Aur´elienAlfonsi∗
∗
DamianoBrigo∗∗
CERMICS,EcoleNationaledesPontsetChauss´ees,6-8avenueBlaisePascal,Cit´e
Descartes,ChampssurMarne,77455Marne-la-vall´ee,France.e-mail:alfonsi@cermics.enpc.fr
∗∗
CreditModels,BancaIMI,SanPaoloIMIGroup,CorsoMatteotti6–20121Mi-
lano,Italy;Fax:+390276019324.e-mail:damiano.brigo@bancaimi.it,webpage:http://www.damianobrigo.it
Firstversion:April1,2003.Thisversion:January30,2004
Abstract
Althoughthereexistsalargevarietyofcopulafunctions,onlyafewarepracticallymanageable,andoftenthechoiceindependencemodelingfallsontheGaussiancopula.Further,mostcopulasareexchangeable,thusimplyingsymmetricdependence.Weintroduceawaytoconstructcopulasbasedonperiodicfunctions.Westudythetwo-dimensionalcasebasedononedepen-denceparameterandthenprovideawaytoextendtheconstructiontothen-dimensionalframework.Wecanthusconstructfamiliesofcopulasindimen-sionnandparameterizedbyn−1parameters,implyingpossiblyasymmetricrelations.Such“periodic”copulascanbesimulatedeasily.
Keywords:DependenceModeling,CopulaFunctions,GaussianCopula,ArchimedeanCopula,PeriodicCopula,Simulation
NewfamiliesofCopulasbasedonperiodicfunctions
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1IntroductionandMotivation
ConsiderarandomvectorX=(X1,...,Xn),andsupposethatwewishtoanalyzethedependencebetweenitscomponents.ThewholeinformationonthedistributionofthevectorisgivenbythejointcumulativedistributionfunctionofX.IfPdenotestheprobabilitymeasureinoursetting,suchfunctioninthepoint(x1,...,xn)isgivenbyP(X1≤x1,...,Xn≤xn).However,thisfunctionmixesinformationonthedependencebetweenthedifferentcomponentsofthevectorwithinformationonthedistributionofthesinglecomponentsthemselves.Copulafunctionshavebeenintroducedinordertoallowaseparationbetweenthemarginalcumulativedistributionfunctions(cdfforshort)andthedependencestructure.Theformerconcernssinglecomponents,takenoneatthetime,andisgivenbythecdf’sFi(x):=P(Xi≤x),i=1,...,n,whichweassumetobecontinuous.Thelatterisentirelyrepresentedbythecopulafunctionweintroducenow.ItiswellknownthatU1=F1(X1),...,Un=Fn(Xn)areuniformlydistributedrandomvariableson[0,1].Thejointcumulativedistributionfunctionof(U1,...,Un),thatwedenoteby
C(u1,...,un)=P(U1≤u1,...,Un≤un),
iscalledthecopulafunctionof(X1,...,Xn)andhasthefollowinglinkwiththemul-tivariatecdf:
P(X1≤x1,...,Xn≤xn)=C(P(X1≤x1),...,P(Xn≤xn)).
Onecaneasilycheckthatacopulahasthefollowingproperties:1.C(u1,..,ui−1,0,ui+1,..,un)=02.C(1,..,1,uk,1,..,1)=uk
3.∂u1...unCisapositivemeasureinthesenseofSchwartzdistributions.ThismeansconcretelythatforanyhypercubeH=[a1,b1]×...×[an,bn]⊂[0,1]n,
P[(U1,..,Un)∈H]≥0.
Whenn=2,thiscanbewrittenas
C(b1,b2)−C(a1,b2)−C(b1,a2)+C(a1,a2)≥0.
(1)
NewfamiliesofCopulasbasedonperiodicfunctions
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Conversely,onecanshowthatanyfunctionthatsatisfiesthesethreeconditionscanbeviewedasthejointcdfofavectorofuniformvariableson[0,1]andisthusacopula.ThisisknownasSklar’stheorem,seeforexampleJoe(1997)orNelsen(1999).Inthefollowing,theexpression”simulatingacopulaC”willdenotethesimulationofarandomvectorofuniformvariables(U1,..,Un)on[0,1]whosejointcdfisC.Amongthedifferentwaystodefinespecificcopulafunctions,therearefollowingtwo.ThefirstoneconsistsinseekingfunctionsCsatisfyingthethreeaboveprop-erties.Archimedeancopulasareanexampleofthisapproach.Indeed,Archimedeancopulascomefromtheremarkthatifϕisaconvexdecreasingfunctionsuchthatϕ(1)=0,then
C(u1,..,un)=1{ϕ(u1)+..+ϕ(un)≤ϕ(0)}ϕ−1(ϕ(u1)+..+ϕ(un))
hastheabovethreepropertiesandisthusacopula.Therefore,byspecifyingfamiliesofdecreasingconvexfunctionsthatvanishin1wespecifyfamiliesofcopulas(e.g.Gumbel,Joe,Frank...),seeBouy´eetal.(2000),Nelsen(1999)andJoe(1997).Thesecondmethodconsistsinworkingdirectlywithjointcdf’sF(x1,...,xn)and
−1−1
(un)).therelatedmarginalcdf’sFi.TheassociatedcopulaisthendefinedasF(F1(u1),...,Fn
Evenifthismethoddoesnotalwaysleadtoanalyticallytractablecopulas,itcanprovideuscopulasthatareeasytosimulate.Indeed,themainexampleofthiskindofconstructionisthewellknownfundamentalfamilyofGaussiancopulas.AGaussiancopulaisdefinedasthecopulaofajointGaussianrandomvectorXwithstandardGaussianmarginalsandcorrelationmatrixρ,andisthusgivenbyNρ(N−1(u1),...,N−1(un))whereNisthecdfofastandardnormalvariableandNρisthejointcdfofX.Thiscopulacannotbecomputedexplicitly.Thesimula-tionishoweverstraightforward:itissufficienttoconsider(N(X1),...,N(Xn))whereX=(X1,...,Xn)isaGaussianvectorwithcorrelationρthatcanbeeasilysimulatedbyresortingtoastandardGaussiansimulatorandtoaCholeskydecompositionofρ.AsimilarapproachleadstoStudent’scopula(seeBouy´eetal.(2000).andGenzandBretz(2002)
ApossiblemajordrawbackofArchimedeanandGaussiancopulasistheirsym-metricproperties.Letusprecisethisbythefollowingdefinition.
Definition1.1.(k-exchangeability).LetusconsideracopulaCthatisthecdfoftherandomvector(U1,..,Un).WewillsaythatthecopulaCisk-exchangeable
NewfamiliesofCopulasbasedonperiodicfunctions
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(2≤k≤n)if,forany1≤i1 Inrecentyears,copulafunctionshavereceivedagreatdealofattention,seeforexamplethepapersofGenzandBretz(2002),H¨urlimann(2002,2003),JuriandW¨uthrich(2002),Nelsenetal.(2001),WeiandHu(2002),andthebooksofJoe(1997)andNelsen(1999).Forfinancialandinsuranceapplications,recentappli-cationsoncopulasincludeforexampleBouy´eetal.(2000),Cherubinietal.(2002),Embrechtsetal.(2001),Jouaninetal.(2001),KlugmanandParsa(1999),Pram-polini(2003),andSch¨onbucherandSchubert(2001). Inthispaper,wewillbuildnewfamiliesofcopulasbasedonthefirstapproach,usingperiodicfunctionsfollowingAlfonsi(2002).Wefirstbegintoworkinthetwo-dimensionalcase,obtainingaone-parametercopula,andthengiveawaytoextendtheresulttothen-dimensionalcasewithn>2,gettingafamilywithn−1parameters,2-exchangeableornot.Finally,weexplainhowsuchcopulascanbesimulated. 2Constructionofcopulasbasedonperiodicfunc-tions 2.1Theconstructionoffamiliesintwodimensions Webeginbydefiningournewcopulafunctionsforbivariatedependence,i.e.forpossibledependencestructuresbetweentworandomvariables.Itishelpfultofirstrecallthreeparticular“limit”copulas.The“middle”one,whichistypicallydenotedbyC⊥,isthecopulaobtainedwhenU1andU2areindependentuniformvariableson NewfamiliesofCopulasbasedonperiodicfunctions 5 [0,1],thatis: C⊥(u1,u2):=u1u2. −+ Thetwoother“limit”copulas,denotedbyCFandCFrespectively,arethetwo Frechetboundsoftheconvexsubsetofcopulas: −+ (u1,u2)=(u1+u2−1)+,(u1,u2)=min(u1,u2),CFCF wherex+=max(x,0)denotesthepositivepartoperator.NamingUauniform + canbeobtainedasthecopulaof(U,U)andcorrespondsrandomvariableon[0,1],CF − obviouslytoperfectpositivedependence,whereasCFisobtainedasthecopulaof (U,1−U)anddescribestotalnegativedependence.Moreover,foranycopulaC,wehave −+ CF(u1,u2)≤C(u1,u2)≤CF(u1,u2),∀(u1,u2)∈[0,1]2. Recalltheabovecharacterizationofacopulafunctionforthebivariatecase:thefunctionCdefinedon[0,1]2isacopulaifandonlyifi)C(u1,0)=0andC(0,u2)=u2,ii)C(u1,1)=u1andC(1,u2)=u2,andiii)ofSchwartzdistributions. Wewillsayinthefollowingthatacopulaadmitsadensitywhenthatcanbewrittenintheform c(u1,u2)=c˜(u1+u2)(resp.c(u1,u2)=c˜(u1−u2)) forafunctionc˜:R→R.Tosatisfypropertiesi),ii)andiii),c˜mustbenonnegativeandverify: u1 ∂2C ∂u1∂u2 ∂2C∂u1∂u2 isapositivemeasureinthesense =c(u1,u2) existsintheordinarysense.Inthispaperweproposecopulasthathaveadensity 1 c˜(x1±x2)dx1dx2=u1,∀u1∈[0,1],c˜(x1±x2)dx1dx2=u2,∀u2∈[0,1]. 010u2 0 0 Differentiatingwithrespecttou1andu2respectively,weobtain1 c˜(u1±x2)dx2=1,∀u1∈[0,1],01 c˜(x1±u2)dx1=1,∀u2∈[0,1]. 1u+1 Differentiatingfurtherthefirstrelation,since0c˜(u1+x2)dx2=u11c˜(x2)dx2(resp.1u1 c˜(u1−x2)dx2=u1−1c˜(x2)dx2),weobtain:0 c˜(u1+1)=c˜(u1)∀u1∈[0,1],(resp.c˜(u1−1)=c˜(u1)∀u1∈[0,1]). 0 NewfamiliesofCopulasbasedonperiodicfunctions 6 Thus,aconsequenceofrequiringc(u1,u2):=c˜(u1±u2)tobethedensityofacopulaisthatc˜hastobe1-periodic(atleaston[−1,2],butitsvalueoutsidethisinterval 1 isirrelevant)andthat0c˜(u)du=1.Conversely,itiseasytoseethatifc˜isa 1 nonnegative1-periodicfunctionsuchthat0c˜(u)du=1,then u1u2u1u2 −(u1,u2):=+(u1,u2):=Cc˜(x1+x2)dx1dx2(resp.Cc˜(x1−x2)dx1dx2) 0 0 0 0 satisfiesconditionsi),ii)andiii),andsoisacopulafunctionthatwecall,withaslightabuseoflanguage,periodiccopula.Wenoteherethatcopulasobtainedwiththesedensitiesformaconvexsetsinceaconvexcombinationof1-periodicnonnegative 1 functionssatisfying0c˜(u)du=1isalsoa1-periodicnonnegativefunctionwithintegral1onaperiod.Noticefurtherthattheuseofthe“-”and“+”signsappearstobecounterintuitive(onewouldexchangetheabovesigns),butthereisareasonforthisthatwillbeclarifiedlateron. Attimes,ratherthancharacterizingcopulasthroughtheirdensities,itisprefer-abletohaveadirectcharacterizationofthecopulaitself.Tocharacterizeperiodiccopulaswithoutexplicitlyreferringtotheirdensities,denotebyϕtheprimitiveofc˜ x thatvanishesat0,andsetΦ(x):=0ϕ(u)du,sothatΦisadoubleprimitiveofc˜.Wecanthenrewritetheaboveperiodiccopulaasfollows: u1u2 −(u1,u2)=Cc˜(x1+x2)dx1dx2=Φ(u1+u2)−Φ(u1)−Φ(u2),(2) 0u10u2 +(u1,u2)=Cc˜(x1−x2)dx1dx2=Φ(u1)+Φ(−u2)−Φ(u1−u2) 0 0 −(u1,u2)=andweseethatthefirstcopulaisalwaysexchangeable(symmetric),inthatC −(u2,u1),whereasthesecondonecanbenonsymmetricifΦisnotpar,i.e.ifC Φ(−x)=Φ(x)forsomex.WehavethuscharacterizedourperiodiccopulasintermsofdoubleprimitivesΦofperiodicfunctionsc˜. Afirstexampleofsuchafunctionwhicharisesnaturallyisc˜(x)=1+sin(2πx+ϕ)whereϕisaparameterthatwecantakein[0,2π).Itgivesrespectivelythefollowingfamiliesofcopulas: −(u1,u2)=u1u2+(sin(2πu1+ϕ)−sin(ϕ)−sin(2π(u1+u2)+ϕ)+sin(2πu2+•Cϕ))/(2π)2 +(u1,u2)=u1u2+(sin(ϕ)−sin(2πu1+ϕ)+sin(2π(u1−u2)+ϕ)−sin(−2πu2+•Cϕ))/(2π)2. NewfamiliesofCopulasbasedonperiodicfunctions 7 Thesecopulas,however,cannotmodelstrongpositiveornegativedependence,since thesecopulascannotapproachneitherC−+ FnorCF.Onthecontrary,itmightbeinterestingtohaveafamilyofcopulaswhichattainsthecopulasC+F ,C⊥andC− Faslimitcasesinordertobeabletodescribealargerangeofdependencestructures.Byexpressingcopulasbymeansof∂2C attainingC+∂u1∂u,F ,C− 2 FandC⊥amountstoattaining µ+=δx1(dx2)⊗dx1,µ−=δ1−x1(dx2)⊗dx1,c⊥=1. withthecopuladensity.Since ∂2C⊥ ∂u1∂u2 existsintheordinarysenseofdifferentiation, ithasadensityc⊥=1thatcorrespondstotheLebesguemeasureonthesquare[0,1]2,µ⊥=dx1⊗dx2.Instead,µ+andµ−aretobeinterpretedinthegeneralizedsense.Moreover,µ+andµ−chargeonlythediagonalsofthesquare[0,1]2,i.e.∆+={(x,x),x∈[0,1]}and∆−={(x,1−x),x∈[0,1]}respectively.Theideaisthentofindafamilyofperiodicfunctionsc˜γindexedbyaparameterγandsuchthatthedensityc˜γ(x1−x2)(resp.c˜γ(x1+x2))concentrateson∆+(resp.∆−)forsomevaluesofγ.Thus,ifwedefinethepiecewise1-periodicfunctionc˜γfor0<γ≤ 12byc˜γ(x):= 1 2γ 1[0,γ](x)+1(1−γ,1)(x)forx∈[0,1),(3) weseethatthefamilyofdensitiesdefinedasc+γ(x1,x2):=˜c γ(x1−x2)verifies:c+⊥+ D 1/2=c,cγ(x1,x2)dx1dx2− γ→→0 µ+→D denotingconvergenceindistribution.Thecorrespondingconvergenceinlawfor randomvariablesisdenotedbyL.TocalculatetheassociatedcopulaCγ +:=C +,itisbesttotryadrawingandseethatitsvaluein(u1,u2)istheareaintheintersectionoftherectangledelimitedby(0,0)and(u1,u2)with{(x1,x2)∈[0,1]2,−γ≤x1−x2≤γorx1−x2≤γ−1orx1−x2≥1−γ}.Weobtain,foru1≤u2whichisnotrestrictive sinceCγ+(u1,u2)=Cγ+ (u2,u1),Cγ+(u1,u2)= 1 [u12γ1u2 +[−((u2−u1−γ)++(u2−γ)+)(min(u1,u2−γ))+−((u1−γ)+)2 2+((u2−1+γ)++(u2−1+γ−u1)+)·(min(u1,u2−1+γ))++((u1−1+γ)+)2]] InordertoobtainafamilythatreachesC− Finstead,weusetheotherfamily,precisely c−γ(x1,x2)=˜c γ(x1+x2).Weobtainc−⊥− 1/2=c,cγ(x1,x2)dx1dxD 2− γ→→0µ−.(4) NewfamiliesofCopulasbasedonperiodicfunctions 8 +− toCγ,thusFortunately,asimplegeometricremarklinkstherelatedcopulaCγ avoidinganewcalculation: −+ (1−u1,u2)(u1,u2)=u2−CγCγ (5) Thus,withthismethod,wehaveobtainedafamilyofcopulaquite”exhaustive”going +− andtakingthein-betweenvalueC⊥.Incidentally,weseenowwhytoCFfromCF +thecopulacomingfromc−thecopulacomingwechosetonameC˜(x1−x2)andC+ andthelatterfromc˜(x1+x2):thisisdonebecauseinourcasetheformerattainsCF− CF. Attimesitcanbehandytohaveasinglenumbermeasuringsomestylizedaspectsofagivencopula.TheSpearman’srhoissuchanumberandisawellknownmeasureofconcordance,seeforexampleEmbrechtsetal.(2001).Whendefinedintermsofcopulafunctions,itisgivenbythefollowingintegralinthecopuladensityc:ρ:=11 −+ copulasrespectively,andCγ1200u1u2c(u1,u2)du1du2−3.WeobtainfortheCγ − ρ+γ=(2γ−1)(γ−1),ργ=(1−2γ)(γ−1). Aninterestingremarkconcernstheconstructionofnon-exchangeable(non-symmetric)copulas(C(u1,u2)=C(u2,u1))throughthismethod.Thiscanberelevantforex-ampleincreditriskwhenmodellingthedefaultdependencebetweentwofirmswithasymmetricrelations.Onemayhaveafirstfirmdependingmoreonasecondonethanthelatterdependsontheformer.Thiscouldbethecaseofalittlefirmthatprovidesgoodstoalargeone.Adefaultofthelargecompanycouldinduceadra-maticeffectonthesmallerone,whereasadefaultofthesmallfirmcouldhavelittlerelevancetothelargeone.Noticethat,inthisrespect,ArchimedeanandGaussiancopulasonlyprovidesymmetricrelationsbetweenthetwofirmsdefaults. Inordertoprovideanexampleofnon-exchangeablecopulaobtainedfromourfamily,weseefrom(2)thatouronlychanceistoselectaperiodicfunctionc˜whose +.ThesimplestsuchdoubleprimitiveΦisnotparandthentaketherelatedCfunctionisc˜:=c¯γdefined,forγ∈[0,1],as c¯γ(x):=(1/γ)1[0,γ](x),forx∈[0,1). (6) γ¯+:=C+Wehavethenc¯γ(x)=c˜γ(x−).Thus,weobtainthefollowingcopulaCγ22NewfamiliesofCopulasbasedonperiodicfunctions 9 associatedwiththedensityc˜(x1−x2):=c¯γ(x1−x2): u1u2u1u2 ¯+(u1,u2)=c¯γ(x1−x2)dx1dx2=c˜γ/2(x1−x2−γ/2)dx1dx2Cγ 0000u1u2+γ/2=c˜γ/2(x1−x2)dx1dx2 = γ 02+ Cγ/2(u1,min(u2 +++ +γ/2,1))−Cγ/2(u1,γ/2)+Cγ/2(u1,(u2+γ/2−1)) Withthisasymmetricperiodiccopulawestillhavegoodasymptoticproperties,in +¯1¯+→C+whenγ→0.CalculatingtheSpearman’srho,wethatC=C⊥andCγF ¯+)=(2γ−1)(γ−1)forγ∈[0,1].Somehowsurprisingly,ρ(C¯+)findagainρ(C γ γ takesnegativevaluebetweenγ=1/2andγ=1,andvanishesat1/2foracopula − differentfromC⊥.Toobtaina(symmetric)familythatreachesCFweneedconsider u1u2 −¯C(u1,u2):=c¯γ(x1+x2)dx1dx2.Wegetγ 0 0 ¯−(u1,u2)=C−(u1,(u2−γ/2)+)+C−(u1,1−(γ/2−u2)+)−C−(u1,1−γ/2)Cγγ/2γ/2γ/2¯−)=−ρ(C¯+)=−(2γ−andcanshow,withatrivialchangeofvariable,thatρ(Cγγ1)(γ−1).Thus,ifwewishtodescribeanegativeasymmetricdependence,itisbest +touseCγwith1/2<γ<1.However,wepointoutthatwecannotdescribenegative−dependencewithanasymmetriccopulaattainingCF. Anotherinterestingsyntheticquantityconcerningcopulasistheupper-tailde-pendence.Thisindicatorisdefinedas 111 c(x1,x2)dx1dx2λ:=lim u→0u1−u1−u whenthecopulahasadensityc.Letusconsiderthegeneralperiodiccase,where 1 asbeforec˜isanonnegative1-periodicfunctionsuchthat0c˜(u)du=1,andϕis 111 itsprimitivethatvanishesat0.Usingtheperiodicity,wehaveu1−u1−uc(x1± 01010 x2)dx1dx2=uc(x±x)dxdx=±(ϕ(x1)−ϕ(x1∓u))dx1→0since1212u−u−u−uu→0101u limu−uϕ(x1)dx1=limu0ϕ(x1)dx1=ϕ(0)=0.Thus,periodiccopulashaveno u→0 u→0 upper-taildependence.However,ifonewishestoobtainacopulawithanupper-taildependenceequaltoλ>0,itisstillpossibletoconsidertheconvexcombination + (1−λ)C+λCFwhereCisapreferredperiodiccopula.Thisconvexcombination canbesimulatedeasilywhenoneknowshowtosimulatethebasicC,aswedofortheperiodiccopulas(withinvertibleϕ,i.e.withastrictlypositiveperiodicfunctionc˜)weintroducedhere. NewfamiliesofCopulasbasedonperiodicfunctions 10 2.2 −+ AsmoothfamilythatreachesCF,C⊥andCF +− AdrawbackofthefamiliesCγandCγisthatthesecopulasareconstantonsome intervalsandcomesfromthe0-1natureofthedensity,andmorepreciselyfromtheexistenceofadomain(withpositivemeasure)wherethedensityvanishes.Thiscausesproblems,especiallywheninneedofsimulatingthecopula.Inordertoavoidthisdrawback,theideaisthentoreplaceγwitharandomvariableΓandthentaketheexpectationofc˜Γ,usingtheconvexityofthesubsetoftheperiodiccopulas.Indeed,ifΓ∼pwherepisthedensityofaprobabilitymeasureon[0,1/2]suchthatp(γ)>0∀γ, 1thencp(x):=E[˜cΓ(x)]=02c˜γ(x)p(γ)dγisapositive1-periodicfunction.Thus,ifwehaveafamilyofrandomvariables(Γα)α≥0concentratedon[0,1/2]withdensities{pα,α∈]0,+∞[}on[0,1/2]andsuchthatΓα−→1/2andΓα−→0,wecandefine α→0 α→∞ L L +:=E[C+]=CαΓα 0 12+−:=E[C−]=Cγpα(γ)dγ,CαΓα 0 12−Cγpα(γ)dγ (7) (thatcorrespondrespectivelytotheperiodicdensitiesc˜(x1−x2)=cpα(x1−x2)andc˜(x1+x2)=cpα(x1+x2)).Wehaveobtainedafamilyofcopulaswithgoodasymptoticproperties,inthat +→C⊥,C+Cαα α→0 α→+∞ +−→C⊥,andC−→CF,Cαα α→0 α→+∞− →CF. 1 Wecanbuildeasilyarandomvariablewithsuitabledensityon[0,2]bytransforming auniformvariableUon[0,1]accordingtoahomeomorphism.Indeed,considerΓα:= 1αU,2 1 α∈]0,+∞[,sothatwegetafamilyofdensitiespαon[0,2]thatfeaturethe desiredasymptoticpropertiesin0and+∞andareimmediatelycomputed: pα(u)=(2α/α)u 1 1−α α. + ThecalculationofCαdoesnotpresentdifficultieseither.Wefirstcalculatethe periodicfunctioncα:=cpα,obtaining cα(x)= 1−α1 [1−(2x)α],α=11−α c1(x)=−ln(2x) for0≤x≤1/2,andcα(x)=cα(1−x)for1/2≤x≤1,sincethesamepropertyholdsforthebasicc˜γ’s.Letuscomputetheprimitiveψαofcαthatvanishesatx=0.Weobtain,for0≤x≤1/2: ψα(x)= 11 [2x−α(2x)α],α=1, 2(1−α) ψ1(x)=x−xln(2x). NewfamiliesofCopulasbasedonperiodicfunctions 11 Usingthesymmetrypropertycα(x)=cα(1−x)weobtain,for1/2≤x≤1,ψα(x)=ψα(1/2)+(ψα(1/2)−ψα(1−x))=1−ψα(1−x),sinceψα(1/2)=1/2.Instead,forx∈[−1,0]weusetheperiodicityofctogetψα(x)=−ψα(−x).Toproceedfurther,weneedtoknowalsotheprimitiveΨαofψα,i.e.thedoubleprimitiveofcα.Wefind, 1forx∈[0,2]: 1α211+α 2 Ψα(x)=[x−2αxα],α=1, 2(1−α)α+1and,forx∈[1,1]:2Ψα(x)=x− 32x2 Ψ1(x)=x−ln(2x), 42 1 +Ψα(1−x),2 andfinallyΨα(x)=Ψα(−x)forx∈[−1,0],sinceψαisanoddfunction.Weare uuu +αnowabletocalculateC(u1,u2)=0201cα(x1−x2)dx1dx2=02(ψα(u1−x2)− u2 ψα(−x2))dx2=0(ψα(u1−x2)+ψα(x2))dx2andsoweget,inagreementwithourearliergeneralresult(2): +(u1,u2)=Ψα(u1)+Ψα(u2)−Ψα(u1−u2).Cα (8) −canthenbecalculatedeasily,sinceC−(u1,u2)=u2−C+(1−u1,u2)ThecopulaCαγγ 1 +(1−u1,u2).We−(u1,u2)=2(u2−C−(1−u1,u2))pα(γ)dγ=u2−CandthereforeCγαα0 1/2 ++canalsoeasilycalculatetheSpearman’srhoofCα,sinceρ(Cα)=0(2γ−1)(γ−1)21/αγ(1−α)/α/αdγsothat +αρ(C)=1− 13 + 2(1+α)2(1+2α) −+ααandwehaveρ(C)=−ρ(C)(thisisageneralrelationbetweentherhoofthe periodiccopulaswithdensityc˜(x1+x2)andc˜(x1−x2)).WecansumupinTable1thevaluesforwhichalimitcopulaisreached.Thefamiliesbuiltpreviouslyare α + Cα−Cα+Cα − CF C⊥ 1212+CF /0/+∞ 0/+∞/ 00 −Cα αTable1:LimitcopulasfortheparameterizationCαandC NewfamiliesofCopulasbasedonperiodicfunctions 12 exchangeable,sincetherelatedc˜areexpectationsoffunctionsleadingtopardoubleprimitivesandthereforeleadthemselvestopardoubleprimitives,sothat(2)yieldssymmetry. However,wecanalsoconstructasmoothfamilyofnonsymmetriccopulasbydefiningc˜astheexpectationofthepreviousnonsymmetricfunction(6)witharandomγ. Indeed,ifZ∼qwhereqisapositivedensityofaprobabilitymeasureon[0,1], 1 thenc¯q:=E[¯cZ]=0c¯γ(x)q(γ)dγisapositive1-periodicfunction.Thus,asbefore,ifwehaveafamilyofrandomvariables(Zα)α≥0concentratedon[0,1]withdensities{qα,α∈]0,+∞[}on[0,1]andsuchthatZα−→1andZα−→0,wecandefine α→0 α→∞ L L ¯:=Cα + ¯+]E[CZα = 0 1 ¯+qα(γ)dγ,Cγ ¯:=Cα − ¯−]E[CZα = 0 1 ¯−qα(γ)dγCγ (9) (thatcorrespondrespectivelytotheperiodicdensitiesc˜(x1−x2)=c¯qα(x1−x2)andc˜(x1+x2)=c¯qα(x1+x2)=E[¯cZα(x1+x2)]). WetakeZα:=Uα,whereUisauniformrandomvariableon[0,1].Itsdensityis qα(u)= 11−α uα,α sothattheassociatedperiodicfunctionc˜=c¯αisgivenin[0,1]by c¯α(x):=c¯qα= 1−α1 [1−xα],α=1,1−α c¯1(x):=c¯q1=−ln(x). Inordertofindanexpressionforthecopula,wecomputetheprimitivegαofc¯αthatvanishesatx=0.Wehave,forx∈[0,1]: gα(x)= 11 [x−αxα],α=1,1−α g1(x)=x−xln(x), andforx∈[−1,0]wehavegα(x)=gα(1+x)−1.DenotebyGαtheprimitiveofgαvanishingat0.Forx∈[0,1]Gαisgivenby 12α21+α1 x−xα,α=1,Gα(x)= 1−α21+αwhereasforx∈[−1,0]wehave Gα(x)=Gα(1+x)−x−Gα(1). ByFubini’stheorem,thecopulasdefinedby(9)arethesameasthecopulasassociatedwiththeperiodicfunctionsc˜=c¯αanddefinedby(2).Wethushave,by(2)andusing 13 G1(x)=x2−x2ln(x), 42 NewfamiliesofCopulasbasedonperiodicfunctions 13 periodicityofc¯(thusreplacingc¯(·)byc¯(·−1),whichishelpfulinsomecomputationalrespects): + ¯Cα(u1,u2)=Gα(u1)+Gα(−u2)−Gα(u1−u2) −¯Cα(u1,u2)=Gα(u1+u2−1)−Gα(u2−1)−Gα(u1−1)+Gα(−1). CalculatingtherelatedSpearmanrhowefind: + ¯ρ(Cα)=1− 23 + 1+α1+2α −+−+¯¯¯¯andρ(Cα)=−ρ(Cα).Asweknowfrom(2),Cα,contrarytoCα,isafamily ofsymmetriccopulas,butthisfamilyishoweverinterestingbecauseitcompletes +¯“naturally”thefamilyCα.InTable2wesumupthevaluesoftheparameterαfor ¯copulasreachthelimitcopulas.whichtheC − CF C⊥1100 +CF ¯+Cα¯−C /0/+∞ 0/+∞/ ¯Cα−¯C α α+ ¯αandC¯αTable2:LimitcopulasfortheparameterizationC 2.3Beyondthebivariatecase Thereareseveralwaystoextendthepreviousconstructiontobuildacopulaindimensionn>2.Indimensionn>2,weseethatifc˜isa1-periodicfunctionsuch 1that0c˜(x)dx=1,thenc˜(ni=1εixi)aredensitiesofcopulaswhenεi∈{−1,1}.However,thesecopulascannotbeobtaineddirectlyintermsofbivariatecopulasandthereforerequirecumbersomecalculations.Inordertokeeptheanalysissimple,weworkinsteadwiththedensitiesalreadydefinedforthebivariatecase.Considerthefollowingproposition. Proposition2.1.Assumethatc1,...,cn−1aredensitiesoftwodimensionalcopulas˜n−1,i.e.cj(x,y)=c˜j(x+εjy)withεj∈builtthroughperiodicdensitiesc˜1,...,c NewfamiliesofCopulasbasedonperiodicfunctions 14 {−1,1}andc˜janonnegative1-periodicfunctionwithunitintegralonaperiod.Setcˇ:=(c1,...,cn−1).Then ˇ1(u1,..,un):=C u1 .. un c1(x1,x2)c2(x2,x3)..cn−1(xn−1,xn)dx1...dxn 0u10un ˇ2(u1,..,un):=C..c1(x1,x2)c2(x1,x3)..cn−1(x1,xn)dx1...dxn 0 0 arecopulas. Theproofisquiteimmediate.Properties1and3inSection1aresatisfiedby uk ˇconstruction.ItremainstoobservethatC(1,..,1,uk,1,..,1)=0dxk=uk,byusingFubini’stheorem,integratingfirstwithrespecttothexi’swithi=k,and 1+x usingthenthepropertyxc˜(u)du=1. ˇ1isconvenientifwewishtoexpressthen-dependenceintermsThefirstcopulaC ˇ2allowsustoofdependencesoftwoconsecutivevariables,whereasthesecondoneC expressthen-dependenceintermsofthedependenceofapreferredvariable(thefirstinourformulation)withallothervariables.Thesecondmethodcouldbereferredtoasa“preferred-”or“main-factor”approach. 3Thesimulationofperiodiccopulas Letusbeginbyrecallinghowtosimulateacopulathatadmitsadensityp(x1,...,xn).Weneedsimulateavectorofuniformvariables(U1,...,Un)thathasthefollowingjointcdf: C(u1,...,un)= 0u1 .. 0 un p(x1,...,xn)dx1...dxn. Thiscanbedoneaccordingtothefollowingsteps. •TosimulatethefirstvariableU1,itsufficestosamplefromauniformrandom ˜1in[0,1].ThiscanbeeasilydoneonaPC.Letuscallu1thevariableUsimulatedsample. •Toobtainasampleu2fromU2consistentlywiththeearliersampledu1,weneedtoknowthelawofU2conditionalonU1=u1.LetusnameF2(.|u1)thecdfofthislaw, F2(u2|u1)=P(U2≤u2|U1=u1)=∂u1C(u1,u2,1,..,1)/∂u1C(u1,1,1,..,1) 1u21 ...p(u1,x2,..xm)dx2..dxm.=∂u1C(u1,u2,1,..,1)= 0 0 0 NewfamiliesofCopulasbasedonperiodicfunctions 15 −1˜˜2isanewuniform-[0,1]sampleindependentWetakeu2=F2(U2|u1)whereU ˜1.ofU •tosimulateUkconsistentlywiththeearliersampledu1,...,uk−1,weneedthelawofUkconditionalonUi=uifori ∂u1,..,uk−1C(u1,..,uk,1,..,1)= ∂u1,..,uk−1C(u1,..,uk−1,1,..,1)uk11 ..0p(u1,..,uk−1,xk,..,xn)dxkdxk+1..dxn 00 =111..0p(u1,..,uk−1,xk,..,xn)dxkdxk+1..dxn00 −1˜˜kisauniform-[0,1]variablewecantakeUk=Fk(Uk|u1,..,uk−1)whereU˜1,...,U˜k−1).independentof(U ˇ1,2,maintainingthenotationofProposition2.1,InthecaseoftheperiodiccopulasC uk 12 wehaverespectivelyFk(uk|u1,..,uk−1)=0ck−1(uk−1,xk)dxkandFk(uk|u1,..,uk−1)=uk ck−1(u1,xk)dxk,wheretheupperindexreferstothecopulaweareconsidering.0 Takingthesmoothfamiliesoftheprevioussection,theseFfunctionscanbeexpressed u intermsofψαandgα(forexample0kcα(u±x)dx=±(ψα(u±uk)−ψα(u)).More-overtheyarestrictlyincreasing,andcanthereforebeinvertedeasilynumerically.Wenoteherethatifwechoosethe“nonsmooth”copulasC±andC,thisinversionisnotfeasiblesincethedensitiesvanishonsomeintervals.Thuswehaveobtainedfamiliesofn-dimensionalcopulasessentiallycharacterizedbyn−1parametersαiplustheflagssgni,symi,fori=1,...,n−1,wheresymiissetaccordingtowhetherwetakeasymmetricfamilyornot(symbolizedherebythebar),andwheresgniistakenfromtheset{−,+}. ± 4Conclusions Thenewfamilyof“periodic”copulasintroducedinthispaperisanattemptatob-tainingpracticallymanageableandpossiblyasymmetriccopulas.Wehavestudiedthetwo-dimensionalcase,basedonasingledependenceparameter,andthenpro-videdameanstoconstructann-dimensionalcopulabuildingonthetwo-dimensionalcase.Weobtainedfamiliesofcopulasindimensionnandparameterizedbyn−1 NewfamiliesofCopulasbasedonperiodicfunctions 16 parameters,implyingpossiblyasymmetricrelations.Weexplainhowsuchcopulascanbesimulated. 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