Copula methods and CDO pricing modelsOlivierGueantContentsI Copulas: aprimer 41 Atheoreticalintroductiontocopulas 41.1 Denitionofcopulafunctionsandprobabilisticinterpretation . . . . . . . . . . 41.2 Frechetboundsanddependencestructures . . . . . . . . . . . . . . . . . . . . . 61.3 Copuladensityandestimations . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Familiesofcopulafunctions 72.1 Gaussiancopulafunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Anotherellipticalcopulafunction: theStudentcopula . . . . . . . . . . . . . . . 92.3 Archimedeancopulafunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Otherusefulcopulasforcreditderivatives . . . . . . . . . . . . . . . . . . . . . 102.4.1 TheMarshall-Olkincopulas . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 Thedouble-tcopula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11II ThereferencecopulaapproachforCDOtranchepricing: fromLitobasecorrelation 133 IntroductiontoCDOs 133.1 Correlationproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 CDOstrancheslegsandthespread . . . . . . . . . . . . . . . . . . . . . . . . . 1414 Thegaussiancopulamodel 174.1 GaussianCopula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Latentvariableviewpointandnumericalmethods . . . . . . . . . . . . . . . . . 184.3 TheLHPmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Impliedcorrelation 215.1 Compoundcorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Basecorrelationandthecorrelationskew. . . . . . . . . . . . . . . . . . . . . . 246 Stochasticrecoveryrates 26III Non-gaussiancopulamodels 297 Towardagoodskewmodel? 297.1 Stochasticcorrelation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.2 Localcorrelationandrandomfactorloadings. . . . . . . . . . . . . . . . . . . . 307.3 Studentcopula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.4 Double-tcopula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.5 NormalinverseGaussiandistributions . . . . . . . . . . . . . . . . . . . . . . . 358 Morecondentialapproaches 368.1 Archimedeancopulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.2 Marshall-Olkincopula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Theimpliedcopulamodel 3810Concludingremarks 40IV Acriticalanalysis 4211Creditderivativespricingmodels? 4312Thedicultytohedge 4413Meetingthepractitioners 442V Areviewofalternativemethods 4514Dynamiccopulamodels 4515Intensitymodels 4516Otherbottom-upmodels 4517Thetop-downapproach 453Introduction4PartICopulas: aprimerMost quantitative analysts know that copula functions are used to model the dependence struc-ture of multidimensional random variables. More exactly, copula functions allow to separate theanalysisofmarginaldistributionsfromtheanalysisofthedependencestructure. Jointdistri-butionsindeedentanglethetwoandtheissueoflinkingrandomvariableswithgivenmarginaldistributionshasbeenmadepossiblebycopulas.1 Atheoreticalintroductiontocopulas1.1 DenitionofcopulafunctionsandprobabilisticinterpretationCopula functions have been introduced through what is known as Sklars Theorem in [136] andallowtowritethejointdistributionof Nrandomvariables(X1, . . . , XN)usingthemarginaldistributions i.e. the respective distributions of the Xis (1 i N) and a function called acopula function. In this section, we are going to dene the latter and the main results of copulatheory: SklarsTheorem,Frechetbounds,canonicalrepresentation,...Denition1. AcopulafunctionindimensionNisafunctionC: [0, 1]N [0, 1]verifyingthefollowinghypotheses:Cisgrounded:1 i N, (x1, . . . , xi1, xi+1, . . . , xN) [0, 1]N1, C(x1, . . . , xi1, 0, xi+1, . . . , xN) = 0MarginsofCareuniform:1 i N, xi [0, 1], C(1, . . . , 1, xi..i, 1, . . . , 1) = xi5CisN-increasing:1 i N, 0 xi yi 1,i,wi{xi,yi}(w)C(w1, . . . , wN)where(w) =___1, ifthenumberofindicesisuchthatwi= yiiseven0, if i, xi= yi1, ifthenumberofindicesisuchthatwi= yiisodd.This denitionallows towriteSklars theoremthat gives theprobabilisticinterpretationofcopulafunctionsandexplaintheirroleinextractingtheverycoreofthedependencestructurefromthejointdistribution:Theorem1(SklarsTheorem). LetF1, . . . , FNbeNcontinuouscumulativedistributionfunc-tions.IfCisacopulafunctionasdenedabove,then:(x1, . . . , xN) RNC(F1(x1), . . . , FN(xN))denesajoint distributionwithmarginscorre-spondingtotheFis.Conversely, if Fisajoint distributionfunctionwithmarginscorrespondingtotheFis, thenthereexistsauniquecopulafunctionCsuchthat:(x1, . . . , xN) RN, F(x1, . . . , xN) = C(F1(x1), . . . , FN(xN))The secondpart of Sklars theoremstates that amultivariate joint distributionF canbeexpressedusingthemargins F1, . . . , FNthroughacopulafunction. This theoremis infactconstructivewhenthemarginal cumulativedistributionfunctionsarestrictlyincreasing. Inthatcaseindeed,wecandene:(u1, . . . , uN) [0, 1]N, C(u1, . . . , uN) = F(F11(u1), . . . , F1N(uN))ItisnoteworthythatSklarstheoremholdsinfactinamoregeneral non-continuoussettingoncecopulasarereplacedbysub-copulas(see[28]foranintroduction)butthecaseofcontin-uousrandomvariablescoversalmostallapplicationsofcopulafunctionsinFinance.61.2 FrechetboundsanddependencestructuresAmongthenumerouspropertiesof copulafunctions(uniformcontinuity, almost everywheredierentiability,...) onepropertyisworthwritingafewlinesonitbecauseitgivesboundsonthe possible dependence structures between random variables. These bounds are referred to asFrechetboundsandaresummedupinthefollowingproposition:Proposition1. LetCbeanN-dimensional copula. Then:(u1, . . . , uN) [0, 1]N, max(u1 +. . . +uN (N 1), 0) C(u1, . . . , uN)andC(u1, . . . , uN) C+(u1, . . . , uN) = min(u1, . . . , uN)Intheaboveproposition, C+isacopulathatcorrespondstothemaximal dependence. IfwehaveindeedthatX1= . . . = XN,thenthecopulaassociatedto(X1, . . . , XN)isC+. Althoughthe lower bound is optimal, it is not a copula for N> 2. For N= 2, it is often referred to as theCcopula corresponding to the maximal negative dependence: if Uis uniformly distributed on[0, 1],then P(U u1, 1 U u2) = P(1 u2 U u1) = max(u1 + u2 1, 0) = C(u1, u2).Thisgivesexamplesofcopulas1andwearegoingtodetailinthenextsectionalargenumberoffamiliesofcopulas.Goingonthemodelingofdependencestructure, itmustbenoticedthatcopulafunctionsareinvariantwithrespecttostrictlyincreasingtransformations. Hence, whenN=2, C+isthecopula function of comonotonic random variables, i.e. random variables where one is the imageof the other through a strictly increasing function. This reinforces the intuition on the fact thatcopulafunctionsmodel thedependencestructureof randomvariablesindependentlyof theirrespectiveprobabilitydistributions.1Another important example of copula function is the one that describes independence, namelyC(u1, . . . , uN) = u1. . . uN.71.3 CopuladensityandestimationsAs presented above,copula functions were a way to relate the generalizedcumulative distribu-tionfunctionofamultivariaterandomvariabletothecumulativedistributionfunctionsofitsmargins. Inmanyapplicationshowever, probabilitydistributionfunctionsmustbeusedandforthatpurposeweintroducewhatisthecalledthedensityofacopula.Denition2(Densityforacopula). GivenacopulaC,onecandenealmosteverywhereon[0, 1]Nthefunction:c(u1, . . . , uN) =NCu1. . . uN(u1, . . . , uN)Thisdenitionallowstodecomposecopulasaccordingtotheirabsolutelycontinuousandsin-gular components2andalsotowrite the probabilitydistributionfunctionof anabsolutelycontinuousrandomvariable(X1, . . . , XN)as:f(x1, . . . , xN) = c(F1(x1), . . . , FN(xN))f1(x1) . . . fN(xN)Thisrepresentationisespeciallyinterestingif onewantstoestimatethejointdistributionofamultivariaterandomvariableusingthedecompositionof thejointdistributionbetweenNmarginsandacopula. Ifonechoosesgivenfamiliesofunivariatedistributionsforthemarginsand a given family of copulas (see next section), then the resulting parametric estimation can bedonethrougheitheraone-steporatwo-steplikelihoodmaximizationusingtheaboveformula(seeforinstanceChap5. of[28]or[47]).2 FamiliesofcopulafunctionsThere exist two major families of copulas: elliptical and archimedean copulas. Elliptical copulasaretheeasiesttointroduceandthewidelyusedgaussiancopulaisaspecial caseofellipticalcopula. Copulathatareneitherelliptical norarchimedeanarealsogoingtobeusedinthistext: the Marshall-Olkin copula and the double-t copula are two instances that have been usedindefaultmodeling(seebelow).2The absolutely continuous component of a copula is dened by (u1, . . . , uN) u10. . .un0c(v1, . . . , vN)dvN . . . dv1.82.1 GaussiancopulafunctionsThe most common intuition behind the practical use of copula functions is linked to the followingmechanism:ConsideratupleofNrandomvariablesY =(Y1, . . . , YN)withmarginsdescribedbyNstrictlyincreasingcumulativedistributionfunctionsF1, . . . , FN.Then (U1, . . . , UN) = (F1(Y1), . . . , FN(YN)) is a tuple of Nrandom variables, each of thembeinguniformlydistributedon[0, 1]Hence, if one picks Ncumulative distribution functions G1, . . . , GN, (G11(U1), . . . , G1N(UN))isamultivariaterandomvariableswithmarginsdescribedbyG1, . . . , GNandadepen-dencestructureinheritedfromthedependencestructureofY .Thismechanismallowstogeneratenon-independentvariableswithgivenmarginsonceweareabletogenerateNnon-independentvariablesandthecopulawillbethesameforthetwosetsofvariables.Therstexample,andthemostusedone,consistsintakingY N(0, )amultivariatenor-mal variable with a correlation matrix (each variable Yibeing then distributed as a standardnormal).Inthatcase,thecopulaiscalledagaussiancopulaandwehave:C(u1, . . . , uN) =F1(u1). . .F1(uN)1(2)Ndet()exp_12x1x_dx1. . . dxNwhereFisherethecumulateddistributionfunctionofastandardnormal.Thiscopulabeingabsolutelycontinuouswecanwriteitsdensity:c(u1, . . . , uN) =1det()exp_12x(1IN)x_, i, xi= F1(ui)As an important example of such a gaussian copula,the 1-factor gaussian copula with uniformcorrelationparametercorrespondstoacaseinwhich:9 =___________1 1...........................1 1___________Inspiteof itsnotoriouslimitations, this1-factorgaussiancopulaisthemostwidelyusedincredit derivativepricing. It canbedescribedas thecopulaof theNvariables (Y1, . . . , YN)where:Yi= Y+1 iwhereY isastandardnormalcommonfactoroftencalledthelatentvariable, whereeachiisastandardnormal,andwherethevariablesY ,1, . . . , Nareindependent.2.2 Anotherellipticalcopulafunction: theStudentcopulaAs intheabovegaussiancase, vectors of correlatedStudent variables canbegeneratedus-ingmultivariategaussianvariables. InthatcasethecopulaiscalledaStudentcopulaorat-copula. This copulashouldnot beconfusedwiththedouble-t copulaalsousedincreditderivativepricingseebelow. WhileaStudentcopulaisthecopulaofamultivariateStudentrandomvariable, adouble-tcopulaisobtainedthroughthegeneralizationoftheabovelatentvariablerepresentationwithY andifollowingStudentdistributions.Theinterestof Studentcopulaliesinthepropertiesof extremeevents. TheStudentcopulasareindeedcharacterizedbyupperandlowertail dependencythataredierentfromnought,contrarytogaussiancopulas.2.3 ArchimedeancopulafunctionsArchimedeancopulafunctionsarecopulasoftheform10C(u1, . . . , uN) = (1(u1) +. . . +1(uN))where,calledthegeneratorofthecopula3veries: C(R+)(0) = 1limt(t) = 0completelymonotonic,i.e. n, t 0, (1)n(n)(t) > 0ThemostfamousarchimedeancopulasaretheClaytoncopulas, theGumbel copulasandtheFrankcopulas. TheCcopulaintroducedaboveisalsoanexampleofarchimedeancopula.AClaytoncopulaischaracterizedby(x) = (1 +x)1, > 0.AGumbelcopulaischaracterizedby(x) = exp_x1_, 1.AFrankcopulaischaracterizedby1(x) = log_exp(x)1exp()1_, R.TheCcopulacorrespondsto(x) = exp(x).Claytoncopulas areoftenusedinorder totakeaccount of lower tail dependency. Gumbelcopulasonlyallowforpositivedependenceandareusedtomodeluppertaildependency.2.4 OtherusefulcopulasforcreditderivativesTheabovecopulas arethemost classical ones andtheyhavebeenusedinvarious elds ofapplied probability and applied statistics in order to model the dependence structure of randomvariables. Theyhavebeenusedinactuarial studiesandtheinterestedreadersmayread[41]and[58]. Innancialriskmanagement,theyareusedinthemodelingofmarketcomovementsespecially to compute risk measures that take account of the non-gaussian distributions of pricereturns. Inassetpricing, optionswithmultiplesunderlyingareoftenmodeledusingcopulafunctions but themainapplicationof thecopulaapproachis evidentlyincredit derivatives3Sometimes1isreferredtoasthegeneratorofthecopula.11pricing. For that purpose, specic copulas have beenintroducedandwe present here twoexamples of copulas that have been used in the literature on CDO pricing (other instances willbepresentedinthenextparts,appliedtocreditderivativespricingissues).2.4.1 TheMarshall-OlkincopulasTherst copulas wepresent, ina2-dimensionframework, aretheMarshall-Olkinfamilyofcopulas. These copula functions appear in multivariate credit derivative pricing when a defaultcan occur either for an idiosyncratic reason or for a systemic reason (all obligors or all obligorsfrom the same sector defaulting at the same time). In a 2-dimension context, the Marshall-Olkincopulasareofthefollowingform:C(u1, u2) = u1u2 min(u11, u22)We will see in part III that this copula appears as the (survival4) copula of default times in theMarshall-Olkinpricingmodel.2.4.2 Thedouble-tcopulaAnother copula that has been used in order to improve models based on gaussian copulas is theso-calleddouble-tcopula. WewilldiscusstheuseofthiscopulainPartIIIanditspropertiesthatinduceaatteningofthebasecorrelationcurve,atleastonpre-crisisdata.This copula has 3 parameters , and andis the copula functionof the Nvariables(Y1, . . . , YN)where:Yi= 2Y+1 2iY standsforacommonfactorandfollowsaStudentdistributionwithdegreesof freedom.EachifollowsaStudentdistributionwithdegreesoffreedom. ThevariablesY ,1, . . . , Nareindependent.SincesumsofStudentvariablesarenotStudentvariables,thiscopulaisnotaStudentcopulabutgeneralizesthe1-factorgaussiancopula, usinganapproachthatiscommoninstructural4The notionof survival copulasimplyrefers tothe case where inthe denitionof copulas, cumulativedistributionfunctionsarereplacedbytailfunctions.12modelswithalatentvariablethegaussiancopulawithuniformcorrelationcorrespondingto= = +.13PartIIThereferencecopulaapproachforCDOtranchepricing: fromLitobasecorrelation3 IntroductiontoCDOs3.1 CorrelationproductsCreditderivativeshaveemergedinthe1990sandmultivariatecreditderivativessuchasDe-fault Baskets or CollateralisedDebt Obligations (CDO) have knowntheir goldenage inaperiodstartingintheearly2000sandendingwiththeoccurrenceofthesubprimecrisis. De-fault Baskets are multi-name contracts in which a protection buyer is guaranteed a payment byaprotectionselleraftertheoccurrenceofaspecicdefaulteventwithinapoolofdefaultableassets. ExamplesofDefaultBasketsareFirst-to-Default(FTD)contracts, Second-to-Default(STD)contracts,...Thesecontractsweretherst leadingmulti-namecredit derivativesbut CollateralisedDebtObligations soon overtook them. CDOs are contracts based on a pool of credits or on a pool ofCDSsthataretranchedinordertoprovidedierentinvestorsdierentexposures. AnequitytrancheofaCDOisacontractinwhichtheprotectionbuyerisguaranteedagainstthelossesofthecreditportfoliosuptoacertainpercentageoflossduringagivenlengthoftime. Thencomethemezzaninetranchesthatarecontractsprotectingtheprotectionbuyeragainstlossesonthecreditportfoliosonceapercentageof losseshavebeenreached(theattachmentpointofthetranche)anduptoanotherpercentageoflosses(thedetachmentpointofthetranche)duringagivenlengthof time. Finally, seniorCDOtranchestypicallyprotecttheprotectionbuyerduringagivenlengthoftimeagainstlossesofthecreditportfoliosonceapercentageoflosseshavebeenreachedandupto100%ofthelosses.In exchange for the protection, the protection buyer typically pays a spread on a quarterly basistotheprotectionsellerandsometimesanupfrontpaymentisaddedtomitigatecounterparty14risk.CDOscanbeofvariousformsandinthecaseofsyntheticCDOs,whichisthecaseofgreatestinterestfromtheacademicliteratureviewpoint, theunderlyingcreditportfolioisreplacedbyaCDSportfoliobutthemechanismisthesame. StandardizedproductsarebasedonindicessuchasiTraxxorCDXandwewilltacklethepricingofiTraxxorCDXtranches.Theseproductssharecharacteristicsthatmakethembepartoftheso-calledcorrelationprod-ucts. Inotherwords,inadditiontobeingexposedtocreditorprotectedagainstsomedefaultevents, thepartiesintheseproductsareexposedtothedependencestructureof thedefaulteventsof theunderlyingcredits. Althoughthevocabularyisseeminglyinsignicant, thisex-posureisalwaysreferredtoasanexposuretocorrelationriskwhereascorrelationisalinearmeasureof dependence. Wewill seethat this remarkonvocabularyis infact rootedtoarealweaknessofthemostcommonlyusedmodelstopriceCDOtranches. Moregenerally,wewill seeinpartIVthatthevocabularyusedwhiledealingwithcreditderivativeportfoliosisresponsibleofmanymisunderstandingsabouttheverynatureofthemodels?3.2 CDOstrancheslegsandthespreadWeconsiderhereandfortherestofthistextasyntheticCDOwhoselossprocessisdenotedL. Inotherwords, everytimeadefaultoccursintheunderlyingportfolioor, inourcaseofasyntheticCDO, everytimeadefaulteventtriggersaCDSintheunderlyingportfolio, Lisincreasedbythelosslinkedtothisevent. For standardizedCDOswefocuson, thislossisnormalizedandthelossprocessisdenedby:L(t) =1MMi=1(1 Ri)1it, t [0, T]where:MisthenumberofunderlyingCDSs.RiistherecoveryratelinkedtotheunderlyingoftheCDSi.iistherandomvariablemodelingthedefaulttimeoftheunderlyingoftheCDSi.15Then, if weconsideratrancheof aCDOwithattachmentpointAanddetachmentpointB(0 A < B 1)5,thenormalizedlossofthetrancheis:L(t, A, B) =L(t) AB A1ABOtherequivalentwaystowritethelossprocessofthetrancheare:L(t, A, B) =min(L(t), B) min(L(t), A)B AorL(t, A, B) =(L(t) A)+ (L(t) B)+B AFromthis, wedenethepresentvalueofthetwolegsof aCDOtranchecontract. First, thedefaultlegpresentvalueisgivenby:DL =T0D(0, t)dL(t, A, B)where:TisthematurityofthecontractD(0, t)isadiscountfactorInturn,theprotectionlegpresentvalueisgivenby6:PL = U+SNj=1(Tj Tj1)D(0, Tj)(1 L(Tj, A, B))where:(Tj)jisasubdivisionoftheperiod[0, T],typicallyaquarterlysubdivision.Uisanupfrontpayment.5If A=0, thenthetrancheisanequity(orjunior)tranche. If B=1, thenthetrancheisasenior(orsuper-senior)tranche.6Infact this is just anapproximationof thereal valuebecausetwoeects must betakenintoaccount.First, therunningnotional maynotbe1 Lbut1 LwhereLisdenedasLwithrecoveryratesequal to0. ThisisparticularlyrelevantwhendealingwithaseniortranchesinceLmaybedierentfrom1whileeverysingle-nameintheportfoliodiddefault. ThesecondissueintheformulaisbasedonthefactthatthenotionaltobetakenintoaccountisnotnecessarilyL(ti, A, B)anddependonthedefaultsthatoccurredinthetimeinterval(Ti1, Ti]. Accrualtermsshouldbeaddedbutwedecidedtoignorethemforexpositionpurposes.16SisaspreadpaidateachdateTj.Fromtheseformulasforthetwolegs, onceanupfrontpaymentisxedoragreedupon, thespreadofthetrancheisxed,underarisk-neutralprobabilitypricing,by:S=E_T0D(0, t)dL(t, A, B)_UE_Nj=1(Tj Tj1)D(0, Tj)(1 L(Tj, A, B))_Assumingindependencebetweeninterestratesanddefaults,thisis:S=T0P(0, t)dE[L(t, A, B)] UNj=1(Tj Tj1)P(0, Tj) (1 E[L(Tj, A, B)])whereP(0, t)isthepriceofazero-couponbondwithmaturityt.Hence, theproblemofcomputingthespreadofaCDOtranchereducestothedeterminationofthefunctionL(t, A, B)andeventuallytothelossfunctionL(t).Inwhatfollows,wearegoingtopresentmodelsthatapplytheso-calledbottom-upapproach.Inotherwords,thelossfunctionwillbemodeledthroughamodelfortheindividualdefaults:starting with a model for the underlying CDSs (bottom),the CDO tranches loss function (up)will becalculatedastheoutcomeof thedefaults. Inparticular, thegaussiancopulamodel,thatisthemostcommonmodel inCDOpricing, isabottom-upmodel. Anotherapproach,whichisoutofthescopeofthistextbutthatwillbediscussedinPartVisatoportop-downapproachinwhichthetranchelossaredirectlymodeledorimpliedfromtheCDOquotesandthenapproximationsaremadetogetdowntoindividualdefaults.Before describingthe gaussiancopulamodel whichis the topic of the next section, let usemphasize the importance of modelingthe dependence betweendefaults inthe bottom-upapproach. A junior CDO tranche or a FtD (First-to-Default) is indeed riskier when the defaultsoftheunderlyingsinglenamesareindependentthanwhentheyarepositivelycorrelated(ortobemoreprecise, comonotone). Theinverseistruefortheseniortrancheof aCDOandweseethat themost important objectiveinmodelingis tohaveagoodrepresentationof thedependencestructurebetweendefaults. Thisiswherewecallcopulasintoplay.174 Thegaussiancopulamodel4.1 GaussianCopulaTobuilduptheloss functionthat is necessarytopriceaCDOtrancheintherisk-neutralframework described above, we need to model the default times linked to the underlying CDSs(1, . . . , M) and their dependence. For that purpose, a seminal approach proposed by Li in [98]consistsinwritingthecumulativedistributionof(1, . . . , M)as:F(t1, . . . , tM) = P(1 t1, . . . , M tM)= C(F1(t1), . . . , FM(tM)) = C (P(1 t1), . . . , P(M tM))where Cis agaussiancopulawithacorrelationmatrixthat has tobe specied. Thiscorrelationmatrixcan, for riskmanagement purposes suchas Value at Riskcomputations(seeCreditMetricsmethodologyforinstance), betakenequaltothecorrelationmatrixoftheunderlyingassets(correlationof stocksforcorporateCDS). However, thepricingtechniquesoftenconsistinspecifyingapredeterminedstructureforthecorrelationmatrixinordertobeable to calibrate it on market prices. The most common case is the case of a one-factor Gaussiancopulainwhichpairwisecorrelationsareassumedtobeallequaltoaconstant: =___________1 1...........................1 1___________In this framework, and for all the models based on copulas, the terms P(i ti) are, for each i,calibratedonthetermstructureofsurvivalprobabilitiesthatdependontheCDSspreadsandusuallyonaCDSpricingmodelthatassumeconstantrecoveryrate.Stopping here, a method to evaluate the spread of a CDO, according to the risk-neutral pricingformulaintroducedabove, couldbeaMonte-Carlosimulation. CopulaapproachesareindeedperfectlysuitedtoMonte-Carlosimulation. OncethecumulativedistributionfunctionFiof18ihasbeencalibratedonCDSspreadsforalli, asampleofdefaulttimes( 1, . . . , M)canbeeasilydrawnby: i= F1i(N(Yi))where (Y1, . . . ,YN) is drawn from a N(0, ) distribution and where Nis the cumulative distri-butionfunctionofastandardnormal..But Monte-Carlo methods, although they can been fastened using variance reduction techniques(see[24] fordetailsonimportancesamplingtechniquesadaptedtoCDOpricing), areusuallyquiteslowandotherapproacheshavebeendevelopedwithintheone-factorgaussiancopulamodel.4.2 LatentvariableviewpointandnumericalmethodsAnother way to think about the one-factor gaussian copula model is indeed to view it as a verypoorstructuralmodelwithonelatentvariable. InthatcaseweintroduceforeachunderlyingiavariableYidenedasYi= Y+1 iwhereY isastandardnormalvariablecommontoeverysingle-nameinthepool,andforthisreasoncalledthesystemicvariable, andwhereiisastandardnormal butidiosyncratic(seeabove).As inastructuralmodel`alaMerton,we assumethat {i t} = {Yi Ai(t)}whereAi(t) isathresholdcalibratedonthesurvivalprobabilityoftheunderlyingi:Ai(t) = N1(P(i ti))Thisisjustanotherwaytoseetheone-factorgaussiancopulamodelbuttherationalebehindthisviewpointisthatY playstheroleofacommonfactoraectingall defaulttimes. IfY islow(abadstate),thentheprobabilityofdefaultforallunderlyingsinthepoolisratherhigh.IfY ishigh(agoodstate),thentheprobabilityofdefaultisratherlow,ceterisparibus.Thispointofviewalsoallowstogofurtherinthedeterminationofthedistributionoftheloss19process L and eventually to semi-analytical pricing approaches that are faster than Monte-Carlosimulations.Wenoticeindeedthat, conditionallyonthecommonfactorY , thedefaulttimesareindepen-dent. This allows, assuming that the recovery rates are not random, to write the characteristicfunctionofL(t):L(t)(u) = E_eiuL(t)= E_E_eiuL(t)|YE_Mk=11 P(Yk Ak(t)|Y ) +P(Yk Ak(t)|Y )eiu(1Rk)M_but P(Yk Ak(t)|Y ) = N_Ak(t)Y1_Hence:L(t)(u) =+Mk=1_1 N_Ak(t) y1 _+N_Ak(t) y1 _eiu(1Rk)M_n(y)dywherenistheprobabilitydistributionfunctionofastandardnormal.Then,usingFouriertransforms(ormoreexactlyFourierinversiontechniques)onecandeducethedistributionofthelossprocessandcomputethespreadaccordinglysee[96].OthermethodshavebeendevelopedandtheinterestedreadermayreadAndersenetal. [8]orHullandWhite[82]forrecursiveandprobabilitybucketingnumericalmethodsthatareoftenused and faster than Fourier inversion techniques. New methods, and especially approximationtechniques, have also been developed more recently, by Yang et al. [144] that propose a saddle-point approximation, byGlassermanandSuchintabandid[70] that provide approximationsbestsuitedtosmall correlationcasesandbyJacksonetal. [87] whocomparethemethodsdevelopedbyAndersenetal.,HullandWhite,arelatedrecursivemethodandtwointerestingapproximations (anapproximationof L(t)(u) as thecharacteristicfunctionof acompoundPoisson process and a Normal-Power approximation). The reader may also see [51] or [119] forothernumericalrecipes.Thesemethodsareoftennotlimitedtogaussiancopulasandcanbeappliedtoothercopula20models. However, theyarenotall developedinaverygeneral frameworkandsomeof themneedtobeadaptedtobeusedinthecaseofnon-homogenousportfolios.Homogenous portfolios, i.e. CDOs whose underlying CDSs have identical probability of defaultandidentical recovery rates,play indeeda very important role inCDO pricing because closed-formexpressions canbeobtainedfor thedistributionof L(t) inthis casewhenMis largeenoughthroughwhatiscalledtheLHPmodel(LargeHomogenousPortfolio).4.3 TheLHPmodelTheLHPmodel canbeseenhasamodel toapproximateCDOsbasedonalargeandratherhomogenouspoolofCDSs(orcredits). WesupposeindeedintheLHPmodelthatthedefaulttimesifollowthesamedistributionandthattherecoveryratesarethesameforall names.Also, wesupposethat M, thenumber of underlyingCDSs, is sucient largetouse, as anapproximation,theasymptoticlimitM +.Withinthissimpliedframework,evaluatingthedistributionofthelossprocessisdoneby:P(L(t) l) = E[P(L(t) l|Y )] M+ P_(1 R)N_A(t) Y1 _ l_whereA(t)=N1(P( t))doesnotdependonaspecicunderlyingCDSinthepoolsincethedefaulttimesareassumedtohavethesamedistribution.Hencewecanapproximate P(L(t) l)by:1 N_1_A(t) 1 N1_l1 R___Fromthisapproximation,onecancomputeinclosed-form(seeforinstance[119])termsoftheform E[min(L(t), K)] for any constant Kand these terms are the terms needed to compute thetranchelossprocessL(t, A, B)andeventuallythespreadoftheCDO.If the LHP model can be used as an approximation for not too heterogenous portfolios, its maininterest, as exemplied by [62] or [113] is linked to the computation of sensitivities. In the LHPmodel,thesensitivityofthetwolegswithrespecttothecorrelationparametercanindeedbecomputedeasily,the samebeingtrueforacommonchange indefaultriskorinrecovery rates.21SlightmodicationsoftheLHPmodelcanalsobeusedtounderstandtheinuenceofanid-iosyncraticshocktooneoftheunderlyingCDSs(eitherontheassociateddefaultprobabilityorontheassociatedrecoveryrate). Forinstance, onecanconsidertheLH+modelpresentedin[72] thatconsidersasetofunderlyingCDSsmadeofalargepool ofhomogenousCDSsinaddition to a single CDS with dierent characteristics, and assigns respective weightsM1Mand1MwhileapplyinganLHP-likeapproximationforthepoolofhomogenousCDSs.All these sensitivity computations are of utmost important when dealing with risk managementofcreditderivativeportfoliosandhedging. Theseriskmanagementandassociatedhedgingis-sueswillbediscussedinamoregeneralcontextinPartIV.To conclude on the LHP model, let us highlight the fact that it has been presented here withinthegaussiancopulaframeworkbutitcaneasilybegeneralizedtootherlatentvariablemodels(seenextpart,[129]and[133]).5 ImpliedcorrelationAt rst, one may see the one-factor gaussian copula model as an ill-adapted framework becauseit reduces a complex dependence structure between dozens of names to one unique gure whichis the correlation coecient . In fact, a parallel is often drawn with the Black-Scholes approachto, at least partially, explain the predominance of the one-factor gaussian copula model todaywith modications that will be discussed below. This parallel which is highly questionable andresponsibleof errorsthatwill bediscussedinPartIV, isattheverycenterof thereferenceapproachtocreditderivativepricing.As for options, we are going to imply parameters from market prices. In the case of call or putoptions, volatilityisimpliedfromoptionprices. Herethecorrelationcoecientisgoingtobe implied from the price of CDO tranches (the idiosyncratic default parameters being alreadyimplied from the CDS spreads), assuming in the rst reference models a constant recovery rate.As for call andput options, asmileis goingtoappear but theanalogystops herebecausemezzanine tranches of a CDO are like call-spread options and not much like call or put options.225.1 CompoundcorrelationTherstapproachthatwasadoptedtocalibratecorrelationincreditderivativespricingwastoconsidereachavailabletranche. Twomainindiceswereavailabletocalibrateupon7: theCDXInvestmentGradeNorthAmericaindex(hereafterCDX)andtheiTraxxEuropeindex(hereafter iTraxx), each of them oering ve tranches (we ignore in this part the maturity issueandconsidersyntheticCDOtrancheswithagivenhorizon,say5years.):FortheCDXindex: ajuniortranche(0%3%),ajuniormezzaninetranche(3%7%),a senior mezzanine tranche (7%10%), a senior tranche (10%15%) and a super seniortranche(15%30%).For the iTraxx index: a junior tranche (0%3%), a junior mezzanine tranche (3%6%),aseniormezzaninetranche(6% 9%), aseniortranche(9% 12%)andasuperseniortranche(12%22%).Thebasicideathatwasintroducedwiththeone-factorgaussiancopulaframeworkistoimplyfromeachtrancheacorrelationwhichis nowreferredtoas compoundcorrelation. Infact,if onewants tocalibrateagaussiancopulaoneachtranche, theonlychoiceis toinfer oneunique parameter per tranche and the one-factor gaussian copula appears much more as a needto mimic the Black-Scholes approach than as a good modeling choice as it will be discussed later.More exactly, the calibration methodology consists in computing the risk-neutral present valueof a long (or short) position in the tranche (A, B) using the quoted spread (and upfront paymentwhenrelevant)PV (A, B) = E_T0D(0, t)dL(t, A, B)_U(A, B) S(A, B)E_Nj=1(Tj Tj1)D(0, Tj)(1 L(Tj, A, B))_and to equate this present value to 0 while moving the correlation parameter of the one-factorgaussiancopulamodelingthedefaults. Thisgives, whenpossible, foreachtranche(A, B)acorrelationparameter(A, B)andeventuallyacompoundcorrelationcurveasonFigure1.7Weareonlypresentingthemostliquidindices.23 0 5 10 15 20 25 30 051015202530implied compound correlationdetachment pointFigure1: ExampleofcompoundcorrelationimpliedfromtheCDXtranches. Eachcompoundcorrelationgurehasbeenarbitrarilyassociatedtothedetachmentpointofitstranche.As for option prices we speak of a correlation smile since the implied correlation is not the sameforalltranchesandtheresultingcurvehappenedtobeU-shapedmostofthetime. However,contrarytotheoptionpricingcase,therearemanyimportantissuesrelatedtothisapproach.First, there is apriori noreasonwhythere wouldbe acorrelationparameter suchthatPV (A, B) = 0. Second,even if there exists an implied correlation , there is a priori no reasonwhywouldbeunique. Asarguedin[138],nocompoundcorrelationcansometimesbefoundfor some tranches and this was not a rare event,even before the crisis. Also,and especially formezzanine tranches, there are many cases in which two implied correlation choices are possible,makingthecalibrationcomplicatedandofteninaccurate.Inadditiontotheaboveissues that disqualifythecompoundcorrelationapproach, andal-though compound correlations are consistent with individual tranches, this approach obviouslyfailstoconservetheexpectedtranchelossasarguedin[119, 120]. Finally, sincewehaveonegureforeachtranche, itisdiculttousecompoundcorrelationtopricebespoketranches,whichisoneoftheverypracticalproblemofCDOpricing.All thesedrawbacksledtoalessintuitivebut better notionof impliedcorrelationwhichisreferredtoasbasecorrelation.245.2 BasecorrelationandthecorrelationskewBasecorrelationwasintroducedin2004in[110]inordertoprovideasolutiontotheproblemsthat arose with the use of compound correlation. Instead of mimicking the usual Black-Scholesapproachandimplyingonecorrelationpertranche, thebasecorrelationconsistsinimplyingfromthemarketpricesacorrelationgureforeachdetachmentpoint. Thisidea,thatwillbemadeclearbelow, isbasedonaremarkmadeaboveforthetranchelossprocess. Weindeednoticedthat:L(t, A, B) =min(L(t), B) min(L(t), A)B A=BL(t, 0, B) AL(t, 0, A)B AHence,onecandecomposeanymezzaninetrancheusingtwoequity-liketranches(thatarenotnecessarily traded) and hence overcome the calibration diculties linked to the call-spread na-tureofmezzaninetranches.Wewill usethisformulatoimplycorrelationrecursively. If thetranchesatstakeare(A1=0, B1),(B1= A2, B2),...,(An= An1, Bn= 1),wecanrstcomputetherisk-neutral expectedpresent value of a long position in the tranche (A1, B1), assuming independence between interestratesanddefaultrisk:PV (A1, B1) =T0P(0, t)dE[L(t, A1, B1)]U(A1, B1) S(A1, B1)Nj=1(Tj Tj1)P(0, Tj) (1 E[L(Tj, A1, B1)])whereP(0, t)standsforthepriceofazero-couponbondwithmaturityt.Equatingthis present value to0, one obtainthe rst base correlation(B1), equal tothecompoundcorrelationofthejuniortranche.Now, coming to the tranche (A2, B2), we do not apply one unique copula to compute the presentvaluebutweratherwrite:PV (A2, B2) =T0P(0, t)dE[L(t, A2, B2)]25U(A2, B2) S(A2, B2)Nj=1(Tj Tj1)P(0, Tj) (1 E[L(Tj, A2, B2)])with E[L(Tj, A2, B2)]anabuseofnotationfor:B2E[L(t, 0, B2)] A2E(B1)[L(t, 0, A2)]B2 A2where E[]standsforanexpectedvalueunderaone-factorgaussiancopulamodelwithcorre-lationforthedefaulttimes.Then, welookforavalueofsuchthatPV (A2, B2)=0, (B1)beingalreadydeterminedatthepreviousstep.In concrete terms it means that we compute two loss processes: the rst one with a correlationcoecient(B1)thatisxedbytheprecedingcalibration, andasecondlossprocesswithacorrelationcoecientthatneedstobecalibratedonmarketprices.Recursively, usingthesameprocess, wethenobtainanimpliedbasecorrelationforeachde-tachmentpoint(seeFigure2). 0 10 20 30 40 50 60 70 80 05101520253035implied base correlationdetachment pointFigure2: ExampleofbasecorrelationimpliedfromtheCDXtranchesThebasecorrelationapproachsolvesthemainissuesraisedbythecompoundcorrelationap-proach. Becauseweareeventuallyttingthecorrelationonjunior-liketranches, webenet26fromthemonotonicrelationbetweencorrelationandjunior expectedtranchepresent value.Hence, uniquenessof impliedbasecorrelationisensured. Also, basecorrelationconservetheexpectedloss of theentireindex. Finally, becauseit oftenexhibits asmoother prole(seeFigure 2. and [109, 120, 138]), the base correlation framework facilitates the pricing of bespoketrancheseitherthroughdirectinterpolationorbyinterpolationoftheexpectedtranchelosses(see[119]). Thislastpointisofutmostinpractice.Basecorrelationhashoweveritsowndrawbacks. First,basecorrelationcurvesoftenexhibitaskew(seeFigure2), whichmeansthattheone-factorgaussiancopulamodel isnottherightone. Also, and we can refer to [18] for a discussion on that topic, the expected loss of a trancheas calculated byBE(B)[L(t,0,B)]AE(A)[L(t,0,A)]BAcan be negative due to the dierence between (A)and(B), violatingtheno-arbitragehypothesis. Thisisamajorissueofthebasecorrelationapproachbutithasremainedastandardinspiteof thispoint. Eventually, acorrelationpa-rameterwassometimesimpossibletondunderaconstant40%recoveryrateassumption,asitwasmostof thetimethecaseinmodelsandimplicitlysupposedabove, inspiteof beingarbitraryandunrealistic.If thebasecorrelationapproachisconsideredastandard, thepresenceof acorrelationskewraisesthequestionofabettermodel. Manycopulaapproacheshaveindeedbeenproposedtoimprovethemodel and/orattentheskew. Wewill reviewtheseothercopulamodelsinthefollowingpartbuttonishontheusualapproach,letusfocusonthelastpointevokedabove:thequestionofrecoveryrates.6 StochasticrecoveryratesUntilnow,wedidnotdiscussthequestionofrecoveryrates. Recoveryrateswereindeedsup-posedtobeconstant intheprecedingapproachesbut thishypothesisisquitequestionable.Firsttheinuenceoftherecoveryrateonpricingisimportantandassumingaconstantuni-formrecoveryrateof40%,asitwasthenorm,isnotadmissible.Second,thechoiceofthe40%gureturnedouttobeincompatiblewiththespreadquotedbythemostseniortranches: evenwith100%correlationbetweendefaults, themaximumspreadobtainedwithanassumedrecoveryrateof40%waslessthantheactualspreadquotedforthe27tranche(15%-30%)oftheCDXindexattheinceptionofthesubprimecrisis. Evenworse,thesimplefactthatsuperseniortranches(60%-100%)startedtoquoteisatoddswiththe40%recoveryrateassumption.Third, there is empirical evidence that aggregate recovery rates are linked to default probabili-ties(seeforinstance[3]forarathercompletestudy). Inparticular,whenthedefaultratesarehigh, one should expect to have defaults with smaller recovery rates. Subsequently, introducinganegativedependencebetweendefaultprobabilitiesandrecoveryrateswillincreasetheprob-abilityofhighlossesandhenceimprovethegaussiancopulamodel.Severalstochasticrecoverymodelshavebeenproposedandwepresentthemostcommonones(seealso[100]foradiscussiononthesemodels).TherstpapertointroducestochasticrecoveryratesinthegaussiancopulamodelisapaperbyAndersenandSidenius [7]. Theauthors proposedtousethesystemicfactor Y inYi=Y+1 itowritetherecoveryrateassociatedtotheithCDSas:Ri= C(ai +biY+cii)whereai,bi> 0,ci> 0areconstant,whereCisanincreasingfunction,andwhereiisanewidiosyncraticnoise,independentoftheothersourcesofrandomness.Withthismodel,inadditiontothestochasticnatureoftherecoveryrate,werecovertheneg-ativedependencebetweendefaultratesandrecoveryrates.Inaslightlydierentspirit,Krekel[92]proposedtousethevariableYiitselftodeterminetheleveloftherecoveryrate. HebasicallyproposedarecoveryratethatisafunctionofYiforYilessthanthedefaultthresholdAi(t). IfYifallsbelowAi0(t) = Ai(t)andwithin[Ai1(t), Ai0(t)]thentheunderlyingi defaultswithagivenrecoveryR1, if YifallsbelowAi0(t)=Ai(t)andwithin[Ai2(t), Ai1(t)]thentheunderlyingidefaultswithagivenrecoveryR2,andsoon. Thisdiscreteframeworkfortherecoveryratecanbegeneralizedtoacontinuousonealthough[92]focusesonthediscretecase.28More recently, Amraoui and Hitier [6] proposed a recovery rate that is a deterministic functionof thesystematic factorYandhence appears as aspecialcaseof the rstsetting we presentedabovewithci= 0. Thisapproachisthemostusedinpractice.Other models include [48]or [124]and they have been somehow made necessary by the impos-sibility to calibrate correlation on senior tranches during the subprime crisis. They also slightlyattenthebasecorrelationskewbutdonotprovideanacceptablesolutiontotheskewissue.29PartIIINon-gaussiancopulamodelsCopulas were introduced in credit derivatives pricing in order to model the dependence betweendefault times. The gaussian copula model and more specically the one-factor gaussian copulamodel becamestandardamongpractitioners andpopularizedthecopulaapproach, limitingthecopulaapproachtoverysimpledependencestructures. Theadvantageshowever, of suchamodelchoicewereevidentfromitsintroduction: tractability,fastnumericalmethodsandarepresentationeasytocommunicateon,throughanequivalentlatentvariablemodel.Othercopulamodelshavebeendevelopedintheliterature, aimingatreplacingthegaussiancopulaone. Mostof themtrytobettertakeaccountof theimportantprobabilityof alargenumber of defaults, thus correctingoneof themaindrawbacks of thegaussiancopulathatisknowntohavenotail dependence. However, mostauthors, toallowcalibrationtomarketprices, useparsimoniouscopulamodelswithonlyafewparameters. Also, itisoftenbetterfortractabilitypurposestouselatentvariablemodelsinwhich, conditionallyonasystemicvariable,thedefaultsareindependent.Wewillreviewthecopulamodelsproposedduringthelastdecadeandstartwithadiscussiononthosethataimedatatteningthecorrelationskew.7 Towardagoodskewmodel?7.1 StochasticcorrelationAsforBlack-Scholesmodelwhoserstimportantextensionistoconsiderastochasticvolatil-ity, werstintroducestochasticcorrelationmodels. Inaverygeneral framework, stochasticcorrelationmodelsgeneralizeone-factorgaussiancopulamodelsbyconsidering:Yi= iY+1 iiwhere iis a random variable taking values in [0, 1] and where the random variables 1, . . . , Mand Yare independent standard normal variables, independent of the random variables 1, . . . , M.30Inotherwords,stochasticcorrelationmodelsintroduceacopulathatisamixtureofgaussiancopulas.Withinthisframework, thereisnothingtochangetotheidiosyncraticcalibrationproceduresinceYiremainsstandardnormal. WhatchangesisthedependencestructurebetweentheYisand hence between the default times. They can nonetheless be described easily using as for thegaussian model, conditional expectations. If indeed we consider that i= is a unique randomvariablewithprobabilitydistributionfunctionh,thenwehave:P(Yk Ak(t)|Y ) =10N_Ak(t) Y1 _h()dandthecharacteristicfunctionL(t)(u) = E_eiuL(t)canbecomputedasinthegaussiancase.Regardingnowtheinterestof thesestochasticcorrelationmodels, itisapriori limitedbothinpracticeandtheoretically, sincethereisnotail dependenceaslongas=1isaectednopositive mass in distribution h. However, a comparative study by Burtschell et al. [20] indicatesthatthettomarketpriceofasimplemixtureofgaussiancopulaisrathergoodandapaperonCDO2[99]indicatesthesamendingsalthoughnosystematictesthavebeencarriedout.Other examples of stochastic correlationmodels are developedin[19] intwospecial cases.Therstoneproposesadiscretedistributionforwithtwovalues1and2andthesecondoneproposestospecify2=1. Inotherwords, itassignsapositiveprobabilitytothetherather extreme fact that the dependence structure of the default times is characterizedbysimultaneous defaults. Related ideas have been developed in other frameworks (see the sectiononMarshall-OlkincopulaandontheCompositeBasketModelofTaveresetal. [137]).7.2 LocalcorrelationandrandomfactorloadingsAsimilarapproachhasbeenintroducedtoenrichtheone-factorgaussiancopulamodel. ThisapproachconsistsinmakingthecorrelationparameterafunctionofthesystemicvariableY .Thisapproachintroducedin[140]simplyconsistsinwriting:Yi=(Y )Y+1 (Y )i31where()isafunctionthattakesvaluesin[0, 1].Then, tomodel theskew, oneneedtorelativelyincreasethecorrelationinbadstates. Thisideahas beenmorepreciselydevelopedwithinaslightlydierent frameworkintroducedbyAndersenandSidenius[7]andcalledtherandomfactorloadingmodel.Inthismodel8:Yi= ai(Y )Y+ii +miwhereY andtheissatisfytheusual hypothesesandwhereiandmiaresuchthatYiisofmean0andvariance1.In these model, Yi is no longer a standard normal and one has to calibrate Ai(t) on CDS survivalcurvesbyaninversionofthefollowingrelationship:P(i t) =+N_Ai(t) ai(y)y mii_n(y)dyNow, thechoiceofthefunctionai()isofutmostimportantand, asalreadyexpressedabove,itshouldbedecreasing. If indeedoneconsidersaseniortranche, heispreoccupiedbylargenumber of defaults and hence low values for Y . Then, when y is low, to reinforce the probabilityofnumerousdefaults, onehastochoosehighvaluesforai(y). Butthen, valuesofai(y)whenyislargeshouldbesmallertocompensate. Inparticular, [7]presentsthecaseofatwo-pointloadingsdistribution, i.e. ai(y) =1y t) = exp((i +s)t)andP(1> t1, . . . , M> tM) = exp(Mi=1iti s max(t1, . . . , tM))ThiscanbewrittenusingtheMarshall-Olkincopula:C(u1, . . . , uM) = u1. . . uMu11. . . uMMmax(u11, . . . , uMM), i=ss +iP(1> t1, . . . , M> tM) = C(P(1> t1), . . . , P(M> tM))This model is a bit extreme and outcomes in terms of spread are at odds with markets (see [21]).Obviously, dierentassumptionscanbemadeandsystemicshockscanindeedbeconsideredthat trigger defaults of some specic underlyings belongingtoagivensector for instance.Animprovement also consists inthe Composite Basket model proposedin[137]: defaultscanhappendue toidiosyncratic reasons, for systemic reasons or because the assets of theunderlyings reached a threshold, exactly as in the above factor models. This model can be seenas mixing the Marshall-Olkin copula model and any of the above factor model described in theabovesection(theone-factorgaussiancopulamodelbeingthereferencecase). TheadvantageofthisCompositeBasket model(withone-factorgaussiancopula)isthatitrequires onlylittleadditional computation compared to the original one-factor copula and indeed atten the skew(see[119]).9 TheimpliedcopulamodelWe have seen above the use of various copulas, either to improve the one-factor gaussian copulamodelortobuildtotallydierentmodels. Thedefaulttimesdependencewasalwaysmodeled39usingaspeciedformof copula, chosenapriori for its properties. Thelast static9copulamethoddoesnotobeythesamerulesincewearegoingtoimplythecopulafromthemarketspreads10.This implied copula approach has been introduced by Hull and White in [84]. The idea behindthisimpliedcopulaapproachcomesfromthemethodologyusedwhendealingwithone-factormodelsasabove. Foranhomogenousportfolio(wepresenttheimpliedcopulaframeworkinthiscaseasintheoriginal paper, althoughtheauthorsproposedmethodstogeneralize), thedefaultswereindependentconditionallyonthesystemicfactorandtheactual probabilityofdefaultwasafunctionofthesystemicfactorvalue(andeventuallyofitsdistribution). Hence,choosingaone-factorcopulamodelissimilartospecifyingasetofhazardratescenarios. Theimpliedcopulamethodologythenconsistsinimplyingfromthemarketasetof hazardratescenarios, that is, givenaset of hazardrates 1, . . . , K, theset of associatedprobabilitiesp = (p1, . . . , pK)thatbesttsthemarketdata.Inconcreteterms, it means that onceaset of hazardrate1