MODELLING DYNAMIC PORTFOLIO CREDIT RISK

MODELLINGDYNAMICPORTFOLIOCREDITRISKEBBEROGGEANDPHILIPPJ.SCHÄONBUCHERDepartmentofMathematics,ImperialCollegeandABNAMROBank,LondonandDepartmentofMathematics,ETHZurich,ZurichApril2002,thisversionFebruary2003Abstract.Inthispaperwepresentamodeltopriceandhedgebasketcreditderivativesandcollateralisedloanobligation.Baseduponthecopula-approachbySchÄonbucherandSchubert(2001)themodelallowsaspeci¯cationofthejointdynamicsofcreditspreadsanddefaultintensities,includingaspeci¯cationoftheinfectiondynamicswhichcausecreditspreadstowidenatdefaultsofotherobligors.Becauseofahighdegreeofanalyticaltractability,jointdefaultandsurvivalprobabilitiesandalsosensitivitiescanbegiveninclosed-formwhichfacilitatesthedevelopmentofhedgingstrategiesbaseduponthemodel.ThemodelusesageneralisationoftheclassofArchimedeancopulafunctionswhichgivesrisetomorerealisticcreditspreaddynamicsthantheGaussiancopulaortheStudent-t-copulawhichareusuallychoseninpractice.Anexamplespeci¯cationusingGamma-distributedfactorsisprovided.1.IntroductionWhilethearrivalofacertainnumberofdefaultsoveragiventimeperiodistobeexpectedduringthenormalcourseofbusiness,majorrisksarisewheneitherthenumberofdefaultsexceedsexpectationsor{evenifthetotalnumberofdefaultsremainslargelyuna®ected{whenthetimingofthedefaultsissuchthatseveraldefaultsoccurcloselyaftereachother.Inordertomanagethisriskanumberofnew¯nancialinstrumentshavebeenintroduced(basketcreditderivativesandcollateraliseddebtobligations)whichareexplicitlydesignedtotradeandmanagetherisksofdefaultdependencies.JELClassi¯cation.G13.Keywordsandphrases.PortfolioCreditRiskModels,Copulafunctions,CreditDerivatives,First-to-DefaultSwap,AssetPricing,RiskManagement.TheauthorsthankDarrellDu±e,MarkDavisandMarkdeVriesforstimulatingdiscussions.ResultsofthispaperwerepresentedattheJourn¶eeRisquedeCreditinEvry,France,February2003;andtheStochasticAnalysisinFinanceandInsuranceMeetinginOberwolfach,Germany,March2003.MuchoftheworkforthispaperwasdonewhileP.SchÄonbucherwasattheDepartmentofStatisticsatBonnUniversity,¯nancialsupportbytheDFGduringthatperiodisgratefullyacknowledgedbyPhilippSchÄonbucher.AtETH,hethanksforFinancialsupportbytheNationalCentreofCompetenceinResearch\FinancialValuationandRiskManagement(NCCRFINRISK),Project5:CreditRisk.TheNCCRFINRISKisaresearchprogramsupportedbytheSwissNationalScienceFoundation.Theviewsexpressedinthispaperaretheauthors'ownanddonotnecessarilyre°ectthoseofABNAMROBank.Allerrorsareourown.Commentsandsuggestionsarewelcome.12EBBEROGGEANDPHILIPPJ.SCHÄONBUCHERInthispaperwepresentamodeltopriceandhedgethesenewinstruments.Baseduponthecopula-approachthemodelallowsaspeci¯cationofthejointdynamicsofcreditspreadsanddefaultintensities,includingaspeci¯cationoftheinfectiondynamicswhichcausecreditspreadstowidenatdefaultsofotherobligors.Becauseofahighdegreeofanalyticaltractability,jointdefaultandsurvivalprobabilitiesandalsosensitivitiescanbegiveninclosed-formwhichfacili-tatesthedevelopmentofhedgingstrategiesbaseduponthemodel.ThemodelisbaseduponageneralisationoftheclassofArchimedeancopulafunctionswhichgivesrisetomuchmorereal-isticdynamicsofthemodelvariablesthantheGaussiancopulaortheStudent-t-copulawhichareusuallychoseninpractice.Defaultcorrelationand(moregenerally)defaultdependencyareatopicofhighinterestinthebankingandinvestmentcommunity.Thisinterestisfurtherincreasedbyotherdevelopments:First,theupcomingBaselIIcapitalaccordallowsinternallydevelopedcreditriskmodelstobeusedforregulatorycapitalallocationpurposes.Butalsointernally,theparadigmofthehandlingofcreditriskinmodernbankinghaschangedsigni¯cantly.Whileonlyafewyearsagotheonlypossibilitytomanagethecreditriskofalargebankwasbymanagingtheoriginationprocess(i.e.theacception/rejectionofnewbusiness),nowcreditriskscanbemanageddirectlybytheuseofcreditderivativesandsecuritisationwithloansandbondsascollateralassets:collateralisedloanobligations(CLOs)collateralisedbondobligations(CBOs)ormoregenerally,collateraliseddebtobligations(CDOs).Inshort,creditriskmanagementhasevolvedfromapassivemeasurementandmonitoringfunctionintotheactivemanagementofthecreditriskexposureofabankwhichusesthenewpossibilitiestobuyandsellexposuresinordertooptimizetherisk-returnpro¯leofthecreditbook.Giventheadvantagesofactivecreditportfoliomanagementitisnotsurprisingthatthemarketfortheinstrumentswhichmakecreditriskstradeable,themarketforcreditderivatives,isinfullstrideandstillgrowingstrongly.AccordingtothelatestsurveybyRiskmagazine(Patel(2003)),thevolumeofthecreditderivativesmarkethasdoubledagainin2002reachinganoutstandingnotionalofmorethan2.3trillionUSDinFebruary2003.Thedevelopmenttowardsactivetradingofcreditriskshasseveralconsequences:Withgrowingliquidityofsingle-namecreditdefaultswaps(CDS),areliablemarking-to-marketofindividualcreditrisksbecomespossible.Thismeans,thatthemarketriskofacreditportfolioisnowmeasurable,andshouldthereforebemanaged{itcannotbeignoredanymore.InsofarascreditspreadsandCDS-spreadscontainthemarket'sopiniononthedefaultriskoftheobligorinquestion,theyprovideanewobjective,market-basedearly-warninginstrumentforchan