Non-linear versus non-gaussian volatility models

Non-linearversusNon-gaussianVolatilityModelsChristianSchittenkopfGeorgDornerEngelbertJ.DocknerReportNo.39Oktober1999Oktober1999SFB‘AdaptiveInformationSystemsandModellinginEconomicsandManagementScience’ViennaUniversityofEconomicsandBusinessAdministrationAugasse2{6,1090Wien,AustriaincooperationwithUniversityofViennaViennaUniversityofTechnology:JournalofEmpiricalFinanceThispieceofresearchwassupportedbytheAustrianScienceFoundation(FWF)undergrantSFB#010(‘AdaptiveInformationSystemsandModellinginEconomicsandManagementScience’).Non-linearversusNon-gaussianVolatilityModelsChristianSchittenkopfAustrianResearchInstituteforArticialIntelligenceGeorgDornerAustrianResearchInstituteforArticialIntelligenceandDept.ofMedicalCyberneticsandArticialIntelligence,UniversityofVienna,AustriaEngelbertJ.DocknerDept.ofBusinessAdministration,UniversityofVienna,AustriaAbstractOneofthemostchallengingtopicsinnancialtimeseriesanalysisisthemod-elingofconditionalvariancesofassetreturns.Althoughconditionalvariancesarenotdirectlyobservabletherearenumerousapproachesintheliteraturetoovercomethisproblemandtopredictvolatilitiesonthebasisofhistoricalassetreturns.ThemostprominentapproachistheclassofGARCHmodelswhereconditionalvari-ancesaregovernedbyalinearautoregressiveprocessofpastsquaredreturnsandvariances.Recentresearchinthiseld,however,hasfocusedonmodelingasymme-triesofconditionalvariancesbymeansofnon-linearmodels.Whilethereisevidencethatsuchanapproachimprovesthettoempiricalassetreturns,mostnon-linearspecicationsassumeconditionalnormaldistributionsandignoretheimportanceofalternativemodels.Concentratingonthedistributionalassumptionsis,however,essentialsinceassetreturnsarecharacterizedbyexcesskurtosisandhencefattailsthatcannotbeexplainedbymodelswithsucientheteroskedasticity.Inthispaperwetakeuptheissueofreturns’distributionsandcontrastitwiththespecicationofnon-linearGARCHmodels.WeusedailyreturnsfortheDowJonesIndustrialAverageoveralargeperiodoftimeandevaluatethepredictivepowerofdierentlinearandnon-linearvolatilityspecicationsunderalternativedis-tributionalassumptions.Ourempiricalanalysissuggeststhatwhilenon-linearitiesdoplayaroleinexplainingthedynamicsofconditionalvariances,thepredictivepowerofthemodelsdoesalsodependonthedistributionalassumptions.Keywords:volatility,neuralnetworks,GARCH,non-linearity,fattails11IntroductionRecentresearchinnancialtimeseriesanalysishasputalotofemphasisonmodelingandforecastingassetreturnvolatilities.Thisinteresthasatleasttworoots.Onestemsfromthefactthatoptionpricesvarywithchangesinvolatilitiesoftheunderlyinginstrumentandhenceanaccuratepredictionoffutureoptionpricesrequiresaforecastofvolatilities.Thesecondoneisrelatedtomarketriskmanagement.Heretheconceptofvalueatriskseemstobetheindustrystandard,whichrequiresaforecastofvolatilitiesoftheriskfactors(likeinterestrates,exchangerates,marketreturns)inordertocalculatethemarketriskofagivenportfolioofsecurities.Manyvolatilitymodelshavebeenproposedinthenanceliteraturebutallofthemcanbegroupedintotwocategories.Thereistheclassofmodelsthatbuildonhistoricalassetreturnsandpredictvolatilitiesonthebasisofdierenttimeseriesanalysistechniques.Andthereistheconceptofimpliedvolatility.Thisiscloselyrelatedtooptionpricesandrequiresanoptionpricingmodelinordertocalculatethemarketdrivenvolatilities1.Whilethechoiceofanoptionvaluationmodelisarbitraryandhencetheconceptofimpliedvolatilityismodeldependent,thereisonthecontraryanadvantageoverhistoricalvolatilities:impliedvolatilitiesdonotrequireanyconceptofconditionalexpectationandhenceareindependentoftimeseriespropertiesofassetreturns.Inthispaperwedonottakeuptheissueofmodelingandderivingimpliedvolatilities,insteadwefocusonpredictinghistoricalones.ThemostprominentvolatilitymodelthatestimatesconditionalvariancesofassetreturnsonthebasisofhistoricalobservationsistheGARCHmodel(Bollerslev,1986;Bollerslevetal.,1992;Engle,1982).Itisabletocaptureseveralimportantstylizedfactsofassetreturns,namelyheteroskedasticity,volatilityclusteringandexcesskurtosis.InthemeantimealargebodyofliteratureexiststhatevaluatesthepredictivepowerofGARCHmodels.Thesestudieshavefoundthatthereexistadditionalempiricalregularitiesbesidesvolatilityclusteringandexcesskurtosis.Theseregularitiesaretheleverageeect,theco-movementofvolatilitiesandthereectionofinformationthataccumulateswhennancialmarketsareclosed.BasedonthesestylizedfactsmanyvariationsofparametricGARCHmodelshavebeendevelopedoverthepastdecadestoaccountforthem.WhileARCHandGARCHmodelscapturefattailedreturnsandvolatilityclusteringtheyarenotwellsuitedtocapturetheleverageeect.Toaccountfortheleverageeectitisnecessarythattheconditionalvarianceattimetdependsonboththelevelandthesignofthelaggedresidualofassetreturns.TheExponentialGARCH(EGARCH)modelofNelson(1991)accountsfortheseeectsasdoesthesign-GARCHmodelbyGlosten,JagannathanandRunkle(1993).Moreover,manypapersfoundthattheau