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FinancialApplicationsofCopulasCREREG,Rennes,11/16/00ThierryRoncalliGroupedeRechercheOp´erationnelleCr´editLyonnaisJointworkwithEricBouy´e,ValdoDurrleman,AshkanNikeghbaliandGa¨elRiboulet.TheWorkingPaper“CopulasForFinance”isavailableonthewebsite:finition1AcopulafunctionCisamultivariateuniformdistribution.Theorem1LetF1;:::;FNbeNunivariatedistributions.ItcomesthatC(F1(x1);:::;Fn(xn);:::;FN(xN))definesamultivariatedistributionswithmarginsF1;:::;FN(becausetheintegraltransformsareuniformdistributions).)Copulasarealsoageneraltooltoconstructmultivariatedistributions,andsomultivariatemodels.FinancialApplicationsofCopulasIntroduction1-12Thedependencefunction²Canonicalrepresentation²Corcondanceorder²MeasureofdependenceFrom1958to1976,virtuallyalltheresultsconcerningcopulaswereobtainedinconnectionwiththestudyanddevelopmentofthetheoryofprobabilisticmetricspaces(Schweizer[1991]).)SchweizerandWolff[1976]=connectionwithrankstatistics.FinancialApplicationsofCopulasThedependencefunction2-12.1CanonicalrepresentationTheorem2(Sklar’stheorem)LetFbeaN-dimensionaldistributionfunctionwithcontinuousmarginsF1;:::;FN.ThenFhasauniquecopularepresentationF(x1;:::;xN)=C(F1(x1);:::;FN(xN)))Copulasarealsoapowerfultool,becausethemodellingproblemcouldbedecomposedintotwosteps:²Identificationofthemarginaldistributions;²Definingtheappropriatecopulafunction.Intermsofthedensity,wehavethefollowingcanonicalrepresentationf(x1;:::;xN)=c(F1(x1);:::;FN(xN))£NYn=1fn(xn).FinancialApplicationsofCopulasThedependencefunction2-2Thecopulafunctionofrandomvariables(X1;:::;XN)isinvariantunderstrictlyincreasingtransformations(@xhn(x)0):CX1;:::;XN=Ch1(X1);:::;hN(XN)...thecopulaisinvariantwhilethemarginsmaybechangedatwill,itfollowsthatispreciselythecopulawhichcapturesthosepropertiesofthejointdistributionwhichareinvariantundera.s.stricklyincreasingtransformations(SchweizerandWolff[1981]).)Copula=dependencefunctionofrandomvariables.ThispropertywasalreadyetablishedbyDeheuvels[1978,1979].FinancialApplicationsofCopulasThedependencefunction2-32.2ExamplesFortheNormalcopula,WehaveC(u1;:::;uN;½)=Φ½³Φ¡1(u1);:::;Φ¡1(uN)´andc(u1;:::;uN;½)=1j½j12expµ¡12&³½¡1¡I´&¶FortheGumbelcopula,WehaveC(u1;u2)=expá³(¡lnu1)±+(¡lnu2)±´1±!Othercopulas:Archimedean,Plackett,Frank,Student,Clayton,etc.FinancialApplicationsofCopulasThedependencefunction2-42.3ConcordanceorderThecopulaC1issmallerthanthecopulaC2(C1ÁC2)if8(u1;:::;uN)2IN;C1(u1;:::;uN)·C2(u1;:::;uN))ThelowerandupperFr´echetboundsC¡andC+areC¡(u1;:::;uN)=max0@NXn=1un¡N+1;01AC+(u1;:::;uN)=min(u1;:::;uN)WecanshowthatthefollowingorderholdsforanycopulaC:C¡ÁCÁC+)TheminimalandmaximaldistributionsoftheFr´echetclassF(F1;F2)arethenC¡(F1(x1);F2(x2))andC+(F1(x1);F2(x2)).ExampleofthebivariateNormalcopula(C?(u1;u2)=u1u2):C¡=C¡1ÁC½0ÁC0=C?ÁC½0ÁC1=C+FinancialApplicationsofCopulasThedependencefunction2-5Mikusi´nski,SherwoodandTaylor[1991]givethefollowinginterpretationofthethreecopulasC¡,C?andC+:²TworandomvariablesX1andX2arecountermonotonic—orC=C¡—ifthereexistsar.v.XsuchthatX1=f1(X)andX2=f2(X)withf1non-increasingandf2non-decreasing;²TworandomvariablesX1andX2areindependentifthedependencestructureistheproductcopulaC?;²TworandomvariablesX1andX2arecomonotonic—orC=C+—ifthereexistsarandomvariableXsuchthatX1=f1(X)andX2=f2(X)wherethefunctionsf1andf2arenon-decreasing;FinancialApplicationsofCopulasThedependencefunction2-62.4MeasuresofassociationordependenceIf·isameasureofconcordance,itsatisfiestheproperties:¡1··C·1;C1ÁC2)·C1··C2;etc.SchweizerandWolff[1981]showthatKendall’stauandSpearman’srhocanbe(re)formulatedintermsofcopulas¿=4ZZI2C(u1;u2)dC(u1;u2)¡1%=12ZZI2u1u2dC(u1;u2)¡3)Thelinear(orPearson)correlationisnotameasureofdependence.FinancialApplicationsofCopulasThedependencefunction2-72.5SomemisinterpretationsofthecorrelationThefollowingstatementsarefalse:1.X1andX2areindependentifandonlyif½(X1;X2)=0;2.Forgivenmargins,thepermissiblerangeof½(X1;X2)is[¡1;1];3.½(X1;X2)=0meansthattherearenorelationshipbetweenX1andX2.²WeconsiderthecubiccopulaofDurrleman,NikeghbaliandRoncalli[2000]C(u1;u2)=u1u2+®[u1(u1¡1)(2u1¡1)][u2(u2¡1)(2u2¡1)]with®2[¡1;2].IfthemarginsF1andF2arecontinousandsymmetric,thePearsoncorrelationiszero.Moreover,if®6=0,therandomvariablesX1andX2arenotindependent.FinancialApplicationsofCopulasThedependencefunction2-8²Wang[1997]showsthatthemin.andmax.correlationsofX1»LN(¹1;¾1)andX2»LN(¹2;¾2)are½¡=e¡¾1¾2¡1µe¾21¡1¶12µe¾22¡1¶12·0½+=e¾1¾2¡1µe¾21¡1¶12µe¾22¡1¶12¸0½¡and½+arenotnecessarilyequalto¡1and1.Examplewith¾1=1and¾2=3:Copula½(X1;X2)¿(X1;X2)%(X1;X2)C¡¡0:008¡1¡1½=¡0:7'0¡0:49¡0:68C?000½=0:7'0:100:490:68C+0:1611FinancialApplicationsofCopulasThedependencefunction2-9²UsinganideaofFerguson[1994],Nelsen[1998]definesthefollowingcopulaC(u1;u2)=8:u10·u1·12u2·1212u20·12u2·u1·1¡12u2u1+u2¡112·1¡12u2·u1·1Wehavecov(U1;U2)=0,butPrfU2=1¡j2U1¡1jg=1,i.e.“thetworandomvariablescanbeuncorrelatedalthoughonecanbepredictedperfectlyfromtheother”.FinancialApplicationsofCopulasThedependencefunction2-103Anopenfieldforriskmanagement²Economiccapitaladequ