The t copula with Multiple Parameters of Degrees o

1ThetcopulawithMultipleParametersofDegreesofFreedom:BivariateCharacteristicsandApplicationtoRiskManagementDrXiaolinLuoPrincipalResearchScientistCSIROMathematicalandInformationSciences,Sydney,Lockedbag17,NorthRyde,NSW,1670,Australia.e-mail:[e-mail]rchScientistCSIROMathematicalandInformationSciences,Sydney,Lockedbag17,NorthRyde,NSW,1670,Australia.e-mail:[e-mail]:(dof)thatmaylimititscapabilitytomodelthetaildependencestructureinamultivariatecase.Toovercomethisproblem,groupedtcopulawasproposedrecentlywhererisksaregroupedaprioriinsuchawaythateachgrouphasastandardtcopulawithitsspecificdofparameter.Inthispaperweproposetheuseofanewtcopula,generalizinggroupedtcopulatohaveeachgroupconsistingofoneriskonly,sothatapriorigroupingisnotrequired.Thecharacteristicsofthiscopulainthebivariatecasearedescribed.Weexplainsimulationandcalibrationproceduresandprovideexamples.Keywords:groupedtcopula,taildependence,riskmanagement.31.IntroductionAppropriatemodelingofdependenciesbetweendifferentfinancialmarketsandriskfactorsisanimportantandchallengingaspectofquantifyingfinancialrisks.Copulafunctionshavebecomepopularandflexiblemodelsinthisfield.Theuseofcopulafunctionsenablesthespecificationofthemarginaldistributionstobedecoupledfromthedependencestructureofvariables,whichinturnhelpswiththetaskofmodelingfinancialrisksunderamorerealisticnon-Gaussianassumption.TheconceptofcopulaswasfirstintroducedbySklarin1959butonlyadecadeagobecamepopularinapplicationtofinancialriskmanagement.Foracomprehensivereviewofcopulainfinancialriskmanagement,seeMcNeiletal(2005).Modellingdependencesinthecaseofmorethantwodependentrisksisachallengingtaskconsideredbymanyresearchers.Wewouldliketomentiontwoflexibleapproaches:paircopulacascadeandnestedArchmediancopulas.BuildingonthepioneeringworkofBedfordandCooke(2001,2002),Aasetal(2007)showedhowthemultivariatedependencecanbemodeledusingacascadeofpair-copulas,actingontwovariablesatatime.Inthemostgeneralform,thispair-copulaproducesmanypossibleconstructions,andmodelselectionbecomescriticalandverychallenging.NestedArchimedeancopula,seeJoe(1997)andMcNeil(2007),isanotherflexiblewaytomodelmultivariatedependence.ThesenestedcopulashavebivariateArchmedianmarginalsandallowfordifferentlevelsofpositivedependenceindifferentbivariatemarginals,however,theyrequireconstraintsonthecopulaparameters.Inpractice,themostpopularcopulasinmodelingmultivariatefinancialdataareperhapsthetcopula,impliedbythemultivariatetdistribution,seeEmbrechtsetal(2001);Fangetal(2002);DemartaandMcNeil(2005).Thisisduetoitssimplicityintermsofsimulationandcalibrationcombinedwithitsabilitytomodeltaildependencewhichisoftenobservedinfinancialreturnsdata.ThisstylizedfactcannotbeadequatelydescribedbythecommonlyusedGaussiancopula.RecentpapersbyMashaletal(2003)andBreymannetal(2003)havedemonstratedthattheempiricalfitofthetcopulaissuperiorinmostcaseswhencomparedtotheGaussiancopula.However,itissometimescriticizedduetotherestrictionofhavingonlyoneparameterforthedegreesoffreedom(dof),thatmaylimititsabilitytomodeltaildependenceinamultivariatecase.Toovercomethisproblem,Dauletal(2003)proposedtousegroupedtcopula,whererisksaregroupedintoclassesandeachclasshasitsowntcopulawithaspecificdof.This,however,requiresanapriorichoiceofclasses.Itisnotalwaysobvioushowtheriskfactorsshouldbedividedintosub-groups.Anadequatechoiceofgroupingrequiressubstantialadditionaleffortifthereisnonaturalgrouping,forexamplebysectororclassofasset.Inthispaper,toovercometheproblemwithapriorichoiceofgroupsinthegroupedtcopula,weproposeanewcopula,hereafterreferredtoasthetcopulawithmultipledofparameters.Thiscopulacanbeviewedasagroupedtcopulawitheachgrouphavingonlyonemember.Forconvenience,denotethenewcopulaasνt~copula,whereνidentifiesthevectorofdofparameters.Whilethedofparametersofthegroupedtcopulacanbeestimatedmarginallybyfittinggroupsseparately,thedofparametersoftheνt~copulacanonlybeestimatedjointlyusinge.g.theMaximumLikelihood(ML)method.Itisworthnotingthat,theνt~copulaisnotameta-tdistributionconsideredbyEmbrechtsetal(2001),Fangetal(2002).Althoughourmainmotivationforstudyingtheνt~copulaismodelingmultivariatecases,forsimplicitythispaperwillconsiderbivariateexamplesindetail.Applicationinthegeneralmultivariatecaseincludingmodelselectionwillbestudiedinourforthcomingpaper.Eveninthebivariatecaseitcanbedemonstratedthattherecouldbesignificantimpactonportfolioriskmeasures(suchasValueatRiskandExpectedShortfall)ifthestandardtcopulaisusedwhenthetruecopulaisνt~copula.4Theorganisationofthispaperisasfollows.Section2describesthemodelandnotationsfortcopulas.Explicitrepresentationsandcalibrationoftheνt~copulaarediscussedinSection3andSection4respectively.Section5presentstheimpo