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1、MultivariateVarianceGammaandGaussiandependence:astudywithcopulas∗ElisaLucianoandPatriziaSemeraroAbstract.ThispaperexploresthedynamicdependencepropertiesofaLevyprocess,the´VarianceGamma,whichhasnon-Gaussianmarginalfeaturesandnon-Gaussiandependence.Bycomput
2、ingthedistancebetweentheGaussiancopulaandtheactualone,weshowthatevenanon-Gaussianprocess,suchastheVarianceGamma,can“converge”tolineardependenceovertime.Empiricalversionsofdifferentdependencemeasuresconfirmtheresultovermajorstockindicesdata.Keywords:multiva
3、riatevarianceGamma,Levyprocess,copulas,non-lineardependence´1IntroductionRiskmeasuresandthecurrentevolutionoffinancialmarketshavespurredtheinterestofthefinancialcommunitytowardsmodelsofassetpriceswhichpresentbothnon-Gaussianmarginalbehaviourandnon-Gaussian,
4、ornon-linear,dependence.Whenchoosingfromtheavailablemenuoftheseprocesses,onelooksforparsimoniousnessofparameters,goodfitofmarketdataand,possibly,abilitytocapturetheirdependenceandtheevolutionofthelatterovertime.Itisdifficulttoencapsulateallofthesefeatures–d
5、ynamicdependence,inparticular–inasinglemodel.ThepresentpaperstudiesanextensionofthepopularVarianceGamma(VG)model,namedα-VG,whichhasnon-Gaussianfeaturesbothatthemarginalandjointlevel,whilesucceedinginbeingbothparsimoniousandaccurateindatafitting.Weshowthatd
6、ependence“converges”towardslineardependenceovertime.Thisrepresentsgoodnewsforempiricalapplications,sinceoverlonghorizonsonecanrelyonstandarddependencemeasures,suchasthelinearcorrelationcoefficient,aswellasonastandardanalyticalcopulaordependencefunction,nam
7、elytheGaussianone,evenstartingfromdatawhichdonotpresentthestandardGaussianfeaturesoftheBlackScholesorlog-normalmodel.Letusputthemodelintheappropriatecontextfirstandthenoutlinethedifficultiesincopulatowardsdynamicdependencedescriptionthen.∗"c2008byElisaLucia
8、noandPatriziaSemeraro.AnyopinionsexpressedherearethoseoftheauthorsandnotthoseofCollegioCarloAlberto.M.Corazzaetal.(eds.),MathematicalandStatisticalMethodsforActuarialSciencesandFinance©Springer-VerlagItalia2010194E.
