcorrespondingtothelinearGARCHspecicationofBollerslev(1986)and/ortheNGARCHofEngleandNg(1993).TheresultingapproximationformulaissimilartoaBlack-ScholesformulabutbeingadjustedforskewnessandkurtosisofthecumulativereturnunderGARCH.TheDGS(1999)approximationperformswellnumerically,especiallyforshorter-termoptions.IncontrasttotheapproachofHestonandNandi1Strictlyspeaking,HestonandNandi(2000)isanumericaltechnique.Theyrstderiveadierenceequationsystemforthecharacteristicfunction.Theythensolvethedierenceequationsystemnumerically.Finally,theyrelyonanumericalFourierinversiontoobtaintheEuropeanoptionprice.2(2000),itisnotlimitedtoaspecicformofGARCH.Indeed,comparableanalyticalapproximationformulascanbeobtainedforotherGARCHspecications,butsuchmodicationsarenottrivialextensions.Inthispaper,wederivevariouscomponentsneededforapplyingtheDGS(1999)approachtotheGARCHspecicationofGlosten,JagannathanandRunkle(1993)(GJR-GARCH)andtotheexponentialGARCHspecicationofNelson(1991)(EGARCH).ThechoiceofthesetwomodelsisjustiedbythefactthatthesetwospecicationsandtheNGARCHmodelallexhibittheleverageeect,animportantfeatureofstockreturndata.IncontrasttoNGARCH,thesetwospecicationsincorporatetheleverageeectnotbyshiftingtheminimumofthenewimpactcurveawayfromzero.Instead,theyaltertheshapeofthenewimpactcurve.Thesedierentspecicationshavebeenshownindierentempiricalstudiestobetterdescribeassetreturnsandmaythusbeusefulinexplainingoptionprices.OurdevelopmentoftheanalyticalapproximationformulascorrespondingtotheGJR-GARCHandEGARCHoptionpricingmodelscanthusfacilitatefutureempiricalresearchusingthesetwomodelsaswellasprovideapracticaltoolforpotentialon-lineapplicationsofthesemodels.Theremainderofthispaperisorganizedasfollows.Insection2,weshowhowtheanalyticalapproximationcanbemodiedspecicallyfortheGJR-GARCHandEGARCHoptionpricingmodels.Wethenexamininsection3thenumericalperformanceoftheseanalyticalapproximations.Finally,section4concludes.2Theanalyticalapproximation2.1GeneralvaluationframeworkWestartbyassumingthattheassetreturndynamic,underthephysicalmeasureP,islnSt+1St=r+pht+112ht+1+pht+1t+1;fort=0;1;2;:::(1)wheretPN(0;1)(2)andht+1=0+1ht+2ht2t+3htmax(0;t)2(3)3fortheGJR-GARCHprocessorln(ht+1)=0+1ln(ht)+4(jtj+t):(4)fortheEGARCHprocess.Notethatristheone-periodcontinuouslycompoundedrisk-freerate,isaconstan