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CREDITRISKMODELINGMATH587:CourseNotesTomaszR.BieleckiDepartmentofAppliedMathematicsIllinoisInstituteofTechnologyChicago,IL60616,USAMoniqueJeanblancD¶epartementdeMath¶ematiquesUniversit¶ed'EvryVald'Essonne¶91025EvryCedex,France¶MarekRutkowskiSchoolofMathematicsandStatisticsUniversityofNewSouthWalesSydney,NSW2052,AustraliaAppliedMathematicsDepartmentIllinoisInstituteofTechnologyFall2008 2 Contents1StructuralApproach91.1BasicAssumptions.....................................91.1.1DefaultableClaims.................................91.1.2Risk-NeutralValuationFormula.........................101.1.3DefaultableZero-CouponBond..........................111.2Merton'sModelofaCorporateDebt...........................121.3PropertiesofFirstPassageTimes.............................141.3.1ProbabilityLawoftheFirstPassageTime...................151.3.2JointProbabilityLawofYand¿.........................171.4BlackandCoxModel...................................211.4.1BlackandCoxFormula..............................221.4.2CorporateCouponBond..............................261.4.3OptimalCapitalStructure.............................281.5ExtensionsoftheBlackandCoxModel.........................291.5.1StochasticInterestRates.............................301.6RandomBarrier......................................311.6.1IndependentBarrier................................322HazardFunctionApproach332.1ElementaryMarketModel.................................332.1.1HazardFunction..................................332.1.2DefaultableZero-CouponBondwithRecoveryatMaturity..........342.1.3DefaultableZero-CouponwithRecoveryatDefault...............372.2MartingaleApproach....................................382.2.1ConditionalExpectations.............................392.2.2MartingalesAssociatedwithDefaultTime....................392.2.3PredictableRepresentationTheorem.......................432.2.4Girsanov'sTheorem................................452.2.5RangeofArbitragePrices.............................472.2.6ImpliedRisk-NeutralDefaultIntensity......................482.2.7DynamicsofPricesofDefaultableClaims....................492.3GenericDefaultableClaims................................503 4CONTENTS2.3.1Buy-and-HoldStrategy..............................512.3.2SpotMartingaleMeasure.............................532.3.3Self-FinancingTradingStrategies.........................542.3.4MartingalePropertiesofPricesofDefaultableClaims.............562.4HedgingofSingleNameCreditDerivatives.......................562.4.1StylizedCreditDefaultSwap...........................562.4.2PricingofaCDS..................................572.4.3MarketCDSRate.................................582.4.4PriceDynamicsofaCDS.............................592.4.5DynamicReplicationofaDefaultableClaim..................602.5DynamicHedgingofBasketCreditDerivatives.....................622.5.1First-to-DefaultIntensities............................632.5.2First-to-DefaultMartingaleRepresentationTheorem..............642.5.3PriceDynamicsoftheithCDS..........................662.5.4Risk-NeutralValuationofaFirst-to-DefaultClaim...............682.5.5DynamicReplicationofaFirst-to-DefaultClaim................692.5.6ConditionalDefaultDistributions.........................712.5.7RecursiveValuationofaBasketClaim......................722.5.8RecursiveReplicationofaBasketClaim.....................752.6ApplicationstoCopula-BasedCreditRiskModels...................752.6.1IndependentDefaultTimes............................762.6.2ArchimedeanCopulas...............................773HazardProcessApproach813.1HazardProcessanditsApplications...........................813.1.1ConditionalExpectations.............................823.1.2StochasticIntensity................................843.1.3ValuationofDefaultableClaims.........................843.1.4MartingalesAssociatedwithDefaultTime....................863.1.5ReductionoftheReferenceFiltration......................883.1.6EnlargementofFiltration.............................903.2Hypothesis(H).......................................903.2.1EquivalentFormulations..............................903.2.2CanonicalConstructionofaDefaultTime....................923.2.3StochasticBarrier.................................923.3PredictableRepresentationTheorem...........................933.4Girsanov'sTheorem....................................943.5InvarianceofHypothesis(H)...............................963.5.1CaseoftheBrownianFiltration..........................973.5.2ExtensiontoOrthogonalMartingales.......................983.6AlternativeApproach...................................100 CONTENTS53.7Single-NameCreditDefaultSwapMarket........................1013.7.1PriceDynamicsinaSingle-NameModel.....................1013.7.2ReplicationofaDefaultableClaim........................1093.8Multi-NameCreditDefaultSwapMarket........................1163.8.1PriceDynamicsinaMulti-NameModel.....................1163.8.2ReplicationofaFirst-to-DefaultClaim......................1214HedgingofDefaultableClaims1234.1SemimartingaleModelwithaCommonDefault.....................1234.1.1DynamicsofAssetPrices.............................1234.2TradingStrategiesinaSemimartingaleSet-up.....................1264.2.1UnconstrainedStrategies.............................1264.2.2ConstrainedStrategies...............................1284.3MartingaleApproachtoValuationandHedging.....................1324.3.1DefaultableAssetwithTotalDefault.......................1324.3.2DefaultableAssetwithNon-ZeroRecovery...................1444.3.3TwoDefaultableAssetswithTotalDefault...................1454.4PDEApproachtoValuationandHedging........................1484.4.1DefaultableAssetwithTotalDefault.......................1484.4.2DefaultableAssetwithNon-ZeroRecovery...................1524.4.3TwoDefaultableAssetswithTotalDefault...................1555DependentDefaultsandCreditMigrations1575.1BasketCreditDerivatives.................................1585.1.1Theith-to-DefaultContingentClaims......................1585.1.2CaseofTwoEntities................................1585.1.3RoleoftheHypothesis(H)............................1595.2ConditionallyIndependentDefaults............................1595.2.1CanonicalConstruction..............................1605.2.2IndependentDefaultTimes............................1605.2.3SignedIntensities..................................1615.2.4ValuationofFTDCandLTDC..........................1625.3Copula-BasedApproaches.................................1635.3.1DirectApproach..................................1635.3.2IndirectApproach.................................1635.4JarrowandYuModel...................................1655.4.1ConstructionandPropertiesoftheModel....................1655.5ExtensionoftheJarrowandYuModel..........................1695.5.1Kusuoka'sConstruction..............................1695.5.2BondValuation...................................1705.6MarkovianModelsofCreditMigrations.........................171 6CONTENTS5.6.1In¯nitesimalGenerator..............................1715.6.2Speci¯cationofCreditRatingsTransitionIntensities..............1735.6.3ConditionallyIndependentMigrations......................1745.6.4ExamplesofMarkovMarketModels.......................1745.6.5ForwardCDS....................................1765.6.6CreditDefaultSwaptions.............................1765.7BasketCreditDerivatives.................................1775.7.1kth-to-DefaultCDS.................................1785.7.2Forwardkth-to-DefaultCDS...........................1795.7.3ModelImplementation...............................1805.7.4StandardCreditBasketProducts.........................1835.7.5ValuationofStandardBasketCreditDerivatives................1866Appendix:ConditionalPoissonProcess1896.1PoissonProcesswithConstantIntensity.........................1896.2PoissonProcesswithDeterministicIntensity......................1946.3ConditionalPoissonProcess................................195 IntroductionThegoaloftheselecturenotesistopresentasurveyofsomerecentdevelopmentsintheareaofmathematicalmodelingofcreditrisk.TheyarelargelybasedonthefollowingpapersbyT.R.Bielecki,M.JeanblancandM.Rutkowski:²Modellingandvaluationofcreditrisk.In:StochasticMethodsinFinance,M.FrittelliandW.Runggaldier,eds.,Springer-Verlag,2004,27{126,²Hedgingofdefaultableclaims.In:Paris-PrincetonLecturesonMathematicalFinance2003,R.Carmonaetal.,eds.Springer-Verlag,2004,1{132,²PDEapproachtovaluationandhedgingofcreditderivatives.QuantitativeFinance5(2005),257{270,²Hedgingofcreditderivativesinmodelswithtotallyunexpecteddefault.In:StochasticPro-cessesandApplicationstoMathematicalFinance,J.Akahorietal.,eds.,WorldScienti¯c,Singapore,2006,35{100,²Hedgingofbasketcreditderivativesincreditdefaultswapmarket.JournalofCreditRisk3(2007),91{132.aswellasonsomesectionsfromthemonographbyT.R.BieleckiandM.Rutkowski:CreditRisk:Modelling,ValuationandHedging,Springer-Verlag,2002.Creditriskembeddedina¯nancialtransactionistheriskthatatleastoneofthepartiesinvolvedinthetransactionwillsu®era¯nanciallossduetodefaultordeclineinthecreditworthinessofthecounter-partytothetransaction,orperhapsofsomethirdparty.Forexample:²Aholderofacorporatebondbearsariskthatthemarketvalueofthebondwilldeclineduetodeclineincreditratingoftheissuer.²Abankmaysu®eralossifabank'sdebtordefaultsonpaymentoftheinterestdueand/ortheprincipalamountoftheloan.²Apartyinvolvedinatradeofacreditderivative,suchasacreditdefaultswap(CDS),maysu®eralossifareferencecrediteventoccurs.²Themarketvalueofindividualtranchesconstitutingacollateralizeddebtobligation(CDO)maydeclineasaresultofchangesinthecorrelationbetweenthedefaulttimesoftheunderlyingdefaultablesecurities(i.e.,thecollateralassets).Themostextensivelystudiedformofcreditriskisthedefaultrisk{thatis,theriskthatacounterpartyina¯nancialcontractwillnotful¯lacontractualcommitmenttomeether/hisobligationsstatedinthecontract.Forthisreason,themaintoolintheareaofcreditriskmodelingisajudiciousspeci¯cationoftherandomtimeofdefault.Alargepartofthepresenttextisdevotedtothisissue.Ourmaingoalistopresentthemostimportantmathematicaltoolsthatareusedforthearbitragevaluationofdefaultableclaims,whicharealsoknownunderthenameofcreditderivatives.Wealsoexaminetheimportantissueofhedgingtheseclaims.7 8CHAPTER0.INTRODUCTIONThesenotesareorganizedasfollows:²InChapter1,weprovideaconcisesummaryofthemaindevelopmentswithintheso-calledstructuralapproachtomodelingandvaluationofcreditrisk.Inparticular,wepresenttheclassicstructuralmodels,putforwardbyMerton[77]andBlackandCox[16],andwementionsomevariantsandextensionsofthesemodels.Wealsostudyverybrie°ythecaseofastructuralmodelwitharandomdefaulttriggeringbarrier.²Chapter2isdevotedtothestudyofasimplemodelofcreditriskwithinthehazardfunctionframework.Wefocushereonthederivationofpricingformulaefordefaultableclaimsandthedynamicsoftheirprices.Wealsodealherewiththeissueofreplicationofsingle-andmulti-namecreditderivativesinthestylizedCDSmarket.Resultsofthischaptershouldbeseenasa¯rststeptowardmorepracticalapproachesthatarepresentedintheforegoingchapters.²Chapter3dealswiththeso-calledreduced-formapproachinwhichthemainmodelingtoolisthehazardrateprocess.Thisapproachisofapurelyprobabilisticnatureand,technicallyspeaking,ithasalotincommonwiththereliabilitytheory.Weexaminepricingformulaeinareduced-formset-upwithstochastichazardrateandweexaminethebehaviourofthestochasticintensitywhenthereference¯ltrationisreduced.Inthesecondpartofthischapter,specialemphasisisputontheso-calledhypothesis(H)anditsinvariancewithrespecttoanequivalentchangeofaprobabilitymeasure.Asanapplicationofmathematicalresults,wepresenthereanextensionofhedgingresultsobtainedinChapter2inthecaseofdeterministicpre-defaultintensitiestothecaseofstochasticdefaultintensities.²Chapter4isdevotedtoastudyofhedgingstrategiesfordefaultableclaimsundertheassump-tionthatsomeprimarydefaultableassetsaretraded.We¯rstdiscusssomepreliminaryresultsonhedginginanabstractsemimartingaleset-up.Subsequently,wedevelopthePDEapproachtothevaluationandhedgingofdefaultableclaimsinaMarkovianframework.Forthesakeofsimplicityofpresentation,wefocusonthecaseofamarketmodelwiththreetradedprimaryassetsandwedealwithasingledefaulttimeonly.However,anextensionofthePDEmethodtothecaseofany¯nitenumberoftradedassetsandseveraldefaulttimesisreadilyavailable.²Chapter5providesanintroductiontotheareaofmodelingdependentdefaultsand,moregenerally,tomodelingofdependentcreditmigrationsforaportfolioofreferencenames.Wepresentsomeapplicationsofthesemodelstothevaluationofreal-lifeexamplesofcreditderiva-tives,suchas:suchas:creditdefaultswapsandswaptions,¯rst-to-defaultswaps,creditdefaultindexswapsandCDOs.²Intheappendix,wepresentsomebasicresultsconcerningthePoissonprocessanditsgener-alizations.LetusmentionthattheproofsofmostresultscanbefoundinBieleckiandRutkowski[12],Bieleckietal.[4,5,8]andJeanblancandRutkowski[58].Wequotesomeoftheseminalpapers;thereadercanalsorefertobooksbyBruyµere[26],Bluhmetal.[19],BieleckiandRutkowski[12],CossinandPirotte[33],Du±eandSingleton[44],Frey,McNeilandEmbrechts[50],Lando[65],orSchÄonbucher[84]formoreinformation.Itshouldbemadeclearthatseveralresults(especiallywithinthereduced-formapproach)wereobtainedindependentlybyvariousauthors,whoworkedunderdi®erentsetofassumptionsand/orwithindi®erentsetups.Forthisreason,wedecidedtoomitthedetailedcredentialsinmostcases.Wehopethatourcolleagueswillacceptourapologyforthisde¯ciency,andwestressthatthisbynomeanssigni¯esthatanyresultgiveninwhatfollowsthatisnotexplicitlyattributedisours.Acknowledgement.The¯naldraftoftheselecturenoteswascompletedduringthevisitofMarekRutkowskitotheCenterfortheStudyofFinanceandInsuranceattheOsakaUniversity.HewouldliketoexpresshisgratitudetoProfessorHideoNagaiforthekindinvitationandhisgreathospitality. Chapter1StructuralApproachInthischapter,wepresentaverybriefoverviewoftheso-calledstructuralapproachtomodelingcreditrisk.Sinceitisbasedonthemodelingofthebehaviorofthetotalvalueofthe¯rm'sassets,itisalsoknownasthevalue-of-the-¯rmapproach.Thismethodologyrefersdirectlytoeconomicfundamentals,suchasthecapitalstructureofacompany,inordertomodelcreditevents(adefaultevent,inparticular).Asweshallseeinwhatfollows,thetwomajordrivingconceptsinthestructuralmodelingare:thetotalvalueofthe¯rm'sassetsandthedefaulttriggeringbarrier.Itisworthnotingthatthiswashistoricallythe¯rstapproachusedinthisarea{itgoesbacktothefundamentalpapersbyBlackandScholes[17]andMerton[77].1.1BasicAssumptionsWe¯xa¯nitehorizondateT¤>0;andwesupposethattheunderlyingprobabilityspace(•;F;P);endowedwithsome(reference)¯ltrationF=(Ft)0·t·T¤;issu±cientlyrichtosupportthefollowingobjects:²Theshort-terminterestrateprocessr;andthusalsoadefault-freetermstructuremodel.²The¯rm'svalueprocessV;whichisinterpretedasamodelforthetotalvalueofthe¯rm'sassets.²Thebarrierprocessv;whichwillbeusedinthespeci¯cationofthedefaulttime¿.²ThepromisedcontingentclaimXrepresentingtheliabilitiestoberedeemedtotheholderofadefaultableclaimatmaturitydateT·T¤.²TheprocessA;whichmodelsthepromiseddividends,i.e.,theliabilitiesstreamthatisredeemedcontinuouslyordiscretelyovertimetotheholderofadefaultableclaim.²TherecoveryclaimXerepresentingtherecoverypayo®receivedattimeTifdefaultoccurspriortoorattheclaim'smaturitydateT.²TherecoveryprocessZ;whichspeci¯estherecoverypayo®attimeofdefaultifitoccurspriortooratthematuritydateT:1.1.1DefaultableClaimsTechnicalassumptions.WepostulatethattheprocessesV;Z;Aandvareprogressivelymeasur-ablewithrespecttothe¯ltrationF;andthattherandomvariablesXandXeareFT-measurable.Inaddition,Aisassumedtobeaprocessof¯nitevariation,withA0=0:Weassumewithoutmentioningthatallrandomobjectsintroducedabovesatisfysuitableintegrabilityconditions.9 10CHAPTER1.STRUCTURALAPPROACHProbabilitiesPandQ.TheprobabilityPisassumedtorepresentthereal-world(orstatistical)probability,asopposedtoamartingalemeasure(alsoknownasarisk-neutralprobability).AnymartingalemeasurewillbedenotedbyQinwhatfollows.Defaulttime.Inthestructuralapproach,thedefaulttime¿willbetypicallyde¯nedintermsofthe¯rm'svalueprocessVandthebarrierprocessv:Weset¿=infft>0:t2TandVt·vtgwiththeusualconventionthatthein¯mumovertheemptysetequals+1:Inmaincases,thesetTisaninterval[0;T](or[0;1)inthecaseofperpetualclaims).In¯rstpassagestructuralmodels,thedefaulttime¿isusuallygivenbytheformula:¿=infft>0:t2[0;T]andVt·v¹(t)g;where¹v:[0;T]!R+issomedeterministicfunction,termedthebarrier.Predictabilityofdefaulttime.Sincetheunderlying¯ltrationFinmoststructuralmodelsisgeneratedbyastandardBrownianmotion,¿willbeanF-predictablestoppingtime(asanystoppingtimewithrespecttoaBrownian¯ltration){thereexistsastrictlyincreasingsequenceofstoppingtimesannouncingthedefaulttime.Recoveryrules.IfdefaultdoesnotoccurbeforeorattimeT;thepromisedclaimXispaidinfullattimeT:Otherwise,dependingonthemarketconvention,either(1)theamountXeispaidatthematuritydateT;or(2)theamountZ¿ispaidattime¿:Inthecasewhendefaultoccursatmaturity,i.e.,ontheeventf¿=Tg,wepostulatethatonlytherecoverypaymentXeispaid.Inageneralsetting,weconsidersimultaneouslybothkindsofrecoverypayo®,andthusagenericdefaultableclaimisformallyde¯nedasaquintuple(X;A;X;Z;¿e):1.1.2Risk-NeutralValuationFormulaSupposethatour¯nancialmarketmodelisarbitrage-free,inthesensethatthereexistsamartingalemeasure(risk-neutralprobability)Q;meaningthatpriceprocessofanytradeablesecurity,whichpaysnocouponsordividends,becomesanF-martingaleunderQ;whendiscountedbythesavingsaccountB;givenas³Zt´Bt=exprudu:(1.1)0WeintroducethejumpprocessHt=1f¿·tg;andwedenotebyDtheprocessthatmodelsallcash°owsreceivedbytheownerofadefaultableclaim.LetusdenoteXd(T)=X1+Xe1:f¿>Tgf¿·TgDe¯nition1.1.1ThedividendprocessDofadefaultablecontingentclaim(X;A;X;Z;¿e);whichsettlesattimeT;equalsZZD=Xd(T)1+(1¡H)dA+ZdH:tft¸Tguuuu]0;t]]0;t]ItisapparentthatDisaprocessof¯nitevariation,andZZ(1¡Hu)dAu=1f¿>ugdAu=A¿¡1f¿·tg+At1f¿>tg:]0;t]]0;t]Notethatifdefaultoccursatsomedatet;thepromiseddividendAt¡At¡;whichisduetobepaidatthisdate,isnotreceivedbytheholderofadefaultableclaim.Furthermore,ifweset¿^t=minf¿;tgthenZZudHu=Z¿^t1f¿·tg=Z¿1f¿·tg:]0;t] 1.1.BASICASSUMPTIONS11Remark1.1.1Inprinciple,thepromisedpayo®Xcouldbeincorporatedintothepromiseddivi-dendsprocessA:However,thiswouldbeinconvenient,sinceinpracticetherecoveryrulesconcerningthepromiseddividendsAandthepromisedclaimXaredi®erent,ingeneral.Forinstance,inthecaseofadefaultablecouponbond,itisfrequentlypostulatedthatincaseofdefaultthefuturecouponsarelost,butastrictlypositivefractionofthefacevalueisusuallyreceivedbythebondholder.Weareinapositiontode¯netheex-dividendpriceStofadefaultableclaim.Atanytimet;therandomvariableStrepresentsthecurrentvalueofallfuturecash°owsassociatedwithagivendefaultableclaim.De¯nition1.1.2Foranydatet2[0;T[;theex-dividendpriceofthedefaultableclaim(X;A;X;Z;¿e)isgivenas³Z¯´¡1¯St=BtEQBudDu¯Ft:(1.2)]t;T]Thediscountedex-dividendpriceS¤=SB¡1;t2[0;T],satis¯esttt³Z¯´Z¤¡1¯¡1St=EQBudDu¯Ft¡BudDu;]0;T]]0;t]andthusitisasupermartingaleunderQifandonlyifthedividendprocessDisincreasing.TheprocessSc,givenbytheformulaZSc=S+BB¡1dD;tttuu]0;t]iscalledthecumulativeprice.1.1.3DefaultableZero-CouponBondAssumethatA=0,Z=0andX=LforsomepositiveconstantL>0:ThenthevalueprocessSrepresentsthearbitragepriceofadefaultablezero-couponbond(alsoknownasthecorporatediscountbond)withthefacevalueLandrecoveryatmaturityonly.Ingeneral,thepriceD(t;T)ofsuchabondequals¡¯¢D(t;T)=BEB¡1(L1+Xe1)¯F:tQTf¿>Tgf¿·TgtItisconvenienttorewritethelastformulaasfollows:¡¯¢D(t;T)=LBEB¡1(1+±(T)1)¯F;tQTf¿>Tgf¿·Tgtwheretherandomvariable±(T)=X=Lerepresentstheso-calledrecoveryrateupondefault.Itisnaturaltoassumethat0·Xe·Lsothat±(T)satis¯es0·±(T)·1:Alternatively,wemayre-expressthebondpriceasfollows:³¡¯¢´D(t;T)=LB(t;T)¡BEB¡1w(T)1¯F;tQTf¿·Tgtwhere¡1B(t;T)=BtEQ(BTjFt)isthepriceofaunitdefault-freezero-couponbond,andw(T)=1¡±(T)isthewritedownrateupondefault.Generallyspeaking,thetime-tvalueofacorporatebonddependsonthejointproba-bilitydistributionunderQofthethree-dimensionalrandomvariable(BT;±(T);¿)or,equivalently,(BT;w(T);¿): 12CHAPTER1.STRUCTURALAPPROACHExample1.1.1Merton[77]postulatesthattherecoverypayo®upondefault(thatis,whenVT0;where·istheconstantpayout(dividend)ratioandtheprocessWisastandardBrownianmotionunderthemartingalemeasureQ:WepresentheretheclassicmodelduetoMerton[77].Basicassumptions.A¯rmhasasingleliabilitywithpromisedterminalpayo®L;interpretedasthezero-couponbondwithmaturityTandfacevalueL>0:Theabilityofthe¯rmtoredeemitsdebtisdeterminedbythetotalvalueVTof¯rm'sassetsattimeT:DefaultmayoccurattimeTonly,andthedefaulteventcorrespondstotheeventfVT0:T¡tThisagreeswiththewell-knownfactthatriskybondshaveanexpectedreturninexcessoftherisk-freeinterestrate.Inotherwords,theyieldsoncorporatebondsarehigherthanyieldsonTreasurybondswithmatchingnotionalamounts.Note,however,thatwhentimetconvergestomaturitydateTthenthecreditspreadinMerton'smodeltendseithertoin¯nityorto0,dependingonwhetherVTL.Formally,ifwede¯netheforwardshortcreditspreadattimeTasFSST=limS(t;T);t"Tthen½0;if!2fVT>Lg,FSST(!)=1;if!2fVT0.LetusnoticethatYinheritsfromWastrongMarkovpropertywithrespecttoF: 1.3.PROPERTIESOFFIRSTPASSAGETIMES151.3.1ProbabilityLawoftheFirstPassageTimeLet¿standforthe¯rstpassagetimetozerobytheprocessY,thatis,¿:=infft2R+:Yt=0g:(1.5)Itiswellknownthatinanarbitrarilysmalltimeinterval[0;t]thesamplepathoftheBrownianmotionstartedat0passesthroughoriginin¯nitelymanytimes.UsingGirsanov'stheoremandthestrongMarkovpropertyoftheBrownianmotion,itisthuseasytodeducethatthe¯rstpassagetimebyYtozerocoincideswiththe¯rstcrossingtimebyYofthelevel0,thatis,withprobability1,¿=infft2R+:Yt<0g=infft2R+:Yt·0g:WewillalsowriteXt=ºt+¾Wtforeveryt2R+:Lemma1.3.1Let¾>0andº2R.Thenforeveryx>0wehaveµ¶µ¶¡¢x¡ºs2º¾¡2x¡x¡ºsQsupXu·x=Np¡eNp(1.6)0·u·s¾s¾sandforeveryx<0µ¶µ¶¡¢¡x+ºs2º¾¡2xx+ºsQinfXu¸x=Np¡eNp:(1.7)0·u·s¾s¾sProof.Toderivethe¯rstequality,wewilluseGirsanov'stheoremandthere°ectionprincipleforaBrownianmotion.Assume¯rstthat¾=1:LetPbetheprobabilitymeasureon(•;Fs)givenby2dP¡ºW¡ºs=es2;Q-a.s.,dQsothattheprocessW¤:=X=W+ºt;t2[0;s],isastandardBrownianmotionunderP.AlsotttdQºW¤¡º2s=es2;P-a.s.dPMoreover,forx>0,³´¡¢¤º2ºWs¡2sQsupXu>x;Xs·x=EPe1fsup0·u·sW¤>x;W¤·xg:us0·u·sWeset¿=infft¸0:W¤=xgandwede¯neanauxiliaryprocess(Wf;t2[0;s])bysettingxttWf=W¤1+(2x¡W¤)1:ttf¿x¸tgtf¿xx;Wf·xg=fW¤¸xg½f¿·sg:ussx0·u·s¡¢LetJ:=Qsup0·u·s(Wu+ºu)·x.Thenweobtain¡¢J=Q(Xs·x)¡QsupXu>x;Xs·x0·u·s³´¤º2ºW¡s=Q(Xs·x)¡EPes21fsupW¤>x;W¤·xg0·u·sus³´2ºWf¡ºs=Q(W+ºs·x)¡Ees21sPfsup0·u·sWfu>x;Wfs·xg³´¤º2º(2x¡Ws)¡2s=Q(Xs·x)¡EPe1fW¤¸xgs³´¤º22ºxºW¡s=Q(Xs·x)¡eEPes21fW¤·¡xgs=Q(X·x)¡e2ºxQ(W+ºs·¡x)ssµ¶µ¶x¡ºs2ºx¡x¡ºs=Np¡eNp:ss 16CHAPTER1.STRUCTURALAPPROACHThisendstheproofofthe¯rstequalityfor¾=1:Forany¾>0wehave¡¢¡¢Qsup(¾W+ºu)·x=Qsup(W+º¾¡1u)·x¾¡1;uu0·u·s0·u·sandthisimplies(1.6).Since¡WisastandardBrownianmotionunderQ,wealsohavethat,foranyx<0,¡¢¡¢Qinf(¾Wu+ºu)¸x=Qsup(¾Wu¡ºu)·¡x;0·u·s0·u·sandthus(1.7)easilyfollowsfrom(1.6).¤Proposition1.3.1The¯rstpassagetime¿givenby(1.5)hastheinverseGaussianprobabilitydistributionunderQ.Speci¯cally,forany00.Letusalsoassumethatthebarrierprocessvisconstantandequalto¹v;wheretheconstant¹vsatis¯es¹vt,ontheeventft<¿g,Ã!Ã!lnv¹¡º(s¡t)³v¹´2alnv¹+º(s¡t)Q(¿·sjF)=NVtp+NVtp;t¾Vs¡tVt¾Vs¡twherewedenoter¡·¡1¾2º2Va==:(1.11)¾2¾2VVThisresultwasusedinLelandandToft[70]. 1.3.PROPERTIESOFFIRSTPASSAGETIMES17Example1.3.2LetthevalueprocessVandtheshort-terminterestraterbeasinExample1.3.1.ForastrictlypositiveconstantKandanarbitrary°2R+,letthebarrierfunctionbede¯nedasv¹(t)=Ke¡°(T¡t)fort2R,sothatthefunction¹v(t)satis¯es+dv¹(t)=°v¹(t)dt;v¹(0)=Ke¡°T:WenowsetY=ln(V=v¹(t)),andthusthecoe±cientsin(1.4)areºe=r¡·¡°¡1¾2and¾=¾.tt2VVWede¯nethedefaulttime¿as¿=infft¸0:Vt·v¹(t)g.FromCorollary1.3.1,weobtainforeverytt,I:=Q(Ys¸y;¿¸sjFt)=Q(Ys¸y;¿>sjFt);where¿isgivenby(1.5).LetusdenotebyMWandmWtherunningmaximumandminimumofaone-dimensionalstandardBrownianmotionW,respectively.Moreexplicitly,MW=supWs0·u·suandmW=infW.Itiswellknownthatforeverys>0wehaves0·u·suQ(MW>0)=1;Q(mW<0)=1:(1.13)ssThefollowingwell-knownresult,commonlyreferredtoasthere°ectionprinciple,isastraightforwardconsequenceofthestrongMarkovpropertyoftheBrownianmotion.Lemma1.3.2Wehavethat,foreverys>0;y¸0andx·y,Q(W·x;MW¸y)=Q(W¸2y¡x)=Q(W·x¡2y):(1.14)ssssWeneedtoexaminetheBrownianmotionwithnon-zerodrift.ConsidertheprocessXthatequalsX=ºt+¾W.WewriteMX=supXandmX=infX.Byvirtueoftts0·u·sus0·u·suGirsanov'stheorem,theprocessXisaBrownianmotion(uptoanappropriaterescaling)underanequivalentprobabilitymeasureandthus,foranys>0,Q(MX>0)=1;Q(mX<0)=1:ssLemma1.3.3Foreverys>0,thejointdistributionof(X;MX)isgivenbytheexpressionss¡2Q(X·x;MX¸y)=e2ºy¾Q(X¸2y¡x+2ºs)(1.15)sssforeveryx;y2Rsuchthaty¸0andx·y:Proof.Since¡¢¡¾¢I:=QX·x;MX¸y=QX¾·x¾¡1;MX¸y¾¡1;sssswhereX¾=W+ºt¾¡1,itisclearthatwemayassume,withoutlossofgenerality,that¾=1.Wettwilluseanequivalentchangeofprobabilitymeasure.FromGirsanov'stheorem,itfollowsthatX 18CHAPTER1.STRUCTURALAPPROACHisastandardBrownianmotionundertheprobabilitymeasureP,whichisgivenon(•;Fs)bytheRadon-Nikod¶ymdensity(recallthat¾=1)2dP¡ºWºss¡=e2;Q-a.s.dQNotealsothatdQºW¤¡º2s=es2;P-a.s.,dPwheretheprocess(W¤=X=W+ºt;t2[0;s])isastandardBrownianmotionunderP.Itisttteasilyseenthat³´³´ºW¤¡º2sºW¤¡º2sI=EPes21fXs·x;MX¸yg=EPes21fW¤·x;MW¤¸yg:sssSinceWisastandardBrownianmotionunderP,anapplicationofthere°ectionprinciple(1.14)gives³´¤º2º(2y¡W)¡sI=EPes21f2y¡W¤·x;MW¤¸ygss³´¤º2º(2y¡Ws)¡2s=EPe1fW¤¸2y¡xgs³´¤º22ºy¡ºW¡s=eEPes21fW¤¸2y¡xg;ssinceclearly2y¡x¸y:Letusde¯neonemoreequivalentprobabilitymeasurePebysettingdPe¡ºW¤¡º2s=es2;P-a.s.dPIsisclearthat³´¤º2I=e2ºyEe¡ºWs¡2s1¤=e2ºyPe(W¤¸2y¡x):PfWs¸2y¡xgsFurthermore,theprocess(Wf=W¤+ºt;t2[0;s])isastandardBrownianmotionunderPeandwetthavethatI=e2ºyPe(Wf+ºs¸2y¡x+2ºs):sThelastequalityeasilyyields(1.15).¤Itisworthwhiletoobservethat(asimilarremarkappliestoallformulaebelow)Q(X·x;MX¸y)=Q(Xy):ssssThefollowingresultisastraightforwardconsequenceofLemma1.3.3.Proposition1.3.2Foranyx;y2Rsatisfyingy¸0andx·y,wehavethatµ¶¡X¢2ºy¾¡2x¡2y¡ºsQXs·x;Ms¸y=eNp:(1.16)¾sHenceµ¶µ¶¡X¢x¡ºs2ºy¾¡2x¡2y¡ºsQXs·x;Ms·y=Np¡eNp(1.17)¾s¾sforeveryx;y2Rsuchthatx·yandy¸0:Proof.Forthe¯rstequality,notethatµ¶x¡2y¡ºsQ(Xs¸2y¡x+2ºs)=Q(¡¾Ws·x¡2y¡ºs)=Np;¾ssince¡¾WhasGaussianlawwithzeromeanandvariance¾2t:For(1.17),itisenoughtoobservetthatQ(X·x;MX·y)+Q(X·x;MX¸y)=Q(X·x)sssssandtoapply(1.16).¤ 1.3.PROPERTIESOFFIRSTPASSAGETIMES19ItisclearthatQ(MX¸y)=Q(X¸y)+Q(X·y;MX¸y)ssssforeveryy¸0;andthus¡2Q(MX¸y)=Q(X¸y)+e2ºy¾Q(X¸y+2ºs):(1.18)sssConsequently,¡2Q(MX·y)=1¡Q(MX¸y)=Q(X·y)¡e2ºy¾Q(X¸y+2ºs):ssssThisleadstothefollowingcorollary.Corollary1.3.2Thefollowingequalityisvalid,foreverys>0andy¸0,µ¶µ¶Xy¡ºs2ºy¾¡2¡y¡ºsQ(Ms·y)=Np¡eNp:(1.19)¾s¾sWewillnowfocusonthedistributionoftheminimalvalueofX.Observethatwehave,foranyy·0,¡¢¡¢¡¢Qsup(¾Wu¡ºu)¸¡y=Qinf(¡¾Wu+ºu)·y=QinfXu·y;0·u·s0·u·s0·u·swherethelastequalityfollowsfromthesymmetryoftheBrownianmotion.Consequently,foreveryXXey·0wehaveQ(ms·y)=Q(Ms¸¡y),wheretheprocessXeequalsXet=¾Wt¡ºt.Itisthusnotdi±culttoestablishthefollowingresult.Proposition1.3.3Thejointprobabilitydistributionof(X;mX)satis¯es,foreverys>0,ssµ¶µ¶X¡x+ºs2ºy¾¡22y¡x+ºsQ(Xs¸x;ms¸y)=Np¡eNp¾s¾sforeveryx;y2Rsuchthaty·0andy·x:Corollary1.3.3Thefollowingequalityisvalid,foreverys>0andy·0,µ¶µ¶X¡y+ºs2ºy¾¡2y+ºsQ(ms¸y)=Np¡eNp:¾s¾sRecallthatwedenoteYt=y0+Xt;whereXt=ºt+¾Wt.WewritemX=infX;mY=infY:susu0·u·s0·u·sCorollary1.3.4Foranys>0andy¸0wehaveµ¶µ¶¡y+y0+ºs¡2º¾¡2y0¡y¡y0+ºsQ(Ys¸y;¿¸s)=Np¡eNp:¾s¾sProof.SinceQ(Y¸y;¿¸s)=Q(Y¸y;mY¸0)=Q(X¸y¡y;mX¸¡y);ssss0s0theassertedformulaisratherobvious.¤Moregenerally,theMarkovpropertyofYjusti¯esthefollowingresult. 20CHAPTER1.STRUCTURALAPPROACHLemma1.3.4Wehavethat,foranyt0and°2R.UsingagainLemma1.3.4,butthistimewithYt=ln(Vt=v¹(t))andy=ln(x=v¹(s));we¯ndthat,foreveryt0;whereWisaBrownianmotion(undertherisk-neutralprobabilityQ),theconstant·¸0representsthepayoutratio,and¾V>0istheconstantvolatility.Theshort-terminterestraterisassumedtobeconstant.Safetycovenants.Safetycovenantsprovidethe¯rm'sbondholderswiththerighttoforcethe¯rmtobankruptcyorreorganizationifthe¯rmisdoingpoorlyaccordingtoasetstandard.ThestandardforapoorperformanceissetbyBlackandCoxintermsofatime-dependentdeterministicbarrier¹v(t)=Ke¡°(T¡t);t2[0;T[;forsomeconstantK>0:Assoonasthevalueof¯rm'sassetscrossesthislowerthreshold,thebondholderstakeoverthe¯rm.Otherwise,defaulttakesplaceatdebt'smaturityornotdependingonwhetherVTKe¡°(T¡t)g;++withtheboundaryconditionu(Ke¡°(T¡t);t)=¯Ke¡°(T¡t)2andtheterminalconditionu(v;T)=min(¯1v;L):Probabilisticapproach.Foranyttg=f¿>t¹g³¯´¡r(T¡t)¯D(t;T)=EQLe1f¿¹¸T;VT¸Lg¯Ft³¯´¡r(T¡t)¯+¯1EQVTe1f¿¹¸T;VT0:Priortodefault,thatis,onthesetf¿>tg;thepriceprocessD(t;T)=u(Vt;t)ofadefaultablebondequals¡¡¢¡2¡¢¢D(t;T)=LB(t;T)Nh(V;T¡t)¡R2b¾Nh(V;T¡t)1tt2t¡¡¡¢¢+¯Ve¡·(T¡t)Nh(V;T¡t))¡Nh(V;T¡t)1t3t4t¡¡¡¢¢+¯Ve¡·(T¡t)R2b+2Nh(V;T¡t))¡Nh(V;T¡t)1tt5t6t¡¡¢¡¢¢µ+³µ¡³+¯2VtRtNh7(Vt;T¡t)+RtNh8(Vt;T¡t);pwhereR=¹v(t)=V;µ=b+1;³=¾¡2m2+2¾2(r¡°)andttln(Vt=L)+º(T¡t)h1(Vt;T¡t)=p;¾T¡tln¹v2(t)¡ln(LV)+º(T¡t)th2(Vt;T¡t)=p;¾T¡tln(L=V)¡(º+¾2)(T¡t)th3(Vt;T¡t)=p;¾T¡tln(K=V)¡(º+¾2)(T¡t)th4(Vt;T¡t)=p;¾T¡tln¹v2(t)¡ln(LV)+(º+¾2)(T¡t)th5(Vt;T¡t)=p;¾T¡tln¹v2(t)¡ln(KV)+(º+¾2)(T¡t)th6(Vt;T¡t)=p;¾T¡tln(¹v(t)=V)+³¾2(T¡t)th7(Vt;T¡t)=p;¾T¡tln(¹v(t)=V)¡³¾2(T¡t)th8(Vt;T¡t)=p:¾T¡tBeforeproceedingtotheproofofProposition1.4.1,wewillproveanelementarylemma.Lemma1.4.1Foranya2Randb>0wehave,foreveryy>0,Zyµ¶µ2¶lnx+a1b2¡alny+a¡bxdN=e2N(1.21)0bbandZµ¶µ¶y2¡lnx+a1b2+a¡lny+a+bxdN=e2N:(1.22)0bbLeta;b;c2Rsatisfyb<0andc2>2a.Thenwehave,foreveryy>0,Zyµ¶axb¡cxd+cd¡cedNp=g(y)+h(y);(1.23)0x2d2dpwhered=c2¡2aandwherewedenoteµ¶µ¶b(c¡d)b¡dyb(c+d)b+dyg(y)=eNp;h(y)=eNp:yyProof.Theproofof(1.21){(1.22)isstandard.For(1.23),observethatZyµ¶Zyµ¶µ¶axb¡cxaxb¡cxbcf(y):=edNp=enp¡¡pdx;0x0x2x3=22x 24CHAPTER1.STRUCTURALAPPROACHwherenistheprobabilitydensityfunctionofthestandardGaussianlaw.NotealsothatÃp!Ãp!0b(c¡pc2¡2a)b¡c2¡2axbc2¡2ag(x)=enp¡¡px2x3=22xµ¶µ¶axb¡cxbd=enp¡¡px2x3=22xandÃp!Ãp!0b(c+pc2¡2a)b+c2¡2axbc2¡2ah(x)=enp¡+px2x3=22xµ¶µ¶axb¡cxbd=enp¡+p:x2x3=22xConsequently,µ¶00axbb¡cxg(x)+h(x)=¡enpx3=2xandµ¶00axdb¡cxg(x)¡h(x)=¡enp:x1=2xHencefcanberepresentedasfollowsZy1¡c¢f(y)=g0(x)+h0(x)+(g0(x)¡h0(x))dx:20dSincelimy!0+g(y)=limy!0+h(y)=0,weconcludethatwehave,foreveryy>0,1cf(y)=(g(y)+h(y))+(g(y)¡h(y)):22dThisendstheproofofthelemma.¤ProofofProposition1.4.1.Weneedtocomputethefollowingconditionalexpectations:D1(t;T)=LB(t;T)Q(VT¸L;¿¹¸TjFt);¡¯¢D2(t;T)=¯1B(t;T)EQVT1fVTr.ForK=Land°>r;itisnaturaltoexpectthatD(t;T)wouldbesmallerthanLB(t;T):Itisalsopossibletoshowthatwhen°tendstoin¯nity(allotherparametersbeing¯xed),thentheBlackandCoxpriceconvergestoMerton'sprice.1.4.2CorporateCouponBondWeshallassumenowthatr>0andthatadefaultablebondof¯xedmaturityTandfacevalueLpayscontinuouslycouponsataconstantratec,sothatAt=ctforeveryt2R+.Thecouponpaymentsstopassoonasdefaultoccurs.Formally,weconsiderhereadefaultableclaimspeci¯edasfollows:X=L;At=ct;Z=¯2V;Xe=¯1VT;¿=infft2[0;T]:Vts¹gds¯Ft=ceeQ(¹¿>sjFt)ds:ttSettingt=0,wethusobtainZTD(0;T)=D(0;T)+ce¡rsQ(¹¿>s)ds=D(0;T)+A(0;T);c0where(recallthatwewrite¾insteadof¾V)µ¶µ¶2eaµ¶ln(V0=v¹(0))+ºsev¹(0)ln(¹v(0)=V0)+ºseQf¿>s¹g=Np¡Np:¾sV0¾sAnintegrationbypartsformulayieldsZT³ZT´¡rs1¡rT¡rseQ(¹¿>s)ds=1¡eQ(¹¿>T)+edQ(¹¿>s):0r0 1.4.BLACKANDCOXMODEL27Weassume,asusual,thatV0>v¹(0),sothatln(¹v(0)=V0)<0.ArguinginasimilarwayasinthelastpartoftheproofofProposition1.4.1(speci¯cally,usingformula(1.23)),weobtainZTµ¶ea+³eµ2¶¡rsv¹(0)ln(¹v(0)=V0)+³¾eTedQf¿>s¹g=¡Np0V0¾Tµ¶ea¡³eµ2¶v¹(0)ln(¹v(0)=V0)¡³¾eT¡Np;V0¾Tpwhereºe=r¡·¡°¡1¾2;ea=º¾e¡2and³e=¾¡2ºe2+2¾2r.Althoughwehavefocusedon2thecasewhent=0,itisclearthatthederivationofthegeneralformulaforanyt0andthepayoutrate·isequaltozero.Thisconditioncanbegivena¯nancialinterpretationastherestrictiononthesaleofassets,asopposedtoissuingofnewequity.Equivalently,wemaythinkaboutasituationinwhichthestockholderswillmakepaymentstothe¯rmtocovertheinterestpayments.However,theyhavetherighttostopmakingpaymentsatanytimeandeitherturnthe¯rmovertothebondholdersorpaythemalumppaymentofc=rperunitofthebond'snotionalamount.RecallthatwedenotebyE(Vt)(D(Vt);resp.)thevalueattimetofthe¯rmequity(debt,resp.),hencethetotalvalueofthe¯rm'sassetssatis¯esVt=E(Vt)+D(Vt):BlackandCox[16]arguethatthereisacriticallevelofthevalueofthe¯rm,denotedasv¤;belowwhichnomoreequitycanbesold.Thecriticalvaluev¤willbechosenbystockholders,whoseaimistominimizethevalueofthebonds(equivalently,tomaximizethevalueoftheequity).Letusobservethatv¤isnothingelsethanaconstantdefaultbarrierintheproblemunderconsideration;theoptimaldefaulttime¿¤thusequals¿¤=infft2R:V·v¤g:+tTo¯ndthevalueofv¤;letus¯rst¯xthebankruptcylevel¹v:TheODEforthepricingfunctionu1=u1(V)ofaconsolbondtakesthefollowingform(recallthat¾=¾)V122111V¾uVV+rVuV+c¡ru=0;2subjecttothelowerboundaryconditionu1(¹v)=min(¹v;c=r)andtheupperboundaryconditionlimu1(V)=0:VV!1Forthelastcondition,observethatwhenthe¯rm'svaluegrowstoin¯nity,thepossibilityofdefaultbecomesmeaningless,sothatthevalueofthedefaultableconsolbondtendstothevaluec=rofthedefault-freeconsolbond.Thegeneralsolutionhasthefollowingform:1c¡®u(V)=+K1V+K2V;rwhere®=2r=¾2andK;Karesomeconstants,tobedeterminedfromboundaryconditions.We12¯ndthatK1=0;and½v¹®+1¡(c=r)¹v®;if¹vtg;µ¶®+1¤c1cD(Vt)=¡®2r®Vtr+¾=2 1.5.EXTENSIONSOFTHEBLACKANDCOXMODEL29or,equivalently,¤c¤¤¤D(Vt)=(1¡qt)+vqt;rwhereµ¤¶®µ¶®¤v1cqt==®2:VtVtr+¾=2WeendthissectionbymentioningthatotherimportantdevelopmentsintheareaofoptimalcapitalstructurewerepresentedinthepapersbyLeland[69],LelandandToft[70],andChristensenetal.[31].ChenandKou[29],Dao[34],HilberinkandRogers[53],andLeCourtoisandQuittard-Pinon[68]studythesameproblem,buttheymodelthe¯rm'svalueprocessasadi®usionwithjumps.Thereasonforthisextensionwastoeliminateanundesirablefeatureofpreviouslyexaminedmodels,inwhichshortspreadstendtozerowhenabondapproachesmaturitydate.1.5ExtensionsoftheBlackandCoxModelTheBlackandCox¯rst-passage-timeapproachwaslaterdevelopedby,amongothers:BrennanandSchwartz[22,23]{ananalysisofconvertiblebonds,Nielsenetal.[79]{arandombarrierandrandominterestrates,Leland[69],LelandandToft[70]{astudyofanoptimalcapitalstructure,bankruptcycostsandtaxbene¯ts,Longsta®andSchwartz[72]{aconstantbarriercombinedwithrandominterestrates,andbyBrigo[24].Ingeneral,onecanstudythebondvaluationproblemforthedefaulttimegivenas¿=infft2R+:Vt·v(t)g,wherev:R+!Risanarbitraryfunctionandthevalueofthe¯rmVismodeledasageometricBrownianmotion.However,thereexistsonlyfewexplicitresults.Moraux[78]proposestomodelthedefaulttimeasaParisianstoppingtime.ForacontinuousprocessVandagivent>0,weintroducearandomvariablegb(V),representingthelastmomenttbeforetwhentheprocessVwasatagivenlevelb,thatis,gb(V)=supf0·s·t:V=bg.ThetsParisianstoppingtimeisthe¯rsttimeatwhichtheprocessVisbelowthelevelbforatimeperiod¡;boflengthgreaterorequaltoaconstantD.Formally,thestoppingtime¿=G(V)isgivenbytheDformula¡;bbGD(V)=infft2R+:(t¡gt(V))1fVtDg,wherewedenoteR+tVtAt=01fVu>bgdu.Theprobabilitydistributionofthisrandomtimeisrelatedtotheso-calledcumulativeoptions.CampiandSbuelz[27]assumethatthedefaulttimeisgivenbya¯rsthittingtimeof0byaCEVprocessandtheystudythedi±cultproblemofpricinganequitydefaultswap.Moreprecisely,theyassumethatthedynamicsunderQofthe¯rm'svalueare³´¯dVt=Vt¡(r¡·)dt+¾VtdWt¡dMt;whereWisaBrownianmotionandMthecompensatedmartingaleofaPoissonprocess(i.e.,Mt=Nt¡¸t),andtheyset¿=infft2R+:Vt·0g.Putanotherway,CampiandSbuelz[27]de¯nethedefaulttimebysetting¿=¿¯^¿N,where¿Nisthe¯rstjumpofthePoissonprocessand¿¯isde¯nedas¿¯=infft2R:X·0g,whereinturntheprocessXobeysthefollowing+tSDE³´¯dXt=Xt¡(r¡·+¸)dt+¾XtdWt:Usingthewell-knownfactthattheCEVprocesscanbeexpressedintermsofatime-changedBesselprocessandresultsonthehittingtimeof0foraBesselprocessofdimensionsmallerthan2,theyobtainclosedfromsolutions. 30CHAPTER1.STRUCTURALAPPROACHZhou[86]examinesthecasewherethedynamicsunderQofthe¯rmare³¡¢´dVt=Vt¡r¡¸ºdt+¾dWt+dXt;whereWisastandardBrownianmotionandXisacompoundPoissonprocess.Speci¯cally,wesetPN¡¢X=teYi¡1,whereNisaPoissonprocesswithaconstantintensity¸,randomvariablesti=1YareindependentandhavetheGaussiandistributionN(a;b2).Wealsosetº=exp(a+b2=2)¡1,isinceforthischoiceofºtheprocessVe¡rtisamartingale.Zhou[86]¯rststudiesMerton'sproblemtinthisframework.Next,hegivesanapproximationforthe¯rstpassagetimeproblemwhenthedefaulttimeisgivenas¿=infft2R+:Vt·Lg.1.5.1StochasticInterestRatesInthissection,theBlackandCoxvaluationformulaforacorporatebondwillbeextendedtothecaseofrandominterestrates.Weassumethattheunderlyingprobabilityspace(•;F;P);endowedwiththe¯ltrationF=(Ft)t2R+;supportstheshort-terminterestrateprocessrandthevalueprocessV:ThedynamicsunderthemartingalemeasureQofthe¯rm'svalueandofthepriceofadefault-freezero-couponbondB(t;T)are¡¢dVt=Vt(rt¡·(t))dt+¾(t)dWtand¡¢dB(t;T)=B(t;T)rtdt+b(t;T)dWtrespectively,whereWisad-dimensionalstandardQ-Brownianmotion.Furthermore,·:[0;T]!R;¾:[0;T]!Rdandb(¢;T):[0;T]!Rdareassumedtobeboundedfunctions.TheforwardvalueFV(t;T)=Vt=B(t;T)ofthe¯rmsatis¯esundertheforwardmartingalemeasureQT¡¢dF(t;T)=¡·(t)F(t;T)dt+F(t;T)¾(t)¡b(t;T)dWT;VVVtRtwheretheprocessWT=W¡b(u;T)du;t2[0;T];isad-dimensionalBrownianmotionundertt0QT:Foranyt2[0;T];wesetRT·¡·(u)duFV(t;T)=FV(t;T)et:Then¡¢dF·(t;T)=F·(t;T)¾(t)¡b(t;T)dWT:VVtFurthermore,itisapparentthatF·(T;T)=F(T;T)=V:Weconsiderthefollowingmodi¯cationVVToftheBlackandCoxapproachX=L;Zt=¯2Vt;X~=¯1VT;¿=infft2[0;T]:Vttg¡¡¢¡¢¢LNbh(F;t;T)¡(F=K)e¡·(t;T)Nbh(F;t;T)1tt2t¡¡¢¡¢¢+¯Fe¡·(t;T)Nbh(F;t;T)¡Nbh(F;t;T)1t3t4t¡¡¢¡¢¢+¯1KNbh5(Ft;t;T)¡Nbh6(Ft;t;T)+¯KJ(F;t;T)+¯Fe¡·(t;T)J(F;t;T);2+t2t¡twherebh(F;t;T)=ln(Ft=L)¡´+(t;T);1t¾(t;T)bh(F;T;t)=2lnK¡ln(LFt)+´¡(t;T);2t¾(t;T)bh(F;t;T)=ln(L=Ft)+´¡(t;T);3t¾(t;T)bh(F;t;T)=ln(K=Ft)+´¡(t;T);4t¾(t;T)bh(F;t;T)=2lnK¡ln(LFt)+´+(t;T);5t¾(t;T)bh(F;t;T)=ln(K=Ft)+´+(t;T);6t¾(t;T)andforany¯xed0·t0wesetZTµ12¶·(u;T)ln(K=Ft)+·(t;T)§2¾(t;u)J§(Ft;t;T)=edN:t¾(t;u)Inthespecialcasewhen·=0;theformulaofProposition1.5.1coversasaspecialcasethevaluationresultestablishedbyBriysanddeVarenne[25].Insomeotherrecentstudiesof¯rstpassagetimemodels,inwhichthetriggeringbarrierisassumedtobeeitheraconstantoranunspeci¯edstochasticprocess,typicallynoclosed-formsolutionforthevalueofacorporatedebtisavailable,andthusanumericalapproachisrequired(see,forinstance,Longsta®andSchwartz[72],Nielsenetal.[79],orSa¶a-RequejoandSanta-Clara[82]).1.6RandomBarrierInthecaseofthefullinformationandtheBrownian¯ltration,the¯rsthittingtimeofadeterministicbarrierisapredictablestoppingtime.Thisisnolongerthecasewhenwedealwithanincompleteinformation(as,e.g.,inDu±eandLando[41]),orwhenanadditionalsourceofrandomnessispresent.Wepresenthereaformulaforcreditspreadsarisinginaspecialcaseofatotallyinaccessibletimeofdefault.ForamoredetailedstudywerefertoBabbsandBielecki[1].Asweshallsee,themethodusedhereisinfactfairlyclosetothegeneralmethodpresentedinChapter3.Wenowpostulatethatthebarrierwhichtriggersdefaultisrepresentedbyarandomvariable´de¯nedontheunderlyingprobabilityspace.Thedefaulttime¿isgivenas¿=infft2R+:Vt·´g,whereVisthevalueofthe¯rmand,forsimplicity,V0=1.Notethatf¿>tg=finfu·tVu>´g.WeshalldenotebymVtherunningminimumofthecontinuousprocessV,thatis,mV=infV.tu·tuWiththisnotation,wehavethatf¿>tg=fmV>´g.NotethatmVismanifestlyadecreasing,tcontinuousprocess. 32CHAPTER1.STRUCTURALAPPROACH1.6.1IndependentBarrierWeassumethat,undertherisk-neutralprobabilityQ,arandomvariable´modelingthebarrierisindependentofthevalueofthe¯rm.WedenotebyF´thecumulativedistributionfunctionof´,thatis,F´(z)=Q(´·z).WeassumethatF´isadi®erentiablefunctionandwedenotebyf´itsderivative(withf´(z)=0forz>V0).Lemma1.6.1LetussetFt=Q(¿·tjFt)and¡t=¡ln(1¡Ft).ThenZtVf´(mu)V¡t=¡Vdmu:0F´(mu)Proof.Ifarandomvariable´isindependentofF1thenF=Q(¿·tjF)=Q(mV·´jF)=1¡F(mV):tttt´tTheprocessmVisdecreasingandthus¡=¡lnF(mV).Weconcludethatt´tZtVf´(mu)V¡t=¡Vdmu;0F´(mu)asrequired.¤Example1.6.1LetV0=1andlet´bearandomvariableuniformlydistributedontheinterval[0;1].Thenmanifestly¡=¡lnmV.ThecomputationoftheexpectedvalueE(e¡Tf(V))requiresttQTtheknowledgeofthejointprobabilitydistributionofthepair(V;mV).TTWenowpostulate,inaddition,thatthevalueprocessVismodeledbyageometricBrownianmotionwithadrift.Speci¯cally,wesetV=eªt,whereª=¹t+¾W.Itisclearthat¿=tttinfft2R:mª·Ãg;whereÃ=ln´andmªistherunningminimumoftheprocessª,thatis,+tmª=inffª:0·s·tg.WechoosetheBrownian¯ltrationasthereference¯ltration,thatis,tswesetF=FW.Thismeansthatweassumethatthevalueofthe¯rmprocessV(hencealsotheprocessª)isperfectlyobserved.ThebarrierÃisnotobserved,however.Weonlypostulatethataninvestorcanobservetheoccurrenceofthedefaulttime.Inotherwords,hecanobservetheprocessHt=1f¿·tg=1fmª·Ãg:WedenotebyHthenatural¯ltrationoftheprocessH:Theinformationtavailabletotheinvestoristhusrepresentedbythe¯ltrationG=F_H.Wealsoassumethatthedefaulttime¿andinterestratesareindependentunderQ:Itisthenispossibletoestablishthefollowingresult(fortheproof,theinterestedreaderisreferredtoGiesecke[51]orBabbsandBielecki[1]).Proposition1.6.1Undertheassumptionsstatedabove,wedealwithaunitcorporatebondwithzerorecovery.ThenthecreditspreadS(t;T)isgivenas,foreveryt2[0;T[,()³ZTª´¯1fÃ(mu)ª¯S(t;T)=¡1ft<¿glnEQexpªdmu¯Ft:T¡ttFÃ(mu)Notethattheprocessmªisdecreasing,sothatthestochasticintegralwithrespecttothisprocesscanbeinterpretedasapathwiseStieltjesintegral.InChapter3,wewillexaminethenotionofahazardprocessofarandomtimewithrespecttoareference¯ltrationF.Itisthusworthmentioningthatforthedefaulttime¿de¯nedabove,theF-hazardprocess¡existsanditisgivenbytheformulaZtªfÃ(mu)ª¡t=¡ªdmu:0FÃ(mu)Sincethisprocessisclearlycontinuous,thedefaulttime¿isinfactatotallyinaccessiblestoppingtimewithrespecttothe¯ltrationG. Chapter2HazardFunctionApproachWeprovideinthischapteradetailedanalysisoftherelativelysimplecaseofthereducedformmethodology,whenthe°owofinformationavailabletoanagentreducestotheobservationsoftherandomtimerepresentingthedefaulteventofsomecreditname.Thefocusisontheevaluationofconditionalexpectationswithrespecttothe¯ltrationgeneratedbyadefaulttimewiththeuseofthehazardfunction.Wealsostudyhedgingstrategiesbasedonstylizedcreditdefaultswaps.Weconcludethischapterbypresentingacreditriskmodelwithseveraldefaulttimes.2.1ElementaryMarketModelWebeginwiththesimplecasewherearisk-freezerocouponbonds,drivenbyadeterministicinterestrate(r(t);t2R+),aretheonlytradedassetsinthedefault-freemarketmodel.Recallthatinthatcase,thepriceattimetofarisk-freezero-couponbondwithmaturityTequals³ZT´B(t)B(t;T)=exp¡r(u)du=;tB(T)³Zt´whereB(t)=expr(u)durepresentsthesavingsaccount.0Defaultoccursattime¿,where¿isassumedtobeapositiverandomvariablede¯nedonaprobabilityspace(•;G;Q).WedenotebyFthecumulativedistributionfunctionoftherandomvariable¿de¯nedasF(t)=Q(¿·t).WeassumethatF(t)<1foranyt2R+.Otherwise,therewouldexista¯nitedatet0forwhichF(t0)=1,sothatthedefaultwouldoccurbeforeoratt0withRtprobability1.Wewillfrequentlypostulate,inaddition,thatF(t)=f(u)duforsomeprobability0densityfunctionf.Weemphasizethattherandompayo®oftheform1fT<¿gcannotbeperfectlyhedgedwithdeterministiczero-couponbonds,whicharetheonlytradeableprimaryassetsinourmodel.Tohedgetherisk,weshalllaterpostulatethatsomedefaultableassetsaretraded,e.g.,adefaultablezero-couponbondoracreditdefaultswap.Inthe¯rststep,wewillpostulatethatthefairvalue"ofadefaultableassetisgivenbytherisk-neutralvaluationformulawithrespecttoQ.Letusnoteinthisregardthatinpractice,therisk-neutraldistributionofdefaulttimeisimpliedfrommarketquotesoftradeddefaultableassets,ratherthanpostulatedapriori.2.1.1HazardFunctionWeintroducethehazardfunction¡:R+!R+of¿bysetting,foranyt2R+,¡(t)=¡ln(1¡F(t)):33 34CHAPTER2.HAZARDFUNCTIONAPPROACHNotethat¡isanon-decreasingfunctionwith¡(0)=0andlimt!+1¡(t)=+1.IfweassumethatthefunctionFisdi®erentiable,thehazardfunctionisdi®erentiableaswellf(t)and¡0(t):=°(t)=.Thismeansthat1¡F(t)³Zt´Q(¿>t)=1¡F(t)=e¡¡(t)=exp¡°(u)du:0Thequantity°(t)iscalledthehazardrateordefaultintensity.Theinterpretationofthehazardrateisthatitrepresentstheconditionalprobabilitythatthedefaultoccursinasmalltimeinterval[t;t+dt]giventhatitdidnotoccurbeforetimet,speci¯cally,1°(t)=limQ(¿·t+hj¿>t):h!0hRemark2.1.1Let¿bethe¯rstjumpofaninhomogeneousPoissonprocesswithdeterministicintensity(¸(t);t2R+).Thentheprobabilitydensityfunctionof¿equalsµZt¶Q(¿2dt)¡¤(t)f(t)==¸(t)exp¡¸(u)du=¸(t)e;dt0Rtwhere¤(t)=¸(s)dsandthusF(t)=Q(¿·t)=1¡e¡¤(t).Hencethehazardfunctionisequal0tothecompensatorofthePoissonprocess,thatis,¡(t)=¤(t).Conversely,if¿isarandomtimewiththedensityf,setting¤(t)=¡ln(1¡F(t))allowsustointerpret¿asthe¯rstjumptimeofaninhomogeneousPoissonprocesswiththeintensityequaltothederivativeof¤.Remark2.1.2Itisnotdi±culttogeneralizethestudypresentedinwhatfollowstothecasewhere¿doesnotadmitadensity,bydealingwiththeright-continuousversionofthecumulativefunction.Thecasewhere¿isboundedcanalsobestudiedalongthesamemethod.Weleavethedetailstothereader.2.1.2DefaultableZero-CouponBondwithRecoveryatMaturityWedenoteby(Ht;t2R+)theright-continuousincreasingprocessHt=1ft¸¿g,referredtoasthedefaultindicatorprocess.LetHstandforthenatural¯ltrationoftheprocessH.Itisclearthatthe¯ltrationHisthesmallest¯ltrationwhichmakes¿astoppingtime.The¾-¯eldHtisgeneratedbytheeventsfs¸¿gfors·t.ThekeyobservationisthatanyHt-measurablerandomvariableXhastheformX=h(¿)1ft¸¿g+c1ft<¿g;whereh:R+!RisaBorelmeasurablefunctionandcisaconstant.Remark2.1.3ItisworthmentioningthatifthecumulativedistributionfunctionFiscontinuousthen¿isknowntobeaH-totallyinaccessiblestoppingtime(seeDellacherieandMeyer[39]IV,Page107).Wewillnotusethisimportantpropertyexplicitly,however.Ourgoalistoderivesomeusefulvaluationformulaefordefaultablebondswithdi®eringrecoveryschemes.Forthesakeofsimplicity,wewill¯rstassumethatabondisrepresentedbyasinglepayo®atitsmaturityT.Therefore,itispossibletovalueabondasaEuropeanclaimXmaturingatT,byapplyingthestandardrisk-neutralvaluationformula³¯´X¯¼t(X)=B(t)EQ¯Ht=B(t;T)EQ(XjHt):B(T)Fortheeaseofnotation,wewillconsideradefaultablebondwiththefacevalueL=1. 2.1.ELEMENTARYMARKETMODEL35CaseofaConstantRecoveryatMaturityAdefaultable(orcorporate)zero-couponbond(DZC,inshort)withmaturityTandrecoveryvalue±paidatmaturity,consistsof:²thepaymentofonemonetaryunitattimeTifdefaulthasnotoccurredbeforeT,i.e.,if¿>T,²thepaymentof±monetaryunits,madeatmaturity,if¿·T,where0·±<1.Thefairvalue"attime0ofthedefaultablezero-couponbondisde¯nedastheexpectationofthediscountedpayo®,sothat¡¢D±(0;T)=B(0;T)E1+±1:QfT<¿gf¿·TgConsequently¡¢D±(0;T)=B(0;T)E1¡(1¡±)1=B(0;T)¡(1¡±)B(0;T)F(T):(2.1)Qf¿·TgThevalueofthedefaultablezero-couponbondisthusequaltothevalueofthedefault-freezero-couponbondminusthediscountedvalueoftheexpectedloss,computedundertherisk-neutralprobability.Obviously,thefairvalueisnotahedgingpricesincethepayo®atmaturityofthedefaultablebondcannotbereplicatedinthepresentsetupbytradinginprimaryassets(thatis,default-freezero-couponbonds).Hencewedealwithanincompletemarketmodelandthepricingformulaforthedefaultablezero-couponbondispostulated.Ofcourse,for±=1werecover,asexpected,thepriceofadefault-freezero-couponbond.Thevalueofthebondattimet2(0;T)dependswhetherornotdefaulthashappenedbeforethistime.Ontheonehand,ifdefaulthasoccurredbeforeorattimet,theconstantpaymentof±willbemadeatmaturitydateT,andthusthepriceoftheDZCisobviously±B(t;T).Ontheotherhand,ifdefaulthasnotyetoccurredbeforeorattimet,theholderstilldoesnotknowexactlythedateofitsoccurrence.Itisthusnaturalinthissituationtode¯nethefairvalue"D±(t;T)oftheDZCastheconditionalexpectationofthediscountedpayo®¡¢B(t;T)1fT<¿g+±1f¿·Tggiventheinformationavailableattimet,thatis,giventheeventsf¿·ugforu·tandtheeventf¿>tg.Inviewofthespeci¯cationofthebondpayo®,weobtainD±(t;T)=1±B(t;T)+1De±(t;T);ft¸¿gft<¿gwherethepre-defaultvalueDe±isde¯nedas¡¯¢De±(t;T)=EB(t;T)(1+±1)¯t<¿:QfT<¿gf¿·TgWethushave³¯´De±(t;T)=B(t;T)1¡(1¡±)Q(¿·T¯t<¿)µ¶Q(t<¿·T)=B(t;T)1¡(1¡±)Q(t<¿)µ¶G(t)¡G(T)=B(t;T)1¡(1¡±);(2.2)G(t)wherewedenoteG(t)=1¡F(t).Letusdenote,fors¸t,³Zs´dG(s)Rt;s=B(t;s)=exp¡(r(u)+°(u))du:G(t)t 36CHAPTER2.HAZARDFUNCTIONAPPROACHThelastformulacanberepresentedasfollowsµ¶De±(t;T)=B(t;T)G(T)+±G(t)¡G(T)G(t)G(t)¡¢=Rd+±B(t;T)¡Rd:t;Tt;TInparticular,for±=0,thatis,forthecaseofthebondwithzerorecovery,weobtaintheequalityDe0(t;T)=Rdsothatt;TD0(t;T)=1De0(t;T)=1R±:ft<¿gft<¿gt;TItisworthnotingthatthevalueoftheDZCisdiscontinuousattime¿sincewehave,ontheeventf¿·Tg,D±(¿;T)¡D±(¿¡;T)=±B(¿;T)¡De±(¿;T)=(±¡1)Rd6=0;¿;Twherethelastinequalityholdsforany±6=1.Recallthatweassumedthat±<1,sincefor±=1theDZCissimplyadefault-freezero-couponbond.Formula(2.2)canberewrittenasfollowsDe±(t;T)=B(t;T)(1¡LGD£DP);wherethelossgivendefault(LGD)isde¯nedas1¡±andtheconditionaldefaultprobability(DP)isde¯nedasfollowsQ(t<¿·T)DP==Q(¿·Tjt<¿):Q(t<¿)If°>0and±2[0;1)areconstant,thecreditspreadequals³´1B(t;T)1°(T¡t)ln=°¡ln1+±(e¡1):T¡tDe±(t;T)T¡tItisthuseasilyseenthatthecreditspreadconvergesto°(1¡±)whentgoestoT.Recallthatfor±=0,wehaveDe0(t;T)=Rt;d.Hencetheshort-terminterestratehassimplyTtobeadjustedbymeansofthecreditspread(equalhereto°)inordertoevaluateDZCswithzerorecovery.Thedefaultableinterestrateisr+°andthusitis,asexpected,greaterthantherisk-freeinterestrater.ThiscorrespondstotheobviousobservationthatthevalueofaDZCwithzerorecoveryisstrictlysmallerthanthevalueofadefault-freezero-couponwiththesamematurity.CaseofaGeneralRecoveryatMaturityLetusnowassumethatthepaymentisadeterministicfunctionofthedefaulttime,denotedas±:R+!R.Thenthevalueattime0ofthisdefaultablezero-couponis¡¢D±(0;T)=B(0;T)E1+±(¿)1QfT<¿gf¿·Tg³ZT´=B(0;T)G(T)+±(s)f(s)ds;0where,asbefore,G(t)=1¡F(t)standsforthesurvivalprobability.Moregenerally,weset¡¯¢D±(t;T)=B(t;T)E1+±(¿)1¯H(2.3)QfT<¿gf¿·Tgtandwehavethefollowingresult. 2.1.ELEMENTARYMARKETMODEL37Lemma2.1.1Thepriceofthebondsatis¯es,fort2[0;T),D±(t;T)=1De±(t;T)+1±(¿)B(t;T);(2.4)ft<¿gft¸¿gwherethepre-defaultvalueDe±(t;T)equals¡¯¢De±(t;T)=B(t;T)E1+±(¿)1¯t<¿QfT<¿gf¿·TgZTG(T)B(t;T)=B(t;T)+±(u)f(u)duG(t)G(t)tdZTdRt;T=Rt;T+±(u)f(u)duG(T)tZTRTdd¡°(v)dv=Rt;T+Rt;T±(u)°(u)eudu:tThedynamicsofDe±(t;T)aredDe±(t;T)=(r(t)+°(t))De±(t;T)dt¡B(t;T)°(t)±(t)dt:(2.5)Theproofofthelemmaisbasedonstraightforwardcomputations.ToderivethedynamicsofDe±(t;T)itisusefultonote,inparticular,thatdRd=(r(t)+°(t))Rddt:t;Tt;TTherisk-neutraldynamicsofthediscontinuousprocessD±(t;T)involvealsotheH-martingaleMintroducedinSection2.2below(seeExercise2.2.1).2.1.3DefaultableZero-CouponwithRecoveryatDefaultUnderthisconvention,adefaultablezero-couponbondwithmaturityTconsistsof:²thepaymentofonemonetaryunitattimeTifdefaulthasnotyetoccurred,²thepaymentof±(¿)monetaryunits,where±isadeterministicfunction,madeattime¿if¿·T.Thevalueattime0ofthisdefaultablezero-couponbondis¡¢D±(0;T)=EB(0;T)1+B(0;¿)±(¿)1QfT<¿gf¿·TgZT=Q(T<¿)B(0;T)+B(0;u)±(u)dF(u)0ZT=G(T)B(0;T)+B(0;u)±(u)f(u)du:(2.6)0Obviously,ifthedefaulthasoccurredbeforetimet,thevalueoftheDZCisnull(thiswasnotthecasefortherecoverypaymentmadeatbond'smaturity),sinceweadoptthroughouttheex-dividendpriceconventionforallassets.Lemma2.1.2Thepriceofthebondsatis¯es,fort2[0;T),D±(t;T)=1De±(t;T);(2.7)ft<¿g 38CHAPTER2.HAZARDFUNCTIONAPPROACHwherethepre-defaultvalueDe±(t;T)equals¡¯¢De±(t;T)=EB(t;T)1+B(t;¿)±(¿)1¯t<¿QfT<¿gf¿·TgZTG(T)1=B(t;T)+B(t;u)±(u)dF(u)G(t)G(t)tZTd1=Rt;T+B(t;u)±(u)f(u)duG(t)tZT=Rd+Rd±(u)°(u)du:t;Tt;utThedynamicsofDe±(t;T)aredDe±(t;T)=(r(t)+°(t))De±(t;T)dt¡±(t)°(t)dt:(2.8)ThedynamicsofthepriceprocessD±(t;T)includeajumpattime¿(seeProposition2.2.2).FractionalRecoveryofParValueThisschemecorrespondstotheconstantrecovery±atdefault.Thepre-defaultvalueisherethesameasthepre-defaultvalueintherecoveryatmaturityschemewiththefunction±B¡1(t;T).Thisfollowsfromasimplereasoning,butcanalsobeseefromformulaeestablishedinLemmas2.1.1and2.1.2.FractionalRecoveryofTreasuryValueThiscasecorrespondstothefollowingrecovery±(t)=±B(t;T)atthemomentofdefault.Thepre-defaultvalueisinthiscasethesameasthepre-defaultvalueofadefaultablebondwithaconstantrecovery±atmaturityT.Onceagain,thisisalsoeasilyseenfromLemmas2.1.1and2.1.2.Underthisconvention,wehavethatRZT±¡T(r(u)+°(u))du±B(t;T)De(t;T)=et+°(u)G(u)duG(t)tZTRu0¡°(v)dv=De(t;T)+±B(t;T)°(u)etdu:tFractionalRecoveryofMarketValueLetusassumeherethattherecoveryis±(t)De±(t;T)where±isadeterministicfunction,thatis,therecoveryis±(¿)D±(¿¡;T).ItcanbeshownthatthedynamicsofDe±(t;T)are¡¢dDe±(t;T)=r(t)+°(t)(1¡±(t))De±(t;T)dt;andthusZZ³TT´De±(t;T)=exp¡r(u)du¡°(u)(1¡±(u))du:tt2.2MartingaleApproachWeshallnowpresentsomegeneralresultsthatareusefulinthepresentsetup.WeshallworkunderthestandingassumptionthatF(t)<1foranyt2R+,butwedoimposeanyfurtherassumptionsonthecumulativedistributionfunctionFof¿underQatthisstage.Inparticular,wedonotpostulatethatFisacontinuousfunction. 2.2.MARTINGALEAPPROACH39De¯nition2.2.1Thehazardfunction¡isde¯nedbysetting,foranyt2R+,¡(t)=¡ln(1¡F(t)):2.2.1ConditionalExpectationsWe¯rstgiveanelementaryformulaforthecomputationoftheconditionalexpectationwithrespecttoHt,aspresented,forinstance,inBr¶emaud[20],Dellacherie[36],orElliott[45].Lemma2.2.1ForanyQ-integrableandG-measurablerandomvariableXwehavethatEQ(X1ft<¿g)1ft<¿gEQ(XjHt)=1ft<¿g:(2.9)Q(t<¿)Proof.TheconditionalexpectationEQ(XjHt)isclearlyHt-measurable.Therefore,itcanberepresentedasfollowsEQ(XjHt)=h(¿)1f¿·tg+c1ft<¿gforsomeBorelmeasurablefunctionh:R+!Randsomeconstantc.Bymultiplyingbothmembersby1ft<¿gandtakingtheexpectation,weobtainEQ[1ft<¿gEQ(XjHt)]=EQ[EQ(X1ft<¿gjHt)]=EQ(X1ft<¿g)=EQ(c1ft<¿g)=cQ(t<¿):EQ(X1ft<¿g)Hencec=,whichyieldsthedesiredresult.¤Q(t<¿)Corollary2.2.1AssumethatXisanH1-measurableandQ-integrablerandomvariable,sothatX=h(¿)forsomeBorelmeasurablefunctionh:R+!RsuchthatEQjh(¿)j<+1.Ifthehazardfunction¡of¿iscontinuousthenZ1E(XjH)=1h(¿)+1h(u)e¡(t)¡¡(u)d¡(u):(2.10)Qtft¸¿gft<¿gtIf,inaddition,¿admitstheintensityfunction°thenZ1Ru¡°(v)dvEQ(XjHt)=1ft¸¿gh(¿)+1ft<¿gh(u)°(u)etdu:tInparticular,wehave,foranyt·s,Rs¡°(v)dvQ(s<¿jHt)=1ft<¿get(2.11)and³R´s¡°(v)dvQ(t<¿s,E(LjH)=e¡(t)E(1jH):QtsQft<¿gsFrom(2.9),weobtain1¡F(t)¡(s)¡¡(t)EQ(1ft<¿gjHs)=1fs<¿g=1fs<¿ge:1¡F(s) 42CHAPTER2.HAZARDFUNCTIONAPPROACHHenceE(LjH)=1e¡(s)=L:Qtsfs<¿gsToestablish(2.17),itsu±cestonotethatL0=1andtoapplytheintegrationbypartsformulaforfunctionsof¯nitevariation.Recallalsothat¡iscontinuous.WethusobtaindL=¡e¡(t)dH+(1¡H)e¡(t)d¡(t)=¡e¡(t)dMtttt=¡1e¡(t)dM=¡LdM:ft·¿gtt¡tAnalternativemethodistoshowdirectlythatListheexponentialmartingaleofM,thatis,ListheuniquesolutionoftheSDEdLt=¡Lt¡dMt;L0=1:Notethatthisstochasticdi®erentialequationcanbesolvedpathwise.¤Proposition2.2.4Assumethat¡isacontinuousfunction.Leth:R+!RbeaBorelmeasurablefunctionsuchthattherandomvariableh(¿)isQ-integrable.Thentheprocess(M¹h;t2R),givent+bytheformulaZt^¿M¹h=1h(¿)¡h(u)d¡(u);(2.18)tft¸¿g0isanH-martingale.Moreover,foreveryt2R+,ZZM¹h=h(u)dM=¡e¡¡(u)h(u)dL:(2.19)tuu]0;t]]0;t]Proof.TheproofgivenbelowprovidesanalternativeproofofCorollary2.2.2.Wewishtoestablish,throughdirectcalculations,themartingalepropertyoftheprocessM¹hgivenbyformula(2.18).Ontheonehand,formula(2.10)inCorollary2.2.1yieldsZs¡¢Eh(¿)1jH=1e¡(t)h(u)e¡¡(u)d¡(u):Qft<¿·sgtft<¿gtOntheotherhand,wenotethat³Zs^¿´¡¢J:=EQh(u)d¡(u)=EQh~(¿)1ft<¿·sg+h~(s)1f¿>sgjHtt^¿Rswherewewriteh~(s)=h(u)d¡(u).Consequently,usingagain(2.10),weobtaint³Zs´J=1e¡(t)h~(u)e¡¡(u)d¡(u)+e¡¡(s)h~(s):ft<¿gtToconcludetheproof,itisenoughtoobservethattheFubinitheoremyieldsZsZue¡¡(u)h(v)d¡(v)d¡(u)+e¡¡(s)h~(s)ttZsZsZs=h(u)e¡¡(v)d¡(v)d¡(u)+e¡¡(s)h(u)d¡(u)tutZs=h(u)e¡¡(u)d¡(u);tasrequired.Theproofofformula(2.19)islefttothereader.¤Corollary2.2.3Assumethat¡isacontinuousfunction.Leth:R+!RbeaBorelmeasurablefunctionsuchthattherandomvariableeh(¿)isQ-integrable.Thentheprocess(Mfh;t2R),givent+bytheformula¡¢Zt^¿Mfh=exp1h(¿)¡(eh(u)¡1)d¡(u);(2.20)tft¸¿g0isanH-martingale. 2.2.MARTINGALEAPPROACH43Proof.Itisenoughtoobservethat¡¢exp1h(¿)=1eh(¿)+1=1(eh(¿)¡1)+1;ft¸¿gft¸¿gft<¿gft¸¿gandtoapplytheprecedingresulttothefunctioneh¡1.¤Proposition2.2.5Assumethat¡isacontinuousfunction.Leth:R+!RbeaBorelmeasurablefunctionsuchthath¸¡1and,foreveryt2R+,Zth(u)d¡(u)<+1:(2.21)0Thentheprocess(Mch;t2R),givenbytheformulat+¡¢³Zt^¿´Mch=1+1h(¿)exp¡h(u)d¡(u);(2.22)tft¸¿g0isanon-negativeH-martingale.Proof.Observethat³Zt´³Z¿´Mch=exp¡(1¡H)h(u)d¡(u)+1h(¿)exp¡(1¡H)h(u)d¡(u)tuft¸¿gu00³Zt´Zt³Zu´=exp¡(1¡Hu)h(u)d¡(u)+h(u)exp¡(1¡Hs)h(s)d¡(s)dHu:000UsingIt^o'sformula,weobtain³Zt´¡¢dMch=exp¡(1¡H)h(u)d¡(u)h(t)dH¡(1¡H)h(t)d¡(t)tutt0³Zt´=h(t)exp¡(1¡Hu)h(u)d¡(u)dMt:0ThisshowsthatMchisanon-negativelocalH-martingaleandthusasupermartingale.ItcanbecheckeddirectlythatE(Mch)=1foreveryt2R.HenceMchisindeedanH-martingale.¤Qt+2.2.3PredictableRepresentationTheoremWeassumeinthissubsectionthatthehazardfunction¡iscontinuous,sothattheprocessMt=Ht¡¡(t^¿)isanH-martingale.ThenextresultisasuitableversionofthepredictablerepresentationtheoremforH-martingaleswithrespecttothebasicmartingaleM.Proposition2.2.6Leth:R+!RbeaBorelmeasurablefunctionsuchthattherandomvariableh(¿)isQ-integrable.ThenthemartingaleMh=E(h(¿)jH)admitstherepresentationtQtZMh=Mh+(h(u)¡g(u))dM;(2.23)t0u]0;t]whereZ11¡(t)g(t)=h(u)dF(u)=eEQ(h(¿)1ft<¿g)=EQ(h(¿)jt<¿):(2.24)G(t)tMoreover,gisacontinuousfunctionandg(t)=Mhonft<¿g,sothattZMh=Mh+(h(u)¡Mh)dM:t0u¡u]0;t] 44CHAPTER2.HAZARDFUNCTIONAPPROACHProof.FromLemma2.2.1,weobtainhEQ(h(¿)1ft<¿g)Mt=h(¿)1ft¸¿g+1ft<¿gQ(t<¿)=h(¿)1+1e¡(t)E(h(¿)1):ft¸¿gft<¿gQft<¿gWe¯rstconsidertheeventft<¿g.Onthisevent,weclearlyhavethatMh=g(t).AnintegrationtbypartsformulayieldsZ1¡¢Mh=g(t)=e¡(t)Eh(¿)1=e¡(t)h(u)dF(u)tQft<¿gtZ1ZtZt=h(u)dF(u)¡e¡(u)h(u)dF(u)+e¡¡(u)g(u)de¡(u)000Z1ZtZt=h(u)dF(u)¡e¡(u)h(u)dF(u)+g(u)d¡(u):000Ontheotherhand,theright-handsideof(2.23)yields,ontheeventft<¿g,ZtZ1ZtZtE(h(¿))¡(h(u)¡g(u))d¡(u)=h(u)dF(u)¡e¡(u)h(u)dF(u)+g(u)d¡(u);Q0000whereweusedtheequalityd¡(u)=e¡(u)dF(u).Henceequality(2.23)isestablishedontheeventft<¿g.Toprovethat(2.23)holdsontheeventft¸¿gaswell,itsu±cestonotethattheprocessMhandtheprocessgivenbytheright-handsideof(2.23)areconstantonthisevent(thatis,theyarestoppedat¿)andthejumpattime¿ofbothprocessesareidentical;speci¯cally,itequalsh(¿)¡g(¿).Thiscompletestheproof.¤Assumethat¿admitsintensity°.Thenanalternativederivationof(2.23)consistsincomputingtheconditionalexpectationZ1Mh=E(h(¿)jH)=h(¿)1+1e¡(t)h(u)dF(u)tQtft¸¿gft<¿gtZtZ1Zt=h(u)dH+(1¡H)e¡(t)h(u)dF(u)=h(u)dH+(1¡H)g(t):utut0t0NotingthatdF(t)=e¡¡(t)d¡(t)=e¡¡(t)°(t)dt;wegetdg(t)=E(h(¿)1)de¡(t)¡e¡(t)h(t)e¡¡(t)°(t)dt=(g(t)¡h(t))°(t)dt:Qft<¿gWethusobtainfromIt^o'sformuladMh=(h(t)¡g(t))dH+(1¡H)(g(t)¡h(t))°(t)dt=(h(t)¡g(t))dM;ttttsinceobviouslydMt=dHt¡°(t)(1¡Ht)dt:ThefollowingcorollarytoProposition2.2.6emphasizestheroleofthebasicmartingaleM.RtCorollary2.2.4AnyH-martingale(Xt;t2R+)canbewrittenasXt=X0+0³sdMs,where(³t;t2R+)isanH-predictableprocess.Exercise2.2.2Assumethatthehazardfunction¡isright-continuous.EstablishthefollowingformulaZt^¿E(h(¿)jH)=E(h(¿))¡e¢¡(u)(g(u)¡h(u))dM;QtQu0where¢¡(u)=¡(u)¡¡(u¡)andthefunctiongisgivenby(2.24). 2.2.MARTINGALEAPPROACH452.2.4Girsanov'sTheoremLet¿beanon-negativerandomvariableonaprobabilityspace(•;G;Q).WedenotebyFthecumulativedistributionfunctionof¿underQ.WeassumethatF(t)<1foreveryt2R+,sothatthehazardfunction¡of¿underQiswellde¯ned.LetPbeanarbitraryprobabilitymeasureon(•;H1);whichisabsolutelycontinuouswithrespecttoQ.Let´standfortheH1-measurableRadon-Nikod¶ymdensityofPwithrespecttoQdP´:==h(¿)¸0;Q-a.s.,(2.25)dQwhereh:R+!R+isaBorelmeasurablefunctionsatisfyingZEQ(h(¿))=h(u)dF(u)=1:(2.26)[0;1[TheprobabilitymeasurePisequivalenttoQifandonlyiftheinequalityin(2.25)isstrictQ-a.s.LetFbbethecumulativedistributionfunctionof¿underP,thatis,ZFb(t):=P(¿·t)=h(u)dF(u):[0;t]WeassumeFb(t)<1foranyt2R+or,equivalently,thatZP(¿>t)=1¡Fb(t)=h(u)dF(u)>0;(2.27)]t;1[sothatthehazardfunction¡ofb¿underPiswellde¯ned(ofcourse,thisalwaysholdsifPisequivalenttoQ).Putanotherway,weassumethatZ¡¢g(t):=e¡(t)E1h(¿)=e¡(t)h(u)dF(u)=e¡(t)P(¿>t)>0:Qf¿>tg]t;1[Our¯rstgoalistoexaminetherelationshipbetweenthehazardfunctions¡(bt)=¡ln(1¡Fb(t))and¡(t)=¡ln(1¡F(t)).The¯rstresultisanimmediateconsequenceofthede¯nitionofthehazardfunction.Lemma2.2.2Wehave,foreveryt2R+,³R´¡(bt)ln]t;1[h(u)dF(u)=:¡(t)ln(1¡F(t))Fromnowon,weassumethatFisacontinuousfunction.ThefollowingresultcanbeseenasacounterpartofthecelebratedGirsanovtheoremforaBrownianmotion.Lemma2.2.3AssumethatthecumulativedistributionfunctionFof¿underQiscontinuous.ThenthecumulativedistributionfunctionFbof¿underPiscontinuousandwehavethat,foreveryt2R+,Zt¡(bt)=bh(u)d¡(u);0wherethenon-negativefunctionbh:R+!Risgivenbytheformulabh(t)=h(t)=g(t).Hencetheprocess(M;tc2R+),givenbyZt^¿Zt^¿Mct:=Ht¡bh(u)d¡(u)=Mt¡(bh(u)¡1)d¡(u)00isanH-martingaleunderP. 46CHAPTER2.HAZARDFUNCTIONAPPROACHProof.Indeed,ifF(andthusFb)iscontinuous,weobtaindFb(t)d(1¡e¡¡(t)g(t))g(t)d¡(t)¡dg(t)d¡(bt)====bh(t)d¡(t);1¡Fb(t)e¡¡(t)g(t)g(t)whereweusedtheequalities1¡Fb(t)=e¡¡(t)g(t)anddg(t)=(g(t)¡h(t))d¡(t).¤Remark2.2.1Since¡isthehazardfunctionofb¿underP,wenecessarilyhaveZ1lim¡(bt)=bh(t)d¡(t)=+1:(2.28)t!+10Conversely,if¡isthecontinuoushazardfunctionof¿underQandbh:R+!R+isaBorelmeasurablefunctionsuchthat,foreveryt2R+,Zt¡(bt):=bh(u)d¡(u)<+1(2.29)0and(2.28)holdsthenwecan¯ndaprobabilitymeasurePabsolutelycontinuouswithrespecttoQsuchthat¡isthehazardfunctionofb¿underP(seeRemark2.2.2).InthespecialcasewhenFisanabsolutelycontinuousfunction,sothattheintensityfunction°of¿underQiswellde¯ned,thecumulativedistributionfunctionFbof¿underPequalsZtFb(t)=h(u)f(u)du;0sothatFbisanabsolutelycontinuousfunctionaswell.Therefore,theintensityfunction°boftherandomtime¿underPexistsanditisgivenbytheformulah(t)f(t)h(t)f(t)°b(t)==R:t1¡Fb(t)1¡h(u)f(u)du0FromLemma2.2.3,itfollowsthat°b(t)=bh(t)°(t).Tore-derivethisresult,observethath(t)f(t)h(t)f(t)h(t)f(t)h(t)f(t)f(t)°b(t)==Rt=R1=¡¡(t)=bh(t)=bh(t)°(t):1¡Fb(t)1¡0h(u)f(u)duth(u)f(u)dueg(t)1¡F(t)LetusnowexaminetheRadon-Nikod¶ymdensityprocess(´t;t2R+),whichisgivenbytheformuladP´t:==EQ(´jHt):dQjHtProposition2.2.7AssumethatFisacontinuousfunction.ThenZ´t=1+´u¡(bh(u)¡1)dMu(2.30)]0;t]or,equivalently,³Z¢´´t=Et(bh(u)¡1)dMu;0whereEstandsfortheDol¶eansexponential. 2.2.MARTINGALEAPPROACH47Proof.Notethat´=MhwhereMh=E(h(¿)jH).UsingProposition2.2.6andnotingthattttQt´=Mh=1,weobtain(cf.(2.23))00ZtZZ´t=´0+(h(u)¡g(u))dMu=1+(h(u)¡´u¡)dMu=1+´u¡(bh(u)¡1)dMu:0]0;t]]0;t]Thesecondstatementfollowsfromthede¯nitionoftheDol¶eansexponential.¤ItisworthnotingthatZ1´=1h(¿)+1h(u)e¡(t)¡¡(u)d¡(u);tft¸¿gft<¿gtbutalso(thiscanbededucedfrom(2.30))¡¢³Zt^¿´´t=1+1ft¸¿g·(¿)exp¡·(u)d¡(u);(2.31)0wherewewrite·(t)=bh(t)¡1.Sincebhisanon-negativefunction,itisclearthattheinequality·¸¡1holds.Remark2.2.2Let·beanyBorelmeasurablefunction·¸¡1(·>¡1,respectively)suchthatRttheinequality·(u)d¡(u)<+1holdsforeveryt2R+.Then,byvirtueofProposition2.2.5,the0process³Z¢´´·:=E·(u)dMttu0isanon-negative(positive,respectively)H-martingaleunderQ.If,inaddition,wehavethatZ1(1+·(u))d¡(u)=+10then´·=E(´·jH),where´·=lim´·.Inthatcase,wemayde¯neaprobabilitymeasuretQ1t1t!1tPon(•;H)bysettingdP=´·dQ.Thehazardfunction¡ofb¿underPsatis¯esd¡(bt)=11(1+·(t))d¡(t).Notealsothatintermsof·,wehave(cf.Proposition3.4.1)Zt^¿Zt^¿Mct:=Mt¡·(u)d¡(u)=Ht¡(1+·(u))d¡(u):002.2.5RangeofArbitragePricesInordertostudythecompletenessofthe¯nancialmarket,we¯rstneedtospecifytheclassofprimarytradedassets.Inthetoymodel,theprimarytradedassetsaretherisk-freezero-couponbondswithdeterministicprices,andthusthereexistsin¯nitelymanyequivalentmartingalemeasures(EMMs).Indeed,thediscountedassetpricesareconstant,andthustheclassQofallEMMscoincidewiththesetofallprobabilitymeasuresequivalenttothehistoricalprobability.Weassumethatunderthehistoricalprobabilitythedefaulttimeisanunboundedrandomvariablewithastrictlypositiveprobabilitydensityfunction.ForanyQ2Q,wedenotebyFQthecumulativedistributionfunctionof¿underQ,thatis,ZTFQ(t)=Q(¿·t)=fQ(u)du:0Therangeofpricesisde¯nedasthesetofpriceswhichdonotinducearbitrageopportunities.Forinstance,inthecaseofaDZCwithaconstantrecovery±2[0;1[paidatmaturity,therangeofarbitragepricesisequaltothesetfB(0;T)EQ(1fT<¿g+±1f¿·Tg);Q2Qg:Itisnotdi±culttocheckthatthissetisexactlytheopeninterval]±B(0;T);B(0;T)[.Thisrangeofarbitrage(orviable)pricesismanifestlytoowideforpracticalpurposes. 48CHAPTER2.HAZARDFUNCTIONAPPROACH2.2.6ImpliedRisk-NeutralDefaultIntensityItiscommontointerprettheabsenceofarbitrageopportunitiesina¯nancialmodelintermsoftheexistenceofanEMM.Ifdefaultablezero-couponbonds(DZCs)aretraded,theirpricesaregivenbythemarket.Therefore,theequivalentmartingalemeasureQtobeusedforpricingpurposesischosenbythemarket.Wewillnowshowthatitispossibletoderivethecumulativedistributionfunctionof¿underQfromthemarketpricesofdefaultablezero-couponbonds.Itisimportanttostressthatinthepresentsetupthereisnospeci¯crelationshipbetweentherisk-neutraldefaultintensityandthehistoricalone.Therisk-neutraldefaultintensitycanbegreaterorsmallerthanthehistoricalone.Thehistoricaldefaultintensitycanbededucedfromobservationofdefaulttimewhereastherisk-neutraloneisobtainedfromthepricesoftradeddefaultableclaims.ZeroRecoveryIfaDZCwithzerorecoveryofmaturityTistradedatsomepriceD0(t;T)belongingtotheinterval]0;B(t;T)[then,underarisk-neutralprobabilityQ,theprocessB(0;t)D0(t;T)isamartingale.Wedonotpostulatethatthemarketmodeliscomplete,sowedonotclaimthatanequivalentmartingalemeasureisunique.ThefollowingequalitiesholdunderamartingalemeasureQ2Q³ZT´B(0;t)D0(t;T)=E(B(0;T)1jH)=B(0;T)1exp¡°Q(u)duQfT<¿gtft<¿gtQfQ(u)where°(u)=.Letusnowconsidert=0.Itiseasilyseenthatifforanymaturity1¡FQ(u)dateTthepriceD0(0;T)belongstotherangeofviableprices]0;B(0;T)[thenthefunction°Qisstrictlypositive,andtheconverseimplicationholdsaswell.Thefunction°Q,whichsatis¯es,foreveryT>0,³ZT´D0(0;T)=B(0;T)exp¡°Q(u)du0istheimpliedrisk-neutraldefaultintensity,thatis,theuniqueQ-intensityof¿thatisconsistentRTwiththemarketdataforDZCs.Moreprecisely,thevalueoftheintegral°Q(s)dsisknownfor0anyTassoonastheDZCbondswithallmaturitiesT>0aretradedattime0.Theuniquerisk-neutralintensitycanbeformallyobtainedfromthepricesofDZCsbydi®eren-tiationwithrespecttomaturityr(t)+°Q(t)=¡@lnD0(0;T)j;TT=tprovided,ofcourse,thatthepartialderivativeintheright-handsideofthelastformulaiswellde¯ned.RecoveryatMaturityAssumethatthepricesofDZCswithdi®erentmaturitiesand¯xedrecovery±atmaturity,areknown.Thenwededucefrom(2.1)B(0;T)¡D±(0;T)FQ(T)=:B(0;T)(1¡±)Hencetheprobabilitydistributionof¿undertheEMMimpliedbythemarketquotesofDZCsisuniquelydetermined.However,asobservedbyHullandWhite[54],extractingrisk-neutraldefaultprobabilitiesfrombondpricesisinpracticemorecomplicated,sincemostcorporatebondsarecoupon-bearingbonds. 2.2.MARTINGALEAPPROACH49RecoveryatDefaultInthiscase,thecumulativedistributionfunctioncanalsobeobtainedbydi®erentiationofthedefaultablezero-couponcurvewithrespecttothematurity.Indeed,denotingby@D±(0;T)theTderivativeofthevalueoftheDZCattime0withrespecttothematurityandassumingthatG=1¡Fisdi®erentiable,weobtainfrom(2.6)@D±(0;T)=g(T)B(0;T)¡G(T)B(0;T)r(T)¡±(T)g(T)B(0;T);Twherewewriteg(t)=G0(t).Bysolvingthisequation,weobtainÃZ!t(K(u))¡1Q(¿>t)=G(t)=K(t)1+@D±(0;u)du;T0B(0;u)(1¡±(u))³Zt´r(u)wherewedenoteK(t)=expdu.01¡±(u)2.2.7DynamicsofPricesofDefaultableClaimsThissectiongivesasummaryofbasicresultsconcerningthedynamicsofpricesofdefaultableclaims.Forthesakeofsimplicity,wepostulateherethattheinterestraterisconstantandweassumethatthedefaultintensity°iswellde¯ned.RecoveryatMaturityLetSbethepriceofanassetthatonlydeliversarecoveryZ(¿)attimeTforsomefunctionZ.Formally,thiscorrespondstothedefaultableclaim(0;0;Z(¿);0;¿),thatis,Xe=Z(¿).WeknowalreadythattheprocessZtMt=Ht¡(1¡Hu)°(u)du0isanH-martingale.Recallthat°(t)=f(t)=G(t),wherefistheprobabilitydensityfunctionof¿.Observethat¡r(T¡t)¡r(T¡t)¡r(T¡t)EQ(Z(¿)1ft<¿·Tg)St=EQ(Z(¿)ejHt)=1ft¸¿geZ(¿)+1ft<¿geG(t)Zt=e¡r(T¡t)Z(u)dH+1e¡r(T¡t)Ze(t);uft<¿g0wherethefunctionZe:[0;T]!RisgivenbytheformulaRTZe(t)=EQ(Z(¿)1ft<¿·Tg)tZ(u)f(u)du=:G(t)G(t)ItiseasilyseenthatRTZ(u)f(u)duZ(t)f(t)f(t)Z(t)f(t)dZe(t)=f(t)tdt¡dt=Ze(t)dt¡dt;G2(t)G(t)G(t)G(t)andthusµ¶¡rt¡rTf(t)¡Ze(t)¡Z(t)¢d(eSt)=eZ(t)dHt+(1¡Ht)dt¡Ze(t¡)dHtG(t)¡¢¡¢=e¡rTZ(t)¡e¡rtSdH¡(1¡H)°(t)dtt¡tt¡¢=e¡rte¡r(T¡t)Z(t)¡SdM:t¡tThediscountedpriceishereanH-martingaleundertherisk-neutralprobabilityQandthepriceSdoesnotvanish(unlessZequalszero). 50CHAPTER2.HAZARDFUNCTIONAPPROACHRecoveryatDefaultAssumenowthattherecoverypayo®isreceivedatdefaulttime.Hencewedealherewiththedefaultableclaim(0;0;0;Z;¿),andthusthepriceofthisclaimisobviouslyequaltozeroafter¿.Ingeneral,wehaveE(e¡r(¿¡t)Z(¿)1)S=E(e¡r(¿¡t)Z(¿)1jH)=1Qft<¿·Tg=1ertZb(t);tQft<¿·Tgtft<¿gft<¿gG(t)wherethefunctionZb:[0;T]!RisgivenbytheformulaZTZb(t)=1Z(u)e¡ruf(u)du:G(t)tNotethatRTZ(u)e¡ruf(u)du¡rtf(t)tdZb(t)=¡Z(t)edt+f(t)dtG(t)G2(t)¡rtf(t)f(t)=¡Z(t)edt+Zb(t)dtG(t)G(t)¡¢=°(t)Zb(t)¡Z(t)e¡rtdt:Consequently,¡¢d(e¡rtS)=(1¡H)°(t)Zb(t)¡Z(t)e¡rtdt¡Zb(t)dHttt¡¢=Z(t)e¡rt¡Zb(t)dM¡Z(t)e¡rt(1¡H)°(t)dttt=e¡rt(Z(t)¡S)dM¡Z(t)e¡rt(1¡H)°(t)dt:t¡ttInthatcase,thediscountedpricee¡rtSisnotanH-martingaleundertherisk-neutralprobability.tBycontrast,theprocessZt^¿Se¡rt+e¡ruZ(u)°(u)dut0isanH-martingale.ItisalsoworthnotingthattherecoverycanbeformallyinterpretedhereasadividendstreampaidattherateZ°uptotime¿^T.2.3GenericDefaultableClaimsLetus¯rstrecallthenotation.Astrictlypositiverandomvariable¿,de¯nedonaprobabilityspace(•;G;Q),istermedarandomtime.Inviewofitsinterpretation,itwillbelaterreferredtoasadefaulttime.WeintroducethedefaultindicatorprocessHt=1f¿·tgassociatedwith¿,andwedenotebyHthe¯ltrationgeneratedbythisprocess.Weassumefromnowonthatwearegiven,inaddition,someauxiliary¯ltrationF,andwewriteG=H_F,meaningthatwehaveGt=¾(Ht;Ft)foreveryt2R+.De¯nition2.3.1ByadefaultableclaimmaturingatTwemeanthequadruple(X;A;Z;¿),whereXisanFT-measurablerandomvariable,AisanF-adaptedprocessof¯nitevariation,ZisanF-predictableprocess,and¿isarandomtime.The¯nancialinterpretationofthecomponentsofadefaultableclaimbecomesclearfromthefollowingde¯nitionofthedividendprocessD,whichdescribesallcash°owsassociatedwithadefaultableclaimoverthelifespan]0;T],thatis,afterthecontractwasinitiatedattime0.Ofcourse,thechoiceof0asthedateofinceptionisarbitrary. 2.3.GENERICDEFAULTABLECLAIMS51De¯nition2.3.2ThedividendprocessDofadefaultableclaimmaturingatTequals,foreveryt2[0;T],ZZDt=X1f¿>Tg1[T;1[(t)+(1¡Hu)dAu+ZudHu:]0;t]]0;t]The¯nancialinterpretationofthede¯nitionabovejusti¯esthefollowingterminology:Xisthepromisedpayo®,Arepresentstheprocessofpromiseddividends,andtheprocessZ,termedtherecoveryprocess,speci¯estherecoverypayo®atdefault.Itisworthstressingthat,accordingtoourconvention,thecashpayment(premium)attime0isnotincludedinthedividendprocessDassociatedwithadefaultableclaim.Whendealingwithacreditdefaultswap,itisnaturaltoassumethatthepremiumpaidattime0equalszero,andtheprocessArepresentsthefee(annuity)paidininstalmentsuptomaturitydateordefault,whichevercomes¯rst.Forinstance,ifAt=¡·tforsomeconstant·>0,thenthe`price'ofastylizedcreditdefaultswapisformallyrepresentedbythisconstant,referredtoasthecontinuouslypaidcreditdefaultrateorpremium(seeSection2.4.1fordetails).Iftheothercovenantsofthecontractareknown(i.e.,thepayo®sXandZaregiven),thevaluationofaswapisequivalentto¯ndingtheleveloftherate·thatmakestheswapvaluelessatinception.Typically,inacreditdefaultswapwehaveX=0,andZisdeterminedinreferencetorecoveryrateofareferencecredit-riskyentity.Inamorerealisticapproach,theprocessAisdiscontinuous,withjumpsoccurringatthepremiumpaymentdates.Inthisnote,weshallonlydealwithastylizedCDSwithacontinuouslypaidpremium.Letusreturntothegeneralsetup.ItisclearthatthedividendprocessDfollowsaprocessof¯nitevariationon[0;T].SinceZZ(1¡Hu)dAu=1f¿>ugdAu=A¿¡1f¿·tg+At1f¿>tg;]0;t]]0;t]itisalsoapparentthatifdefaultoccursatsomedatet,the`promiseddividend'At¡At¡thatisduetobereceivedorpaidatthisdateisdisregarded.Ifwedenote¿^t=min(¿;t)thenwehaveZZudHu=Z¿^t1f¿·tg=Z¿1f¿·tg:]0;t]LetusstressthattheprocessDu¡Dt;u2[t;T],representsallcash°owsfromadefaultableclaimreceivedbyaninvestorwhopurchasesitattimet.Ofcourse,theprocessDu¡Dtmaydependonthepastbehavioroftheclaim(e.g.,throughsomeintrinsicparameters,suchascreditspreads)aswellasonthehistoryofthemarketpriortot.Thepastdividendsarenotvaluedbythemarket,however,sothatthecurrentmarketvalueattimetofaclaim(i.e.,thepriceatwhichittradesattimet)dependsonlyonfuturedividendstobepaidorreceivedoverthetimeinterval]t;T].Supposethatourunderlying¯nancialmarketmodelisarbitrage-free,inthesensethatthereexistsaspotmartingalemeasureQ(alsoreferredtoasarisk-neutralprobability),meaningthatQisequivalenttoQon(•;GT),andthepriceprocessofanytradeablesecurity,payingnocouponsordividends,followsaG-martingaleunderQ,whendiscountedbythesavingsaccountB,givenbyµZ¶tBt=exprudu;8t2R+:(2.32)02.3.1Buy-and-HoldStrategyWewriteSi;i=1;:::;ktodenotethepriceprocessesofkprimarysecuritiesinanarbitrage-free¯nancialmodel.WemakethestandardassumptionthattheprocessesSi;i=1;:::;k¡1followsemimartingales.Inaddition,wesetSk=BsothatSkrepresentsthevalueprocessofthesavingsttaccount.Thelastassumptionisnotnecessary,however.Wecanassume,forinstance,thatSkisthe 52CHAPTER2.HAZARDFUNCTIONAPPROACHpriceofaT-maturityrisk-freezero-couponbond,orchooseanyotherstrictlypositivepriceprocessasasnum¶eraire.Forthesakeofconvenience,weassumethatSi;i=1;:::;k¡1arenon-dividend-payingassets,andweintroducethediscountedpriceprocessesSi¤bysettingSi¤=Si=B.Allprocessesaretttassumedtobegivenona¯lteredprobabilityspace(•;G;Q),whereQisinterpretedasthereal-life(i.e.,statistical)probabilitymeasure.Letusnowassumethatwehaveanadditionaltradedsecuritythatpaysdividendsduringitslifespan,assumedtobethetimeinterval[0;T],accordingtoaprocessof¯nitevariationD,withD0=0.LetSdenotea(yetunspeci¯ed)priceprocessofthissecurity.Inparticular,wedonotpostulateapriorithatSfollowsasemimartingale.ItisnotnecessarytointerpretSasapriceprocessofadefaultableclaim,thoughwehaveherethisparticularinterpretationinmind.LetaG-predictable,Rk+1-valuedprocessÁ=(Á0;Á1;:::;Ák)representagenerictradingstrat-egy,whereÁjrepresentsthenumberofsharesofthejthassetheldattimet.WeidentifyhereS0twithS,sothatSisthe0thasset.Inordertoderiveapricingformulaforthisasset,itsu±cestoexamineasimpletradingstrategyinvolvingS,namely,thebuy-and-holdstrategy.Supposethatoneunitofthe0thassetwaspurchasedattime0,attheinitialpriceS,andit0washolduntiltimeT.Weassumealltheproceedsfromdividendsarere-investedinthesavingsaccountB.Morespeci¯cally,weconsiderabuy-and-holdstrategyÃ=(1;0;:::;0;Ãk),whereÃkisaG-predictableprocess.TheassociatedwealthprocessV(Ã)equalsV(Ã)=S+ÃkB;8t2[0;T];(2.33)ttttsothatitsinitialvalueequalsV(Ã)=S+Ãk.000De¯nition2.3.3WesaythatastrategyÃ=(1;0;:::;0;Ãk)isself-¯nancingifdV(Ã)=dS+dD+ÃkdB;tttttormoreexplicitly,foreveryt2[0;T],ZV(Ã)¡V(Ã)=S¡S+D+ÃkdB:(2.34)t0t0tuu]0;t]WeassumefromnowonthattheprocessÃkischoseninsuchaway(withrespecttoS;DandB)thatabuy-and-holdstrategyÃisself-¯nancing.Also,wemakeastandingassumptionthattheRrandomvariableY=B¡1dDisQ-integrable.]0;T]uuLemma2.3.1ThediscountedwealthV¤(Ã)=B¡1V(Ã)ofanyself-¯nancingbuy-and-holdtradingtttstrategyÃsatis¯es,foreveryt2[0;T],ZV¤(Ã)=V¤(Ã)+S¤¡S¤+B¡1dD:(2.35)t0t0uu]0;t]Hencewehave,foreveryt2[0;T],ZV¤(Ã)¡V¤(Ã)=S¤¡S¤+B¡1dD:(2.36)TtTtuu]t;T]Proof.Wede¯neanauxiliaryprocessVb(Ã)bysettingVb(Ã)=V(Ã)¡S=ÃkBfort2[0;T].Intttttviewof(2.34),wehaveZVb(Ã)=Vb(Ã)+D+ÃkdB;t0tuu]0;t] 2.3.GENERICDEFAULTABLECLAIMS53andsotheprocessVb(Ã)followsasemimartingale.AnapplicationofIt^o'sproductruleyields¡¢dB¡1Vb(Ã)=B¡1dVb(Ã)+Vb(Ã)dB¡1tttttt=B¡1dD+ÃkB¡1dB+ÃkBdB¡1tttttttt¡1=BtdDt;¡1¡1wherewehaveusedtheobviousidentity:BtdBt+BtdBt=0.Integratingthelastequality,weobtainZ¡¢¡¢B¡1V(Ã)¡S=B¡1V(Ã)¡S+B¡1dD;ttt000uu]0;t]andthisimmediatelyyields(2.35).¤ItisworthnotingthatLemma2.3.1remainsvalidiftheassumptionthatSkrepresentsthesavingsaccountBisrelaxed.Itsu±cestoassumethatthepriceprocessSkisanum¶eraire,thatis,astrictlypositivecontinuoussemimartingale.Forthesakeofbrevity,letuswriteSk=¯.WesaythatÃ=(1;0;:::;0;Ãk)isself-¯nancingitthewealthprocessV(Ã)=S+Ãk¯;8t2[0;T];ttttsatis¯es,foreveryt2[0;T],ZV(Ã)¡V(Ã)=S¡S+D+Ãkd¯:t0t0tuu]0;t]Lemma2.3.2TherelativewealthV¤(Ã)=¯¡1V(Ã)ofaself-¯nancingtradingstrategyÃsatis¯es,tttforeveryt2[0;T],ZV¤(Ã)=V¤(Ã)+S¤¡S¤+¯¡1dD;t0t0uu]0;t]whereS¤=¯¡1S.ttProof.Theproofproceedsalongthesamelinesasbefore,notingthat¯1d¯+¯d¯1+dh¯;¯1i=0.¤2.3.2SpotMartingaleMeasureOurnextgoalistoderivetherisk-neutralvaluationformulafortheex-dividendpriceSt.Tothisend,weassumethatourmarketmodelisarbitrage-free,meaningthatitadmitsa(notnecessarilyunique)martingalemeasureQ,equivalenttoQ,whichisassociatedwiththechoiceofBasanum¶eraire.De¯nition2.3.4WesaythatQisaspotmartingalemeasureifthediscountedpriceSi¤ofanynon-dividendpayingtradedsecurityfollowsaQ-martingalewithrespecttoG.ItiswellknownthatthediscountedwealthprocessV¤(Á)ofanyself-¯nancingtradingstrat-egyÁ=(0;Á1;Á2;:::;Ák)isalocalmartingaleunderQ.Inwhatfollows,weshallonlyconsideradmissibletradingstrategies,thatis,strategiesforwhichthediscountedwealthprocessV¤(Á)isamartingaleunderQ.Amarketmodelinwhichonlyadmissibletradingstrategiesareallowedisarbitrage-free,thatis,therearenoarbitrageopportunitiesinthismodel.Followingthislineofarguments,wepostulatethatthetradingstrategyÃintroducedinSection2.3.1isalsoadmissible,sothatitsdiscountedwealthprocessV¤(Ã)followsamartingaleunderQwithrespecttoG.Thisassumptionisquitenaturalifwewishtopreventarbitrageopportunitiestoappearintheextendedmodelofthe¯nancialmarket.Indeed,sincewepostulatethatSistraded,thewealthprocessV(Ã)canbeformallyseenasanadditionalnon-dividendpayingtradeablesecurity. 54CHAPTER2.HAZARDFUNCTIONAPPROACHToderiveapricingformulaforadefaultableclaim,wemakeanaturalassumptionthatthemarketvalueattimetofthe0thsecuritycomesexclusivelyfromthefuturedividendsstream,thatis,fromthecash°owsoccurringintheopeninterval]t;T[.SincethelifespanofSis[0;T],thisamountstopostulatethatS=S¤=0.Toemphasizethisproperty,weshallrefertoSastheTTex-dividendpriceofthe0thasset.De¯nition2.3.5AprocessSwithS=0istheex-dividendpriceofthe0thassetifthediscountedTwealthprocessV¤(Ã)ofanyself-¯nancingbuy-and-holdstrategyÃfollowsaG-martingaleunderQ.Asaspecialcase,weobtaintheex-dividendpriceadefaultableclaimwithmaturityT.Proposition2.3.1Theex-dividendpriceprocessSassociatedwiththedividendprocessDsatis¯es,foreveryt2[0;T],³Z¯´¡1¯St=BtEQBudDu¯Gt:(2.37)]t;T]Proof.ThepostulatedmartingalepropertyofthediscountedwealthprocessV¤(Ã)yields,foreveryt2[0;T],¡¯¢EV¤(Ã)¡V¤(Ã)¯G=0:QTttTakingintoaccount(2.36),wethusobtain³Z¯´¤¤¡1¯St=EQST+BudDu¯Gt:]t;T]Since,byvirtueofthede¯nitionoftheex-dividendpricewehaveS=S¤=0,thelastformulaTTyields(2.37).¤Itisnotdi±culttoshowthattheex-dividendpriceSsatis¯esSt=1ft<¿gSetfort2[0;T],wheretheprocessSerepresentstheex-dividendpre-defaultpriceofadefaultableclaim.ThecumulativepriceprocessScassociatedwiththedividendprocessDisgivenbytheformula,foreveryt2[0;T],³Z¯´c¡1¯St=BtEQBudDu¯Gt:(2.38)]0;T]Thecorrespondingdiscountedcumulativepriceprocess,Sb:=B¡1Sc,isaG-martingaleunderQ.ThesavingsaccountBcanbereplacedbyanarbitrarynum¶eraire¯.Thecorrespondingvaluationformulabecomes,foreveryt2[0;T],³Z¯´¡1¯St=¯tEQ¯¯udDu¯Gt;(2.39)]t;T]whereQ¯isamartingalemeasureon(•;G)associatedwithanum¶eraire¯,thatis,aprobabilityTmeasureon(•;GT)givenbytheformuladQ¯¯T=;Q-a.s.dQ¯0BT2.3.3Self-FinancingTradingStrategiesLetusnowexamineageneraltradingstrategyÁ=(Á0;Á1;:::;Ák)withG-predictablecomponents.Pkii0TheassociatedwealthprocessV(Á)equalsVt(Á)=i=0ÁtSt,where,asbeforeS=S.AstrategyÁissaidtobeself-¯nancingifVt(Á)=V0(Á)+Gt(Á)foreveryt2[0;T];wherethegainsprocessG(Á)isde¯nedasfollows:ZXkZG(Á)=Á0dD+ÁidSi:tuuuu]0;t]i=0]0;t] 2.3.GENERICDEFAULTABLECLAIMS55Corollary2.3.1LetSk=B.Thenforanyself-¯nancingtradingstrategyÁ,thediscountedwealthprocessV¤(Á)=B¡1V(Á)followsamartingaleunderQ:ttProof.SinceBisacontinuousprocessof¯nitevariation,It^o'sproductrulegivesdSi¤=SidB¡1+B¡1dSitttttfori=0;1;:::;k,andsodV¤(Á)=V(Á)dB¡1+B¡1dV(Á)ttttt³Xk´=V(Á)dB¡1+B¡1ÁidSi+Á0dDttttttti=0Xk¡¢=ÁiSidB¡1+B¡1dSi+Á0B¡1dDtttttttti=0kX¡1¡¢kX¡1=ÁidSi¤+Á0dS¤+B¡1dD=ÁidSi¤+Á0dSc;¤;tttttttttti=1i=1wheretheauxiliaryprocessSc;¤isgivenbythefollowingexpression:ZSc;¤=S¤+B¡1dD:ttuu]0;t]Toconclude,itsu±cestoobservethatinviewof(2.37)theprocessSc;¤satis¯es³Z¯´c;¤¡1¯St=EQBudDu¯Gt;(2.40)]0;T]andthusitfollowsamartingaleunderQ:¤c;¤ItisworthnotingthatSt,givenbyformula(2.40),representsthediscountedcumulativepriceattimetofthe0thasset,thatis,thearbitragepriceattimetofallpastandfuturedividendsassociatedwiththe0thassetoveritslifespan.Tocheckthis,letusconsiderabuy-and-holdstrategysuchthatÃk=0.Then,inviewof(2.36),theterminalwealthattimeTofthisstrategyequals0ZV(Ã)=BB¡1dD:(2.41)TTuu]0;T]ItisclearthatVT(Ã)representsalldividendsfromSintheformofasinglepayo®attimeT.Thearbitrageprice¼t(Yb)attimet0foreveryt2[0;T]. 2.4.HEDGINGOFSINGLENAMECREDITDERIVATIVES57Sincewestartwithonlyonetradeableassetinourmodel(thesavingsaccount),itisclearthatanyprobabilitymeasureQon(•;HT)equivalenttoQcanbechosenasaspotmartingalemeasure.ThechoiceofQisre°ectedinthecumulativedistributionfunctionF(inparticular,inthedefaultintensityifFadmitsadensityfunction).Inpracticalapplicationsofreduced-formmodels,thechoiceofFisdonebycalibration.2.4.2PricingofaCDSSincetheex-dividendpriceofaCDSisthepriceatwhichitisactuallytraded,weshallrefertotheex-dividendpriceasthepriceinwhatfollows.Recallthatwealsointroducedtheso-calledcumulativeprice,whichencompassesalsopastdividendsreinvestedinthesavingsaccount.Lets2[0;T]standsforsome¯xeddate.WeconsiderastylizedT-maturityCDScontractwithaconstantrate·anddefaultprotectionfunction±,initiatedattimesandmaturingatT.ThedividendprocessofaCDSequalsZZDt=±(u)dHu¡·(1¡Hu)du(2.42)]0;t]]0;t]andthus,inviewof(2.37),thepriceofthisCDSisgivenbytheformula³¯¯´³¡¢¯¯´St(·;±;T)=EQ1ft<¿·Tg±(¿)¯Ht¡EQ1ft<¿g·(¿^T)¡t¯Ht(2.43)wherethe¯rstconditionalexpectationrepresentsthecurrentvalueofthedefaultprotectionstream(ortheprotectionleg),andthesecondisthevalueofthesurvivalannuitystream(orthefeeleg).Toalleviatenotation,weshallwriteSt(·)insteadofSt(·;±;T)inwhatfollows.Lemma2.4.1Thepriceattimet2[s;T]ofacreditdefaultswapstartedats,withrate·andprotectionpayment±(¿)atdefault,equalsÃZZ!TT1St(·)=1ft<¿g¡±(u)dG(u)¡·G(u)du:(2.44)G(t)ttProof.Wehave,onthesetft<¿g,RÃR!TT±(u)dG(u)¡udG(u)+TG(T)S(·)=¡t¡·t¡ttG(t)G(t)ÃZZ!T³T´1=¡±(u)dG(u)¡·TG(T)¡tG(t)¡udG(u):G(t)ttSinceZZTTG(u)du=TG(T)¡tG(t)¡udG(u);(2.45)ttweconcludethat(2.44)holds.¤Thepre-defaultpriceisde¯nedastheuniquefunctionSe(·)suchthatwehave(seeLemma2.5.1withn=1)St(·)=1ft<¿gSet(·);8t2[0;T]:(2.46)Combining(2.44)with(2.46),we¯ndthatthepre-defaultpriceoftheCDSequals,fort2[s;T],ÃZZ!TT1Set(·)=¡±(u)dG(u)¡·G(u)du=±e(t;T)¡·Ae(t;T)(2.47)G(t)tt 58CHAPTER2.HAZARDFUNCTIONAPPROACHwhereZT1±e(t;T)=¡±(u)dG(u)G(t)tisthepre-defaultpriceattimetoftheprotectionleg,andZT1Ae(t;T)=G(u)duG(t)trepresentsthepre-defaultpriceattimetofthefeelegfortheperiod[t;T]peroneunitofspread·.WeshallrefertoAe(t;T)astheCDSannuity.NotethatSe(·)isacontinuousfunction,underourassumptionthatGiscontinuous.2.4.3MarketCDSRateACDSthathasnullvalueatitsinceptionplaysanimportantroleasabenchmarkCDS,andthusweintroduceaformalde¯nition,inwhichitisimplicitlyassumedthatarecoveryfunction±ofaCDSisgivenandthatweareontheeventfs<¿g.De¯nition2.4.2AmarketCDSstartedatsistheCDSinitiatedattimeswhoseinitialvalueisequaltozero.TheT-maturitymarketCDSrate(alsoknownasthefairCDSspread)attimesisthe¯xedleveloftherate·=·(s;T)thatmakestheT-maturityCDSstartedatsvaluelessatitsinception.ThemarketCDSrateattimesisthusdeterminedbytheequationSes(·(s;T))=0whereSes(·)isgivenby(2.47).Underthepresentassumptions,byvirtueof(2.47),theT-maturitymarketCDSrate·(s;T)equals,foreverys2[0;T],RT±e(s;T)±(u)dG(u)·(s;T)==¡s:(2.48)Ae(s;T)RTG(u)dusExample2.4.1Assumethat±(t)=±isconstant,andF(t)=1¡e¡°tforsomeconstantdefaultintensity°>0underQ.Inthatcase,thevaluationformulaeforaCDScanbefurthersimpli¯ed.InviewofLemma2.4.1,theex-dividendpriceofa(spot)CDSwithrate·equals,foreveryt2[0;T],³´S(·)=1(±°¡·)°¡11¡e¡°(T¡t):tft<¿gThelastformula(orthegeneralformula(2.48))yields·(s;T)=±°foreveryst)=exp¡°(u)du;8t2[0;T];0wherethedefaultintensity°(t)underQisastrictlypositivedeterministicfunction.RecallthattheprocessM,givenbytheformulaZtMt=Ht¡(1¡Hu)°(u)du;8t2[0;T];(2.51)0isanH-martingaleunderQ.We¯rstfocusondynamicsofthepriceofaCDSwithrate·startedatsomedatesTg+Z(¿)1f¿·Tg:(2.59)Remark2.4.1Alternatively,onemaysaythataself-¯nancingtradingstrategyÁ=(Á;0)(i.e.,atradingstrategywithC=0)replicatesadefaultableclaim(X;A;Z;¿)ifandonlyifV¿^T(Á)=Yb,wherewesetYb=X1f¿>Tg+A(¿^T)+Z(¿)1f¿·Tg:(2.60)However,inthecaseofnon-zero(possiblyrandom)interestrates,itismoreconvenienttode¯nereplicationofadefaultableclaimviaDe¯nition2.4.3,sincetherunningpayo®sspeci¯edbyAaredistributedovertimeandthus,inprinciple,theyneedtobediscountedaccordingly(thisdoesnotshowin(2.60),sinceitisassumedherethatr=0).Letusdenote,foreveryt2[0;T],ÃZ!T1Ze(t)=XG(T)¡Z(u)dG(u)(2.61)G(t)t 2.4.HEDGINGOFSINGLENAMECREDITDERIVATIVES61andZ1Ae(t)=G(u)dA(u):(2.62)G(t)]t;T]Let¼and¼ebetherisk-neutralvalueandthepre-defaultrisk-neutralvalueofadefaultableclaimunderQ,sothat¼t=1ft<¿g¼e(t)foreveryt2[0;T].Also,let¼bstandforitsrisk-neutralcumulativeprice.Itisclearthat¼e(0)=¼(0)=¼b(0)=EQ(Yb)Proposition2.4.2Thepre-defaultrisk-neutralvalueofadefaultableclaim(X;A;Z;¿)equals¼e(t)=Ze(t)+Ae(t)foreveryt2[0;T[(obviously,¼e(T)=0).Therefore,fort2[0;T[,d¼e(t)=°(t)(¼e(t)¡Z(t))dt¡dA(t):(2.63)Moreoverd¼t=¡¼e(t¡)dMt¡°(t)(1¡Ht)Z(t)dt¡dA(t^¿)(2.64)andd¼bt=(Z(t)¡¼e(t¡))dMt:(2.65)Proof.Theproofofequality¼e(t)=Ze(t)+Ae(t)issimilartothederivationofformula(2.47).Wehave,fort2[0;T[,³¯´¯¼t=EQ1ft<¿gY+A(¿^T)¡A(¿^t)¯Ht³ZT´Z11=1ft<¿gXG(T)¡Z(u)dG(u)+1ft<¿gG(u)dA(u)G(t)tG(t)]t;T]=1ft<¿g(Ze(t)+Ae(t))=1ft<¿g¼e(t):Byelementarycomputation,weobtaindZe(t)=°(t)(Ze(t)¡Z(t))dt;dAe(t)=°(t)Ae(t)dt¡dA(t);andthus(2.63)holds.Formula(2.64)followseasilyfrom(2.63)andtheintegrationbypartsformulaappliedtotheequality¼t=(1¡Ht)¼e(t)(seetheproofofLemma2.4.2forsimilarcomputations).Thelastformulaisalsoclear.¤Thenextpropositionshowsthattherisk-neutralvalueofadefaultableclaimisalsoitsreplicationprice,thatis,adefaultableclaimderivesitsvaluefromthepriceofthetradedCDS.Theorem2.1AssumethattheinequalitySe(·)6=±(t)holdsforeveryt2[0;T].LetÁ1=Áe(¿^t),tt1wherethefunctionÁe1:[0;T]!RisgivenbytheformulaZ(t)¡¼e(t¡)Áe1(t)=;8t2[0;T];(2.66)±(t)¡Set(·)andletÁ0=V(Á;A)¡Á1S(·),wheretheprocessV(Á;A)isgivenbytheformulattttZV(Á;A)=¼e(0)+Áe(u)dSc(·)¡A(t^¿):(2.67)t1u]0;¿^t]Thenthetradingstrategy(Á0;Á1;A)replicatesadefaultableclaim(X;A;Z;¿).Proof.Assume¯rstthatatradingstrategyÁ=(Á0;Á1;C)isareplicatingstrategyfor(X;A;Z;¿).Byvirtueofcondition(i)inDe¯nition2.4.3wehaveC=Aandthus,bycombining(2.67)with(2.53),weobtaindV(Á;A)=Á1(±(t)¡Se(·))dM¡dA(¿^t)tttt 62CHAPTER2.HAZARDFUNCTIONAPPROACHForÁ1givenby(2.66),wethusobtaindVt(Á;A)=(Z(t)¡¼e(t¡))dMt¡dA(¿^t):ItisthusclearthatifwetakeÁ1=Áe(¿^t)withÁegivenby(2.66),andtheinitialconditiont11V0(Á;A)=¼e(0)=¼0,thenwehavethatVt(Á;A)=¼e(t)foreveryt2[0;T[ontheeventft<¿g.Byexamining,inparticular,thejumpofthewealthprocessV(Á;A)atthemomentofdefault,onemaycheckthatallconditionsofDe¯nition2.4.3areindeedsatis¯ed.¤Remark2.4.2Ofcourse,ifwetakeas(X;A;Z;¿)aCDSwithrate·andrecoveryfunction±,thenwehaveZ(t)=±(t)and¼e(t¡)=¼e(t)=Se(·),sothatÁ1=1foreveryt2[0;T].tt2.5DynamicHedgingofBasketCreditDerivativesInthissection,weshallexaminehedgingof¯rst-to-defaultbasketclaimswithsinglenamecreditdefaultswapsontheunderlyingncreditnames,denotedas1;2;:::;n.Ourstandingassumption(A)ismaintainedthroughoutthissection.Lettherandomtimes¿1;¿2;:::;¿ngivenonacommonprobabilityspace(•;G;Q)representthedefaulttimesofwithncreditnames.Wedenoteby¿(1)=¿1^¿2^:::^¿n=min(¿1;¿2;:::;¿n)themomentofthe¯rstdefault,sothatnodefaultsareobservedontheeventf¿(1)>tg.LetF(t1;t2;:::;tn)=Q(¿1·t1;¿2·t2;:::;¿n·tn)bethejointprobabilitydistributionfunctionofdefaulttimes.Weassumethattheprobabilitydistributionofdefaulttimesisjointlycontinuous,andwewritef(t1;t2;:::;tn)todenotethejointprobabilitydensityfunction.Also,letG(t1;t2;:::;tn)=Q(¿1>t1;¿2>t2;:::;¿n>tn)standforthejointprobabilitythatthenames1;2;:::;nhavesurviveduptotimest1;t2;:::;tn.Inparticular,thejointsurvivalfunctionequalsG(t;:::;t)=Q(¿1>t;¿2>t;:::;¿n>t)=Q(¿(1)>t)=G(1)(t):Foreachi=1;2;:::;n,weintroducethedefaultindicatorprocessHi=1andthecorre-tf¿i·tgsponding¯ltrationHi=(Hi)whereHi=¾(Hi:u·t).WedenotebyGthejoint¯ltrationtt2R+tugeneratedbydefaultindicatorprocessesH1;H2;:::;Hn,sothatG=H1_H2_:::_Hn.Itisclearthat¿(1)isaG-stoppingtimeasthein¯mumofG-stoppingtimes.Finally,wewriteH(1)=1andH(1)=(H(1))whereH(1)=¾(H(1)tf¿(1)·tgtt2R+tu:u·t).SinceweassumethatQ(¿i=¿j)=0foranyi6=j;i;j=1;2;:::;n,wealsohavethatXn(1)(1)iHt=Ht^¿(1)=Ht^¿(1):i=1WemakethestandingassumptionQ(¿(1)>T)=G(1)(T)>0.Foranyt2[0;T],theeventf¿(1)>tgisanatomofthe¾-¯eldGt.Hencethefollowingsimple,butuseful,result.Lemma2.5.1LetXbeaQ-integrablestochasticprocess.ThenEQ(XtjGt)1f¿(1)>tg=Xe(t)1f¿(1)>tg 2.5.DYNAMICHEDGINGOFBASKETCREDITDERIVATIVES63wherethefunctionXe:[0;T]!Risgivenbytheformula¡¢Xe(t)=EQXt1f¿(1)>tg;8t2[0;T]:G(1)(t)IfXisaG-adapted,Q-integrablestochasticprocessthenXt=Xt1f¿(1)·tg+Xe(t)1f¿(1)>tg;8t2[0;T]:Byconvention,thefunctionXe:[0;T]!Riscalledthepre-defaultvalueofX.2.5.1First-to-DefaultIntensitiesInthissection,weintroducetheso-called¯rst-to-defaultintensities.Thisnaturalconceptwillproveusefulinthevaluationandhedgingofthe¯rst-to-defaultbasketclaims.De¯nition2.5.1Thefunction¸ei:R+!R+givenby1¸ei(t)=limQ(t<¿i·t+hj¿(1)>t)(2.68)h#0hiscalledtheith¯rst-to-defaultintensity.Thefunction¸e:R!Rgivenby++1¸e(t)=limQ(t<¿(1)·t+hj¿(1)>t)(2.69)h#0hiscalledthe¯rst-to-defaultintensity.Letusdenote@G(t1;t2;:::;tn)@iG(t;:::;t)=¯:@ti¯t1=t2=:::=tn=tThenwehavethefollowingelementarylemmasummarizingthepropertiesofthe¯rst-to-defaultintensity.Lemma2.5.2Theith¯rst-to-defaultintensity¸esatis¯es,fori=1;2;:::;n,iR1R1¸et:::tf(u1;:::;ui¡1;t;ui+1;:::;un)du1:::dui¡1dui+1:::duni(t)=G(t;:::;t)R1R1t:::tF(du1;:::;dui¡1;t;dui+1;:::;dun)@iG(t;:::;t)==¡:G(1)(t)G(1)(t)The¯rst-to-defaultintensity¸esatis¯es¸e(t)=¡1dG(1)(t)f(1)(t)=(2.70)G(1)(t)dtG(1)(t)Pn¸ewheref(1)(t)istheprobabilitydensityfunctionof¿(1).Theequality¸e(t)=i=1i(t)holds.Proof.ClearlyR1Rt+hR1¸e1t:::t:::tf(u1;:::;ui;:::;un)du1:::dui:::duni(t)=limh#0hG(t;:::;t) 64CHAPTER2.HAZARDFUNCTIONAPPROACHandthusthe¯rstassertedequalityfollows.Thesecondequalityfollowsdirectlyfrom(2.69)andthePn¸ede¯nitionofG(1).Finally,equality¸e(t)=i=1i(t)isequivalenttotheequalityXn11limQ(t<¿i·t+hj¿(1)>t)=limQ(t<¿(1)·t+hj¿(1)>t);h#0hh#0hi=1whichinturniseasytoestablish.¤Remarks2.5.1Theith¯rst-to-defaultintensity¸eshouldnotbeconfusedwiththe(marginal)iintensityfunction¸iof¿i,whichisde¯nedasfi(t)¸i(t)=;8t2R+;Gi(t)wherefiisthe(marginal)probabilitydensityfunctionof¿i,thatis,Z1Z1fi(t)=:::f(u1;:::;ui¡1;t;ui+1;:::;un)du1:::dui¡1dui+1:::dun;00R1andGi(t)=1¡Fi(t)=tfi(u)du.Indeed,wehavethat¸ei6=¸i,ingeneral.However,if¿1;:::;¿naremutuallyindependentunderQthen¸ei=¸i,thatis,the¯rst-to-defaultandmarginaldefaultintensitiescoincide.Itisalsoratherclearthatthe¯rst-to-defaultintensity¸eisnotequaltothesumofmarginalPndefaultintensities,thatis,wehavethat¸e(t)6=i=1¸i(t),ingeneral.2.5.2First-to-DefaultMartingaleRepresentationTheoremWenowstateanintegralrepresentationtheoremforaG-martingalestoppedat¿(1)withrespecttosomebasicprocesses.Tothisend,wede¯ne,fori=1;2;:::n,Zt^¿(1)Mci=Hi¡¸e(u)du;8t2R:(2.71)tt^¿(1)i+0Thenwehavethefollowing¯rst-to-defaultmartingalerepresentationtheorem.Proposition2.5.1ConsidertheG-martingaleMct=EQ(YjGt);t2[0;T],whereYisaQ-integrablerandomvariablegivenbytheexpressionXnY=Zi(¿i)1f¿i·T;¿i=¿(1)g+X1f¿(1)>Tg(2.72)i=1forsomefunctionsZi:[0;T]!R;i=1;2;:::;nandsomeconstantX.ThenMcadmitsthefollowingrepresentationXnZMc=E(Y)+h(u)dMci(2.73)tQiui=1]0;t]wherethefunctionshi;i=1;2;:::;naregivenbyhi(t)=Zi(t)¡Mct¡=Zi(t)¡Mf(t¡);8t2[0;T];(2.74)whereMfistheuniquefunctionsuchthatMct1f¿(1)>tg=Mf(t)1f¿(1)>tgforeveryt2[0;T].ThefunctionMfsatis¯esMf0=EQ(Y)andXn¡¢dMf(t)=¸ei(t)Mf(t)¡Zi(t)dt:(2.75)i=1 2.5.DYNAMICHEDGINGOFBASKETCREDITDERIVATIVES65Moreexplicitly(Zt)Ztn(Zt)XMf(t)=EQ(Y)exp¸e(s)ds¡¸ei(s)Zi(s)exp¸e(u)duds:00i=1sProof.Toalleviatenotation,weprovidetheproofofthisresultinabivariatesettingonly.Inthatcase,¿=¿^¿andG=H1_H2.Westartbynotingthat(1)12tttMct=EQ(Z1(¿1)1f¿1·T;¿2>¿1gjGt)+EQ(Z2(¿2)1f¿2·T;¿1>¿2gjGt)+EQ(X1f¿(1)>TgjGt);andthus(seeLemma2.5.1)X31Mc=1Mf(t)=1Yei(t)f¿(1)>tgtf¿(1)>tgf¿(1)>tgi=1wheretheauxiliaryfunctionsYei:[0;T]!R;i=1;2;3,aregivenbyRTR1RTR1Ye1tduZ1(u)udvf(u;v)2tdvZ2(v)vduf(u;v)3XG(1)(T)(t)=;Ye(t)=;Ye(t)=:G(1)(t)G(1)(t)G(1)(t)ByelementarycalculationsandusingLemma2.5.2,weobtainR1RTR1dYe1(t)Z1(t)dvf(t;v)duZ1(u)dvf(u;v)dG(1)(t)=¡t¡tudtG(1)(t)G2(t)dt(1)R1tdvf(t;v)Ye1(t)dG(1)(t)=¡Z1(t)¡G(1)(t)G(1)(t)dt=¡Z(t)¸e(t)+Ye1(t)(¸e(t)+¸e(t));(2.76)1112andthus,bysymmetry,dYe2(t)=¡Z(t)¸e(t)+Ye2(t)(¸e(t)+¸e(t)):(2.77)2212dtMoreoverdYe3(t)XG(1)(T)dG(1)(t)=¡=Ye3(t)(¸e(t)+¸e(t)):(2.78)dtG2(t)dt12(1)P3HencerecallingthatMf(t)=Yei(t),weobtainfrom(2.76)-(2.78)i=1¡¢¡¢dMf(t)=¡¸e1(t)Z1(t)¡Mf(t)dt¡¸e2(t)Z2(t)¡Mf(t)dt(2.79)Consequently,sincethefunctionMfiscontinuous,wehave,ontheeventf¿(1)>tg,¡¢¡¢dMct=¡¸e1(t)Z1(t)¡Mct¡dt¡¸e2(t)Z2(t)¡Mct¡dt:Weshallnowcheckthatbothsidesofequality(2.73)coincideattime¿(1)ontheeventf¿(1)·Tg.Tothisend,weobservethatwehave,ontheeventf¿(1)·Tg,Mc¿(1)=Z1(¿1)1f¿(1)=¿1g+Z2(¿2)1f¿(1)=¿2g;whereastheright-handsidein(2.73)isequaltoZZMc+h(u)dMc1+h(u)dMc201u2u]0;¿(1)[]0;¿(1)[ZZ+1h(u)dH1+1h(u)dH2f¿(1)=¿1g1uf¿(1)=¿2g2u[¿(1)][¿(1)]¡¢¡¢=Mf(¿(1)¡)+Z1(¿1)¡Mf(¿(1)¡)1f¿(1)=¿1g+Z2(¿2)¡Mf(¿(1)¡)1f¿(1)=¿2g=Z1(¿1)1f¿(1)=¿1g+Z2(¿2)1f¿(1)=¿2g 66CHAPTER2.HAZARDFUNCTIONAPPROACHasMf(¿(1)¡)=Mc¿(1)¡.Sincetheprocessesonbothsidesofequality(2.73)arestoppedat¿(1),weconcludethatequality(2.73)isvalidforeveryt2[0;T].Formula(2.75)wasalsoestablishedintheproof(seeformula(2.79)).¤ThenextresultshowsthatthebasicprocessesMciareinfactG-martingales.Theywillbereferredtoasthebasic¯rst-to-defaultmartingales.Corollary2.5.1Foreachi=1;2;:::;n,theprocessMcigivenbytheformula(2.71)isaG-martingalestoppedat¿(1).Proof.Letus¯xk2f1;2;:::;ng.ItisclearthattheprocessMckisstoppedat¿.LetMfk(t)=(1)Rt¡¸ei(u)dubetheuniquefunctionsuchthat01Mci=1Mfk(t);8t2[0;T]:f¿(1)>tgtf¿(1)>tgLetustakehk(t)=1andhi(t)=0foranyi6=kinformula(2.73),orequivalently,letussetZ(t)=1+Mfk(t);Z(t)=Mfk(t);i6=k;kiinthede¯nition(2.72)oftherandomvariableY.Finally,lettheconstantXin(2.72)bechoseninsuchawaythattherandomvariableYsatis¯esE(Y)=Mck.Thenwemaydeducefrom(2.73)Q0thatMck=McandthusweconcludethatMckisaG-martingale.¤2.5.3PriceDynamicsoftheithCDSAstradedassetsinourmodel,wetaketheconstantsavingsaccountandafamilyofsingle-nameCDSswithdefaultprotections±iandrates·i.Forconvenience,weassumethattheCDSshavethesamematurityT,butthisassumptioncanbeeasilyrelaxed.TheithtradedCDSisformallyde¯nedbyitsdividendprocessZDi=±(u)dHi¡·(t^¿);8t2[0;T]:tiuii(0;t]Consequently,thepriceattimetoftheithCDSequals¡¡¢¯¢Si(·)=E(1±(¿)jG)¡·E1(¿^T)¡t¯G:(2.80)tiQft<¿i·TgiitiQft<¿igitToreplicatea¯rst-to-defaultclaim,weonlyneedtoexaminethedynamicsofeachCDSontheinterval[0;¿(1)^T].Thefollowinglemmawillproveusefulinthisregard.Lemma2.5.3Wehave,ontheeventf¿(1)>tg,³X¯´ii¯St(·i)=EQ1ft<¿(1)=¿i·Tg±i(¿(1))+1ft<¿(1)=¿j·TgS¿(1)(·i)¡·i1ft<¿(1)g(¿(1)^T¡t)¯Gt:j6=iProof.We¯rstnotethatthepriceSi(·)canberepresentedasfollows,ontheeventf¿>tg,ti(1)³XSi(·)=E1±(¿)+1(1±(¿^T)tiQft<¿(1)=¿i·Tgi(1)ft<¿(1)=¿j·Tgf¿(1)<¿i·Tgiij6=i¯¯´¡¯¢¡·i1f¿(1)<¿ig(¿i¡¿(1)))¯Gt¡·iEQ1ft<¿(1)g(¿(1)^T¡t)¯Gt:Byconditioning¯rstonthe¾-¯eldG¿,weobtaintheclaimedformula.¤(1) 2.5.DYNAMICHEDGINGOFBASKETCREDITDERIVATIVES67RepresentationestablishedinLemma2.5.3isbynomeanssurprising;itmerelyshowsthatinordertocomputethepriceofaCDSpriortothe¯rstdefault,wecaneitherdothecomputationsinasinglestep,byconsideringthecash°owsoccurringon]t;¿i^T],orwecancompute¯rstthepriceofthecontractattime¿(1)^T,andsubsequentlyvalueallcash°owsoccurringon]t;¿(1)^T].However,italsoshowsthatwecanconsiderfromnowonnottheoriginalithCDSbuttheassociatedCDScontractwithrandommaturity¿i^T.SimilarlyasinSection2.4.2,wewriteSi(·)=1Sei(·)wherethepre-defaultpriceSei(·)tift<¿(1)gtitisatis¯esSei(·)=±ei(t;T)¡·Aei(t;T)(2.81)tiiwhere±ei(t;T)and·Aei(t;T)arepre-defaultvaluesoftheprotectionlegandthefeelegrespectively.Foranyj6=i,wede¯neafunctionSi(·):[0;T]!R,whichrepresentsthepriceoftheithtjjiCDSattimetontheeventf¿(1)=¿j=tg.Formally,thisquantityisde¯nedastheuniquefunctionsatisfying1Si(·)=1Si(·)f¿(1)=¿j·Tg¿(1)jjif¿(1)=¿j·Tg¿(1)isothatX1Si(·)=1Si(·):f¿(1)·Tg¿(1)if¿(1)=¿j·Tg¿(1)jjij6=iLetusexaminethecaseoftwonames.ThenthefunctionS1(·);t2[0;T],representsthepricetj21ofthe¯rstCDSattimetontheeventf¿(1)=¿2=tg.Lemma2.5.4ThefunctionS1(·);v2[0;T];equalsvj21RTRTR11vR±1(u)f(u;v)duvRduudzf(z;v)Svj2(·1)=1¡·11:(2.82)f(u;v)duf(u;v)duvvProof.Notethattheconditionalc.d.f.of¿1giventhat¿1>¿2=vequalsRuf(z;v)dzQ(¿·uj¿>¿=v)=F(u)=Rv;8u2[v;1];112¿1j¿1>¿2=v1f(z;v)dzvsothattheconditionaltailequalsR1f(z;v)dzG(u)=1¡F(u)=Ru;8u2[v;1]:(2.83)¿1j¿1>¿2=v¿1j¿1>¿2=v1f(z;v)dzvLetJbetheright-handsideof(2.82).ItisclearthatZTZTJ=¡±1(u)dG¿1j¿1>¿2=v(u)¡·1G¿1j¿1>¿2=v(u)du:vvCombiningLemma2.4.1withthefactthatS1(·)isequaltotheconditionalexpectationwith¿(1)irespectto¾-¯eldGofthecash°owsoftheithCDSon]¿_¿;¿^T],weconcludethatJ¿(1)(1)iicoincideswithS1(·),thepriceofthe¯rstCDSontheeventf¿=¿=vg.¤vj21(1)2ThefollowingresultextendsLemma2.4.2.Lemma2.5.5Thedynamicsofthepre-defaultpriceSei(·)areti³Xn´dSei(·)=¸e(t)Sei(·)dt+·¡±(t)¸e(t)¡Si(·)¸e(t)dt(2.84)titiiiitjjiij6=i 68CHAPTER2.HAZARDFUNCTIONAPPROACHPn¸ewhere¸e(t)=i=1i(t),orequivalently,¡¢X¡¢dSei(·)=¸e(t)Sei(·)¡±(t)dt+¸e(t)Sei(·)¡Si(·)dt+·dt:(2.85)tiitiijtitjjiij6=iThecumulativepriceoftheithCDSstoppedat¿satis¯es(1)ZtXZtSc;i(·)=Si(·)+±(u)dHi+Si(·)dHj¡·(¿^t);(2.86)titiiu^¿(1)ujjiu^¿(1)i(1)00j6=iandthus¡¢X¡¢dSc;i(·)=±(t)¡Sei(·)dMci+Si(·)¡Sei(·)dMcj:(2.87)tiit¡ittjjit¡itj6=iProof.Weshallconsiderthecasen=2.UsingtheformuladerivedinLemma2.5.3,weobtainRTR1RT1R1±e1tdu±1(u)udvf(u;v)tdvSvj2(·1)vduf(u;v)(t;T)=+:(2.88)G(1)(t)G(1)(t)Byadaptingequality(2.76),weget³´d±e1(t;T)=(¸e(t)+¸e(t))ge(t)¡¸e(t)±(t)¡¸e(t)S1(·)dt:(2.89)121112tj21Toestablish(2.84)-(2.85),weneedalsotoexaminethefeeleg.Itspriceequals³¡¢¯¯´E1·(¿^T)¡t¯G=1·Aei(t;T);Qft<¿(1)g1(1)tft<¿(1)g1Toevaluatetheconditionalexpectationabove,itsu±cestousethec.d.f.F(1)oftherandomtime¿(1).AsinSection2.4.1(seetheproofofLemma2.4.1),weobtainZTAei(t;T)=1G(u)du;(2.90)(1)G(1)(t)tandthus¡¢dAei(t;T)=1+(¸e(t)+¸e(t))Aei(t;T)dt:12SinceSe1(·)=±ei(t;T)¡·Aei(t;T),theformulae(2.84)-(2.85)follow.Formula(2.86)isratherclear.t1iFinally,dynamics(2.87)canbeeasilydeducedfrom(2.85)and(2.86)¤2.5.4Risk-NeutralValuationofaFirst-to-DefaultClaimInthissection,weshallanalyzetherisk-neutralvaluationof¯rst-to-defaultclaimsonabasketofncreditnames.De¯nition2.5.2A¯rst-to-defaultclaim(FTDC)withmaturityTisadefaultableclaim(X;A;Z;¿(1))whereXisaconstantamountpayableatmaturityifnodefaultoccurs,A:[0;T]!RwithA0=0isafunctionofboundedvariationrepresentingthedividendstreamupto¿(1),andZ=(Z1;Z2;:::;Zn)isthevectoroffunctionsZi:[0;T]!RwhereZi(¿(1))speci¯estherecoveryreceivedattime¿(1)iftheithnameisthe¯rstdefaultedname,thatis,ontheeventf¿=¿·Tg.i(1)Wede¯netherisk-neutralvalue¼ofanFTDCbysettingXn³ZT¯´(1)¯¼t=EQZi(¿i)1ft<¿(1)=¿i·Tg+1ft<¿(1)g(1¡Hu)dA(u)+X1f¿(1)>Tg¯Gt;i=1t 2.5.DYNAMICHEDGINGOFBASKETCREDITDERIVATIVES69andtherisk-neutralcumulativevalue¼bofanFTDCbytheformulaXn³ZT¯´(1)¯¼bt=EQZi(¿i)1ft<¿(1)=¿i·Tg+1ft<¿(1)g(1¡Hu)dA(u)¯Gti=1tXnZtZt+E(X1jG)+Z(u)dHi+(1¡H(1))dA(u)Qf¿(1)>Tgtiu^¿(1)ui=100wherethelasttwotermsrepresentthepastdividends.Letusstressthattherisk-neutralvaluationofanFTDCwillbelatersupportedbyreplicationarguments(seeTheorem2.2),andthusrisk-neutralvalue¼ofanFTDCwillbeshowntobeitsreplicationprice.Bythepre-defaultrisk-neutralvalueassociatedwithaG-adaptedprocess¼,wemeanthefunction¼esuchthat¼t1f¿(1)>tg=¼e(t)1f¿(1)>tgforeveryt2[0;T].Directcalculationsleadtothefollowingresult,whichcanalsobededucedfromProposition2.5.1.Lemma2.5.6Thepre-defaultrisk-neutralvalueofanFTDCequalsXnZTG(T)ªi(t)1(1)¼e(t)=+G(1)(u)dA(u)+X(2.91)G(1)(t)G(1)(t)tG(1)(t)i=1whereZTZ1Z1Z1Z1ªi(t)=::::::Zi(ui)F(du1;:::;dui¡1;dui;dui+1;:::;dun):ui=tu1=uiui¡1=uiui+1=uiun=uiThenextresultextendsProposition2.4.2tothemulti-namesetup.ItsproofissimilartotheproofofLemma2.5.5,andthusitisomitted.Proposition2.5.2Thepre-defaultrisk-neutralvalueofanFTDCsatis¯esX¡¢d¼e(t)=¸ei(t)¼e(t)¡Zi(t)dt¡dA(t):i=1Moreover,therisk-neutralvalueofanFTDCsatis¯esXnd¼=¡¼e(t¡)dMci¡dA(¿^t);(2.92)tu(1)i=1andtherisk-neutralcumulativevalue¼bofanFTDCsatis¯esXnd¼b=(Z(t)¡¼e(t¡))dMci:tiui=12.5.5DynamicReplicationofaFirst-to-DefaultClaimLetB=1andsingle-nameCDSswithpricesS1(·);:::;Sn(·)betradedassets.Wesaythata1nG-predictableprocessÁ=(Á0;Á1;:::;Án)andafunctionCof¯nitevariationwithC(0)=0de¯neaself-¯nancingstrategywithdividendstreamCifthewealthprocessV(Á;C),de¯nedasXnV(Á;C)=Á0+ÁiSi(·);ttttii=1satis¯esXn¡¢XndV(Á;C)=ÁidSi(·)+dDi¡dC(t)=ÁidSc;i(·)¡dC(t)(2.93)tttitttii=1i=1whereSi(·)(Sc;i(·),respectively)istheprice(cumulativeprice,respectively)oftheithCDS.ii 70CHAPTER2.HAZARDFUNCTIONAPPROACHDe¯nition2.5.3Wesaythatatradingstrategy(Á;C)replicatesanFTDC(X;A;Z;¿(1))if:(i)theprocessesÁ=(Á0;Á1;:::;Án)andV(Á;C)arestoppedat¿^T,(1)(ii)C(¿(1)^t)=A(¿(1)^t)foreveryt2[0;T],(iii)theequalityV¿(1)^T(Á;C)=Yholds,wheretherandomvariableYequalsXnY=X1f¿(1)>Tg+Zi(¿(1))1f¿i=¿(1)·Tg:i=1WearenowinapositiontoextendTheorem2.1tothecaseofa¯rst-to-defaultclaimonabasketofncreditnames.Theorem2.2AssumethatdetN(t)6=0foreveryt2[0;T],where23±(t)¡Se1(·)S2(·)¡Se2(·):::Sn(·)¡Sen(·)1t1tj12t2tj1ntn6S1(·)¡Se1(·)±(t)¡Se2(·):::Sn(·)¡Sen(·)76tj21t12t2tj2ntn7N(t)=66...77......4...5S1(·)¡Se1(·)S2(·)¡Se2(·):::±(t)¡Sen(·)tjn1t1tjn1t1ntnLetÁe(t)=(Áe1(t);Áe2(t);:::;Áen(t))betheuniquesolutiontotheequationN(t)Áe(t)=h(t)whereh(t)=(h1(t);h2(t);:::;hn(t))withhi(t)=Zi(t)¡¼e(t¡)andwhere¼eisgivenbyLemma2.5.6.Moreexplicitly,thefunctionsÁe1;Áe2;:::;Áensatisfy,fort2[0;T]andi=1;2;:::;n,¡¢X¡¢Áe(t)±(t)¡Sei(·)+Áe(t)Sj(·)¡Sej(·)=Z(t)¡¼e(t¡):(2.94)iitijtjijtjij6=iLetussetÁi=Áe(¿^t)fori=1;2;:::;nandletti(1)XnÁ0=V(Á;A)¡ÁiSi(·);8t2[0;T];(2.95)ttttii=1wheretheprocessV(Á;A)isgivenbytheformulaXnZV(Á;A)=¼e(0)+Áe(u)dSc;i(·)¡A(¿^t):(2.96)tiui(1)i=1]0;¿(1)^t]Thenthetradingstrategy(Á;A)replicatesanFTDC(X;A;Z;¿(1)).Proof.TheproofisbasedonsimilarargumentsastheproofofTheorem2.1.Itsu±cestocheckthatundertheassumptionofthetheorem,foratradingstrategy(Á;A)stoppedat¿(1),weobtainfrom(2.93)and(2.87)thatXn³¡¢X¡¢´dV(Á;A)=Ái±(t)¡Sei(·)dMci+Si(·)¡Sei(·)dMcj¡dA(¿^t):ttit¡ittjjit¡it(1)i=1j6=iForÁi=Áe(¿^t),wherethefunctionsÁe;Áe;:::;Áesolve(2.94),wethusobtainti(1)12nXndV(Á;A)=(Z(t)¡¼e(t¡))dMci¡dA(¿^t):tit(1)i=1Bycomparingthelastformulawith(2.92),weconcludethatif,inaddition,V0(Á;A)=¼0=¼e0andÁ0isgivenby(2.95),thenthestrategy(Á;A)replicatesanFTDC(X;A;Z;¿).¤(1) 2.5.DYNAMICHEDGINGOFBASKETCREDITDERIVATIVES712.5.6ConditionalDefaultDistributionsInthecaseof¯rst-to-defaultclaims,itwasenoughtoconsidertheunconditionaldistributionofdefaulttimes.Asexpected,inordertodealwithageneralbasketdefaultableclaim,weneedtoanalyzeconditionaldistributionsofdefaulttimes.Itispossibletofollowtheapproachpresentedinprecedingsections,andtoexplicitlyderivethedynamicsofallprocessesofinterestonthetimeinterval[0;T].However,sincewedealherewithasimplemodelofjointdefaults,itsu±cestomakeanon-restrictiveassumptionthatweworkonthecanonicalspace•=Rn,andtousesimpleargumentsbasedonconditioningwithrespecttopastdefaults.Supposethatknamesoutofatotalofnnameshavealreadydefaulted.Tointroduceaconvenientnotation,weadopttheconventionthatthen¡knon-defaultednamesareintheiroriginalorderj1<:::uk,¡¢F(t1;:::;tn¡kj¿i1=u1;:::;¿ik=uk)=Q¿j1·t1;:::;¿jn¡k·tn¡kj¿i1=u1;:::;¿ik=uk:Thejointconditionalsurvivalfunctionofdefaulttimes¿j1;:::;¿jn¡kisgivenbytheexpression¡¢G(t1;:::;tn¡kj¿i1=u1;:::;¿ik=uk)=Q¿j1>t1;:::;¿jn¡k>tn¡kj¿i1=u1;:::;¿ik=ukforeveryt1;:::;tn¡k>uk.Asexpected,theconditional¯rst-to-defaultintensitiesarede¯nedusingthejointconditionaldistributions,insteadofthejointunconditionaldistribution.WewriteG(1)(tjDk)=G(t;:::;tjDk).De¯nition2.5.5GiventheeventDk,foranyjl2fj1;:::;jn¡kg,theconditional¯rst-to-defaultintensityofasurvivingnamejlisdenotedby¸ejl(tjDk)=¸ejl(tj¿i1=u1;:::;¿ik=uk),andisgivenbytheformulaR1R1R1¸ett:::tdF(t1;:::;tl¡1;t;tl+1;:::;tn¡kjDk)jl(tjDk)=;8t2[uk;T]:G(1)(tjDk)InSection2.5.3,weintroducedtheprocessesSi(·)representingthevalueoftheithCDSattjjjtimetontheeventf¿(1)=¿j=tg.Accordingtothenotationintroducedabove,wethusdealtwiththeconditionalvalueoftheithCDSwithrespecttoD=f¿=tg.ItisclearthattovalueaCDS1jforeachsurvivingnamewecanproceedaspriortothe¯rstdefault,exceptthatnowweshouldusetheconditionaldistributionF(t1;:::;tn¡1jD1)=F(t1;:::;tn¡1j¿j=j);8t1;:::;tn¡12[t;T];ratherthantheunconditionaldistributionF(t1;:::;tn)employedinProposition2.5.6.ThesameargumentcanbeappliedtoanydefaulteventDk.ThecorrespondingconditionalversionofPropo-sition2.5.6israthereasytoformulateandprove,andthuswefeelthereisnoneedtoprovideanexplicitconditionalpricingformulahere.Theconditional¯rst-to-defaultintensitiesintroducedinDe¯nition2.5.5willallowustoconstructtheconditional¯rst-to-defaultmartingalesinasimilarwayaswede¯nedthe¯rst-to-defaultmar-tingalesMiassociatedwiththe¯rst-to-defaultintensities¸e.However,sinceanynamecandefaultiatanytime,weneedtointroduceanentirefamilyofconditionalmartingales,whosecompensatorsarebasedonintensitiesconditionedontheinformationstructureofpastdefaults. 72CHAPTER2.HAZARDFUNCTIONAPPROACHDe¯nition2.5.6GiventhedefaulteventDk=f¿i1=u1;:::;¿ik=ukg,foreachsurvivingnamej2fj;:::;jg,wede¯nethebasicconditional¯rst-to-defaultmartingaleMcjlbysettingl1n¡ktjDkZtMcjl=Hjl¡1¸e(ujD)du;8t2[u;T]:(2.97)tjDkt^¿(k+1)fu<¿(k+1)gjlkkukTheprocessMcjl;t2[u;T],isamartingaleundertheconditionedprobabilitymeasureQjD,tjDkkkthatis,theprobabilitymeasureQconditionedontheeventDk,andwithrespecttothe¯ltrationgeneratedbydefaultprocessesofthesurvivingnames,thatis,the¯ltrationGDk:=Hj1_:::_Hjn¡ktttfort2[uk;T].SinceweconditionontheeventDk,wehave¿(k+1)=¿j1^¿j2^:::^¿jn¡k,sothat¿(k+1)isthe¯rstdefaultforallsurvivingnames.Formula(2.97)isthusaratherstraightforwardgeneralizationofformula(2.71).Inparticular,fork=0weobtainMci=Mci;t2[0;T],foranyi=1;2;:::;n.tjD0tThemartingalepropertyoftheprocessMcjl,statedinDe¯nition2.5.6,followsfromPropositiontjDk2.5.3(itcanalsobeseenasaconditionalversionofCorollary2.5.1).Weareinapositiontostatetheconditionalversionofthe¯rst-to-defaultmartingalerepre-sentationtheoremofSection2.5.2.Formally,thisresultisnothingelsethanarestatementofthemartingalerepresentationformulaofProposition2.5.1intermsofconditional¯rst-to-defaultinten-sitiesandconditional¯rst-to-defaultmartingales.Letus¯xaneventDwriteGDk=Hj1_:::_Hjn¡k.kProposition2.5.3LetYbearandomvariablegivenbytheformulanX¡kY=ZjljDk(¿jl)1f¿jl·T;¿jl=¿(k+1)g+X1f¿(k+1)>Tg(2.98)l=1forsomefunctionsZjljDk:[uk;T]!R;l=1;2;:::;n¡k,andsomeconstantX(possiblydependentonDk).Letusde¯neMc=E(YjGDk);8t2[u;T]:(2.99)tjDkQjDktkThenMc;t2[u;T],isaGDk-martingalewithrespecttotheconditionedprobabilitymeasuretjDkkQjDkanditadmitsthefollowingrepresentation,fort2[uk;T],nX¡kZMc=Mc+h(ujD)dMcjltjDk0jDkjlkujDkl=1]uk;t]wheretheprocesseshjlaregivenbyhjl(tjDk)=ZjljDk(t)¡Mct¡jDk;8t2[uk;T]:Proof.TheproofreliesonadirectextensionofargumentsusedintheproofofProposition2.5.1tothecontextofconditionaldefaultdistributions.Therefore,itislefttothereader.¤2.5.7RecursiveValuationofaBasketClaimWearereadyextendtheresultsdevelopedinthecontextof¯rst-to-defaultclaimstovalueandhedgegeneralbasketclaims.Agenericbasketclaimisanycontingentclaimthatpaysaspeci¯edamountoneachdefaultfromabasketofncreditnamesandaconstantamountatmaturityTifnodefaultshaveoccurredpriortooratT. 2.5.DYNAMICHEDGINGOFBASKETCREDITDERIVATIVES73De¯nition2.5.7Abasketclaimassociatedwithafamilyofncreditnamesisgivenas(X;A;Z;¹¿¹)whereXisaconstantamountpayableatmaturityonlyifnodefaultoccurspriortooratT,thevector¹¿=(¿1;:::;¿n)representsdefaulttimes,andthetime-dependentmatrixZ¹representsthepayo®satdefaults,speci¯cally,23Z1(tjD0)Z2(tjD0):::Zn(tjD0)66Z1(tjD1)Z2(tjD1):::Zn(tjD1)77Z¹=6.....7:4.......5Z1(tjDn¡1)Z2(tjDn¡1):::Zn(tjDn¡1)NotethattheabovematrixZ¹ispresentedintheshorthandnotation.Infact,ineachrowweneedtospecify,foranarbitrarychoiceoftheeventDk=f¿i1=u1;:::;¿ik=ukgandanynamej2f=i;:::;ig,theconditionalpayo®functionatthemomentofthe(k+1)thdefault:l1kZjl(tjDk)=Zjl(tj¿i1=u1;:::;¿ik=uk);8t2[uk;T]:Inthe¯nancialinterpretation,thefunctionZjl(tjDk)determinestherecoverypaymentatthedefaultofthenamejl,conditionalontheeventDk,ontheeventf¿jl=¿(k+1)=tg,thatis,assumingthatthenamejlisthe¯rstdefaultingnameamongallsurvivingnames.Inparticular,Z(tjD):=Z(t)representstherecoverypaymentatthedefaultoftheithnameattimet2[0;T],i0igiventhatnodefaultshaveoccurredpriortot,thatis,atthemomentofthe¯rstdefault(notethatthesymbolD0meansmerelythatweconsiderasituationofnodefaultspriortot).Example2.5.1Letusconsiderthekth-to-defaultclaimforsome¯xedk2f1;2;:::;ng.Assumethatthepayo®atthekthdefaultdependsonlyonthemomentofthekthdefaultandtheidentityofthekth-to-defaultname.ThenallrowsofthematrixZ¹areequaltozero,exceptforthekthrow,whichis[Z1(tjk¡1);Z2(tjk¡1);:::;Zn(tjk¡1)]fort2[0;T].Wewriteherek¡1,ratherthanDk¡1,inordertoemphasizethattheknowledgeoftimingsandidentitiesofthekdefaultednamesisnotrelevantunderthepresentassumptions.Moregenerally,foragenericbasketclaiminwhichthepayo®attheithdefaultdependsonthetimeoftheithdefaultandidentityoftheithdefaultingnameonly,therecoverymatrixZ¹reads23Z1(t)Z2(t):::Zn(t)66Z1(tj1)Z2(tj1):::Zn(tj1)77Z¹=6.....74.......5Z1(tjn¡1)Z2(tjn¡1):::Zn(tjn¡1)whereZ(tjk¡1)representsthepayo®atthemoment¿=tofthekthdefaultifjisthekthj(k)defaultingname,thatis,ontheeventf¿j=¿(k)=tg.Thisshowsthatinseveralpracticallyimportantexamplesofbasketcreditderivatives,thematrixZ¹willhaveasimplestructure.Itisclearthatanybasketclaimcanberepresentedasastaticportfolioofkth-to-defaultclaimsfork=1;2;:::;n.However,thisdecompositiondoesnotseemtobeadvantageousforourpurposes.Inwhatfollows,weprefertorepresentabasketclaimasasequenceofconditional¯rst-to-defaultclaims,withthesamevaluebetweenanytwodefaultsasourbasketclaim.Inthatway,wewillbeabletodirectlyapplyresultsdevelopedforthecaseof¯rst-to-defaultclaimsandthustoproduceasimpleiterativealgorithmforthevaluationandhedgingofabasketclaim.Insteadofstatingaformalresult,usingaratherheavynotation,wepreferto¯rstfocusonthecomputationalprocedureforvaluationandhedgingofabasketclaim.Theimportantconceptinthisprocedureistheconditionalpre-defaultpriceZe(tjDk)=Ze(tj¿i1=u1;:::;¿ik=uk);8t2[uk;T]; 74CHAPTER2.HAZARDFUNCTIONAPPROACHofaconditional¯rst-to-defaultclaim".ThefunctionZe(tjDk);t2[uk;T],isde¯nedastherisk-neutralvalueofaconditionalFTDConn¡ksurvivingnames,withthefollowingrecoverypayo®suponthe¯rstdefaultatanydatet2[uk;T]Zbjl(tjDk)=Zjl(tjDk)+Ze(tjDk;¿jl=t):(2.100)Assumeforthemomentthatforanynamejm2f=i1;:::;ik;jlgtheconditionalrecoverypayo®Zbjm(tj¿i1=u1;:::;¿ik=uk;¿jl=uk+1)uponthe¯rstdefaultafterdateuk+1isknown.ThenwecancomputethefunctionZe(tj¿i1=u1;:::;¿ik=uk;¿jl=uk+1);8t2[uk+1;T];asinLemma2.5.6,butusingconditionaldefaultdistribution.Theassumptionthattheconditionalpayo®sareknownisinfactnotrestrictive,sincethefunctionsappearinginright-handsideof(2.100)areknownfromthepreviousstepinthefollowingrecursivepricingalgorithm.²Firststep.We¯rstderivethevalueofabasketclaimassumingthatallbutonedefaultshavealreadyoccurred.LetDn¡1=f¿i1=u1;:::;¿in¡1=un¡1g.Foranyt2[un¡1;T],wedealwiththepayo®sZbj1(tjDn¡1)=Zj1(tjDn¡1)=Zj1(tj¿i1=u1;:::;¿in¡1=un¡1);forj12f=i1;:::;in¡1gwheretherecoverypaymentfunctionZj1(tjDn¡1);t2[un¡1;T],isgivenbythespeci¯cationofthebasketclaim.Hencewecanevaluatethepre-defaultvalueZe(tjDn¡1)atanytimet2[un¡1;T],asavalueofaconditional¯rst-to-defaultclaimwiththesaidpayo®,usingtheconditionaldistributionunderQjDn¡1oftherandomtime¿j1=¿inontheinterval[un¡1;T].²Secondstep.Inthisstep,weassumethatallbuttwonameshavealreadydefaulted.LetDn¡2=f¿i1=u1;:::;¿in¡2=un¡2g.Foreachsurvivingnamej1;j22f=i1;:::;in¡2g,thepayo®Zbjl(tjDn¡2);t2[un¡2;T],ofabasketclaimatthemomentofthenextdefaultfor-mallycomprisestherecoverypayo®fromthedefaultednamejlwhichisZjl(tjDn¡2)andthepre-defaultvalueZe(tjDn¡2;¿jl=t);t2[un¡2;T],whichwascomputedinthe¯rststep.Therefore,wehaveZbjl(tjDn¡2)=Zjl(tjDn¡2)+Ze(tjDn¡2;¿jl=t);8t2[un¡2;T]:To¯ndthevalueofabasketclaimbetweenthe(n¡2)thand(n¡1)thdefault,itsu±cestocomputethepre-defaultvalueoftheconditionalFTDCassociatedwiththetwosurvivingnames,j1;j22f=i1;:::;in¡2g.Sincetheconditionalpayo®sZbj1(tjDn¡2)andZbj2(tjDn¡2)areknown,wemaycomputetheexpectationundertheconditionalprobabilitymeasureQjDn¡2inorderto¯ndthepre-defaultvalueofthisconditionalFTDCforanyt2[un¡2;T].²Generalinductionstep.Wenowassumethatexactlykdefaulthaveoccurred,thatis,weassumethattheeventDk=f¿i1=u1;:::;¿ik=ukgisgiven.>Fromtheprecedingstep,weknowthefunctionZe(tjDk+1)whereDk=f¿i1=u1;:::;¿ik=uk;¿jl=uk+1g.InordertocomputeZe(tjDk),wesetZbjl(tjDk)=Zjl(tjDk)+Ze(tjDk;¿jl=t);8t2[uk;T];(2.101)foranyj1;:::;jn¡k2f=i1;:::;ikg,andwecomputeZe(tjDk);t2[uk;T],astherisk-neutralvalueunderQjDkattimeoftheconditionalFTDCwiththepayo®sgivenby(2.101).Weareinapositionstatethevaluationresultforabasketclaim,whichcanbeformallyprovedusingthereasoningpresentedabove. 2.6.APPLICATIONSTOCOPULA-BASEDCREDITRISKMODELS75Proposition2.5.4Therisk-neutralvalueattimet2[0;T]ofabasketclaim(X;A;Z;¹¿¹)equalsnX¡1¼t=Ze(tjDk)1[¿(k)^T;¿(k+1)^T[(t);8t2[0;T];k=0whereDk=Dk(!)=f¿i1(!)=u1;:::;¿ik(!)=ukgfork=1;2;:::;n,andD0meansthatnodefaultshaveyetoccurred.2.5.8RecursiveReplicationofaBasketClaimFromthediscussionoftheprecedingsection,itisclearthatabasketclaimcanbeconvenientlyinterpretedasaspeci¯csequenceofconditional¯rst-to-defaultclaims.Henceitiseasytoguessthatthereplicationofabasketclaimshouldrefertohedgingoftheunderlyingsequenceofconditional¯rst-to-defaultclaims.Inthenextresult,wedenote¿(0)=0.Theorem2.3Foranyk=0;1;:::;n,thereplicatingstrategyÁforabasketclaim(X;A;Z;¹¿¹)onthetimeinterval[¿k^T;¿k+1^T]coincideswiththereplicatingstrategyfortheconditionalFTDCwithpayo®sZb(tjD)givenby(2.101).ThereplicatingstrategyÁ=(Á0;Áj1;:::;Ájn¡k;A),kcorrespondingtotheunitsofsavingsaccountandunitsofCDSoneachsurvivingnameattimet,hasthewealthprocessXn¡iV(Á;A)=Á0+ÁjlSjl(·)ttttjll=1whereprocessesÁjl;l=1;2;:::;n¡kcanbecomputedbytheconditionalversionofTheorem2.2.Proof.Weknowthatthebasketclaimcanbedecomposedintoaseriesofconditional¯rst-to-defaultclaims.So,atanygivenmomentoftimet2[0;T],assumingthatkdefaultshavealreadyoccurred,ourbasketclaimisequivalenttotheconditionalFTDCwithpayo®sZb(tjDk)andthepre-defaultvalueZe(tjDk).ThisconditionalFTDCisaliveuptothenextdefault¿(k+1)ormaturityT,whichevercomes¯rst.Henceitisclearthatthereplicatingstrategyofabasketclaimovertherandominterval[¿k^T;¿k+1^T]needtocoincidewiththereplicatingstrategyforthisconditional¯rst-to-defaultclaim,andthusitcanbefoundalongthesamelinesasinTheorem2.2,usingtheconditionaldistributionunderQjDkofdefaultsforsurvivingnames.¤2.6ApplicationstoCopula-BasedCreditRiskModelsInthissection,wewillapplyourpreviousresultstosomespeci¯cmodels,inwhichsomecommoncopulasareusedtomodeldependencebetweendefaulttimes(see,forinstance,Cherubinietal.[30],Embrechtsetal.[48],LaurentandGregory[67],Li[71]orMcNeiletal.[76]).Itisfairtoadmitthatcopula-basedcreditriskmodelsarenotfullysuitableforadynamicalapproachtocreditrisk,sincethefuturebehaviorofcreditspreadscanbepredictedwithcertainty,uptotheobservationsofdefaulttimes.Hencetheyareunsuitableforhedgingofoption-likecontractsoncreditspreads.Ontheotherhand,however,thesemodelsareofacommonuseinpracticalvaluationcreditderivativesandthuswedecidedtopresentthemhere.Ofcourse,ourresultsaremoregeneral,sothattheycanbeappliedtoanarbitraryjointdistributionofdefaulttimes(i.e.,notnecessarilygivenbysomecopulafunction).Also,inafollow-upwork,wewillextendtheresultsofthisworktoafullydynamicalsetup.Forsimplicityofexpositionandinordertogetmoreexplicitformulae,weonlyconsiderthebivariatesituationandwemakethefollowingstandingassumptions.Assumptions(B).Weassumefromnowonthat:(i)wearegivenanFTDC(X;A;Z;¿(1))whereZ=(Z1;Z2)forsomeconstantsZ1;Z2andX, 76CHAPTER2.HAZARDFUNCTIONAPPROACH(ii)thedefaulttimes¿1and¿2haveexponentialmarginaldistributionswithparameters¸1and¸2,(ii)therecovery±oftheithCDSisconstantand·=¸±fori=1;2(seeExample2.4.1).iiiiBeforeproceedingtocomputations,letusnotethatZTZ1ZTG(du;dv)=¡G(du;u)u=tv=utandthus,assumingthatthepair(¿1;¿2)hasthejointprobabilitydensityfunctionf(u;v),ZTZ1ZTdudvf(u;v)=¡@1G(u;u)dututandZb¡¢dvf(u;v)du=G(a;dv)¡G(b;dv)=dv@2G(b;v)¡@2G(a;v)aZTZ1ZTdudzf(z;v)=¡@2G(u;v)du:vuv2.6.1IndependentDefaultTimesLetus¯rstconsiderthecasewherethedefaulttimes¿1and¿2areindependent(thiscorrespondstotheproductcopulaC(u;v)=uv).Inviewofindependence,themarginalintensitiesandthe¯rst-to-defaultintensitiescanbeeasilyshowntocoincide.Wehave,fori=1;2G(u)=Q(¿>u)=e¡¸iuiiandthusthejointsurvivalfunctionequalsG(u;v)=G(u)G(v)=e¡¸1ue¡¸2v:12ConsequentlyF(du;dv)=G(du;dv)=¸¸e¡¸1ue¡¸2vdudv=f(u;v)dudv12andG(du;u)=¡¸e¡(¸1+¸2)udu.1Proposition2.6.1Assumethatthedefaulttimes¿1and¿2areindependent.ThenthereplicatingstrategyforanFTDC(X;0;Z;¿(1))isgivenasÁe1(t)=Z1¡¼e(t);Áe2(t)=Z2¡¼e(t)±1±2where(Z1¸1+Z2¸2)¡(¸1+¸2)(T¡t)¡(¸1+¸2)(T¡t)¼e(t)=(1¡e)+Xe:¸1+¸2Proof.Fromthepreviousremarks,weobtainRTR1RTR1Z1tudF(u;v)Z2tvdF(u;v)G(T;T)¼e(t)=++XG(t;t)G(t;t)G(t;t)Z¸RTe¡(¸1+¸2)uduZ¸RTe¡(¸1+¸2)vdv11t22tG(T;T)=++Xe¡(¸1+¸2)te¡(¸1+¸2)tG(t;t)Z1¸1¡(¸1+¸2)(T¡t)Z2¸2¡(¸1+¸2)(T¡t)G(T;T)=(1¡e)+(1¡e)+X(¸1+¸2)(¸1+¸2)G(t;t)(Z1¸1+Z2¸2)¡(¸1+¸2)(T¡t)¡(¸1+¸2)(T¡t)=(1¡e)+Xe:¸1+¸2 2.6.APPLICATIONSTOCOPULA-BASEDCREDITRISKMODELS77Undertheassumptionofindependenceofdefaulttimes,wealsohavethatSi(·)=Sei(·)fortjjitii;j=1;2andi6=j.FurthermorefromExample2.4.1,wehavethatSei(·)=0fort2[0;T]andtithusthematrixN(t)inTheorem2.2reducesto·¸±10N(t)=:0±2ThereplicatingstrategycanbefoundeasilybysolvingthelinearequationN(t)Áe(t)=h(t)whereh(t)=(h1(t);h2(t))withhi(t)=Zi¡¼e(t¡)=Zi¡¼e(t)fori=1;2:¤Asanotherimportantcaseofa¯rst-to-defaultclaim,wetakea¯rst-to-defaultswap(FTDS).ForastylizedFTDSwehaveX=0;A(t)=¡·(1)twhere·(1)istheswapspread,andZi(t)=±i2[0;1)forsomeconstants±i;i=1;2:HenceanFTDSisformallygivenasanFTDC(0;¡·(1)t;(±1;±2);¿(1)).Underthepresentassumptions,weeasilyobtain¸T³´1¡e¼0=¼e(0)=(¸1±1+¸2±2)¡·(1)¸where¸=¸1+¸2.TheFTDSmarketspreadisthelevelof·(1)thatmakestheFTDSvaluelessatinitiation.Henceinthiselementaryexamplethisspreadequals¸1±1+¸2±2.Inaddition,itcanbeshownthatunderthepresentassumptionswehavethat¼e(t)=0foreveryt2[0;T].SupposethatwewishtohedgetheshortpositionintheFTDSusingtwoCDSs,sayCDSi,i=1;2,withrespectivedefaulttimes¿i,protectionpayments±iandspreads·i=¸i±i.Recallthatinthepresentsetupwehavethat,fort2[0;T],Si(·)=Sei(·)=0;i;j=1;2;i6=j:(2.102)tjjitiConsequently,wehaveherethathi(t)=¡Zi(t)=¡±iforeveryt2[0;T].ItthenfollowsfromequationN(t)Áe(t)=h(t)thatÁe(t)=Áe(t)=1foreveryt2[0;T]andthusÁ0=0forevery12tt2[0;T].Thisresultisbynomeanssurprising:wehedgeashortpositionintheFTDSbyholdingastaticportfoliooftwosingle-nameCDSssince,underthepresentassumptions,theFTDSisequivalenttosuchaportfolioofcorrespondingsinglenameCDSs.Ofcourse,onewouldnotexpectthatthisfeaturewillstillholdinageneralcaseofdependentdefaulttimes.The¯rstequalityin(2.102)isduetothestandingassumptionofindependenceofdefaulttimes¿1and¿2andthusitwillnolongerbetrueforothercopulas(seeforegoingsubsections).Thesecondequalityfollowsfromthepostulatethattherisk-neutralmarginaldistributionsofdefaulttimesareexponential.Inpractice,therisk-neutralmarginaldistributionsofdefaulttimeswillbeobtainedby¯ttingthemodeltomarketdata(i.e.,marketpricesofsinglenameCDSs)andthustypicallytheywillnotbeexponential.Toconclude,bothequalitiesin(2.102)areunlikelytoholdinanyreal-lifeimplementation.Hencethisexampleshowbeseenasthesimplestillustrationofgeneralresultsandwedonotpretendthatithasanypracticalmerits.2.6.2ArchimedeanCopulasWenowproceedtothecaseofexponentiallydistributed,butdependent,defaulttimes.Theirinterdependencewillbespeci¯edbyachoiceofsomeArchimedeancopula.RecallthatabivariateArchimedeancopulaisde¯nedasC(u;v)='¡1('(u);'(v)),where'iscalledthegeneratorofacopula.ClaytonCopulaRecallthatthegeneratoroftheClaytoncopulaisgivenas'(s)=s¡µ¡1;s2R,forsomestrictly+positiveparameterµ.HencethebivariateClaytoncopulacanberepresentedasfollowsClayton¡µ¡µ¡1C(u;v)=C(u;v)=(u+v¡1)µ:µ 78CHAPTER2.HAZARDFUNCTIONAPPROACHUnderthepresentassumptions,thecorrespondingjointsurvivalfunctionG(u;v)equals¸uµ¸vµ¡1G(u;v)=C(G(u);G(v))=(e1+e2¡1)µ12sothatG(u;dv)¸1¡12vµ¸1uµ¸2vµ¡=¡¸2e(e+e¡1)µdvandG(du;dv)¸uµ+¸vµ¸uµ¸vµ¡1¡2f(u;v)==(µ+1)¸¸e12(e1+e2¡1)µ:12dudvProposition2.6.2Letthejointdistributionof(¿1;¿2)begivenbytheClaytoncopulawithµ>0.ThenthereplicatingstrategyforanFTDC(X;0;Z;¿(1))isgivenbytheexpressions±(Z¡¼e(t))+S2(·)(Z¡¼e(t))Áe21tj1221(t)=12;(2.103)±1±2¡Stj2(·1)Stj1(·2)±(Z¡¼e(t))+S1(·)(Z¡¼e(t))Áe12tj2112(t)=12;(2.104)±1±2¡Stj2(·1)Stj1(·2)whereRe¸1µT¸21Re¸2µT¸11(s+s¸1¡1)¡µ¡1ds(s+s¸2¡1)¡µ¡1dse¸1µte¸2µt¼e(t)=Z11+Z21µ(e¸1µt+e¸2µt¡1)¡µµ(e¸1µt+e¸2µt¡1)¡µ¸11µT¸2µT¡(e+e¡1)µ+X;¡1(e¸1µt+e¸2µt¡1)µ¸µT¸µT¡1¡1¸µv¸µv¡1¡1[(e1+e2¡1)µ¡(e1+e2¡1)µ]S1(·)=±vj211¸1µv¸2µv¡1¡1(e+e¡1)µRT¸1µu¸2µv¡1¡1(e+e¡1)µdu¡·v;11(e¸1µv+e¸2µv¡1)¡µ¡1and1¡1¸1¸1µT¸2µT¡1µu¸2µu¡¡12[(e+e¡1)µ¡(e+e¡1)µ]Suj1(·2)=±2(e¸1µu+e¸2µu¡1)¡1=µ¡1RT¸1µu¸2µv¡1¡1(e+e¡1)µdv¡·u:21(e¸1µu+e¸2µu¡1)¡µ¡1Proof.UsingtheobservationthatZTZ1ZT¸uµ¸uµ¸uµ¡1¡1duf(u;v)dv=¸e1(e1+e2¡1)µdu1tutZe¸1µT1¸21=(s+s¸1¡1)¡µ¡1dsµe¸1µtandthusbysymmetryZTZ1Ze¸2µT1¸11dvf(u;v)du=(s+s¸2¡1)¡µ¡1ds:tvµe¸2µt 2.6.APPLICATIONSTOCOPULA-BASEDCREDITRISKMODELS79WethusobtainRTR1RTR1Z1tudG(u;v)Z2tvdG(u;v)G(T;T)¼e(t)=++XG(t;t)G(t;t)G(t;t)Re¸1µT¸21Re¸2µT¸11(s+s¸1¡1)¡µ¡1ds(s+s¸2¡1)¡µ¡1dse¸1µte¸2µt=Z11+Z21µ(e¸1µt+e¸2µt¡1)¡µµ(e¸1µt+e¸2µt¡1)¡µ¸11µT¸2µT¡(e+e¡1)µ+X:¡1(e¸1µt+e¸2µt¡1)µWeareinapositiontodeterminethereplicatingstrategy.Underthestandingassumptionthat·=¸±fori=1;2westillhavethatSei(·)=0fori=1;2andfort2[0;T].HencethematrixiiitiN(t)reducesto"#±¡S2(·)1tj12N(t)=1¡Stj2(·1)±2whereRTRTR1f(u;v)duf(z;v)dzduS1(·)=±Rv¡·vRuvj211111f(u;v)duf(u;v)duvv¸111µT¸2µT¡¡1¸1µv¸2µv¡¡1[(e+e¡1)µ¡(e+e¡1)µ]=±11(e¸1µv+e¸2µv¡1)¡µ¡1RT¸1µu¸2µv¡1¡1(e+e¡1)µdu¡·v:11(e¸1µv+e¸2µv¡1)¡µ¡1TheexpressionforS2(·)canbefoundbyanalogouscomputations.Bysolvingtheequationuj12N(t)Áe(t)=h(t),weobtainthedesiredexpressions(2.103)-(2.104).¤Similarcomputationscanbedoneforthevaluationandhedgingofa¯rst-to-defaultswap.GumbelCopulaAsanotherofanArchimedeancopula,weconsidertheGumbelcopulawiththegenerator'(s)=(¡lns)µ;s2R,forsomeµ¸1.ThebivariateGumbelcopulacanthusbewrittenas+1µµGumbel¡[(¡lnu)+(¡lnv)]µC(u;v)=Cµ(u;v)=e:Underourstandingassumptions,thecorrespondingjointsurvivalfunctionG(u;v)equals1µµµµ¡(¸u+¸v)µG(u;v)=C(G1(u);G2(v))=e12:ConsequentlydG(u;v)µµ¡1µµµµ1¡1=¡G(u;v)¸2v(¸1u+¸2v)µdvanddG(u;v)µµ¡1µµµµ1¡2¡µµµµ1¢=G(u;v)(¸1¸2)(uv)(¸1u+¸2v)µ(¸1u+¸2v)µ+µ¡1:dudvProposition2.6.3Letthejointdistributionof(¿1;¿2)begivenbytheGumbelcopulawithµ¸1.ThenthereplicatingstrategyforanFTDC(X;0;Z;¿(1))isgivenby(2.103)-(2.104)with¼e(t)=(Z¸µ+Z¸µ)¸¡µ(e¡¸t¡e¡¸T)+Xe¡¸(T¡t);1122 80CHAPTER2.HAZARDFUNCTIONAPPROACH1µµµµ1¡(¸T+¸v)µµµµµ¡1¡¸v1¡µ1¡µ1e12(¸1T+¸2v)µ¡e¸vSvj2(·1)=±1e¡¸v¸1¡µv1¡µRTµµµµ11¡(¸1T+¸2v)µµµµµ¡1e(¸1u+¸2v)µdu¡·v;1e¡¸v¸1¡µv1¡µ1µµµµ1¡(¸1u+¸2T)µµµµµ¡1¡¸v1¡µ1¡µ2e(¸1u+¸2T)µ¡e¸uSuj1(·2)=±2e¡¸v¸1¡µu1¡µRTµµµµ11¡(¸u+¸T)µµµµµ¡1e12(¸1u+¸2v)µdv¡·u:2e¡¸v¸1¡µu1¡µProof.WehaveZTZ1ZT11µµµµµ¡1¡(¸1+¸2)µudG(u;v)=¸1(¸1+¸2)µedutut=(¡¸µ¸¡µe¡¸u)ju=T=¸µ¸¡µ(e¡¸t¡e¡¸T)1u=t1µµ1where¸=(¸1+¸2)µ.SimilarlyZTZ1dG(u;v)=¸µ¸¡µ(e¡¸t¡e¡¸T):2tvFurthermoreG(T;T)=e¡¸TandG(t;t)=e¡¸t.HenceRTR1RTR1dG(u;v)dG(u;v)G(T;T)¼e(t)=Ztu+Ztv+X12G(t;t)G(t;t)G(t;t)=Z¸µ¸¡µ(e¡¸t¡e¡¸T)+Z¸µ¸¡µ(e¡¸t¡e¡¸T)+Xe¡¸(T¡t)1122=(Z¸µ+±Zµ)¸¡µ(e¡¸t¡e¡¸T)+Xe¡¸(T¡t):1122Inorderto¯ndthereplicatingstrategy,weproceedasintheproofofProposition2.6.2.Underthepresentassumptions,wehaveRTRTR1f(u;v)duf(z;v)dzduS1(·)=±Rv¡·vRuvj211111f(u;v)duf(u;v)duvv1µµµµ1¡(¸T+¸v)µµµµµ¡1¡¸v1¡µ1¡µe12(¸1T+¸2v)µ¡e¸v=±1e¡¸v¸1¡µv1¡µRTµµµµ11¡(¸1T+¸2v)µµµµµ¡1e(¸1u+¸2v)µdu¡·v:1e¡¸v¸1¡µv1¡µThiscompletestheproof.¤ Chapter3HazardProcessApproachInthegeneralreduced-form(orhazardprocess)approach,wedealwithtwokindsofinformation:theinformationconveyedbytheassetsprices,denotedasF=(Ft)0·t·T¤,andtheinformationabouttheoccurrenceofthedefaulttime,thatis,theknowledgeofthetimewherethedefaultoccurredinthepast,ifthedefaulthasindeedalreadyhappen.Aswealreadyknow,thelatterinformationismodeledbythe¯ltrationHgeneratedbythedefaultprocessH.Attheintuitivelevel,thereference¯ltrationFisgeneratedbypricesofsomeassets,orbyothereconomicfactors(suchas,e.g.,interestrates).This¯ltrationcanalsobeasub¯ltrationofthe¯ltrationgeneratedbytheassetprices.ThecasewhereFisthetrivial¯ltrationisexactlywhatwehavestudiedinthepreviouschapter.ThoughinatypicalexampleFischosentobetheBrownian¯ltration,mosttheoreticalresultsdonotrelyonaparticularchoiceofthereference¯ltrationF.WedenotebyGt=Ft_Htthefull¯ltration(sometimesreferredtoastheenlarged¯ltration).Specialattentionwillbepaidinthischaptertotheso-calledhypothesis(H),which,inthepresentcontext,postulatestheinvarianceofthemartingalepropertywithrespecttotheenlargementofFbytheobservationsofthedefaulttime.Inordertoexaminetheexactmeaningofmarketcompletenessinadefaultableworldandtoderivethehedgingstrategiesforcreditderivatives,weshallalsoestablishasuitableversionofthepredictablerepresentationtheorem.MostresultspresentedinSections3.1-3.6canbefound,forinstance,insurveypapersbyBieleckiatal.[8]andJeanblancandRutkowski[58,59].Sections3.7-3.8arebasedonBieleckiatal.[11].3.1HazardProcessanditsApplicationsTheconceptsintroducedinthepreviouschapterwillnowbeextendedtoamoregeneralset-up,whenallowanceforanadditional°owofinformation{formallyrepresentedhereafterbysomereference¯ltrationF{ismade.Wedenoteby¿anon-negativerandomvariableonaprobabilityspace(•;G;Q);satisfyingQ(¿=0)=0andQ(¿>t)>0foranyt2R+.Weintroducetheright-continuousprocessHbysettingHt=1f¿·tgandwedenotebyHtheassociated¯ltration,sothatHt=¾(Hu:u·t)foreveryt2R+.Weassumethatwearegivenanauxiliaryreference¯ltrationFsuchthatG=H_F,thatis,Gt=Ht_Ftforanyt2R+.Foreacht2R+,thetotalinformationavailableattimetiscapturedbythe¾-¯eldGt.All¯ltrationsconsideredinwhatfollowsareimplicitlyassumedtosatisfythe`usualconditions'ofright-continuityandcompleteness.Forthesakeofsimplicity,weassumethatthe¾-¯eldF0istrivial(sothatG0isatrivial¾-¯eldaswell).81 82CHAPTER3.HAZARDPROCESSAPPROACHTheprocessHisobviouslyG-adapted,butitisnotnecessarilyF-adapted.Inotherwords,therandomtime¿isaG-stoppingtime,butitmayfailtobeanF-stoppingtime.Lemma3.1.1Assumethatthe¯ltrationGsatis¯esG=H_F.ThenGµG¤,whereG¤=(G¤)withtt2R+©ªG¤:=A2G:9B2F;Af¿>tg=Bf¿>tg:ttProof.ItisratherclearthattheclassG¤isasub-¾-¯eldofG:Therefore,itisenoughtocheckthattHµG¤andFµG¤foreveryt2R:Putanotherway,weneedtoverifythatifeitherA=f¿·ugtttt+forsomeu·torA2Ft;thenthereexistsaneventB2FtsuchthatAf¿>tg=Bf¿>tg.Intheformercase,wemaytakeB=;andinthelatterB=A:¤Foranyt2R+;wewriteFt=Qf¿·tjFtg;andwedenotebyGtheF-survivalprocessof¿withrespecttothe¯ltrationF;givenas:Gt:=1¡Ft=Q(¿>tjFt);8t2R+:Noticethatforany0·t·swehavef¿·tgµf¿·sg;andso¡¯¢EQ(FsjFt)=EQQ(¿·sjFs)¯Ft=Q(¿·sjFt)¸Q(¿·tjFt)=Ft:ThisshowsthattheprocessF(G;resp.)followsabounded,non-negativeF-submartingale(F-supermartingale,resp.)underQ:Wemaythusdealwiththeright-continuousmodi¯cationofF(ofG)with¯niteleft-handlimits.Thenextde¯nitionisaratherstraightforwardgeneralizationoftheconceptofthehazardfunction(seeDe¯nition2.2.1).De¯nition3.1.1AssumethatFt<1fort2R+:TheF-hazardprocessof¿underQ;denotedby¡;isde¯nedthroughtheformula1¡F=e¡¡t:Equivalently,¡=¡lnG=¡ln(1¡F)foreveryttttt2R+:SinceG0=1¡F0=1,itisclearthat¡0=0.Itisalsoeasytocheckthatlimt!1¡t=+1.Forthesakeofconciseness,weshallreferbrie°yto¡astheF-hazardprocess,ratherthantheF-hazardprocessunderQ;unlessthereisadangerofconfusion.Throughoutthischapter,wewillworkunderthestandingassumptionthattheinequalityFt<1holdsforeveryt2R+;sothattheF-hazardprocess¡iswellde¯ned.Hencethecasewhen¿isanF-stoppingtime(thatis,thecasewhenF=G)isnotdealtwithhere.3.1.1ConditionalExpectationsWeshall¯rstfocusontheconditionalexpectationEQ(1ft<¿gXjGt);whereXisaQ-integrablerandomvariable.Westartbythefollowingresult,whichextendsLemma2.2.1.Lemma3.1.2ForanyG-measurableandQ-integrablerandomvariableXwehave,foranyt2R+,EQ(1ft<¿gXjFt)EQ(1ft<¿gXjGt)=1ft<¿gEQ(XjGt)=1ft<¿g:(3.1)Q(t<¿jFt)Inparticular,foranyt·sQft<¿·sjFtg¡t¡¡sQft<¿·sjGtg=1ft<¿g=1ft<¿gEQ(1¡ejFt):(3.2)Q(t<¿jFt)Proof.SinceFtµGt,itsu±cestocheckthat¡¯¢¡¯¢EQ1CXQ(CjFt)¯Gt=EQ1CEQ(1CXjFt)¯Gt 3.1.HAZARDPROCESSANDITSAPPLICATIONS83wherewedenoteC=ft<¿g.Putanotherway,weneedtoshowthatforanyA2GtwehaveZZ1CXQ(CjFt)dQ=1CEQ(1CXjFt)dQ:AAInviewofLemma3.1.1,foranyA2GtwehaveAC=BCforsomeeventB2Ft.Therefore,ZZZ1CXQ(CjFt)dQ=XQ(CjFt)dQ=XQ(CjFt)dQAZACZBC=1CXQ(CjFt)dQ=EQ(1CXjFt)Q(CjFt)dQZBBZ=EQ(1CEQ(1CXjFt)jFt)dQ=EQ(1CXjFt)dQZBZBC=EQ(1CXjFt)dQ=1CEQ(1CXjFt)dQACAasdesired.¤ThefollowingcorollarytoLemma3.1.2isratherstraightforward(notethattoobtainthesecondequalityin(3.3),itsu±cestoconditiononthe¾-¯eldFs).Corollary3.1.1LetXbeanFs-measurableandQ-integrablerandomvariable.Then,foreveryt·s,E(X1jG)=1EQ(X1fs<¿gjFt)=1E(Xe¡t¡¡sjF):(3.3)Qfs<¿gtft<¿gft<¿gQtEQ(1ft<¿gjFt)Theproofofthenextresult,basedonanapproximationoftheprocessZbyasuitablesequenceofstep-wiseprocesses,islefttothereader.Proposition3.1.1AssumethatZisanF-predictableprocesssuchthattherandomvariableZ¿1f¿·sgisQ-integrable.Then,foreveryt·s,³Z¯´¡t¯1ft<¿gEQ(Z¿1f¿·sgjGt)=1ft<¿geEQZudFu¯Ft:(3.4)]t;s]LetF=N+CbetheDoob-MeyerdecompositionofF,whereNisanF-martingale,withN0=0,andCisanF-predictableincreasingprocess,withC0=0.Then,foreveryt·s,³Z¯´¡t¯1ft<¿gEQ(Z¿1f¿·sgjGt)=1ft<¿geEQZudCu¯Ft:]t;s]Inparticular,ifFisanincreasing,continuousprocessthenF=C=1¡e¡¡tsothatdF=e¡¡td¡.ttConsequently,³Zs¯´¡t¡¡u¯1ft<¿gEQ(Z¿1f¿·sgjGt)=1ft<¿gEQZued¡u¯Ft:(3.5)tThenextresultappearstobeusefulinvaluationofdefaultablesecuritiesthatpaydividendspriortothedefaulttime.Theproofisalsolefttothereaderasanexercise.Proposition3.1.2AssumethatAisabounded,F-predictableprocessof¯nitevariation.Then,foreveryt·s,³Z¯´³Z¯´¯¡t¯EQ(1¡Hu)dAu¯Gt=1ft<¿geEQ(1¡Fu)dAu¯Ft]t;s]]t;s]or,equivalently,³Z¯´³Z¯´¯¡t¡¡u¯EQ(1¡Hu)dAu¯Gt=1ft<¿gEQedAu¯Ft:]t;s]]t;s] 84CHAPTER3.HAZARDPROCESSAPPROACH3.1.2StochasticIntensityRtAssumethattheprocessFisabsolutelycontinuous,thatis,Ft=0fuduforsomeF-progressivelymeasurable,non-negativeprocessf.ThennecessarilyFisanincreasingprocessandthus¡isanabsolutelycontinuousandincreasingprocessaswell.Speci¯cally,itiseasytocheckthat¡admitsthestochasticintensity(orhazardrate)°,thatis,Zt¡t=°udu0whereinturntheF-progressivelymeasurable,non-negativeprocess°isgivenbytheformulaft°t=:1¡Ft3.1.3ValuationofDefaultableClaimsOurnextgoalistoestablishaconvenientrepresentationforthepre-defaultvalueofadefaultableclaimintermsofthehazardprocess¡ofthedefaulttime.WepostulatethatQrepresentsamartingalemeasureassociatedwiththechoiceofthesavingsaccountBasadiscountfactor(ornumeraire).Hence,inthepresentset-up,therisk-neutralvaluationformulareads³Z¯´¡1¯St=BtEQBudDu¯Gt:(3.6)]t;T]whereSistheex-dividendpriceprocessandDisthedividendprocessassociatedwithadefaultableclaim(seeSection1.1.2),thatis,ZZD=Xd(T)1+(1¡H)dA+ZdH:(3.7)tft¸Tguuuu]0;t]]0;t]Finally,Bstandsforthesavingsaccountprocess,thatis,³Zt´Bt=exprudu0wheretheF-progressivelymeasurableprocessrmodelstheshort-terminterestrate.Forthesakeofconciseness,wewillwrite³Z¯´¡1¯It=BtEQBu(1¡Hu)dAu¯Gt;]t;T]¡¯¢J=BE1B¡1Z¯G;ttQft<¿·Tg¿¿tand¡¯¢K=BEB¡1X1¯G:ttQTfT<¿gtInviewof(3.6){(3.7),itisclearthattheex-dividendpriceofadefaultableclaim(X;A;0;Z;¿)satis¯esSt=It+Jt+Kt.Proposition3.1.3Theex-dividendpriceofadefaultableclaim(X;A;0;Z;¿)admitsthefollowingrepresentation,foreveryt2[0;T],³Z¯´¡1¡1¡1¯St=1ft<¿gGtBtEQBu(GudAu¡ZudGu)+GTBTX¯Ft:]t;T]IfF(andthusalso¡)isanincreasing,continuousprocessthen³Z¯´S=1BEB¡1e¡t¡¡u(dA+Zd¡)+B¡1Xe¡t¡¡T¯:tft<¿gtQuuuuT¯Ft]t;T] 3.1.HAZARDPROCESSANDITSAPPLICATIONS85RProof.ByapplyingProposition3.1.2totheprocessof¯nitevariationB¡1dA,weobtain]0;t]uu³Z¯´¡1¡1¯It=1ft<¿gGtBtEQBuGudAu¯Ft]t;T]or,equivalently,³Z¯´¡1¡t¡¡u¯It=1ft<¿gBtEQBuedAu¯Ft:]t;T]Furthermore,formula(3.4)ofProposition3.1.1yields³Z¯´¡t¡1¯Jt=1ft<¿geBtEQBuZudFu¯Ft:]t;T]If,inaddition,thehazardprocess¡isanincreasingcontinuousprocessthen³ZT¯´¡1¡t¡¡u¯Jt=1ft<¿gBtEQBueZud¡u¯Ft:tFinally,itfollowsfrom(3.1)thatK=1e¡tBE(1B¡1XjF):tft<¿gtQf¿>TgTtSincetherandomvariablesXandBTareFT-measurable,wealsohave(see(3.3))¡¢K=1e¡tBE(GB¡1XjF)=1BEB¡1Xe¡t¡¡TjF:tft<¿gtQTTtft<¿gtQTtBothformulaeofthepropositionareobtaineduponsummation.¤LetusnotethatSt=1ft<¿gSet,wheretheF-adaptedprocessSerepresentsthepre-defaultvalueoftheclaim(X;A;0;Z;¿).ThefollowingresultisanimmediateconsequenceofProposition3.1.3.Corollary3.1.2AssumethatF(andthusalso¡)isanincreasing,continuousprocess.Thenthepre-defaultvalueofadefaultableclaim(X;A;0;Z;¿)coincideswiththepre-defaultvalueofaRtdefaultableclaim(X;A;b0;0;¿);whereAbt=At+0Zud¡u:Remark3.1.1WehaveomittedinProposition3.1.3therecoverypayo®X;esincetheexpressionbasedonthehazardprocessofthedefaulttimedoesnoteasilycoverthecaseofageneralFT-measurablerandomvariable.However,inthespecialcasewhenXe=±forsomeconstant±;itsu±cestosubstituteXewithanequivalentpayo®±B(¿;T)attimeofdefault.Letusreturntothecaseofthedefaulttimethatadmitsthestochasticintensity°.ThesecondformulaofProposition3.1.3nowtakesthefollowingform³ZR¯´¡u(r+°)dv¯S=1Eetvv(dA+°Zdu)¯Fttft<¿gQuuu]t;T]³R¯´¡T(r¯v+°v)dv+1ft<¿gEQetX¯Ft:Togetamoreconciserepresentationforthelastexpression,weintroducethedefault-risk-adjustedinterestratere=r+°andtheassociateddefault-risk-adjustedsavingsaccountBe,givenbytheformula³Zt´Bet=expreudu:(3.8)0AlthoughtheprocessBedoesnotrepresentthepriceofatradeablesecurity,ithassimilarfeaturesasthesavingsaccountB.Inparticular,BeisanF-adapted,continuousprocessof¯nitevariation.IntermsoftheprocessBe,wehave³ZZT¯´S=1BeEBe¡1dA+Be¡1Z°du+Be¡1X¯¯F:(3.9)tft<¿gtQuuuuuTt]t;T]tItisnoteworthythatthedefaulttime¿doesnotappearexplicitlyintheconditionalexpectationontheright-handsideof(3.9). 86CHAPTER3.HAZARDPROCESSAPPROACH3.1.4MartingalesAssociatedwithDefaultTimeWewillnowexaminesomeimportantmartingalesassociatedwiththedefaulttime¿.Proposition3.1.4(i)TheprocessL=(1¡H)e¡tisaG-martingale.tt(ii)IfXisanF-martingaleandtheprocessXLisintegrablethenitisaG-martingale.(iii)IftheprocessF(or,equivalently,¡)isincreasingandcontinuousthentheprocessMt=Ht¡¡(t^¿)isaG-martingale.Proof.(i)FromLemma3.1.2,weobtain,foranyt·s,E(LjG)=1e¡tE(1e¡sjF)=1e¡t=L;Qstft<¿gQfs<¿gtft<¿gtsincethetowerruleyieldsE(1e¡sjF)=E(E(1jF)e¡sjF)=1:Qfs<¿gtQQfs<¿gst(ii)UsingagainLemma3.1.2,weget,foranyt·s,EQ(LsXsjGt)=EQ(1fs<¿gLsXsjGt)¡¯¢=1e¡tE1e¡¡sX¯Fft<¿gQfs<¿gst¡¯¢=1e¡tEE(1jF)e¡¡sX¯Fft<¿gQQfs<¿gsst=LtXt:(iii)NotethatHisaprocessof¯nitevariationand¡anincreasingcontinuousprocess.Hencefromtheintegrationbypartsformula,weobtaindL=(1¡H)e¡td¡¡e¡tdH:ttttMoreover,theprocessMt=Ht¡¡(t^¿)canberepresentedasfollowsZZZM=dH¡(1¡H)d¡=¡e¡¡udL;tuuuu]0;t]]0;t]]0;t]andthusitisaG-martingale,sinceLisG-martingaleande¡¡tisaboundedprocess.Itshouldbestressedthatifthehazardprocess¡isnotincreasingthenthecomputationoftheIt^odi®erentialde¡tismorecomplicatedandpart(iii)ofthepropositionisnolongertrue,ingeneral.¤NotethattheprocessF(or,equivalently,¡)isnotnecessarilyof¯nitevariation.Hencepart(iii)inProposition3.1.4doesnotyieldthegeneralformoftheDoob-MeyerdecompositionofthesubmartingaleH.Forsimplicity,inthenextresultweshallassumethatFisacontinuousprocess.Itisworthnotingthatpart(iii)inProposition3.1.4isalsoaconsequenceofProposition3.1.5,sinceforacontinuousandincreasingprocessFwehavethatF=C=1¡e¡¡t.Proposition3.1.5AssumethatFisacontinuousprocesswiththeDoob-MeyerdecompositionF=N+C.ThentheprocessM=(Mt;t2R+),givenbytheformulaZt^¿dCuMt=Ht¡;(3.10)01¡FuisaG-martingale.Proof.Wesplittheproofintotwosteps.Firststep.Weshallprovethat,foranyt·s,E(HjG)=H+1e¡tE(C¡CjF):(3.11)Qsttft<¿gQstt 3.1.HAZARDPROCESSANDITSAPPLICATIONS87Indeed,wehavethatE(HjG)=1¡Q(s<¿jG)=1¡1e¡tE(1¡FjF)Qsttft<¿gQst=1¡1e¡tE(1¡N¡CjF)ft<¿gQsst¡¢=1¡1e¡t1¡N¡C¡E(C¡CjF)ft<¿gttQstt¡¢=1¡1e¡t1¡F¡E(C¡CjF)ft<¿gtQstt=1+1e¡tE(C¡CjF):ft¸¿gft<¿gQsttSecondstep.LetusdenoteZtZtdCu¡uUt=(1¡Hu)=(1¡Hu)edCu:01¡Fu0Weshallprovethat,foranyt·s,E(UjG)=U+1e¡tE(C¡CjF):Qs^¿tt^¿ft<¿gQsttFromLemma3.1.1,weobtainµZ1¯¶¡t¯EQ(Us^¿jGt)=Ut^¿1ft¸¿g+1ft<¿geEQUs^udFu¯FttµZsZ1¯¶¡t¯=Ut^¿1ft¸¿g+1ft<¿geEQUudFu+UsdFu¯FttsµZs¯¶¡t¯=Ut^¿1ft¸¿g+1ft<¿geEQUudFu+Us(1¡Fs)¯Ft:tUsingtheintegrationbypartsformulaandthefactthatUisof¯nitevariationandcontinuous,weobtaind(Ut(1¡Ft))=¡UtdFt+(1¡Ft)dUt=¡UtdFt+dCt:Consequently,ZsUudFu+Us(1¡Fs)=¡Us(1¡Fs)+Ut(1¡Ft)+Cs¡Ct+Us(1¡Fs)t=Ut(1¡Ft)+Cs¡Ct:Itfollowsthat,foranyt·s,¡¢E(UjG)=1U+1e¡tEU(1¡F)+C¡CjFQs^¿tft¸¿gt^¿ft<¿gQttstt=U+1e¡tE(C¡CjF):t^¿ft<¿gQsttBycombiningtheformulaabovewith(3.11),weconcludethattheprocessMgivenby(3.10)isaG-martingale.¤Letusassume,inaddition,thatCisabsolutelycontinuouswithrespecttotheLebesguemeasure,RtsothatCt=0cuduforsomeF-adaptedprocessc.ThenwemaydeducetheexistenceofanF-adaptedprocess°,calledthedefaultintensitywithrespecttoF,suchthattheprocessZt^¿ZtMt=Ht¡°udu=Ht¡(1¡Hu)°udu00isaG-martingale.Moreexplicitly,wehavethat°=ctforeveryt2R.t1¡Ft+Lemma3.1.3Theintensityprocess°satis¯esalmosteverywhere1Q(t<¿tg=f»tjF)=Q(»tjF)=EQ(¿>tjF)jF=e¡¡t;(3.17)ttQ1tandsoFisanF-adapted,right-continuous,increasingprocess.Itisalsoclearthattheprocess¡representstheF-hazardprocessof¿underQ:Asanimmediateconsequenceof(3.16)and(3.17),weobtainthefollowingpropertyofthecanonicalconstructionofthedefaulttime(cf.(3.14))Q(¿·tjF1)=Q(¿·tjFt);8t2R+:(3.18)Tosumup,wehavethatQ(¿·tjF)=Q(¿·tjF)=Q(¿·tjF)=e¡¡t(3.19)1utforanytwodates0·t·u:3.2.3StochasticBarrierSupposethatQ(¿·tjF)=Q(¿·tjF)=1¡e¡¡t;t1where¡isacontinuous,strictlyincreasing,F-adaptedprocess.Ourgoalistoshowthatthereexistsarandomvariable£,independentofF1;withexponentiallawofparameter1,suchthat¿=infft¸0:¡t>£g:Letusset£:=¡¿:Thenft<£g=ft<¡¿g=fCt<¿g;whereCistherightinverseof¡,sothat¡Ct=t:ThereforeQ(£>ujF)=e¡¡Cu=e¡u:1Wehavethusestablishedtherequiredproperties,namely,theprobabilitylawof£anditsindepen-denceofthe¾-¯eldF1:Furthermore,¿=infft:¡t>¡¿g=infft:¡t>£g. 3.3.PREDICTABLEREPRESENTATIONTHEOREM933.3PredictableRepresentationTheoremKusuoka[63]establishedthefollowingrepresentationtheoreminwhichthereference¯ltrationFisgeneratedbyaBrownianmotion.Theorem3.1Assumethatthehypothesis(H)holds.Thenanysquare-integrablemartingalewithrespecttoGadmitsarepresentationasthesumofastochasticintegralwithrespecttotheBrownianmotionandastochasticintegralwithrespecttothediscontinuousmartingaleMassociatedwith¿.Weassume,forsimplicity,thatFiscontinuousandF<1foreveryt2R+.Sincethehypothesist(H)holds,Fisanincreasingprocess.ThenclearlydF=e¡¡td¡;de¡t=e¡td¡:(3.20)tttProposition3.3.1Supposethathypothesis(H)holdsunderQandthatanyF-martingaleiscon-tinuous.ThenthemartingaleMh=E(hjG),wherehisanF-predictableprocesssuchthattQ¿tEQjh¿j<1,admitsthefollowingdecompositioninthesumofacontinuousmartingaleandadiscontinuousmartingaleZt^¿ZMh=mh+e¡udmh+(h¡Mh)dM;(3.21)t0uuu¡u0]0;t^¿]wheremhisthecontinuousF-martingalegivenby³Z1¯´h¯mt=EQhudFu¯Ft0andMisthediscontinuousG-martingalede¯nedasMt=Ht¡¡t^¿.Proof.Westartbynotingthat³Z1¯´h¡t¯Mt=EQ(h¿jGt)=1ft¸¿gh¿+1f¿>tgeEQhudFu¯Ftt³Zt´=1h+1e¡tmh¡hdF:(3.22)ft¸¿g¿f¿>tgtuu0Wewillsketchtwoslightlydi®erentderivationsof(3.21).Firstderivation.LettheprocessJbegivenbytheformula,fort2R+,³Zt´J=e¡tmh¡hdF:ttuu0Notingthat¡isacontinuousincreasingprocessandmhisacontinuousmartingale,wededucefromtheIt^ointegrationbypartsformulathat³Zt´dJ=e¡tdmh¡e¡thdF+mh¡hdFe¡td¡tttttuut0=e¡tdmh¡e¡thdF+Jd¡:tttttTherefore,from(3.20),dJ=e¡tdmh+(J¡h)d¡tttttor,intheintegratedform,ZtZtJ=Mh+e¡udmh+(J¡h)d¡:t0uuuu00 94CHAPTER3.HAZARDPROCESSAPPROACHNotethatJ=Mh=Mhontheeventft<¿g.Therefore,ontheeventft<¿g,ttt¡Zt^¿Zt^¿Mh=Mh+e¡udmh+(Mh¡h)d¡:t0uu¡uu00From(3.22),thejumpofMhattime¿ish¡J=h¡Mh=Mh¡Mh.Equality(3.21)now¿¿¿¿¡¿¿¡follows.Secondderivation.Equality(3.22)canbere-writtenasfollowsZt³Zt´Mh=hdH+(1¡H)e¡tmh¡hdF:tuuttuu00HencetheresultcanbeobtaineddirectlybytheIt^ointegrationbypartsformula.¤3.4Girsanov'sTheoremWenowstartbyde¯ningarandomtime¿onaprobabilityspace(•;G;Q)andwepostulatethatitadmitsthecontinuousF-hazardprocess¡underQ.Hence,fromProposition3.1.5,weknowthattheprocessMt=Ht¡¡t^¿isaG-martingale.Wepostulatethatthehypothesis(H)holdsunderQ.Finally,wepostulatethatthereference¯ltrationFisgeneratedbyanF-(hencealsoG-)BrownianmotionunderQ.Letus¯xT>0.ForaprobabilitymeasurePequivalenttoQon(•;GT)weintroducetheG-martingale´t;t·T;bysetting¯dP¯´t:=¯Gt=EQ(XjGt);Q-a.s.,(3.23)dQwhereXisaGT-measurablerandomvariable,suchthatQ(X>0)=1andEQX=1.UsingTheorem3.1,wededucethattheRadon-Nikod¶ymdensityprocess´admitsthefollowingrepresentationZtZ´t=1+»udWu+³udMu;0]0;t]where»and³areG-predictablestochasticprocesses.Since´isastrictlypositiveprocess,wegetZ¡¢´t=1+´u¡µudWu+·udMu(3.24)]0;t]whereµand·areG-predictableprocesses,with·>¡1.Moreexplicitly,theprocess´equalsµZ¶µZ¶¢¢(1)(2)´t=EtµudWuEt·udMu=´t´t;(3.25)00wherewewriteµZ¢¶µZtZt¶(1)12´t=EtµudWu=expµudWu¡µudu;0020andµZ¢¶µZtZt^¿¶(2)´t=Et·udMu=expln(1+·u)dHu¡·u°udu:(3.26)000ThenwehavethefollowingextensionofGirsanov'stheorem.Proposition3.4.1LetPbeaprobabilitymeasureon(•;GT)equivalenttoQ.IftheRadon-Nikod¶ymdensityofPwithrespecttoQisgivenby(3.23)with´satisfying(3.24)thentheprocessZtWct=Wt¡µudu;8t2[0;T];(3.27)0 3.4.GIRSANOV'STHEOREM95isaBrownianmotionwithrespecttoGunderPandtheprocessZt^¿Zt^¿Mct:=Mt¡·ud¡u=Ht¡(1+·u)d¡u;8t2[0;T];(3.28)00isaG-martingaleunderPorthogonaltoW:cProof.Note¯rstthatfort·Twehaved(´tWct)=Wctd´t+´t¡dWct+d[W;´c]t=Wctd´t+´t¡dWt¡´t¡µtdt+´t¡µtd[W;W]t=Wctd´t+´t¡dWt:ThisshowsthatWcisaG-martingaleunderP.SincethequadraticvariationofWcunderPequals[W;cWc]t=tandWciscontinuous,byvirtueofL¶evy'stheoremitisclearthatWcisaBrownianmotionunderP.Similarly,fort·Td(´tMct)=Mctd´t+´t¡dMct+d[M;´c]t=Mctd´t+´t¡dMt¡´t¡·td¡t^¿+´t¡·tdHt=Mctd´t+´t¡(1+·t)dMt:WeconcludethatMcisaG-martingaleunderP.ToconcludeitisenoughtoobservethatWcisacontinuousprocessandMcfollowsaprocessof¯nitevariation.¤Theproofofthenextresulthingesonthepredictablerepresentationtheorem.Corollary3.4.1LetYbeaG-martingaleunderP.ThenYadmitsthefollowingdecompositionZtZY=Y+»¤dWc+³¤dMc;(3.29)t0uuuu0]0;t]where»¤and³¤areG-predictablestochasticprocesses.Proof.ConsidertheprocessYegivenbytheformulaZZYe=´¡1d(´Y)¡´¡1Yd´:tu¡uuu¡u¡u]0;t]]0;t]ItisclearthatYeisaG-localmartingaleunderQ.NoticealsothatIt^o'sformulayields¡1¡1¡1´u¡d(´uYu)=dYu+´u¡Yu¡d´u+´u¡d[Y;´]u;andthusZ¡1Yt=Y0+Yet¡´u¡d[Y;´]u:(3.30)]0;t]Fromthepredictablerepresentationtheorem,weknowthatZtZYet=Y0+»eudWu+³eudMu(3.31)0]0;t]forsomeG-predictableprocesses»eand³e.Therefore,¡1dYt=»etdWt+³etdMt¡´t¡d[Y;´]tandthusdY=»edWc+³e(1+·)¡1dMctttttt 96CHAPTER3.HAZARDPROCESSAPPROACHsince(3.24)combinedwith(3.30)-(3.31)yield´¡1d[Y;´]=»eµdt+³e·(1+·)¡1dH:t¡tttttttToderivethelastequalityweobserve,inparticular,thatinviewof(3.30)wehave(wetakeintoaccountcontinuityof¡)¢[Y;´]t=´t¡³et·tdHt¡·t¢[Y;´]t:WeconcludethatYsatis¯es(3.29)withtheprocesses»¤=»eand³¤=³e(1+·)¡1,whereinturn»eand³earegivenby(3.31).¤3.5InvarianceofHypothesis(H)Kusuoka[63]showsbymeansofacounter-example(seeExample3.5.1)thatthehypothesis(H)isnotinvariantwithrespecttoanequivalentchangeoftheunderlyingprobabilitymeasure,ingeneral.Itisworthnotingthathiscounter-exampleisbasedontwo¯ltrations,H1andH2,generatedbythetworandomtimes¿1and¿2,andhechoosesH1toplaytheroleofthereference¯ltrationF.WeshallarguethatinthecasewhereFisgeneratedbyaBrownianmotion,theabove-mentionedinvariancepropertyisvalidundermildtechnicalassumptions.Letus¯rstexamineageneralsetupinwhichG=F_H,whereFisanarbitrary¯ltrationandHisgeneratedbythedefaultprocessH.WesaythatQislocallyequivalenttoPifQisequivalenttoPon(•;Gt)foreveryt2R+.ThenthereexiststheRadon-Nikod¶ymdensityprocess´suchthatdQjGt=´tdPjGt;8t2R+:(3.32)Forpart(i)inLemma3.5.1,werefertoBlanchet-ScallietandJeanblanc[18]orProposition2.2inJamshidian[55].Forpart(ii),seeJeulinandYor[60].Inthissection,wewillworkunderthestandingassumptionthatthehypothesis(H)isvalidunderP.Lemma3.5.1(i)LetQbeaprobabilitymeasureequivalenttoPon(•;Gt)foreveryt2R+,withtheassociatedRadon-Nikod¶ymdensityprocess´.Ifthedensityprocess´isF-adaptedthenwehaveQ(¿·tjFt)=P(¿·tjFt)foreveryt2R+.Hencethehypothesis(H)isalsovalidunderQandtheF-intensitiesof¿underQandunderPcoincide.(ii)AssumethatQisequivalenttoPon(•;G)anddQ=´1dP,sothat´t=EP(´1jGt).Thenthehypothesis(H)isvalidunderQwheneverwehave,foreveryt2R+,EP(´1HtjF1)EP(´tHtjF1)=:(3.33)EP(´1jF1)EP(´tjF1)Proof.Toprove(i),assumethatthedensityprocess´isF-adapted.Wehaveforeacht·s2R+EP(´t1f¿·tgjFt)Q(¿·tjFt)==P(¿·tjFt)=P(¿·tjFs)=Q(¿·tjFs);EP(´tjFt)wherethelastequalityfollowsbyanotherapplicationoftheBayesformula.Theassertionnowfollowsfrompart(i)inLemma3.2.1.Toprovepart(ii),itsu±cestoestablishtheequalityFbt:=Q(¿·tjFt)=Q(¿·tjF1);8t2R+:(3.34)Notethatsincetherandomvariables´t1f¿·tgand´tareP-integrableandGt-measurable,usingtheBayesformula,part(v)inLemma3.2.1,andassumedequality(3.33),weobtainthefollowingchainofequalitiesEP(´t1f¿·tgjFt)EP(´t1f¿·tgjF1)EP(´11f¿·tgjF1)Q(¿·tjFt)====Q(¿·tjF1):EP(´tjFt)EP(´tjF1)EP(´1jF1)Weconcludethatthehypothesis(H)holdsunderQifandonlyif(3.33)isvalid.¤ 3.5.INVARIANCEOFHYPOTHESIS(H)97Unfortunately,straightforwardveri¯cationofcondition(3.33)israthercumbersome.Forthisreason,weshallprovidealternativesu±cientconditionsforthepreservationofthehypothesis(H)underalocallyequivalentprobabilitymeasure.3.5.1CaseoftheBrownianFiltrationLetWbeaBrownianmotionunderPandFitsnatural¯ltration.Sinceweworkunderthestandingassumptionthatthehypothesis(H)issatis¯edunderP,theprocessWisalsoaG-martingale,whereG=F_H.HenceWisaBrownianmotionwithrespecttoGunderP.Ourgoalistoshowthatthehypothesis(H)isstillvalidunderQ2QforalargeclassQof(locally)equivalentprobabilitymeasures.Wepostulatethat¿admitsthehazardrate°withrespecttoFunderP.LetQbeanarbitraryprobabilitymeasurelocallyequivalenttoPon(•;G).ThepredictablerepresentationtheoremimpliesthatthereexistG-predictableprocessesµand·>¡1suchthattheRadon-Nikod¶ymdensity´ofQwithrespecttoPsatis¯esthefollowingSDE¡¢d´t=´t¡µtdWt+·tdMtwiththeinitialvalue´0=1,sothat´isgivenby(3.25).ByvirtueofasuitableversionoftheGirsanovtheorem,thefollowingprocessesWcandMcareG-martingalesunderQZtZtWct=Wt¡µudu;Mct=Mt¡1fu<¿g°u·udu:00Proposition3.5.1Assumethatthehypothesis(H)holdsunderP.LetQbeaprobabilitymeasurelocallyequivalenttoPwiththeassociatedRadon-Nikod¶ymdensityprocess´givenbyformula(3.25).IftheprocessµisF-adaptedthenthehypothesis(H)isvalidunderQandtheF-intensityof¿underQequals°bt=(1+·et)°t,where·eistheuniqueF-predictableprocesssuchthattheequality·et1ft·¿g=·t1ft·¿gholdsforeveryt2R+.Proof.LetPebetheprobabilitymeasurelocallyequivalenttoPon(•;G),givenbyµZ¶¢(2)dPejGt=Et·udMudPjGt=´tdPjGt:(3.35)0Weclaimthatthehypothesis(H)holdsunderPe.FromGirsanov'stheorem,theprocessWfollowsaBrownianmotionunderPewithrespecttobothFandG.Moreover,fromthepredictablerepre-sentationpropertyofWunderPe,wededucethatanyF-localmartingaleLunderPecanbewrittenasastochasticintegralwithrespecttoW.Speci¯cally,thereexistsanF-predictableprocess»suchthatZtLt=L0+»udWu:0ThisshowsthatLisalsoaG-localmartingale,andthusthehypothesis(H)holdsunderPe.SinceµZ¶¢dQjGt=EtµudWudPejGt;0byvirtueofpart(i)inLemma3.5.1,thehypothesis(H)isvalidunderQaswell.Thelastclaiminthestatementofthelemmacanbededucedfromthefactthatthehypothesis(H)holdsunderQand,byGirsanov'stheorem,theprocessZtZtMct=Mt¡1fu<¿g°u·udu=Ht¡1fu<¿g(1+·eu)°udu00isaQ-martingale.¤ 98CHAPTER3.HAZARDPROCESSAPPROACHWeclaimthattheequalityPe=Pholdsonthe¯ltrationF.Indeed,wehavedPejFt=´etdPjFt,(2)wherewewrite´et=EP(´tjFt),andµµZ¶¯¶¢(2)¯EP(´tjFt)=EPEt·udMu¯F1=1;8t2R+;(3.36)0wherethe¯rstequalityfollowsfrompart(v)inLemma3.2.1.Toestablishthesecondequalityin(3.36),we¯rstnotethatsincetheprocessMisstoppedat¿,wemayassume,withoutlossofgenerality,that·=·ewheretheprocess·eisF-predictable.Moreover,theconditionalcumu-lativedistributionfunctionof¿givenF1hastheform1¡exp(¡¡t(!)).Hence,forarbitrarilyselectedsamplepathsofprocesses·and¡,theclaimedequalitycanbeseenasaconsequenceofthemartingalepropertyoftheDol¶eansexponential.Formally,itcanbeprovedbyfollowingelementarycalculations,wherethe¯rstequalityisaconsequenceof(3.26),µµZ¢¶¯¶µ¡¢³Zt^¿´¯¶¯¯EPEt·eudMu¯F1=EP1+1ft¸¿g·e¿exp¡·eu°udu¯F100µZ1¡¢³Zt^u´R¯¶¡u°dv¯=E1+1·eexp¡·e°dv°e0vdu¯F1Pft¸uguvvu00µZt¡¢³Zu´¯¶¯=EP1+·eu°uexp¡(1+·ev)°vdvdu¯F100³Zt´µZ1R¯¶¡u°dv¯+exp¡·e°dvE°e0vdu¯F1vvPu0tZt¡¢³Zu´=1+·eu°uexp¡(1+·ev)°vdvdu00³Zt´Z1Ru¡°vdv+exp¡·ev°vdv°ue0du0t³Zt´³Zt´³Zt´=1¡exp¡(1+·ev)°vdv+exp¡·ev°vdvexp¡°vdv=1;000wherethepenultimateequalityfollowsbyanapplicationofthechainrule.3.5.2ExtensiontoOrthogonalMartingalesEquality(3.36)suggeststhatProposition3.5.1canbeextendedtothecaseofarbitraryorthogonallocalmartingales.Suchageneralizationisconvenient,ifwewishtocoverthesituationconsideredinKusuoka'scounterexample.LetNbealocalmartingaleunderPwithrespecttothe¯ltrationF.ItisalsoaG-localmartingale,sincewemaintaintheassumptionthatthehypothesis(H)holdsunderP.LetQbeanarbitraryprobabilitymeasurelocallyequivalenttoPon(•;G).WeassumethattheRadon-Nikod¶ymdensityprocess´ofQwithrespecttoPequals¡¢d´t=´t¡µtdNt+·tdMt(3.37)forsomeG-predictableprocessesµand·>¡1(thepropertiesoftheprocessµdepend,ofcourse,onthechoiceofthelocalmartingaleN).ThenextresultcoversthecasewhereNandMareorthogonalG-localmartingalesunderP,sothattheproductMNfollowsaG-localmartingale.Proposition3.5.2Assumethatthefollowingconditionshold:(a)NandMareorthogonalG-localmartingalesunderP,(b)NhasthepredictablerepresentationpropertyunderPwithrespecttoF,inthesensethatanyF-localmartingaleLunderPthereexistsanF-predictableprocess»suchthatZtLt=L0+»udNu;8t2R+;0 3.5.INVARIANCEOFHYPOTHESIS(H)99(c)Peisaprobabilitymeasureon(•;G)suchthat(3.35)holds.Thenwehave:(i)thehypothesis(H)isvalidunderPe,(ii)iftheprocessµisF-adaptedthenthehypothesis(H)isvalidunderQ.Lemma3.5.2UndertheassumptionsofProposition3.5.2,wehave:(i)NisaG-localmartingaleunderPe,(ii)NhasthepredictablerepresentationpropertyforF-localmartingalesunderPe.Proof.Inviewof(c),wehavedPej=´(2)dPj,wherethedensityprocess´(2)isgivenby(3.26),GttGtsothatd´(2)=´(2)·dM.FromtheassumedorthogonalityofNandM,itfollowsthatNand´(2)tt¡ttareorthogonalG-localmartingalesunderP,andthusN´(2)isaG-localmartingaleunderPaswell.ThismeansthatNisaG-localmartingaleunderPe,sothat(i)holds.Toestablishpart(ii)inthelemma,we¯rstde¯netheauxiliaryprocess´ebysetting´et=(2)EP(´tjFt).ThenmanifestlydPejFt=´etdPjFt,andthusinordertoshowthatanyF-localmar-tingaleunderPeisanF-localmartingaleunderP,itsu±cestocheckthat´et=1foreveryt2R+,sothatPe=PonF.Tothisend,wenotethatµµZ¶¯¶¢(2)¯EP(´tjFt)=EPEt·udMu¯F1=1;8t2R+;0wherethe¯rstequalityfollowsfrompart(v)inLemma3.2.1,andthesecondonecanestablishedsimilarlyasthesecondequalityin(3.36).Weareinapositiontoprove(ii).LetLbeanF-localmartingaleunderPe.ThenitfollowsalsoanF-localmartingaleunderPandthus,byvirtueof(b),itadmitsanintegralrepresentationwithrespecttoNunderPandPe.ThisshowsthatNhasthepredictablerepresentationpropertywithrespecttoFunderPe.¤ProofofProposition3.5.2.WeshallarguealongthesimilarlinesasintheproofofProposition3.5.1.Toprove(i),notethatbypart(ii)inLemma3.5.2weknowthatanyF-localmartingaleunderPeadmitstheintegralrepresentationwithrespecttoN.But,bypart(i)inLemma3.5.2,NisaG-localmartingaleunderPe.WeconcludethatLisaG-localmartingaleunderPe,andthusthehypothesis(H)isvalidunderPe.Assertion(ii)nowfollowsfrompart(i)inLemma3.5.1.¤Example3.5.1Kusuoka[63]presentsacounter-examplebasedonthetwoindependentrandomiiRt^¿itimes¿1and¿2givenonsomeprobabilityspace(•;G;P).WewriteMt=Ht¡0°i(u)du,whereHi=1and°isthedeterministicintensityfunctionof¿underP.LetussetdQj=´dPj,tft¸¿igiiGttGt(1)(2)where´t=´t´tand,fori=1;2andeveryt2R+,ZtµZ¢¶´(i)=1+´(i)·(i)dMi=E·(i)dMitu¡uutuu00forsomeG-predictableprocesses·(i);i=1;2,whereG=H1_H2.WesetF=H1andH=H2.Manifestly,thehypothesis(H)holdsunderP.Moreover,inviewofProposition3.5.2,itisstillvalid(2)undertheequivalentprobabilitymeasurePegivenbydPejGt=´tdPjGt.ItisclearthatPe=PonF,sinceµµZ¶¯¶¢(2)(2)2¯1EP(´tjFt)=EPEt·udMu¯Ht=1;8t2R+:0However,thehypothesis(H)isnotnecessarilyvalidunderQiftheprocess·(1)failstobeF-adapted.InKusuoka'scounter-example,theprocess·(1)waschosentobeexplicitlydependentonbothrandomtimes,anditwasshownthatthehypothesis(H)doesnotholdunderQ.ForanalternativeapproachtoKusuoka'sexample,throughanabsolutelycontinuouschangeofaprobabilitymeasure,theinterestedreadermayconsultCollin-Dufresneetal.[32]. 100CHAPTER3.HAZARDPROCESSAPPROACH3.6AlternativeApproachInthealternativeapproachtomodelingofdefaulttime,thestartingpointistheknowledgeofdefaulttime¿andsome¯ltrationGsuchthat¿isaG-stoppingtime.Themartingaleintensity"isthende¯nedasanynon-negativeandG-predictableprocess¸,suchthattheprocessM=(Mt;t2R+),givenasZt^¿Mt:=Ht¡¸sds;0isaG-martingaleunderQ.Theexistenceoftheintensity¸hingesonthefactthatHisanincreasingprocess,thereforeasub-martingale,andthusitcanbewrittenasasumofamartingaleMandaG-predictable,increasingprocessA,whichisstoppedat¿.Inthecasewhere¿isapredictablestoppingtime,obviouslyA=H.Infact,themartingaleintensity"existsonlyif¿isatotallyinaccessiblestoppingtimewithrespecttoG:Inthepresentset-up,thedefaultintensityisnotwellde¯nedaftertime¿.Speci¯cally,if¸isamartingaleintensity"thenforanynon-negative,G-predictableprocessgtheprocess¸et=¸t1ft·¿g+gt1ft>¿gisalsoamartingaleintensity".LetRtuswrite¤t=0¸udu.ThefollowingresultisacounterpartofLemma3.1.4(i).Lemma3.6.1TheprocessL=1e¤t;t2R,isaG-martingale.tft<¿g+Proof.Fromtheintegrationbypartsformula,weget¡¢dL=e¤t(1¡H)¸dt¡dH=¡e¤tdM:tttttThisshowsthatLisaG-martingale.¤ThefollowingresultisduetoDu±eetal.[42].¡¯¢Proposition3.6.1Letusde¯netheprocessYbysettingY=EXe¡¤T¯G,foreveryt2R,tQt+ThenforanyGT-measurableandQ-integrablerandomvariableXwehaveE(X1jG)=1e¤tE(Xe¡¤TjG)¡E(1¢Ye¤¿jG):QfT<¿gtft<¿gQtQft<¿·Tg¿tProof.LetusdenoteU=LY.TheIt^ointegrationbypartsformulayieldsdUt=Lt¡dYt+Yt¡dLt+d[L;Y]t=Lt¡dYt+Yt¡dLt+¢Lt¢Yt:SinceLandYareG-martingales,weobtain¡¢¡¢E(UjG)=EX1jG=U¡E1¢Ye¤¿jG:QTtQfT<¿gttQft<¿·Tg¿tConsequently,¡¢¡¢E(X1jG)=1e¤tEXe¡¤TjG¡E1e¤¿¢YjG;QfT<¿gtft<¿gQtQft<¿·Tg¿tasrequired.¤Itisworthtocomparethenextresultwithformula(3.3)inCorollary3.1.1.¡¯¢Corollary3.6.1AssumethattheprocessY=EXe¡¤T¯Giscontinuousattime¿,thatis,tQt¢Y¿=0.Thenwehave,foranyGT-measurableandQ-integrablerandomvariableX,¡¯¢E(X1jG)=1EXe¤t¡¤T¯G:(3.38)QfT<¿gtft<¿gQtItshouldbestressedthatthecontinuityoftheprocessYattime¿dependsonthechoiceof¸aftertime¿anditisratherdi±culttoverify,ingeneral.Moreover,thejumpsize¢Y¿isusuallydi±culttocompute.Therefore,thepracticalusefulnessofthealternativeapproachisratherlimited. 3.7.SINGLE-NAMECREDITDEFAULTSWAPMARKET1013.7Single-NameCreditDefaultSwapMarketAstrictlypositiverandomvariable¿,de¯nedonaprobabilityspace(•;G;Q),istermedarandomtime.Inviewofits¯nancialinterpretation,wewillrefertoitasadefaulttime.Wede¯nethedefaultindicatorprocessHt=1ft¸¿gandwedenotebyHthe¯ltrationgeneratedbythisprocess.Weassumethatwearegiven,inaddition,someauxiliary¯ltrationFandwewriteG=H_F,meaningthatwehaveGt=¾(Ht;Ft)foreveryt2R+.The¯ltrationGisreferredtoastothefull¯ltration.Itisclearthat¿isanH-stoppingtime,aswellasaG-stoppingtime(butnotnecessarilyanF-stoppingtime).Allprocessesarede¯nedonthespace(•;G;Q),whereQistobeinterpretedasthereal-life(i.e.,statistical)probabilitymeasure.Unlessotherwisestated,allprocessesconsideredinwhatfollowsareassumedtobeG-adaptedandwithcµadlµagsamplepaths.3.7.1PriceDynamicsinaSingle-NameModelWeassumethattheunderlyingmarketmodelisarbitrage-free,meaningthatitadmitsaspotmar-tingalemeasureQ(notnecessarilyunique)equivalenttoQ.AspotmartingalemeasureisassociatedwiththechoiceofthesavingsaccountBasanum¶eraire,inthesensethatthepriceprocessofanytradeablesecurity,whichpaysnocouponsordividends,isaG-martingaleunderQ,whenitisdiscountedbythesavingsaccountB.Asusual,Bisgivenby³Zt´Bt=exprudu;8t2R+;0wheretheshort-termrisassumedtofollowanF-progressivelymeasurablestochasticprocess.Thechoiceofasuitabletermstructuremodelisarbitraryanditisnotdiscussedinthepresentwork.LetusdenotebyGt=Q(¿>tjFt)thesurvivalprocessof¿withrespecttoa¯ltrationF.WepostulatethatG0=1andGt>0foreveryt2R+(hencethecasewhere¿isanF-stoppingtimeisexcluded)sothatthehazardprocess¡=¡lnGof¿withrespecttothe¯ltrationFiswellde¯ned.ForanyQ-integrableandFT-measurablerandomvariableY,thefollowingclassicformulaholds¡1EQ(1fT<¿gYjGt)=1ft<¿gGtEQ(GTYjFt):(3.39)Clearly,theprocessGisaboundedG-supermartingaleandthusitadmitstheuniqueDoob-MeyerdecompositionG=¹¡º,where¹isamartingalepartandºisapredictableincreasingprocess.Weshallworkthroughoutunderthefollowingstandingassumption.Assumption3.7.1WepostulatethatGisacontinuousprocessandtheincreasingprocessºinitsDoob-MeyerdecompositionisabsolutelycontinuouswithrespecttotheLebesguemeasure,sothatdºt=ÀtdtforsomeF-progressivelymeasurable,non-negativeprocessÀ.Wedenoteby¸the¡1F-progressivelymeasurableprocessde¯nedas¸t=GtÀt.LetusnoteforthefurtherreferencethatunderAssumption3.7.1wehavedGt=d¹t¡¸tGtdt,wheretheF-martingale¹iscontinuous.Moreover,inviewoftheLebesguedominatedconvergencetheorem,continuityofGimpliesthattheexpectedvalueEQ(Gt)=Q(¿>t)isacontinuousfunction,andthusQ(¿=t)=0forany¯xedt2R+.Finally,wealreadyknowthatunderAssumption3.7.1theprocessM,givenbyZt^¿ZtMt=Ht¡¤t^¿=Ht¡¸udu=Ht¡(1¡Hu)¸udu;(3.40)00isaG-martingale,wheretheincreasing,absolutelycontinuous,F-adaptedprocess¤isgivenbyZtZt¤=G¡1dº=¸du:(3.41)tuuu00TheF-progressivelymeasurableprocess¸iscalledthedefaultintensitywithrespecttoF. 102CHAPTER3.HAZARDPROCESSAPPROACHDefaultableClaimsWeareinapositiontointroducetheconceptofadefaultableclaim.Ofcourse,weworkherewithinasingle-nameframework,sothat¿isthemomentofdefaultofthereferencecreditname.De¯nition3.7.1ByadefaultableclaimmaturingatTwemeanthequadruple(X;A;Z;¿),whereXisanFT-measurablerandomvariable,A=(At)t2[0;T]isanF-adapted,continuousprocessof¯nitevariationwithA0=0,Z=(Zt)t2[0;T]isanF-predictableprocess,and¿isarandomtime.The¯nancialinterpretationofcomponentsofadefaultableclaimbecomesclearfromthefollowingde¯nitionofthedividendprocessD,whichdescribesallcash°owsassociatedwithadefaultableclaimoveritslifespan]0;T],thatis,afterthecontractwasinitiatedattime0(ofcourse,thechoiceof0astheinceptiondateismerelyaconvention).Thedividendprocessmighthavebeencalledthetotalcash°owprocess;wehavechosentheterm`dividendprocess'forthesakeofbrevity.De¯nition3.7.2ThedividendprocessD=(Dt)t2R+oftheabovedefaultableclaimmaturingatTequals,foreveryt2R+,ZZDt=X1fT<¿g1[T;1[(t)+(1¡Hu)dAu+ZudHu:]0;t^T]]0;t^T]ItisclearthatthedividendprocessDisaprocessof¯nitevariationon[0;T].The¯nancialinterpretationofDisasfollows:Xisthepromisedpayo®,ArepresentstheprocessofpromiseddividendsandtheprocessZ,termedtherecoveryprocess,speci¯estherecoverypayo®atdefault.Itisworthstressingthat,accordingtoourconvention,thecashpayment(premium)attime0isnotincludedinthedividendprocessDassociatedwithadefaultableclaim.PriceDynamicsofaDefaultableClaimForany¯xedt2[0;T],theprocessDu¡Dt;u2[t;T],representsallcash°owsfromadefaultableclaimreceivedbyaninvestorwhopurchaseditattimet.Ofcourse,theprocessDu¡Dtmaydependonthepastbehavioroftheclaimaswellasonthehistoryofthemarketpriortot.Thepastdividendsarenotvaluedbythemarket,however,sothatthecurrentmarketvalueattimet2[0;T]ofadefaultableclaim(i.e.,thepriceatwhichittradesattimet)re°ectsonlyfuturecash°owstobepaid/receivedoverthetimeinterval]t;T].Thisleadstothefollowingde¯nitionoftheex-dividendpriceofadefaultableclaim.De¯nition3.7.3Theex-dividendpriceprocessSofadefaultableclaim(X;A;Z;¿)equals,foreveryt2[0;T],³Z¯´¡1¯St=BtEQBudDu¯Gt:(3.42)]t;T]Obviously,ST=0foranydividendprocessD.Weworkthroughoutunderthenaturalintegra-bilityassumptions:¯Z¯¡1¯¡1¯¡1EQjBTXj<1;EQ¯Bu(1¡Hu)dAu¯<1;EQjB¿^TZ¿^Tj<1;]0;T]whichensurethattheex-dividendpriceStiswellde¯nedforanyt2[0;T]:Wewilllaterneedthefollowingtechnicalassumption³ZT´E(B¡1Z)2dh¹i<1:(3.43)Quuu0We¯rstderiveaconvenientrepresentationfortheex-dividendpriceSofadefaultableclaim. 3.7.SINGLE-NAMECREDITDEFAULTSWAPMARKET103Proposition3.7.1Theex-dividendpriceofthedefaultableclaim(X;A;Z;¿)equals,fort2[0;T[,³ZT¡¢¯´Bt¡1¡1¯St=1ft<¿gEQBTGTX+BuGuZu¸udu+dAu¯Ft:(3.44)GttProof.Foranyt2[0;T[,theex-dividendpriceisgivenbytheconditionalexpectation³ZT^¿¯´¡1¡1¡1¯St=BtEQBTX1fT<¿g+BudAu+B¿Z¿1ft<¿·Tg¯Gt:t^¿Letus¯xtandletusintroducetwoauxiliaryprocessesY=(Yu)u2[t;T]andR=(Ru)u2[t;T]bysettingZZuuY=B¡1dA;R=B¡1Z+B¡1dA=B¡1Z+Y:uvvuuuvvuuuttThenStcanberepresentedasfollows³¯´¡1¯St=BtEQBTX1fT<¿g+1fT<¿gYT+R¿1ft<¿·Tg¯Gt:Weusedirectlyformula(3.39)inordertoevaluatetheconditionalexpectations³¯´³¯´¡1¯Bt¡1¯BtEQ1fT<¿gBTX¯Gt=1ft<¿gEQBTGTX¯Ft;Gtand³¯´³¯´¯Bt¯BtEQ1fT<¿gYT¯Gt=1ft<¿gEQGTYT¯Ft:GtInaddition,wewilluseofthefollowingformula³ZT¯´1¯EQ(1ft<¿·TgR¿jGt)=¡1ft<¿gEQRudGu¯Ft;(3.45)GttwhichisknowntobevalidforanyF-predictableprocessRsuchthatEQjR¿j<1.Wethusobtain,foranyt2[0;T[,³ZT¯´Bt¡1¡1¯St=1ft<¿gEQBTGTX+GTYT¡(BuZu+Yu)dGu¯Ft;GttMoreover,sincedGt=d¹t¡¸tGtdt,where¹isanF-martingale,wealsoobtain³ZT¯´³ZT¯´Bt¡1¯Bt¡1¯1ft<¿gEQ¡BuZudGu¯Ft=1ft<¿gEQBuGuZu¸udu¯Ft;GttGttwherewehaveused(3.43).Tocompletetheproof,itremainstoobservethatGisacontinuoussemimartingaleandYisacontinuousprocessof¯nitevariationwithYt=0,sothattheIt^ointegrationbypartsformulayieldsZTZTZTGY¡YdG=GdY=B¡1GdA;TTuuuuuuutttwherethesecondequalityfollowsfromthede¯nitionofY.Weconcludethat(3.44)holdsforanyt2[0;T[,asrequired.¤Formula(3.44)impliesthattheex-dividendpriceSsatis¯es,foreveryt2[0;T],St=1ft<¿gSetforsomeF-adaptedprocessSe,whichistermedtheex-dividendpre-defaultpriceofadefaultableclaim.NotethatSmaynotbecontinuousattimeT,inwhichcaseST¡6=ST=0. 104CHAPTER3.HAZARDPROCESSAPPROACHDe¯nition3.7.4ThecumulativepriceprocessScassociatedwiththedividendprocessDisde¯nedbysetting,foreveryt2[0;T],³Z¯´Zc¡1¯¡1St=BtEQBudDu¯Gt=St+BtBudDu:(3.46)]0;T]]0;t]NotethatthediscountedcumulativepriceB¡1ScisaG-martingaleunderQ.Itfollowsimme-diatelyfrom(3.44)and(3.46)thatthefollowingcorollarytoProposition3.7.1isvalid.Corollary3.7.1Thecumulativepriceofthedefaultableclaim(X;A;Z;¿)equals,fort2[0;T],³ZT¡¢¯´ZcBt¡1¡1¯¡1St=1ft<¿gEQBTGTX1ft0,theconventiondictatesthatoneownsat0timetthemarketCDSwithspread·i,butthatithasalreadypaidthecumulativedividendsgiventby(3.83).Inthisway,weavoidanyproblemwithconsideringtheshort-salepositions:whatwouldbeashort-salepositioninanon-the-run(i.e.,non-market)CDSbecomesashortpositioninthecorrespondingmarketCDS.Thismathematicalconventionisactuallyconsistentwiththemarketpracticewheredefaultprotectionisboughtorsoldandthennulli¯ed,thatis,CDSsarelongedorshortedandthenunwound,asneeded.De¯nition3.7.9TheTi-maturitysyntheticmarketCDSisaTi-maturitydefaultableclaimwiththedividendprocessequaltoD¹i=D¹(·;±i;T;¿)where,foreveryt2[0;T],iiiZD¹i=Bd(B¡1Si(·))+Di:(3.83)tuuuit]0;t]Ofcourse,wemaychoose·=·iinDe¯nition3.7.9.Thenextlemmashowsthattheex-dividendi0priceoftheTi-maturitymarketCDSequalszeroatanydate.Lemma3.7.4Theex-dividendpriceS¹ioftheT-maturitysyntheticmarketCDSequalszeroforianyt2[0;Ti]: 3.7.SINGLE-NAMECREDITDEFAULTSWAPMARKET113Proof.WehaveZZB¡1dD¹i=B¡1Si(·)¡Si(·)+B¡1dDi:(3.84)uutti0iuu]0;t]]0;t]Hencetheex-dividendpriceoftheT-maturitysyntheticmarketCDSsatis¯es(recallthatSi(·)=iTii0)Z³Z¯´³¯´S¹i=BEB¡1dD¹i¯=E¡Si(·)+BB¡1dDi¯¯G=0ttQuu¯GtQtituut]t;Ti]]t;Ti]foreveryt2[0;Ti]:¤Todescribetheself-¯nancingtradingstrategiesinthesavingsaccountBandsyntheticmarketCDSswithex-dividendpricesS¹i,wewilluseDe¯nition3.7.7.InviewofLemma3.7.4,S¹i=0fortanyt2[0;Ti]andthusDe¯nition3.7.7takesthefollowingform.De¯nition3.7.10AstrategyÁ=(Á0;:::;Ák)inthesavingsaccountBandsyntheticmarketCDSswithdividendprocessesD¹i;i=1;:::;k,issaidtobeself-¯nancingifthewealthV(Á)=Á0Btttsatis¯esVt(Á)=V0(Á)+Gt(Á)foreveryt2[0;T];wherethegainsprocessG(Á)isde¯nedasfollowsZXkZG(Á)=Á0dB+ÁidD¹i:tuuuu]0;t]i=1]0;t]Using(3.74),weobtainthefollowingconditionsatis¯edbythediscountedwealthprocessofanyself-¯nancingstrategyÁinthesenseofDe¯nition3.7.10Xkd(B¡1V(Á))=ÁiB¡1dD¹i:ttttti=1Lemma3.7.5LetÁbeaself-¯nancingstrategyinthesavingsaccountBandex-dividendpricesSi(·);i=1;:::;k.ThenthestrategyÃ=(Ã0;:::;Ãk)whereÃi=Áifori=1;:::;kandiÃ0=B¡1V(Á)isaself-¯nancingstrategyinthesavingsaccountBandsyntheticmarketCDSswithtttdividendprocessesD¹ianditswealthprocesssatis¯esV(Ã)=V(Á).Proof.LetÁbeaself-¯nancingstrategyinthesavingsaccountBandex-dividendpricesSi(·);i=i1;:::;k.FromtheproofofLemma3.7.2,weknowthatXk³´d(B¡1V(Á))=Áid(B¡1Si(·))+B¡1dDi:tttttitti=1Inviewof(3.83),weobtainXkd(B¡1V(Á))=ÁiB¡1dD¹ittttti=1asrequired.¤Using(3.84),wededucethatthecumulativepriceoftheTi-maturitysyntheticmarketCDSsatis¯es(see(3.46)and(3.71))ZZS¹c;i=S¹i+BB¡1dD¹i=1(·i¡·)Ae(t;T)¡BSi(·)+BB¡1dDi:tttuuft<¿gtit0ituu]0;t]]0;t]Ifwechoose·=·ithenmanifestlyi0ZS¹c;i=1(·i¡·i)Ae(t;T)+BB¡1dDi=Sc;i(·i):tft<¿gt0tuut0]0;t]WethushavethefollowingimmediatecorollarytoTheorem3.2. 114CHAPTER3.HAZARDPROCESSAPPROACHCorollary3.7.5AssumethatthereexistF-predictableprocessesÁ1;:::;Áksatisfyingthefollowingconditions,foranyt2[0;T],Xk¡¢XkÁi±i¡Sei(·)=Z¡Se;Ái³i=»:tttittttti=1i=1LettheprocessV(Á)begivenby(3.78)withtheinitialconditionV(Á)=ScandletÁ0begivenby,00fort2[0;T],Á0=B¡1V(Á):tttThentheself-¯nancingtradingstrategyÁ=(Á0;:::;Ák)inthesavingsaccountBandsyntheticmarketCDSswithdividendprocessesD¹i;i=1;:::;kreplicatesthedefaultableclaim(X;A;Z;¿).Su±cientConditionsforHedgeabilityThe¯rstequalityin(3.79)eliminatesthejumprisk,whereasthesecondoneisusedtoeliminatethespreadrisk.Ingeneral,theexistenceofÁ1;:::;Áksatisfying(3.79)isnotensuredanditiseasytogiveanexamplewhenasolutionto(3.79)failstoexist.InExample3.7.1below,wedealwitha(admittedlysomewhatarti¯cial)situationwhenthejumpriskcanbeperfectlyhedged,butthepricesoftradedCDSsaredeterministicpriortodefault,sothatthespreadriskofadefaultableclaimisnon-hedgeable.Ingeneral,thesolvabilityof(3.79)dependsonseveralfactors,suchas:thenumberoftradedassets,thedimensionofthedrivingBrownianmotion,therandomcharacterofdefaultintensity°andrecoverypayo®s±iandthefeaturesofadefaultableclaimthatwewishtohedge.Example3.7.1Letr=0andk=2.Assumethat·16=·2arenon-zeroconstants,T16=T2,andlet±1=±2=Z=0.Assumealsothatthedefaultintensity°(t)>0isdeterministicandthepromisedpayo®Xisanon-constantFT-measurablerandomvariable.Wethushave(cf.(3.51))³Zt´mt=EQ(GTXjFt)=GTEQ(XjFt)=GTEQ(X)+»udWu0forsomenon-vanishingprocess».However,since°isdeterministic,itisalsoeasytodeducefromiP2iSei(3.66)that³t=0;i=1;2foreverytR2[0;T].The¯rstconditionin(3.79)readsi=1Átt(·i)=Set,iTandsincemanifestlySet(·i)=·iGttGudu6=0foreveryt2[0;T],nodi±cultymayarisehere.P2iiHowever,thesecondequality,i=1Át³t=»tcannotbesatis¯edforeveryt2[0;T],sincetheleft-handsidevanishesforeveryt2[0;T].Weshallnowprovidesu±cientconditionsfortheexistenceanduniquenessofareplicatingstrategyforanydefaultableclaiminthepracticallyappealingcaseofCDSswithconstantprotectionpayments.We¯rstaddressthisissueinthespecialcasewherek=2andthemodelisdrivenbyaone-dimensionalBrownianmotionW.Inaddition,weassumethatthetwotradedCDSshavethesamematurity,T1=T2=U;thisassumptionismadehereforsimplicityofpresentationonly,anditwillberelaxedinProposition3.7.5below.Letusdenote³ZU¯´³ZU¯´Pe=BG¡1EB¡1G¸du¯;Ae=BG¡1EB¡1Gdu¯¯FtttQuuu¯FttttQuutttsothatSei(·)=±Pe¡·Aefori=1;2.Similarly(cf.(3.77))ni=±m1¡·m2,wherewesettiitittitit³ZU¯´³ZU¯´1¡1¯2¡1¯mt=EQBuGu¸udu¯Ft;mt=EQBuGudu¯Ft:00jjBythepredictablerepresentationpropertyoftheBrownianmotion,dmt=ÃtdWtforj=1;2forsomeF-predictable,real-valuedprocessesÃ1andÃ2. 3.7.SINGLE-NAMECREDITDEFAULTSWAPMARKET115Proposition3.7.4AssumethatT1=T2=Uandtheconstantprotectionpayments±1and±2aresuchthat±·6=±·and(1¡Pe)Ã26=AeÃ1foralmosteveryt2[0;T].Thenforanydefaultable1221ttttclaim(X;A;Z;¿)thereexistsauniquesolution(Á1;Á2)to(3.79).Proof.Inviewof(3.77),weobtain³i=±Ã1¡·Ã2.Hencethematchingconditions(3.79)becometititX2¡¢X2¡¢Ái±Pb+·Ae=Z¡Se;Ái±Ã1¡·Ã2=»;(3.85)titittttititti=1i=1where,forconciseness,wedenotedPbt=1¡Pet.Auniquesolutionto(3.85)existsprovidedthattherandommatrix·¸±1Pbt+·1Aet±2Pbt+·2AetNt=1212±1Ãt¡·1Ãt±2Ãt¡·2Ãtisnon-singularforalmosteveryt2[0;T],thatis,wheneverdetN=(±·¡±·)(PbÃ2¡AeÃ1)6=0t2112ttttforalmosteveryt2[0;T].¤Equality±2·1¡±1·2=0wouldpracticallymeanthatwedealwithasingleCDSratherthantwodistinctCDSs.Notethatwehaveheretwosourcesofuncertainty,thediscontinuousmartingaleMandtheBrownianmotionW;henceitwasnaturaltoexpectthatthenumberofassetsrequiredtospanthemarketequals3Ifthemodelisdrivenbyad-dimensionalBrownianmotion,itisnaturaltoexpectthatonewillneedatleastd+2assets(thesavingsaccountandd+1distinctCDSs,say)toreplicateanydefaultableclaim,thatis,toensurethemodel'scompleteness.Thisquestionisexaminedinthenextresult,inwhichwedenote³ZTi¯´³ZTi¯´Pei=BG¡1EB¡1G¸du¯;Aei=BG¡1EB¡1Gdu¯¯FtttQuuu¯FttttQuutttsothatSei(·)=±Pei¡·Aeifori=1;:::;k.Similarly(cf.(3.77))ni=±m1i¡·m2i,wherewetiitittititset³ZTi¯´³ZTi¯´1i¡1¯2i¡1¯mt=EQBuGu¸udu¯Ft;mt=EQBuGudu¯Ft:00jijiBythepredictablerepresentationpropertyoftheBrownianmotion,dmt=ÃtdWtforj=1;2andi=1;:::;kandsomeF-predictable,Rd-valuedprocessesÃji=(Ãji1;:::;Ãjid).Theproofofthenextresultreliesonaratherstraightforwardveri¯cationof(3.79).Proposition3.7.5AssumethatthenumberoftradedCDSisk=d+1andthemodelisdrivenbyad-dimensionalBrownianmotion.Thenconditions(3.79)canberepresentedbythelinearequationNÁb=»bwiththeRk-valuedprocessÁb=(Á1;:::;Ák)t,theRk-valuedprocess»b=(Z¡ttttttttSe;»1;:::;»d)tandthek£krandommatrixNisgivenbytttt23±1Pbt1¡·1Ae1t:::±kPbtk¡·kAekt6±Ã111¡·Ã211:::±Ã1k1¡·Ã2k1761t1tktkt7Nt=6......7;4...5±Ã11d¡·Ã21d:::±Ã1kd¡·Ã2kd1t1tktktwherePbi=1¡Pei.Foranydefaultableclaim(X;A;Z;¿)thereexistsauniquesolution(Á1;:::;Ák)ttto(3.79)ifandonlyifdetNt6=0foralmostallt2[0;T]. 116CHAPTER3.HAZARDPROCESSAPPROACH3.8Multi-NameCreditDefaultSwapMarketInthissection,weshalldealwithamarketmodeldrivenbyaBrownian¯ltrationinwhicha¯nitefamilyofCDSswithdi®erentunderlyingnamesistraded.3.8.1PriceDynamicsinaMulti-NameModelOur¯rstgoalistoextendthepricingresultsofSection3.7.1tothecaseofamulti-namecreditriskmodelwithstochasticdefaultintensities.JointSurvivalProcessWeassumethatwearegivennstrictlypositiverandomtimes¿1;:::;¿n,de¯nedonacommonprobabilityspace(•;G;Q),andreferredtoasdefaulttimesofncreditnames.Wepostulatethatthisspaceisendowedwithareference¯ltrationF,whichsatis¯esAssumption3.7.2.Inordertodescribedynamicjointbehaviorofdefaulttimes,weintroducetheconditionaljointsurvivalprocessG(u1;:::;un;t)bysetting,foreveryu1;:::;un;t2R+,G(u1;:::;un;t)=Q(¿1>u1;:::;¿n>unjFt):Letusset¿(1)=¿1^:::^¿nandletusde¯netheprocessG(1)(t;t);t2R+bysettingG(1)(t;t)=G(t;:::;t;t)=Q(¿1>t;:::;¿n>tjFt)=Q(¿(1)>tjFt):ItiseasytocheckthatG(1)isaboundedsupermartingale.ItthusadmitstheuniqueDoob-MeyerdecompositionG(1)=¹¡º.WeshallworkthroughoutunderthefollowingcounterpartofAssumption3.7.1.Assumption3.8.1WeassumethattheprocessG(1)iscontinuousandtheincreasingprocessºisabsolutelycontinuouswithrespecttotheLebesguemeasure,sothatdºt=ÀtdtforsomeF-progressivelymeasurable,non-negativeprocessÀ.Wedenoteby¸etheF-progressivelymeasurable¡1processde¯nedas¸et=G(1)(t;t)Àt;wewillreferto¸easthe¯rst-to-defaultintensity.WedenoteHi=1andweintroducethefollowing¯ltrationsHi;HandGtf¿i·tgHi=¾(Hi;s2[0;t]);H=H1_:::_Hn;G=F_H;tsttttttWeassumethattheusualconditionsofcompletenessandright-continuityaresatis¯edbythese¯ltrations.ArguingasinSection3.7.1,weseethattheprocessZt^¿(1)ZtMc=H(1)¡¤e=H(1)¡¸edu=H(1)¡(1¡H(1))¸edu;ttt^¿(1)tutuu00(1)Rt¸eisaG-martingale,wherewedenoteHt=1f¿(1)·tgand¤et=0udu.Notethatthe¯rst-to-defaultintensity¸esatis¯es¸e1Q(t<¿(1)·t+hjFt)11t=lim=lim(ºt+h¡ºt):h#0hQ(¿(1)>tjFt)G(1)(t;t)h#0hSincewenowworkinamulti-nameset-upAssumption3.8.1isnotsu±cientforourfurtherpurposes.We¯nditconvenienttomakethefollowingstandingassumption,inwhich,forany¯xedi=1;:::;n,wedenotebyGithe¯ltrationH1_:::_Hi¡1_Hi+1_:::_Hn_FandwealsodenoteG(t;t)=Q(¿>tjGi).TheprocessGismanifestlyaboundedsupermartingale,andthusiitiitadmitstheuniqueDoob-MeyerdecompositionG=¹i¡ºi.i 3.8.MULTI-NAMECREDITDEFAULTSWAPMARKET117Assumption3.8.2WeassumethateachprocessGiscontinuousandtheincreasingprocessºiiisabsolutelycontinuouswithrespecttotheLebesguemeasure,sothatdºi=ÀidtforsomeGi-ttprogressivelymeasurable,non-negativeprocessÀi.Wedenoteby¸itheGi-progressivelymeasurableprocessde¯nedas¸i=G¡1(t;t)Ài.titAsusual,itcanbeveri¯edthatforany¯xedi=1;:::;n,theprocessZtMi=Hi¡(1¡Hi)¸iduttsu0isaG-martingale,whereHi=1.Moreover,theGi-intensityprocess¸iof¿canalsobetf¿i·tgirepresentedas1Q(t<¿·t+hjGi)¸i=limit:tih#0hQ(¿i>tjGt)Wehavethefollowingauxiliaryresult,inwhichweintroducethe¯rst-to-defaultintensity¸eiandtheassociatedmartingaleMciforeachcreditnamei=1;:::;n.Lemma3.8.1Foranyi=1;:::;n,theprocess¸eigivenby¸ei1Q(t<¿i·t+h;¿(1)>tjFt)t=limh#0hQ(¿(1)>tjFt)iswellde¯nedandtheprocessMci,givenbytheformulaZt^¿(1)Mci=Hi¡¸eidu;tt^¿(1)u0isaG-martingale.Proof.Letusset¿i=¿^:::^¿^¿^:::^¿.Ontheeventf¿i>tg,whichbelongsto(1)1i¡1i+1n(1)Gi,wehavetQ(t<¿·t+h;t<¿ijF)ii(1)tQ(t<¿i·t+h;¿(1)>tjFt)Q(t<¿i·t+hjGt)=i=iQ(¿(1)>tjFt)Q(¿(1)>tjFt)andQ(¿>t;¿i>tjF)ii(1)tQ(¿(1)>tjFt)Q(¿i>tjGt)=i=i:Q(¿(1)>tjFt)Q(¿(1)>tjFt)Hence¸i=¸eiontheeventf¿>tg.Toconcludetheproof,itsu±cestoobservethatMci=Mitt(1)tt^¿(1)foreveryt2RandthusMciisaG-martingaleaswell.¤+Pn¸ePnItisworthnotingthat,asexpected,theequalitiesi=¸eandMc=Mciarevalid.i=1i=1PriceDynamicsofaFirst-to-DefaultClaimWewillnowanalyzetherisk-neutralvaluationof¯rst-to-defaultclaimsonabasketofncreditnames.Asbefore,¿1;:::;¿narerespectivedefaulttimesand¿(1)=¿1^:::^¿nstandsforthemomentofthe¯rstdefault.De¯nition3.8.1A¯rst-to-defaultclaimwithmaturityTassociatedwith¿1;:::;¿nisadefaultableclaim(X;A;Z;¿(1)),whereXisanFT-measurableamountpayableatmaturityTifnodefaultoccurspriortooratT,anF-adapted,continuousprocessof¯nitevariationA:[0;T]!RwithA0=0representsthedividendstreamupto¿,andZ=(Z1;:::;Zn)isthevectorofF-predictable,(1)real-valuedprocesses,whereZispeci¯estherecoveryreceivedattime¿ifdefaultoccursprior¿(1)(1)tooratTandtheithnameisthe¯rstdefaultedname,thatis,ontheeventf¿i=¿(1)·Tg. 118CHAPTER3.HAZARDPROCESSAPPROACHThenextde¯nitionextendsDe¯nition3.7.2tothecaseofa¯rst-to-defaultclaim.Recallthat(1)wedenoteHt=1f¿(1)·tgforeveryt2[0;T].De¯nition3.8.2ThedividendprocessD=(Dt)t2R+ofa¯rst-to-defaultclaimmaturingatTequals,foreveryt2R+,ZZXnD=X11(t)+(1¡H(1))dA+1ZidH(1):tfT<¿(1)g[T;1[uuf¿(1)=¿iguu]0;t^T]]0;t^T]i=1Weareinapositiontoexaminethepricesofthe¯rst-to-defaultclaim.Notethat1Sc=1Sec;1S=1Se;ft<¿(1)gtft<¿(1)gtft<¿(1)gtft<¿(1)gtwhereSecandSearepre-defaultvaluesofScandS,wherethepriceprocessesScandSaregivenbyDe¯nitions3.7.3and3.7.4,respectively.Wepostulatethat,fori=1;:::;n,¯Z¯¡1¯¡1(1)¯¡1iEQjBTXj<1;EQ¯Bu(1¡Hu)dAu¯<1;EQjB¿(1)^TZ¿(1)^Tj<1;]0;T]sothatthattheex-dividendpriceS(andthusalsocumulativepriceSc)iswellde¯nedforanytt2[0;T]:Inthenextauxiliaryresult,wedenoteYi=B¡1Zi.HenceYiisareal-valued,F-predictableprocesssuchthatEjYij<1.Q¿(1)^TLemma3.8.2Wehavethat³Xn¯´B³ZTXn¯´BE1Yi¯=1tEYi¸eiG(u;u)du¯¯F:tQft<¿(1)=¿i·Tg¿(1)¯Gtft<¿(1)gG(t;t)Quu(1)t(1)ti=1i=1Proof.Letus¯xiandletusconsidertheprocessYi=11(u)forsome¯xeddatet·stjH2)andG(u;v)=Q(¿>t1t1u;¿2>v).Itistheneasytoprovethatµ¶µ¶1j2@2G(t;t)G(t;t)222@1G(t;t)dGt=¡dMt+Ht@1h(t;¿2)+(1¡Ht)dt;@2G(0;t)G(0;t)G(0;t)whereh(t;u)=@2G(t;u)andM2istheH2-martingalegivenby@2G(0;u)Zt^¿222@2G(0;u)Mt=Ht+du:0G(0;u) 120CHAPTER3.HAZARDPROCESSAPPROACHIfHypothesis(H)holdsbetweenH2andH1_H2thenthemartingalepartintheDoob-MeyerdecompositionofG1j2vanishes.WethusseethatHypothesis(H)isnotalwaysvalid,sinceclearly@2G(t;t)G(t;t)¡@2G(0;t)G(0;t)doesnotvanish,ingeneral.Onecannotethatinthespecialcasewhen¿2<¿1,themartingalepartintheabove-mentioneddecompositiondisappearsandthusHypothesis(H)holdsbetweenH2andH1_H2.Fromnowon,weshallworkunderAssumption3.8.3.Inthatcase,thedynamicsofpriceprocessesobtainedinProposition3.8.1simplify,asthefollowingresultshows.Corollary3.8.1Thepre-defaultex-dividendpriceSeofa¯rst-to-defaultclaim(X;A;Z;¿(1))sat-is¯esXndSe=(r+¸e)Sedt¡¸eiZidt¡dA+BG¡1(t;t)dm;tttttttt(1)ti=1wherethecontinuousF-martingalemisgivenby(3.86).ThecumulativepriceScofa¯rst-to-defaultclaim(X;A;Z;¿(1))isgivenbytheexpression,fort2[0;T^¿(1)],XndSc=rScdt+(Zi¡Se)dMci+BG¡1(t;t)dm:(3.87)ttttttt(1)ti=1Equivalently,fort2[0;T^¿(1)],XndSc=rScdt+(Zi¡Se)dMci+BG¡1(t;t)dm·;(3.88)ttttttt(1)ti=1wherem·isaG-martingalegivenbym·t=mt^¿(1)foreveryt2[0;T].Inwhatfollows,weassumethatFisgeneratedbyaBrownianmotion.ThenthereexistsanF-predictableprocess»forwhichdmt=»tdWtsothatformula(3.88)yieldsthefollowingresult.Corollary3.8.2Thediscountedcumulativepriceofa¯rst-to-defaultclaim(X;A;Z;¿(1))satis¯es,fort2[0;T^¿(1)],XndSc;¤=B¡1(Zi¡Se)dMci+G¡1(t;t)»dW:ttttt(1)tti=1PriceDynamicsofaCDSBytheithCDSwemeanthecreditdefaultswapwrittenontheithreferencename,withthematuritydateT,theconstantspread·andtheprotectionprocess±i,asspeci¯edbyDe¯nition3.7.5.iiLetSi(·)standfortheex-dividendpriceattimetoftheithCDSontheevent¿=¿=ttjji(1)jforsomej6=i.ThisvaluecanberepresentedusingasuitableextensionofProposition3.8.1,butwedecidedtoomitthederivationofthispricingformula.AssumingthatwehavealreadycomputedSi(·),theithCDScanbeseen,ontherandominterval[0;T^¿],asa¯rst-to-defaultclaimtjjii(1)(X;A;Z;¿)withX=0;Z=(Si(·);:::;±i;:::;Si(·))andA=¡·t.Thisobservation(1)tj1itjnitiappliesalsototherandominterval[0;T^¿(1)]forany¯xedT·Ti.LetusdenotebynithefollowingF-martingaleÃZ!XnTi³Xn´¯ni=EB¡1G(u;u)±i¸ei+Si(·)¸ej¡·du¯¯F:tQu(1)uuujjiuiti=10j=1;j6=iThefollowingresultcanbeeasilydeducedfromProposition3.8.1. 3.8.MULTI-NAMECREDITDEFAULTSWAPMARKET121Corollary3.8.3ThecumulativepriceoftheithCDSsatis¯es,fort2[0;Ti^¿(1)],XndSc;i(·)=rSc;i(·)dt+(±i¡Sei(·))dMci+(Si(·)¡Sei(·))dMcj+BG¡1(t;t)dni:tittittittjjititt(1)tj=1;j6=iConsequently,thediscountedcumulativepriceoftheithCDSsatis¯es,fort2[0;Ti^¿(1)],XndSc;i;¤(·)=B¡1(±i¡Sei(·))dMci+B¡1(Si(·)¡Sei(·))dMcj+G¡1(t;t)³idW;titttitttjjitit(1)ttj=1;j6=iwhere³iisanF-predictableprocesssuchthatdni=³idW.tttNotethattheF-martingalenicanbereplacedbytheG-martingale·ni=ni.tt^¿(1)3.8.2ReplicationofaFirst-to-DefaultClaimOur¯nalgoalistoextendTheorem3.2ofSection2.2.7tothecaseofseveralcreditnamesinahazardprocessmodelinwhichcreditspreadsaredrivenbyamulti-dimensionalBrownianmotion.Weconsideraself-¯nancingtradingstrategyÁ=(Á0;:::;Ák)withG-predictablecomponents,asde¯nedinSection3.7.2.The0thtradedassetisthusthesavingsaccount;theremainingkprimaryassetsaresingle-nameCDSswithdi®erentunderlyingcreditnamesand/ormaturities.Asbefore,foranyl=1;:::;kwewillusetheshort-handnotationSl(·)andSc;l(·)todenotetheex-dividendllandcumulativepricesofCDSswithrespectivedividendprocessesD(·;±l;T;¿~)givenbyformulalll(3.61).Notethathere~¿l=¿jforsomej=1;:::;n.Wewillthuswrite~¿l=¿jlinwhatfollows.Remark3.8.1Notethat,typically,wewillhavek=n+dsothatthenumberoftradedassetswillbeequalton+d+1.Recallthatthecumulativepriceofa¯rst-to-defaultclaim(X;A;Z;¿)isdenotedasSc.We(1)adoptthefollowingnaturalde¯nitionofreplicationofa¯rst-to-defaultclaim.De¯nition3.8.3Wesaythataself-¯nancingstrategyÁ=(Á0;:::;Ák)replicatesa¯rst-to-defaultclaim(X;A;Z;¿)ifitswealthprocessV(Á)satis¯estheequalityV(Á)=Scforany(1)t^¿(1)t^¿(1)t2[0;T].Whendealingwithreplicatingstrategiesinthesenseofthede¯nitionabove,wemayanddoassume,withoutlossofgenerality,thatthecomponentsoftheprocessÁareF-predictableprocesses.Thisisratherobvious,sincepriortodefaultanyG-predictableprocessisequaltotheuniqueF-predictableprocess.ThefollowingresultisacounterpartofLemma3.7.3.ItsprooffollowseasilyfromLemma3.7.2combinedwithCorollary3.8.3,andthusitisomitted.Lemma3.8.3Wehave,foranyt2[0;T^¿(1)],Xk³¡¢Xn¡¢´dV¤(Á)=ÁlB¡1±l¡Sel(·)dMcjl+B¡1Sl(·)¡Sel(·)dMcj+G¡1(t;t)dnl;tttttltttjjltlt(1)tl=1j=1;j6=jlwhereÃ!ZTl³Xn´¯nl=EB¡1G(u;u)±l¸ejl+Sl(·)¸ej¡·du¯¯F:tQu(1)uuujjlult0j=1;j6=jl 122CHAPTER3.HAZARDPROCESSAPPROACHWearenowinapositiontoextendTheorem3.2tothecaseofa¯rst-to-defaultclaimonabasketofncreditnames.Recallthat»and³l;l=1;:::;kareF-predictable,Rd-valuedprocessessuchthatdm=»dWtttanddnl=³ldW.tttTheorem3.3AssumethattheprocessesÁe1;:::;Áensatisfy,fort2[0;T]andi=1;:::;nXk¡¢Xk¡¢Áel±l¡Sel(·)+ÁelSl(·)¡Sel(·)=Zi¡Setttlttjiltlttl=1;jl=il=1;jl6=iPkÁelliiandl=1t³t=»t.LetussetÁt=Áe(t^¿(1))fori=1;:::;kandt2[0;T].LettheprocessV(Á)begivenbyLemma3.8.3withtheinitialconditionV(Á)=ScandletÁ0begivenby00XkV(Á)=Á0B+ÁlSl(·):tttttll=1Thentheself-¯nancingstrategyÁ=(Á0;:::;Ák)replicatesthe¯rst-to-defaultclaim(X;A;Z;¿).(1)Proof.TheproofgoesalongthesimilarlinesastheproofofTheorem3.2.Itsu±cestoexaminereplicatingstrategyontherandominterval[0;T^¿(1)].InviewofLemma3.8.3,thewealthprocessofaself-¯nancingstrategyÁsatis¯eson[0;T^¿(1)]Xk³¡¢Xn¡¢´dV¤(Á)=ÁelB¡1±l¡Sel(·)dMcjl+B¡1Sl(·)¡Sel(·)dMcj+G¡1(t;t)³ldWtttttltttjjltlt(1)ttl=1j=1;j6=jlwhereasthediscountedcumulativepriceofa¯rst-to-defaultclaim(X;A;Z;¿(1))satis¯esontheinterval[0;T^¿(1)](cf.(3.87))XndSc¤=B¡1(Zi¡S)dMci+(1¡H(1))G¡1(t;t)»dW:tttt¡tt(1)tti=1Acomparisonofthelasttwoformulaeleadsdirectlytothestatedconditions.Itthensu±cestoverifythatthestrategyÁ=(Á0;:::;Ák)introducedinthestatementofthetheoremreplicatesa¯rst-to-defaultclaiminthesenseofDe¯nition3.8.3.Sincethisveri¯cationisratherstandard,itislefttothereader.¤ Chapter4HedgingofDefaultableClaimsInthischapter,weshallstudyhedgingstrategiesforcreditderivativesunderassumptionthatsomeprimarydefaultable(aswellasnon-defaultable)assetsaretraded,andthustheycanbeusedinreplicationofnon-tradedcontingentclaims.WefollowherethepaperbyBieleckietal.[8].4.1SemimartingaleModelwithaCommonDefaultInwhatfollows,we¯xa¯nitehorizondateT>0.Forthepurposeofthischapter,itisenoughtoformallyde¯neagenericdefaultableclaimthroughthefollowingde¯nition.De¯nition4.1.1AdefaultableclaimwithmaturitydateTisrepresentedbyatriplet(X;Z;¿);where:(i)thedefaulttime¿speci¯estherandomtimeofdefault,andthusalsothedefaulteventsf¿·tgforeveryt2[0;T],(ii)thepromisedpayo®X2FTrepresentstherandompayo®receivedbytheowneroftheclaimattimeT;providedthattherewasnodefaultpriortoorattimeT;theactualpayo®attimeTassociatedwithXthusequalsX1fT<¿g,(iii)theF-adaptedrecoveryprocessZspeci¯estherecoverypayo®Z¿receivedbytheownerofaclaimattimeofdefault(oratmaturity),providedthatthedefaultoccurredpriortooratmaturitydateT.Inpractice,hedgingofacreditderivativeafterdefaulttimeisusuallyofminorinterest.Also,inamodelwithasingledefaulttime,hedgingafterdefaultreducestoreplicationofanon-defaultableclaim.Itisthusnaturaltode¯nethereplicationofadefaultableclaiminthefollowingway.De¯nition4.1.2Wesaythataself-¯nancingstrategyÁreplicatesadefaultableclaim(X;Z;¿)ifitswealthprocessV(Á)satis¯esVT(Á)1fT<¿g=X1fT<¿gandV¿(Á)1fT¸¿g=Z¿1fT¸¿g.Whendealingwithreplicatingstrategies,inthesenseofDe¯nition4.1.2,wewillalwaysassume,withoutlossofgenerality,thatthecomponentsoftheprocessÁareF-predictableprocesses.4.1.1DynamicsofAssetPricesWeassumethatwearegivenaprobabilityspace(•;G;P)endowedwitha(possiblymulti-dimensional)standardBrownianmotionWandarandomtime¿admittingtheF-intensity°underP,whereFisthe¯ltrationgeneratedbyW.123 124CHAPTER4.HEDGINGOFDEFAULTABLECLAIMSSincethedefaulttimeisassumedtoadmittheF-intensity,itisnotanF-stoppingtime.Indeed,anystoppingtimewithrespecttoaBrownian¯ltrationisknowntobepredictable.Weinterpret¿asthecommondefaulttimeforalldefaultableassetsinourmodel.Forsimplicity,weassumethatonlythreeprimaryassetsaretradedinthemarket,andthedynamicsunderthehistoricalprobabilityPoftheirpricesare,fori=1;2;3andt2[0;T],¡¢dYi=Yi¹dt+¾dW+·dM;(4.1)tt¡i;ti;tti;ttRtwhereMt=Ht¡0(1¡Hs)°sdsisamartingale,orequivalently,¡¢dYi=Yi(¹¡·°1)dt+¾dW+·dH:(4.2)tt¡i;ti;ttft<¿gi;tti;ttTheprocesses(¹i;¾i;·i)=(¹i;t;¾i;t;·i;t;t2R+);i=1;2;3;areassumedtobeG-adapted,whereG=F_H.Inaddition,weassumethat·¸¡1foranyi=1;2;3;sothatYiarenonnegativeiprocesses,andtheyarestrictlypositivepriorto¿.Notethat,inthecaseofconstantcoe±cientswehavethat2Yi=Yie¹ite¾iWt¡¾it=2e¡·i°i(t^¿)(1+·)Ht:t0iAccordingtoDe¯nition4.1.2,replicationreferstothebehaviorofthewealthprocessV(Á)ontherandominterval[[0;¿^T]]only.Therefore,forthepurposeofreplicationofdefaultableclaimsoftheform(X;Z;¿),itissu±cienttoconsiderpricesofprimaryassetsstoppedat¿^T.ThisimpliesthatinsteadofdealingwithG-adaptedcoe±cientsin(4.1),itsu±cestofocusonF-adaptedcoe±cientsofstoppedpriceprocesses.However,forthesakeofcompleteness,weshallalsodealwithT-maturityclaimsoftheformY=G(Y1;Y2;Y3;H)(seeSection4.4below).TTTTPre-DefaultValuesAswillbecomeclearinwhatfollows,whendealingwithdefaultableclaimsoftheform(X;Z;¿),wewillbemainlyconcernedwiththeso-calledpre-defaultprices.Thepre-defaultpriceYeioftheithassetisanF-adapted,continuousprocess,givenbytheequation,fori=1;2;3andt2[0;T],¡¢dYei=Yei(¹¡·°)dt+¾dW(4.3)tti;ti;tti;ttwithYei=Yi.Putanotherway,YeiistheuniqueF-predictableprocesssuchthatYei1=00tft·¿gYi1fort2R.Whendealingwiththepre-defaultprices,wemayanddoassume,withouttft·¿g+lossofgenerality,thattheprocesses¹i;¾iand·iareF-predictable.Itisworthstressingthatthehistoricallyobserveddriftcoe±cientequals¹i;t¡·i;t°t,ratherthan¹i;t.Thedriftcoe±cientdenotedby¹i;tisalreadycredit-riskadjustedinthesenseofourmodel,anditisnotdirectlyobserved.Thisconventionwaschosenhereforthesakeofsimplicityofnotation.Italsolendsitselftothefollowingintuitiveinterpretation:ifÁiisthenumberofunitsoftheithassetheldinourportfolioattimetthenthegains/lossesfromtradesinthisasset,priortodefaulttime,canberepresentedbythedi®erential¡¢ÁidYei=ÁiYei¹dt+¾dW¡ÁiYei·°dt:tttti;ti;tttti;ttThelasttermmaybehereseparated,andformallytreatedasane®ectofcontinuouslypaiddividendsatthedividendrate·i;t°t.However,thisinterpretationmaybemisleading,sincethisquantityisnotdirectlyobserved.Infact,theestimationofthedriftcoe±cientindynamics(4.3)isnotpractical.Still,ifthisformalinterpretationisadopted,itissometimespossiblemakeuseofthestandardresultsconcerningthevaluationofderivativesofdividend-payingassets.Itis,ofcourse,adelicateissuehowtoseparateinpracticebothcomponentsofthedriftcoe±cient.Weshallarguebelowthatalthoughthedividend-basedapproachisformallycorrect,amorepertinentandsimplerwayofdealingwithhedgingreliesontheassumptionthatonlythee®ectivedrift¹i;t¡·i;t°tisobservable.Inpracticalapproachtohedging,thevaluesofdriftcoe±cientsindynamicsofassetpricesplaynoessentialrole,sothattheyarenotconsideredtobemarketobservables. 4.1.TRADINGSTRATEGIES125MarketObservablesTosummarize,weassumethroughoutthatthemarketobservablesare:thepre-defaultmarketpricesofprimaryassets,theirvolatilitiesandcorrelations,aswellasthejumpcoe±cients·i;t(the¯nancialinterpretationofjumpcoe±cientsisexaminedinthenextsubsection).Tosummarize,wepostulatethatunderthestatisticalprobabilityPwehave¡¢dYi=Yi¹edt+¾dW+·dH(4.4)tt¡i;ti;tti;ttwherethedriftterms¹ei;tarenotobserved,butwecanobservethevolatilities¾i;t(andthustheassetscorrelations)andwehaveanaprioriassessmentofjumpcoe±cients·i;t.Inthisgeneralsetup,themostnaturalassumptionisthatthedimensionofadrivingBrownianmotionWequalsthenumberoftradableassets.However,forthesakeofsimplicityofpresentation,weshallfrequentlyassumethatWisone-dimensional.Oneofourgoalswillbetoderiveclosed-formsolutionsforreplicatingstrategiesforderivativesecuritiesintermsofmarketobservablesonly(wheneverreplicationofagivenclaimisactuallyfeasible).Toachievethisgoal,weshallcombineageneraltheoryofhedgingdefaultableclaimswithinacontinuoussemimartingalesetup,withajudiciousspeci¯cationofparticularmodelswithdeterministicvolatilitiesandcorrelations.RecoverySchemesItisclearthatthesamplepathsofpriceprocessesYiarecontinuous,exceptforapossiblediscon-tinuityattime¿.Speci¯cally,wehavethat¢Yi:=Yi¡Yi=·Yi;¿¿¿¡i;¿¿¡sothatYi=Yi(1+·)=Yei(1+·).¿¿¡i;¿¿¡i;¿AprimaryassetYiistermedadefault-freeasset(defaultableasset,respectively)if·=0(·6=0,iirespectively).Inthespecialcasewhen·=¡1,wesaythatadefaultableassetYiissubjecttoaitotaldefault,sinceitspricedropstozeroattime¿andstaysthereforever.Suchanassetceasestoexistafterdefault,inthesensethatitisnolongertradedafterdefault.Thisfeaturemakesthecaseofatotaldefaultquitedi®erentfromothercases,asweshallseeinourstudybelow.Inmarketpractice,itiscommonforacreditderivativetodeliverapositiverecovery(forinstance,aprotectionpayment)incaseofdefault.Formally,thevalueofthisrecoveryatdefaultisdeterminedasthevalueofsomeunderlyingprocess,thatis,itisequaltothevalueattime¿ofsomeF-adaptedrecoveryprocessZ.Forexample,theprocessZcanbeequalto±,where±isaconstant,ortog(t;±Yt)wheregisadeterministicfunctionand(Yt;t2R+)isthepriceprocessofsomedefault-freeasset.Typically,therecoveryispaidatdefaulttime,butitmayalsohappenthatitispostponedtothematuritydate.LetusobservethatthecasewhereadefaultableassetYipaysapre-determinedrecoveryatdefaultiscoveredbyoursetupde¯nedin(4.1).Forinstance,thecaseofaconstantrecoverypayo®±¸0atdefaulttime¿correspondstotheprocess·=±(Yi)¡1¡1.Underthisconvention,theii;tit¡priceYiisgovernedunderPbytheSDE¡¢dYi=Yi¹dt+¾dW+(±(Yi)¡1¡1)dM:(4.5)tt¡i;ti;ttit¡tIftherecoveryisproportionaltothepre-defaultvalueYi,andispaidatdefaulttime¿(thisscheme¿¡isknownasthefractionalrecoveryofmarketvalue),wehave·i;t=±i¡1and¡¢dYi=Yi¹dt+¾dW+(±¡1)dM:(4.6)tt¡i;ti;ttitR¢RThroughoutthischapter,wewritetodenote.0]0;¢] 126CHAPTER4.HEDGINGOFDEFAULTABLECLAIMS4.2TradingStrategiesinaSemimartingaleSet-upWeconsidertradingwithinthetimeinterval[0;T]forsome¯nitehorizondateT>0.Forthesakeofexpositionalclarity,werestrictourattentiontothecasewhereonlythreeprimaryassetsaretraded.ThegeneralcaseofktradedassetswasexaminedbyBieleckietal.[7,9].Inthissection,weconsiderafairlygeneralsetup.Inparticular,processesYi;i=1;2;3;areassumedtobenonnegativesemi-martingalesonaprobabilityspace(•;G;P)endowedwithsome¯ltrationG.Weassumethattheyrepresentspotpricesoftradedassetsinourmodelofthe¯nancialmarket.Neithertheexistenceofasavingsaccount,northemarketcompletenessareassumed,ingeneral.Ourgoalistocharacterizecontingentclaimswhicharehedgeable,inthesensethattheycanbereplicatedbycontinuouslyrebalancedportfoliosconsistingofprimaryassets.Here,byacon-tingentclaimwemeananarbitraryGT-measurablerandomvariable.Weworkunderthestandardassumptionsofafrictionlessmarket.4.2.1UnconstrainedStrategiesLetÁ=(Á1;Á2;Á3)beatradingstrategy;inparticular,eachprocessÁiispredictablewithrespecttothe¯ltrationG.ThewealthofÁequalsX3V(Á)=ÁiYi;8t2[0;T];ttti=1andatradingstrategyÁissaidtobeself-¯nancingifX3ZtV(Á)=V(Á)+ÁidYi;8t2[0;T]:t0uui=10Let©standfortheclassofallself-¯nancingtradingstrategies.Weshall¯rstprovethataself-¯nancingstrategyisdeterminedbyitsinitialwealthandthetwocomponentsÁ2;Á3.Tothisend,wepostulatethatthepriceofY1followsastrictlypositiveprocess,andwechooseY1asanum¶eraireasset.Weshallnowanalyzetherelativevalues:V1(Á):=V(Á)(Y1)¡1;Yi;1:=Yi(Y1)¡1:ttttttLemma4.2.1(i)ForanyÁ2©,wehaveX3ZtV1(Á)=V1(Á)+ÁidYi;1;8t2[0;T]:t0uui=20(ii)Conversely,letXbeaGT-measurablerandomvariable,andletusassumethatthereexistsx2RandG-predictableprocessesÁi;i=2;3suchthatÃZ!X3TX=Y1x+ÁidYi;1:(4.7)Tuui=20ThenthereexistsaG-predictableprocessÁ1suchthatthestrategyÁ=(Á1;Á2;Á3)isself-¯nancingandreplicatesX.Moreover,thewealthprocessofÁ(i.e.thetime-tpriceofX)satis¯esVt(Á)=V1Y1,wherettX3ZtV1=x+ÁidYi;1;8t2[0;T]:(4.8)tuui=20 4.2.TRADINGSTRATEGIES127Proof.Inthecaseofcontinuoussemimartingales,(itisawell-knownresult;fordiscontinuousprocesses,theproofisnotmuchdi®erent.Wereproduceithereforthereader'sconvenience.Letus¯rstintroducesomenotation.Asusual,[X;Y]standsforthequadraticcovariationofthetwosemi-martingalesXandY,asde¯nedbytheintegrationbypartsformula:ZtZtXtYt=X0Y0+Xu¡dYu+Yu¡dXu+[X;Y]t:00Foranycµadlµag(i.e.,RCLL)processY,wedenoteby¢Yt=Yt¡Yt¡thesizeofthejumpattimet.LetV=V(Á)bethevalueofaself-¯nancingstrategy,andletV1=V1(Á)=V(Á)(Y1)¡1beitsvaluerelativetothenum¶eraireY1.TheintegrationbypartsformulayieldsdV1=Vd(Y1)¡1+(Y1)¡1dV+d[(Y1)¡1;V]:tt¡tt¡ttP3iiFromtheself-¯nancingcondition,wehavedVt=i=1ÁtdYt.Hence,usingelementaryrulestocomputethequadraticcovariation[X;Y]ofthetwosemi-martingalesX;Y,weobtaindV1=Á1Y1d(Y1)¡1+Á2Y2d(Y1)¡1+Á3Y3d(Y1)¡1ttt¡ttt¡ttt¡t+(Y1)¡1Á1dY1+(Y1)¡1Á2dY1+(Y1)¡1Á3dY1t¡ttt¡ttt¡tt+Á1d[(Y1)¡1;Y1]+Á2d[(Y1)¡1;Y2]+Á3d[(Y1)¡1;Y1]tttttt¡¢=Á1Y1d(Y1)¡1+(Y1)¡1dY1+d[(Y1)¡1;Y1]tt¡tt¡tt¡¢+Á2Y2d(Y1)¡1+(Y1)¡1dY1+d[(Y1)¡1;Y2]tt¡tt¡t¡t¡¢+Á3Y3d(Y1)¡1+(Y1)¡1dY1+d[(Y1)¡1;Y3]:tt¡tt¡t¡tWenowobservethatY1d(Y1)¡1+(Y1)¡1dY1+d[(Y1)¡1;Y1]=d(Y1(Y1)¡1)=0t¡tt¡ttttandYid(Y1)¡1+(Y1)¡1dYi+d[(Y1)¡1;Yi]=d((Y1)¡1Yi):t¡tt¡ttttConsequently,dV1=Á2dY2;1+Á3dY3;1;tttttaswasclaimedinpart(i).Wenowproceedtotheproofofpart(ii).Weassumethat(4.7)holdsforsomeconstantxandprocessesÁ2;Á3,andwede¯netheprocessV1bysetting(cf.(4.8))X3ZtV1=x+ÁidYi;1;8t2[0;T]:tuui=20Next,wede¯netheprocessÁ1asfollows:X3³X3´Á1=V1¡ÁiYi;1=(Y1)¡1V¡ÁiYi;tttttttti=2i=2whereV=V1Y1.SincetttX3dV1=ÁidYi;1;ttti=2weobtaindV=d(V1Y1)=V1dY1+Y1dV1+d[Y1;V1]tttt¡tt¡ttX3¡¢=V1dY1+ÁiY1dYi;1+d[Y1;Yi;1]:t¡ttt¡tti=2 128CHAPTER4.HEDGINGOFDEFAULTABLECLAIMSFromtheequalitydYi=d(Yi;1Y1)=Yi;1dY1+Y1dYi;1+d[Y1;Yi;1];tttt¡tt¡ttitfollowsthatX3¡¢³X3´X3dV=V1dY1+ÁidYi¡Yi;1dY1=V1¡ÁiYi;1dY1+ÁidYi;tt¡tttt¡tt¡tt¡ttti=2i=2i=2P3iiandouraimistoprovethatdVt=i=1ÁtdYt.ThelastequalityholdsifX3X3Á1=V1¡ÁiYi;1=V1¡ÁiYi;1;(4.9)ttttt¡tt¡i=2i=21P3ii;11i.e.,if¢Vt=i=2Át¢Yt,whichisthecasefromthede¯nition(4.8)ofV.Notealsothatfromthesecondequalityin(4.9)itfollowsthattheprocessÁ1isindeedG-predictable.Finally,thewealthprocessofÁsatis¯esV(Á)=V1Y1foreveryt2[0;T],andthusV(Á)=X.¤tttTWesaythataself-¯nancingstrategyÁreplicatesaclaimX2GTifX3X=ÁiYi=V(Á);TTTi=1orequivalently,X3ZTX=V(Á)+ÁidYi:0tti=10SupposethatthereexistsanEMMforsomechoiceofanum¶eraireasset,andletusrestrictourattentiontotheclassofalladmissibletradingstrategies,sothatourmodelisarbitrage-free.AssumethataclaimXcanbereplicatedbysomeadmissibletradingstrategy,sothatitisattainable(orhedgeable).Then,byde¯nition,thearbitragepriceattimetofX,denotedas¼t(X),equalsVt(Á)foranyadmissibletradingstrategyÁthatreplicatesX.InthecontextofLemma4.2.1,itisnaturaltochooseasanEMMaprobabilitymeasureQ1equivalenttoPon(•;G)andsuchthatthepricesYi;1;i=2;3;areG-martingalesunderQ1.IfaTcontingentclaimXishedgeable,thenitsarbitragepricesatis¯es11¡1¼t(X)=YtEQ1(X(YT)jGt):WeemphasizethatevenifanEMMQ1isnotunique,thepriceofanyhedgeableclaimXisgivenbythisconditionalexpectation.Thatistosay,incaseofahedgeableclaimtheseconditionalexpectationsundervariousequivalentmartingalemeasurescoincide.InthespecialcasewhereY1=B(t;T)isthepriceofadefault-freezero-couponbondwithtmaturityT(abbreviatedasZCBinwhatfollows),Q1iscalledT-forwardmartingalemeasure,anditisdenotedbyQT.SinceB(T;T)=1,thepriceofanyhedgeableclaimXnowequals¼t(X)=B(t;T)EQT(XjGt).4.2.2ConstrainedStrategiesInthissection,wemakeanadditionalassumptionthatthepriceprocessY3isstrictlypositive.LetÁ=(Á1;Á2;Á3)beaself-¯nancingtradingstrategysatisfyingthefollowingconstraint:X2ÁiYi=Z;8t2[0;T];(4.10)tt¡ti=1 4.2.TRADINGSTRATEGIES129forapredetermined,G-predictableprocessZ.Inthe¯nancialinterpretation,equality(4.10)meansthataportfolioÁisrebalancedinsuchawaythatthetotalwealthinvestedinassetsY1;Y2matchesapredeterminedstochasticprocessZ.Forthisreason,theconstraintgivenby(4.10)isreferredtoasthebalancecondition.Our¯rstgoalistoextendpart(i)inLemma4.2.1tothecaseofconstrainedstrategies.Let©(Z)standfortheclassofall(admissible)self-¯nancingtradingstrategiessatisfyingthebalancecondition(4.10).Theywillbesometimesreferredtoasconstrainedstrategies.SinceanystrategyP3iiÁ2©(Z)isself-¯nancing,fromdVt(Á)=i=1ÁtdYt,weobtainX3X3¢V(Á)=Ái¢Yi=V(Á)¡ÁiYi:tttttt¡i=1i=1Bycombiningthisequalitywith(4.10),wededucethatX3V(Á)=ÁiYi=Z+Á3Yi:t¡tt¡ttt¡i=1LetuswriteYi;3=Yi(Y3)¡1;Z3=Z(Y3)¡1:ttttttThefollowingresultextendsLemma1.7inBieleckietal.[4]fromthecaseofcontinuoussemi-martingalestothegeneralcase(seealso[7,9]).ItisapparentfromProposition4.2.1thatthewealthprocessV(Á)ofastrategyÁ2©(Z)dependsonlyonasinglecomponentofÁ,namely,Á2.Proposition4.2.1TherelativewealthV3(Á)=V(Á)(Y3)¡1ofanytradingstrategyÁ2©(Z)tttsatis¯esZÃ2;3!ZtYtZ3V3(Á)=V3(Á)+Á2dY2;3¡u¡dY1;3+udY1;3:(4.11)t0uu1;3u1;3u0Yu¡0Yu¡Proof.LetusconsiderdiscountedvaluesofpriceprocessesY1;Y2;Y3,withY3takenasanum¶eraireasset.Byvirtueofpart(i)inLemma4.2.1,wethushaveX2ZtV3(Á)=V3(Á)+ÁidYi;3:(4.12)t0uui=10Thebalancecondition(4.10)impliesthatX2ÁiYi;3=Z3;tt¡ti=1andthus³´Á1=(Y1;3)¡1Z3¡Á2Y2;3:(4.13)tt¡ttt¡Byinserting(4.13)into(4.12),wearriveatthedesiredformula(4.11).¤Thenextresultwillproveparticularlyusefulforderivingreplicatingstrategiesfordefaultableclaims.Proposition4.2.2LetaGT-measurablerandomvariableXrepresentacontingentclaimthatsettlesattimeT.Weset2;3¤2;3Yt¡1;32;32;11;3dYt=dYt¡1;3dYt=dYt¡Yt¡dYt;(4.14)Yt¡ 130CHAPTER4.HEDGINGOFDEFAULTABLECLAIMSwhere,byconvention,Y¤=0.AssumethatthereexistsaG-predictableprocessÁ2,suchthat0ÃZZ!TTZ3X=Y3x+Á2dY¤+tdY1;3:(4.15)Ttt1;3t00Yt¡ThenthereexistG-predictableprocessesÁ1andÁ3suchthatthestrategyÁ=(Á1;Á2;Á3)belongsto©(Z)andreplicatesX.ThewealthprocessofÁequals,foreveryt2[0;T];ÃZZ!ttZ3V(Á)=Y3x+Á2dY¤+udY1;3:(4.16)ttuu1;3u00Yu¡Proof.Asexpected,we¯rstset(notethattheprocessÁ1isaG-predictableprocess)³´1122Át=1Zt¡ÁtYt¡(4.17)Yt¡andZZttZ3V3=x+Á2dY¤+udY1;3:tuu1;3u00Yu¡ArguingalongthesamelinesasintheproofofProposition4.2.1,weobtainX2ZtV3=V3+ÁidYi;3:t0uui=10Now,wede¯neX2³X2´Á3=V3¡ÁiYi;3=(Y3)¡1V¡ÁiYi;tttttttti=1i=1whereV=V3Y3.AsintheproofofLemma4.2.1,wecheckthattttX2Á3=V3¡ÁiYi;3;tt¡tt¡i=1andthustheprocessÁ3isG-predictable.ItisclearthatthestrategyÁ=(Á1;Á2;Á3)isself-¯nancinganditswealthprocesssatis¯esVt(Á)=Vtforeveryt2[0;T].Inparticular,VT(Á)=X,sothatÁreplicatesX.Finally,equality(4.17)implies(4.10),andthusÁbelongstotheclass©(Z).¤Notethatequality(4.15)isanecessary(byLemma4.2.1)andsu±cient(byProposition4.2.2)conditionfortheexistenceofaconstrainedstrategythatreplicatesagivencontingentclaimX.SyntheticAssetP2iiLetustakeZ=0,sothatÁ2©(0).Thenthebalanceconditionbecomesi=1ÁtYt¡=0,andformula(4.11)reducestoÃ!2;3322;3Yt¡1;3dVt(Á)=ÁtdYt¡1;3dYt:(4.18)Yt¡TheprocessY¹2=Y3Y¤,whereY¤isde¯nedin(4.14)iscalledasyntheticasset.Itcorrespondstoaparticularself-¯nancingportfolio,withthelongpositioninY2andtheshortpositionofY2;1t¡numberofsharesofY1,andsuitablyre-balancedpositionsinthethirdassetsothattheportfolioisself-¯nancing,asinLemma4.2.1.Itcanbeshown(seeBieleckietal.[7,9])thattradinginprimaryassetsY1;Y2;Y3isformallyequivalenttotradinginassetsY1;Y¹2;Y3.ThisobservationsupportsthenamesyntheticassetattributedtotheprocessY¹2.Note,however,thatthesyntheticassetprocessmaytakenegativevalues. 4.2.TRADINGSTRATEGIES131CaseofContinuousAssetPricesInthecaseofcontinuousassetprices,therelativepriceY¤=Y¹2(Y3)¡1ofthesyntheticassetcanbegivenanalternativerepresentation,asthefollowingresultshows.Recallthatthepredictablebracketofthetwocontinuoussemi-martingalesXandY,denotedashX;Yi,coincideswiththeirquadraticcovariation[X;Y].Proposition4.2.3AssumethatthepriceprocessesY1andY2arecontinuous.Thentherelativepriceofthesyntheticassetsatis¯esZtY¤=(Y3;1)¡1e®udYb;tuu0whereYb:=Y2;1e¡®tandttZt®:=hlnY2;1;lnY3;1i=(Y2;1)¡1(Y3;1)¡1dhY2;1;Y3;1i:(4.19)ttuuu0IntermsoftheauxiliaryprocessYb,formula(4.11)becomesZtZt3V3(Á)=V3(Á)+ÁbdYb+ZudY1;3;(4.20)t0uu1;3u00Yu¡whereÁb=Á2(Y3;1)¡1e®t:tttProof.Itsu±cestogivetheproofforZ=0.Theproofreliesontheintegrationbypartsformulastatingthatforanytwocontinuoussemi-martingales,sayXandY,wehave¡¢¡1¡1¡1¡1YtdXt¡YtdhX;Yit=d(XtYt)¡XtdYt;providedthatYisstrictlypositive.AnapplicationofthisformulatoprocessesX=Y2;1andY=Y3;1leadsto¡¢(Y3;1)¡1dY2;1¡(Y3;1)¡1dhY2;1;Y3;1i=d(Y2;1(Y3;1)¡1)¡Y2;1d(Y3;1)¡1:ttttttttTherelativewealthV3(Á)=V(Á)(Y3)¡1ofastrategyÁ2©(0)satis¯estttZtV3(Á)=V3(Á)+Á2dY¤t0uu0Zt=V3(Á)+Á2(Y3;1)¡1e®udYb;0uuu0Zt=V3(Á)+ÁbdYb0uu0wherewedenoteÁb=Á2(Y3;1)¡1e®t:tttRemark4.2.1The¯nancialinterpretationoftheauxiliaryprocessYbwillbestudiedbelow.LetusonlyobserveherethatifY¤isalocalmartingaleundersomeprobabilityQthenYbisaQ-localmartingale(andviceversa,ifYbisaQb-localmartingaleundersomeprobabilityQbthenY¤isaQb-localmartingale).Nevertheless,forthereader'sconvenience,weshallusetwosymbolsQandQb,sincethisequivalenceholdsforcontinuousprocessesonly.ItisthusworthstressingthatwewillapplyProposition4.2.3topre-defaultvaluesofassets,ratherthandirectlytoassetprices,withinthesetupofasemimartingalemodelwithacommondefault,asdescribedinSection4.1.1.Inthismodel,theassetpricesmayhavediscontinuities,buttheirpre-defaultvaluesfollowcontinuousprocesses. 132CHAPTER4.HEDGINGOFDEFAULTABLECLAIMS4.3MartingaleApproachtoValuationandHedgingOurgoalistoderivequasi-explicitconditionsforreplicatingstrategiesforadefaultableclaiminafairlygeneralsetupintroducedinSection4.1.1.Inthissection,weonlydealwithtradingstrategiesbasedonthereference¯ltrationFandtheunderlyingpriceprocesses(thatis,pricesofdefault-freeassetsandpre-defaultvaluesofdefaultableassets)areassumedtobecontinuous.Therefore,ourargumentswillhingeonProposition4.2.3,ratherthanonamoregeneralProposition4.2.1.WeshallalsoadaptProposition4.2.2toourcurrentpurposes.Tosimplifythepresentation,wemakeastandingassumptionthatallcoe±cientprocessesaresuchthattheSDEsappearingbelowadmituniquestrongsolutions,andallstochasticexponentials(usedasRadon-Nikod¶ymderivatives)aretruemartingalesunderrespectiveprobabilities.4.3.1DefaultableAssetwithTotalDefaultInthissection,weshallexamineinsomedetailaparticularmodelwherethetwoassets,Y1andY2,aredefault-freeandsatisfy¡¢dYi=Yi¹dt+¾dW;i=1;2;tti;ti;ttwhereWisaone-dimensionalBrownianmotion.Thethirdassetisadefaultableassetwithtotaldefault,sothat¡¢dY3=Y3¹dt+¾dW¡dM:tt¡3;t3;tttSincewewillbeinterestedinreplicatingstrategiesinthesenseofDe¯nition4.1.2,wemayanddoassume,withoutlossofgenerality,thatthecoe±cients¹i;t;¾i;t;i=1;2;areF-predictable,ratherthanG-predictable.Recallthat,ingeneral,thereexistF-predictableprocesses¹e3and¾e3suchthat¹e3;t1ft·¿g=¹3;t1ft·¿g;¾e3;t1ft·¿g=¾3;t1ft·¿g:(4.21)WeassumethroughoutthatYi>0foreveryi,sothatthepriceprocessesY1;Y2arestrictly0positive,andtheprocessY3isnonnegative,andhasstrictlypositivepre-defaultvalue.Default-FreeMarketItisnaturaltopostulatethatthedefault-freemarketwiththetwotradedassets,Y1andY2,isarbitrage-free.Moreprecisely,wechooseY1asanum¶eraire,andwerequirethatthereexistsaprobabilitymeasureP1,equivalenttoPon(•;F),andsuchthattheprocessY2;1isaP1-martingale.TThedynamicsofprocesses(Y1)¡1andY2;1are¡¢d(Y1)¡1=(Y1)¡1(¾2¡¹)dt¡¾dW;(4.22)tt1;t1;t1;ttand¡¢2;12;1dYt=Yt(¹2;t¡¹1;t+¾1;t(¾1;t¡¾2;t))dt+(¾2;t¡¾1;t)dWt;respectively.HencethenecessaryconditionfortheexistenceofanEMMP1istheinclusionAµB,whereA=f(t;!)2[0;T]£•:¾1;t(!)=¾2;t(!)gandB=f(t;!)2[0;T]£•:¹1;t(!)=¹2;t(!)g.Thenecessaryandsu±cientconditionfortheexistenceanduniquenessofanEMMP1reads½µZ¶¾¢EPETµudWu=1(4.23)0wheretheprocessµisgivenbytheformula(byconvention,0=0=0)¹1;t¡¹2;tµt=¾1;t¡;8t2[0;T]:(4.24)¾1;t¡¾2;tNotethatinthecaseofconstantcoe±cients,if¾1=¾2thenthemodelisarbitrage-freeonlyinthetrivialcasewhen¹2=¹1. 4.3.MARTINGALEAPPROACH133Remark4.3.1SincethemartingalemeasureP1isunique,thedefault-freemodel(Y1;Y2)iscom-plete.However,thisisnotanecessaryassumptionandthusitcanberelaxed.Asweshallseeinwhatfollows,itistypicallymorenaturaltoassumethatthedrivingBrownianmotionWismulti-dimensional.Arbitrage-FreePropertyLetusnowconsideralsoadefaultableassetY3.Ourgoalisnowto¯ndamartingalemeasureQ1(ifitexists)forrelativepricesY2;1andY3;1.Recallthatwepostulatethatthehypothesis(H)holdsunderPfor¯ltrationsFandG=F_H.ThedynamicsofY3;1underParen¡¢o3;13;1dYt=Yt¡¹3;t¡¹1;t+¾1;t(¾1;t¡¾3;t)dt+(¾3;t¡¾1;t)dWt¡dMt:LetQ1beanyprobabilitymeasureequivalenttoPon(•;G),andlet´betheassociatedTRadon-Nikod¶ymdensityprocess,sothatdQ1j=´dPj;(4.25)GttGtwheretheprocess´satis¯esd´t=´t¡(µtdWt+³tdMt)(4.26)forsomeG-predictableprocessesµand³,and´isaG-martingaleunderP.FromGirsanov'stheorem,theprocessesWcandMc,givenbyZtZtWct=Wt¡µudu;Mct=Mt¡1fu<¿g°u³udu;(4.27)00areG-martingalesunderQ1.ToensurethatY2;1isaQ1-martingale,wepostulatethat(4.23)and(4.24)arevalid.Consequently,fortheprocessY3;1tobeaQ1-martingale,itisnecessaryandsu±cientthat³satis¯es¹1;t¡¹2;t°t³t=¹3;t¡¹1;t¡(¾3;t¡¾1;t):¾1;t¡¾2;tToensurethatQ1isaprobabilitymeasureequivalenttoP,werequirethat³>¡1.TheuniquetmartingalemeasureQ1isthengivenbytheformula(4.25)where´solves(4.26),sothatµZ¶µZ¶¢¢´t=EtµudWuEt³udMu:00Weareinapositiontoformulatethefollowingresult.Proposition4.3.1Assumethattheprocessµgivenby(4.24)satis¯es(4.23),andµ¶1¹1;t¡¹2;t³t=¹3;t¡¹1;t¡(¾3;t¡¾1;t)>¡1:(4.28)°t¾1;t¡¾2;tThenthemodelM=(Y1;Y2;Y3;©)isarbitrage-freeandcomplete.ThedynamicsofrelativepricesundertheuniquemartingalemeasureQ1are2;12;1dYt=Yt(¾2;t¡¾1;t)dWct;¡¢3;13;1dYt=Yt¡(¾3;t¡¾1;t)dWct¡dMct:Sincethecoe±cients¹i;t;¾i;t;i=1;2,areF-adapted,theprocessWcisanF-martingale(hence,aBrownianmotion)underQ1.Therefore,byvirtueofProposition3.5.1,thehypothesis(H)holdsunderQ1,andtheF-intensityofdefaultunderQ1equalsµ¶¹1;t¡¹2;t°bt=°t(1+³t)=°t+¹3;t¡¹1;t¡(¾3;t¡¾1;t):¾1;t¡¾2;t 134CHAPTER4.HEDGINGOFDEFAULTABLECLAIMSExample4.3.1Wepresentanexamplewherethecondition(4.28)doesnothold,andthusarbitrageopportunitiesarise.Assumethatthecoe±cientsareconstantandsatisfy:¹1=¹2=¾1=0;¹3<¡°foraconstantdefaultintensity°>0.Thenµ¶µ¶3312312Yt=1ft<¿gY0exp¾3Wt¡¾3t+(¹3+°)t·Y0exp¾3Wt¡¾3t=Vt(Á);22whereV(Á)representsthewealthofaself-¯nancingstrategy(Á1;Á2;0)withÁ2=¾3.Hencethe¾2arbitragestrategywouldbetoselltheassetY3,andtofollowthestrategyÁ.Remark4.3.2Letusstressonceagain,thattheexistenceofanEMMisanecessaryconditionforviabilityofa¯nancialmodel,buttheuniquenessofanEMMisnotalwaysaconvenientconditiontoimposeonamodel.Infact,whenconstructingamodel,weshouldbemostlyconcernedwithits°exibilityandabilitytore°ectthepertinentriskfactors,ratherthanwithitsmathematicalcompleteness.Inthepresentcontext,itisnaturaltopostulatethatthedimensionoftheunderlyingBrownianmotionequalsthenumberoftradeableriskyassets.Inaddition,eachparticularmodelshouldbetailoredtoprovideintuitiveandhandysolutionsforapredeterminedfamilyofcontingentclaimsthatwillbepricedandhedgedwithinitsframework.HedgingaSurvivalClaimWe¯rstfocusonreplicationofasurvivalclaim(X;0;¿),thatis,adefaultableclaimrepresentedbytheterminalpayo®X1fT<¿g,whereXisanFT-measurablerandomvariable.Forthemoment,wemaintainthesimplifyingassumptionthatWisone-dimensional.Asweshallseeinwhatfollows,itmayleadtocertainpathologicalfeaturesofamodel.If,onthecontrary,thedrivingnoiseismulti-dimensional,mostoftheanalysisremainsvalid,exceptthatthemodelcompletenessisnolongerensured,ingeneral.RecallthatYe3standsforthepre-defaultpriceofY3,de¯nedas(see(4.3))¡¢dYe3=Ye3(¹e+°)dt+¾edW(4.29)tt3;tt3;ttwithYe3=Y3.Thisstrictlypositive,continuous,F-adaptedprocessenjoysthepropertythatY3=00t1Ye3.Letusdenotethepre-defaultvaluesinthenum¶eraireYe3byYei;3=Yi(Ye3)¡1;i=1;2,ft<¿gttttandletusintroducethepre-defaultrelativepriceYe¤ofthesyntheticassetY¹2bysettingYe2;3³¡¢´dYe¤:=dYe2;3¡tdYe1;3=Ye2;3¹¡¹+¾e(¾¡¾)dt+(¾¡¾)dW;ttYe1;3tt2;t1;t3;t1;t2;t2;t1;tttandletusassumethat¾1;t¡¾2;t6=0.ItisalsousefultonotethattheprocessYb,de¯nedinProposition4.2.3,satis¯es³¡¢´dYbt=Ybt¹2;t¡¹1;t+¾e3;t(¾1;t¡¾2;t)dt+(¾2;t¡¾1;t)dWt:Weshallshowthatinthecase,where®givenby(4.19)isdeterministic,theprocessYbhasanice¯nancialinterpretationasacredit-riskadjustedforwardpriceofY2relativetoY1.Therefore,itismoreconvenienttoworkwiththeprocessYe¤whendealingwiththegeneralcase,buttousetheprocessYbwhenanalyzingamodelwithdeterministicvolatilities.ConsideranF-predictableself-¯nancingstrategyÁsatisfyingthebalanceconditionÁ1Y1+ttÁ2Y2=0,andthecorrespondingwealthprocessttX3V(Á):=ÁiYi=Á3Y3:ttttti=1 4.3.MARTINGALEAPPROACH135LetVe(Á):=Á3Ye3:SincetheprocessVe(Á)isF-adapted,weseethatthisisthepre-defaultpricetttprocessoftheportfolioÁ,thatis,wehave1f¿>tgVt(Á)=1f¿>tgVet(Á);weshallcallthisprocessthepre-defaultwealthofÁ:Consequently,theprocessVe3(Á):=Ve(Á)(Ye3)¡1=Á3istermedtherelativettttpre-defaultwealth.UsingProposition4.2.1,withsuitablymodi¯ednotation,we¯ndthattheF-adaptedprocessVe3(Á)satis¯es,foreveryt2[0;T],ZtVe3(Á)=Ve3(Á)+Á2dYe¤:t0uu0De¯neanewprobabilityQ¤on(•;F)bysettingTdQ¤=´¤dP;Twhered´¤=´¤µ¤dW;andtttt¤¹2;t¡¹1;t+¾e3;t(¾1;t¡¾2;t)µt=:(4.30)¾1;t¡¾2;tTheprocessYe¤;t2[0;T],isa(local)martingaleunderQ¤drivenbyaBrownianmotion.Weshalltrequirethatthisprocessisinfactatruemartingale;asu±cientconditionforthisisthatZT³´2Ye2;3EQ¤t(¾2;t¡¾1;t)dt<1:0Fromthepredictablerepresentationtheorem,itfollowsthatforanyX2F,suchthatX(Ye3)¡1isTTsquare-integrableunderQ,thereexistsaconstantxandanF-predictableprocessÁ2suchthatÃZ!TX=Ye3x+Á2dYe¤:(4.31)Tuu0WenowdeducefromProposition4.2.2thatthereexistsaself-¯nancingstrategyÁwiththepre-defaultwealthVe(Á)=Ye3Ve3foreveryt2[0;T],wherewesettttZtVe3=x+Á2dYe¤:(4.32)tuu0Moreover,itsatis¯esthebalanceconditionÁ1Y1+Á2Y2=0foreveryt2[0;T].SinceclearlyttttVeT(Á)=X,wehavethatV(Á)=Á3Y3=1Á3Ye3=1Ve(Á)=1X;TTTfT<¿gTTfT<¿gTfT<¿gandthusthisstrategyreplicatesthesurvivalclaim(X;0;¿).Infact,wehavethatVt(Á)=0ontherandominterval[[¿;T]]:De¯nition4.3.1Wesaythatasurvivalclaim(X;0;¿)isattainableiftheprocessVe3givenby(4.32)isamartingaleunderQ¤.Thefollowingresultisanimmediateconsequenceof(4.31)and(4.32).Corollary4.3.1LetX2FbesuchthatX(Ye3)¡1issquare-integrableunderQ¤.ThentheTTsurvivalclaim(X;0;¿)isattainable.Moreover,thepre-defaultprice¼et(X;0;¿)oftheclaim(X;0;¿)isgivenbytheconditionalexpectation3Ye3¡1¼et(X;0;¿)=YetEQ¤(X(T)jFt);8t2[0;T]:(4.33)Theprocess¼e(X;0;¿)(Ye3)¡1isanF-martingaleunderQ. 136CHAPTER4.HEDGINGOFDEFAULTABLECLAIMSProof.SinceX(Ye3)¡1issquare-integrableunderQ,weknowfromthepredictablerepresentationT³R´2T22¤3theoremthatÁin(4.31)issuchthatEQ¤0(Át)dhYeit<1;sothattheprocessVegivenby(4.32)isatruemartingaleunderQ:Weconcludethat(X;0;¿)isattainable.Now,letusdenoteby¼t(X;0;¿)thetime-tpriceoftheclaim(X;0;¿).SinceÁisahedgingportfoliofor(X;0;¿)wethushaveVt(Á)=¼t(X;0;¿)foreacht2[0;T]:Consequently,VeYe3Ve31f¿>tg¼et(X;0;¿)=1f¿>tgt(Á)=1f¿>tgtEQ¤(TjFt)Ye3Ye3¡1=1f¿>tgtEQ¤(X(T)jFt)foreacht2[0;T].Thisprovesequality(4.33).¤Inviewofthelastresult,itisjusti¯edtorefertoQasthepricingmeasurerelativetoY3forattainablesurvivalclaims.Remark4.3.3Itcanbeprovedthatthereexistsauniqueabsolutelycontinuousprobabilitymea-sureQ¹on(•;GT)suchthatwehaveµ¯¶Ã¯!31f¿>TgX¯Ye3X¯YtEQ¹3¯Gt=1f¿>tgtEQ¤¯Ft:YYe3TTHowever,thisprobabilitymeasureisnotequivalenttoQ,sinceitsRadon-Nikod¶ymdensityvanishesafter¿(forarelatedresult,seeCollin-Dufresneetal.[32]).Example4.3.2Weprovidehereanexplicitcalculationofthepre-defaultpriceofasurvivalclaim.Forsimplicity,weassumethatX=1,sothattheclaimrepresentsadefaultablezero-couponbond.Also,weset°t=°=const;¹i;t=0;and¾i;t=¾i;i=1;2;3.Straightforwardcalculationsyieldthefollowingpricingformula3¡(°+1¾2)T¼e0(1;0;¿)=Y0e23:Weseethatherethepre-defaultprice¼e0(1;0;¿)dependsexplicitlyontheintensity°,orrather,onthedrifttermindynamicsofpre-defaultvalueofdefaultableasset.Indeed,fromthepracticalviewpoint,theinterpretationofthedriftcoe±cientindynamicsofY2asthereal-worlddefaultintensityisquestionable,sincewithinoursetupthedefaultintensityneverappearsasanindependentvariable,butismerelyacomponentofthedrifttermindynamicsofpre-defaultvalueofY3.Notealsothatwedealherewithamodelwiththreetradeableassetsdrivenbyaone-dimensionalBrownianmotion.Nowonderthatthemodelenjoyscompleteness,butasadownside,ithasanunde-sirablepropertythatthepre-defaultvaluesofallthreeassetsareperfectlycorrelated.Consequently,thedrifttermsindynamicsoftradedassetsarecloselylinkedtoeachother,inthesense,thattheirbehaviorunderanequivalentchangeofaprobabilitymeasureisquitespeci¯c.Asweshallseelater,iftradedprimaryassetsarejudiciouslychosenthen,typically,thepre-defaultprice(andhencetheprice)ofasurvivalclaimwillnotexplicitlydependontheintensityprocess.Remark4.3.4Generallyspeaking,webelievethatonecanclassifya¯nancialmodelas`realistic'ifitsimplementationdoesnotrequireestimationofdriftparametersin(pre-default)prices,atleastforthepurposeofhedgingandvaluationofasu±cientlylargeclassof(defaultable)contingentclaimsofinterest.Itisworthrecallingthatthedriftcoe±cientsarenotassumedtobemarketobservables.Sincethedefaultintensitycanformallyinterpretedasacomponentofthedrifttermindynamicsofpre-defaultprices,inarealisticmodelthereisnoneedtoestimatethisquantity.Fromthisperspective,themodelconsideredinExample4.3.2mayserveasanexampleofan`unrealistic'model,sinceitsimplementationrequirestheknowledgeofthedriftparameterinthedynamicsofY3.Wedonotpretendherethatitisalwayspossibletohedgederivativeassetswithoutusingthedriftcoe±cientsindynamicsoftradeableassets,butitseemstousthatagoodideaistodevelopmodelsinwhichthisknowledgeisnotessential. 4.3.MARTINGALEAPPROACH137Ofcourse,agenericsemimartingalemodelconsidereduntilnowprovidesonlyaframeworkforaconstructionofrealisticmodelsforhedgingofdefaultrisk.Achoiceoftradeableassetsandspeci¯cationoftheirdynamicsshouldbeexaminedonacase-by-casebasis,ratherthaninageneralsemimartingalesetup.Weshalladdressthisimportantissueintheforegoingsections,inwhichweshalldealwithparticularexamplesofpracticallyinterestingdefaultableclaims.HedgingaRecoveryProcessLetusnowbrie°ystudythesituationwherethepromisedpayo®equalszero,andtherecoverypayo®ispaidattime¿andequalsZ¿forsomeF-adaptedprocessZ.Putanotherway,weconsideradefaultableclaimoftheform(0;Z;¿).Onceagain,wemakeuseofPropositions4.2.1and4.2.2.Inviewof(4.15),weneedto¯ndaconstantxandanF-predictableprocessÁ2suchthatZTZTZt1;32¤ÃT:=¡1dYet=x+ÁtdYet:(4.34)0Yt0Similarlyasbefore,weconcludethat,undersuitableintegrabilityconditionsonÃ;thereexistsÁ2T2¤suchthatdÃt=ÁtdYt,whereÃt=EQ¤(ÃTjFt).WenowsetZtZTZe3Ve3=x+Á2dY¤+udYe1;3;tuuYe1;3u00usothat,inparticular,Ve3=0.Thenitispossibleto¯ndprocessesÁ1andÁ3suchthatthestrategyTÁisself-¯nancinganditsatis¯es:Ve(Á)=Ve3Ye3andV(Á)=Z+Á3Y3foreveryt2[0;T].ItistttttttthusclearthatV¿(Á)=Z¿onthesetf¿·TgandVT(Á)=0onthesetf¿>Tg.BondMarketForthesakeofconcreteness,weassumethatY1=B(t;T)isthepriceofadefault-freeZCBwithtmaturityT,andY3=D(t;T)isthepriceofadefaultableZCBwithzerorecovery,thatis,anassettwiththeterminalpayo®Y3=1.WepostulatethatthedynamicsunderPofthedefault-freeTfT<¿gZCBare¡¢dB(t;T)=B(t;T)¹(t;T)dt+b(t;T)dWt(4.35)forsomeF-predictableprocesses¹(t;T)andb(t;T).WechoosetheprocessY1=B(t;T)asatnum¶eraire.Sincethepricesoftheothertwoassetsarenotgivenapriori,wemaychooseanyprobabilitymeasureQequivalenttoPon(•;G)toplaytheroleofQ1.TInsuchacase,anEMMQ1isreferredtoastheforwardmartingalemeasureforthedateT,andisdenotedbyQT.HencetheRadon-Nikod¶ymdensityofQTwithrespecttoPisgivenby(4.26)forsomeF-predictableprocessesµand³,andtheprocessZtWT=W¡µdu;8t2[0;T];ttu0isaBrownianmotionunderQT.UnderQTthedefault-freeZCBisgovernedby¡¢dB(t;T)=B(t;T)¹b(t;T)dt+b(t;T)dWTtwhere¹b(t;T)=¹(t;T)+µtb(t;T).Let¡standforthebF-hazardprocessof¿underQT,sothat¡bt=¡ln(1¡Fbt),whereFbt=QT(¿·tjFt).Assumethatthehypothesis(H)holdsunderQTsothat,inparticular,theprocess¡isincreasing.Wede¯nethepriceprocessofadefaultableZCBbwithzerorecoverybytheformula¡¡bt¡¡bT¯¯F¢D(t;T):=B(t;T)EQT(1fT<¿gjGt)=1ft<¿gB(t;T)EQTet; 138CHAPTER4.HEDGINGOFDEFAULTABLECLAIMSItisthenclearthatY3;1=D(t;T)(B(t;T))¡1isaQ-martingale,andthepre-defaultpriceDe(t;T)tTequals¡¡b¡b¯¢De(t;T)=B(t;T)Eet¡T¯F:QTtThenextresultexaminesthebasicpropertiesoftheauxiliaryprocess¡(bt;T)givenas,foreveryt2[0;T],¡(bt;T)=Ye3;1¡1¡¡bt¡¡bT¯¯F¢t=De(t;T)(B(t;T))=EQTet:Thequantity¡(bt;T)canbeinterpretedastheconditionalprobability(underQT)thatdefaultwillnotoccurpriortothematuritydateT,giventhatweobserveFtandweknowthatthedefaulthasnotyethappened.Wewillbemoreinterested,however,initsvolatilityprocess¯(t;T)asde¯nedinthefollowingresult.Lemma4.3.1AssumethattheF-hazardprocess¡bof¿underQTiscontinuous.Thentheprocess¡(bt;T);t2[0;T],isacontinuousF-submartingaleand¡¢d¡(bt;T)=¡(bt;T)d¡b+¯(t;T)dWT(4.36)ttforsomeF-predictableprocess¯(t;T).Theprocess¡(bt;T)isof¯nitevariationifandonlyifthehazardprocess¡bisdeterministic.Inthiscase,wehave¡(bt;T)=e¡bt¡¡bT:Proof.Wehave¡(bt;T)=E¡¡bt¡¡bT¢¡btQTejFt=eLt;¡¢wherewesetL=Ee¡¡bTjF.Hence¡(bt;T)isequaltotheproductofastrictlypositive,tQTt¡bincreasing,right-continuous,F-adaptedprocesset,andastrictlypositive,continuousF-martingaleL.Furthermore,thereexistsanF-predictableprocess¯b(t;T)suchthatLsatis¯esdL=L¯b(t;T)dWTttt¡¢withtheinitialconditionL=Ee¡¡bT.Formula(4.36)nowfollowsbyanapplicationofIt^o's0QTformula,bysetting¯(t;T)=e¡¡bt¯b(t;T).Tocompletetheproof,itsu±cestorecallthatacontinuousmartingaleisneverof¯nitevariation,unlessitisaconstantprocess.¤Remark4.3.5Itcanbecheckedthat¯(t;T)isalsothevolatilityoftheprocess¡¯¢¡(t;T)=Ee¡t¡¡T¯F:PtRtAssumethat¡bt=0°buduforsomeF-predictable,nonnegativeprocess°b.Thenwehavethefollowingauxiliaryresult,whichgives,inparticular,thevolatilityofthedefaultableZCB.Corollary4.3.2ThedynamicsunderQTofthepre-defaultpriceDe(t;T)equals³¡¢¡¢´dDe(t;T)=De(t;T)¹b(t;T)+b(t;T)¯(t;T)+°bdt+b(t;T)+¯(t;T)de(t;T)dWT:ttEquivalently,thepriceD(t;T)ofthedefaultableZCBsatis¯esunderQT³¡¢´dD(t;T)=D(t;T)¹b(t;T)+b(t;T)¯(t;T)dt+de(t;T)dWT¡dM:ttwherewesetde(t;T)=b(t;T)+¯(t;T).Notethattheprocess¯(t;T)canbeexpressedintermsofmarketobservables,sinceitissimplythedi®erenceofvolatilitiesde(t;T)andb(t;T)ofpre-defaultpricesoftradeableassets. 4.3.MARTINGALEAPPROACH139Credit-Risk-AdjustedForwardPriceAssumethatthepriceY2satis¯esunderthestatisticalprobabilityP¡¢dY2=Y2¹dt+¾dW(4.37)tt2;ttt2¡12withF-predictablecoe±cients¹and¾.LetFY2(t;T)=Yt(B(t;T))betheforwardpriceofYT.Foranappropriatechoiceofµ(see4.30),weshallhavethat¡¢TdFY2(t;T)=FY2(t;T)¾t¡b(t;T)dWt:Therefore,thedynamicsofthepre-defaultsyntheticassetYe¤underQTaret¡¢¡¢dYe¤=Ye2;3¾¡b(t;T)dWT¡¯(t;T)dt;ttttandtheprocessYb=Y2;1e¡®t(seeProposition4.2.3forthede¯nitionof®)satis¯estt¡¢¡¢dYb=Yb¾¡b(t;T)dWT¡¯(t;T)dt:ttttLetQbbeanequivalentprobabilitymeasureon(•;G)suchthatYb(or,equivalently,Ye¤)isaTQb-martingale.ByvirtueofGirsanov'stheorem,theprocessWcgivenbytheformulaZtWc=WT¡¯(u;T)du;8t2[0;T];tt0QbisaBrownianmotionunderQb.Thus,theforwardpriceFY2(t;T)satis¯esunder¡¢¡¢dFY2(t;T)=FY2(t;T)¾t¡b(t;T)dWct+¯(t;T)dt:(4.38)Itappearsthatthevaluationresultsareeasiertointerpretwhentheyareexpressedintermsofforwardpricesassociatedwithvulnerableforwardcontracts,ratherthanintermsofspotpricesofprimaryassets.Forthisreason,weshallnowexaminecredit-risk-adjustedforwardpricesofdefault-freeanddefaultableassets.De¯nition4.3.2LetYbeaGT-measurableclaim.AnFt-measurablerandomvariableKiscalledthecredit-risk-adjustedforwardpriceofYifthepre-defaultvalueattimetofthevulnerableforwardcontractrepresentedbytheclaim1fT<¿g(Y¡K)equals0.Lemma4.3.2Thecredit-risk-adjustedforwardpriceFbY(t;T)ofanattainablesurvivalclaim(X;0;¿),representedbyaG-measurableclaimY=X1,equals¼e(X;0;¿)(De(t;T))¡1,where¼e(X;0;¿)TfT<¿gttisthepre-defaultpriceof(X;0;¿).TheprocessFbY(t;T);t2[0;T],isanF-martingaleunderQb.Proof.Theforwardpriceisde¯nedasanFt-measurablerandomvariableKsuchthattheclaim1fT<¿g(X1fT<¿g¡K)=X1fT<¿g¡KD(T;T)isworthlessattimetonthesetft<¿g.Itisclearthatthepre-defaultvalueattimetofthisclaimequals¼e(X;0;¿)¡KDe(t;T).Consequently,weobtainFe(t;T)=¼e(X;0;¿)(De(t;T))¡1.¤tYtLetusnowfocusondefault-freeassets.Manifestly,thecredit-risk-adjustedforwardpriceofthebondB(t;T)equals1.To¯ndthecredit-risk-adjustedforwardpriceofY2,letuswriteFb®T¡®t2;1®T¡®tY2(t;T):=FY2(t;T)e=Yte;(4.39)where®isgivenby(see(4.19))Zt¡¢Zt¡¢¡¢®t=¾u¡b(u;T)¯(u;T)du=¾u¡b(u;T)de(u;T)¡b(u;T)du:(4.40)00 140CHAPTER4.HEDGINGOFDEFAULTABLECLAIMSLemma4.3.3Assumethat®givenby(4.40)isadeterministicfunction.Thenthecredit-risk-2adjustedforwardpriceofYequalsFbY2(t;T)(de¯nedin4.39)foreveryt2[0;T].Proof.AccordingtoDe¯nition4.3.2,thepriceFbY2(t;T)isanFt-measurablerandomvariableK,whichmakestheforwardcontractrepresentedbytheclaimD(T;T)(Y2¡K)worthlessonthesetTft<¿g.AssumethattheclaimY2¡Kisattainable.SinceDe(T;T)=1,fromequation(4.33)itTfollowsthatthepre-defaultvalueofthisclaimisgivenbytheconditionalexpectation¡¯¢De(t;T)EY2¡K¯F:QbTtConsequently,¡¯¢¡¯¢Fb2¯F¯F®T¡®tY2(t;T)=EQbYTt=EQbFY2(T;T)t=FY2(t;T)e;aswasclaimed.¤ItisworthnotingthattheprocessFbY2(t;T)isa(local)martingaleunderthepricingmeasureQb,sinceitsatis¯esFbdFbY2(t;T)=Y2(t;T)(¾t¡b(t;T))dWct:(4.41)Underthepresentassumptions,theauxiliaryprocessYbintroducedinProposition4.2.3andthecredit-risk-adjustedforwardpriceFbY2(t;T)arecloselyrelatedtoeachother.Indeed,wehaveFbYb®TY2(t;T)=te,sothatthetwoprocessesareproportional.VulnerableOptiononaDefault-FreeAssetWeshallnowanalyzeavulnerablecalloptionwiththepayo®Cd=1(Y2¡K)+:TfT<¿gTHereKisaconstant.Ourgoalisto¯ndareplicatingstrategyforthisclaim,interpretedasasurvivalclaim(X;0;¿)withthepromisedpayo®X=C=(Y2¡K)+,whereCisthepayo®TTTofanequivalentnon-vulnerableoption.Themethodpresentedbelowisquitegeneral,however,sothatitcanbeappliedtoanysurvivalclaimwiththepromisedpayo®X=G(Y2)forsomefunctionTG:R!Rsatisfyingtheusualintegrabilityassumptions.WeassumethatY1=B(t;T);Y3=D(t;T)andthepriceofadefault-freeassetY2isgovernedttby(4.37).ThenCd=1(Y2¡K)+=1(Y2¡KY1)+:TfT<¿gTfT<¿gTT2;1WearegoingtoapplyProposition4.2.3.Inthepresentsetup,wehaveYt=FY2(t;T)andYb¡®tt=FY2(t;T)e.Sinceavulnerableoptionisanexampleofasurvivalclaim,inviewofLemma4.3.2,itscredit-risk-adjustedforwardpricesatis¯esFbCed(De(t;T))¡1.Cd(t;T)=tProposition4.3.2Supposethatthevolatilities¾;band¯aredeterministicfunctions.Thenthecredit-risk-adjustedforwardpriceofavulnerablecalloptionwrittenonadefault-freeassetY2equalsFbFbCd(t;T)=Y2(t;T)N(d+(FbY2(t;T);t;T))¡KN(d¡(FbY2(t;T);t;T))(4.42)wherelnz¡lnK§1v2(t;T)d(z;t;T)=2§v(t;T)andZTv2(t;T)=(¾¡b(u;T))2du:utThereplicatingstrategyÁinthespotmarketsatis¯esforeveryt2[0;T],onthesetft<¿g,Á1B(t;T)=¡Á2Y2;Á2=De(t;T)(B(t;T))¡1N(d(t;T))e®T¡®t;Á3De(t;T)=Ced;tttt+ttwhered+(t;T)=d+(FbY2(t;T);t;T). 4.3.MARTINGALEAPPROACH141Proof.Inthe¯rststep,weestablishthevaluationformula.Assumeforthemomentthattheoptionisattainable.Thenthepre-defaultvalueoftheoptionequals,foreveryt2[0;T],¡¯¢¡¯¢Ced+¯F+¯Ft=De(t;T)EQb(FY2(T;T)¡K)t=De(t;T)EQb(FbY2(T;T)¡K)t:(4.43)Inviewof(4.41),theconditionalexpectationabovecanbecomputedexplicitly,yieldingthevaluationformula(4.42).To¯ndthereplicatingstrategy,andestablishattainabilityoftheoption,weconsidertheIt^odi®erentialdFbCd(t;T)andweidentifytermsin(4.32).Itappearsthat®TdFbCd(t;T)=N(d+(t;T))dFbY2(t;T)=N(d+(t;T))edYbt(4.44)=N(d(t;T))Ye3;1e®T¡®tdYe¤;+ttsothattheprocessÁ2in(4.31)equalsÁ2=Ye3;1N(d(t;T))e®T¡®t:tt+Moreover,Á1issuchthatÁ1B(t;T)+Á2Y2=0andÁ3=Ced(De(t;T))¡1.Itiseasilyseenthatthistttttprovesalsotheattainabilityoftheoption.¤Letusexaminethe¯nancialinterpretationofthelastresult.First,equality(4.44)showsthatitiseasytoreplicatetheoptionusingvulnerableforwardcontracts.Indeed,wehaveCedZTFb0Cd(T;T)=X=+N(d+(t;T))dFbY2(t;T)De(0;T)0andthusitisenoughtoinvestthepremiumCed=CdindefaultableZCBsofmaturityT,andtakeat00anyinstanttpriortodefaultN(d+(t;T))positionsinvulnerableforwardcontracts.ItisunderstoodthatifdefaultoccurspriortoT,alloutstandingvulnerableforwardcontractsbecomevoid.Second,itisworthstressingthatneitherthearbitrageprice,northereplicatingstrategyforavulnerableoption,dependexplicitlyonthedefaultintensity.Thisremarkablefeatureisduetothefactthatthedefaultriskofthewriteroftheoptioncanbecompletelyeliminatedbytradingindefaultablezero-couponbondwiththesameexposuretocreditriskasavulnerableoption.Infact,sincethevolatility¯isinvariantwithrespecttoanequivalentchangeofaprobabilitymeasure,andsoarethevolatilities¾andb(t;T),theformulaeofProposition4.3.2arevalidforanychoiceofaforwardmeasureQTequivalenttoP(and,ofcourse,theyarevalidunderPaswell).TheonlywayinwhichthechoiceofaforwardmeasureQTimpactstheseresultsisthroughthepre-defaultvalueofadefaultableZCB.Weconcludethatwedealherewiththevolatilitybasedrelativepricingadefaultableclaim.Thisshouldbecontrastedwithmorepopularintensity-basedrisk-neutralpricing,whichiscommonlyusedtoproduceanarbitrage-freemodeloftradeabledefaultableassets.Recall,however,thatiftradeableassetsarenotchosencarefullyforagivenclassofsurvivalclaims,thenbothhedgingstrategyandpre-defaultpricemaydependexplicitlyonvaluesofdriftparameters,whichcanbelinkedinoursetuptothedefaultintensity(seeExample4.3.2).Remark4.3.6AssumethatX=G(Y2)forsomefunctionG:R!R.Thenthecredit-risk-Tadjustedforwardpriceofasurvivalclaimsatis¯esFbX(t;T)=v(t;FbY2(t;T)),wherethepricingfunctionvsolvesthePDE122@tv(t;z)+(¾t¡b(t;T))z@zzv(t;z)=02withtheterminalconditionv(T;z)=G(z).ThePDEapproachisstudiedinSection4.4below. 142CHAPTER4.HEDGINGOFDEFAULTABLECLAIMSRemark4.3.7Proposition4.3.2isstillvalidifthedrivingBrownianmotionistwo-dimensional,ratherthanone-dimensional.Inanextendedmodel,thevolatilities¾t;b(t;T)and¯(t;T)takevaluesinR2andtherespectiveproductsareinterpretedasinnerproductsinR3.Equivalently,onemayprefertodealwithreal-valuedvolatilities,butwithcorrelatedone-dimensionalBrownianmotions.VulnerableSwaptionInthissection,werelaxtheassumptionthatY1isthepriceofadefault-freebond.WenowletY1andY2tobearbitrarydefault-freeassets,withdynamics¡¢dYi=Yi¹dt+¾dW;i=1;2:tti;ti;ttWestilltakeD(t;T)tobethethirdasset,andwemaintaintheassumptionthatthemodelisarbitrage-free,butwenolongerpostulateitscompleteness.Inotherwords,wepostulatetheexis-tenceanEMMQ1,asde¯nedinsubsectiononarbitragefreeproperty,butnottheuniquenessofQ1.Wetakethe¯rstassetasanum¶eraire,sothatallpricesareexpressedinunitsofY1.Inparticular,Y1;1=1foreveryt2R,andtherelativepricesY2;1andY3;1satisfyunderQ1(cf.Propositiont+4.3.1)2;12;1dYt=Yt(¾2;t¡¾1;t)dWct;¡¢3;13;1dYt=Yt¡(¾3;t¡¾1;t)dWct¡dMct:ItisnaturaltopostulatethatthedrivingBrowniannoiseistwo-dimensional.Insuchacase,wemayrepresentthejointdynamicsofY2;1andY3;1underQ1asfollows2;12;11dYt=Yt(¾2;t¡¾1;t)dWt;¡¢3;13;12dYt=Yt¡(¾3;t¡¾1;t)dWt¡dMct;whereW1;W2areone-dimensionalBrownianmotionsunderQ1,suchthatdhW1;W2i=½dtforttadeterministicinstantaneouscorrelationcoe±cient½takingvaluesin[¡1;1].Weassumefromnowonthatthevolatilities¾i;i=1;2;3aredeterministic.LetussetZt®=hlnYe2;1;lnYe3;1i=½(¾¡¾)(¾¡¾)du;(4.45)ttu2;u1;u3;u1;u0andletQbbeanequivalentprobabilitymeasureon(•;G)suchthattheprocessYb=Y2;1e¡®tTttisaQb-martingale.Toclarifythe¯nancialinterpretationoftheauxiliaryprocessYbinthepresentcontext,weintroducetheconceptofcredit-risk-adjustedforwardpricerelativetothenum¶eraireY1.De¯nition4.3.3LetYbeaGT-measurableclaim.AnFt-measurablerandomvariableKiscalledthetime-tcredit-risk-adjustedY1-forwardpriceofYifthepre-defaultvalueattimetofavulnerableforwardcontract,representedbytheclaim1(Y1)¡1(Y¡KY1)=1(Y(Y1)¡1¡K);fT<¿gTTfT<¿gTequals0.1Thecredit-risk-adjustedY-forwardpriceofYisdenotedbyFbYjY1(t;T),anditisalsointerpretedasanabstractdefaultableswaprate.Thefollowingauxiliaryresultsareeasytoestablish,alongthesamelinesasLemmas4.3.2and4.3.3.Lemma4.3.4Thecredit-risk-adjustedY1-forwardpriceofasurvivalclaimY=(X;0;¿)equalsFb1¡1YjY1(t;T)=¼et(X;0;¿)(De(t;T))whereX1=X(Y1)¡1isthepriceofXinthenum¶eraireY1,and¼e(X1;0;¿)isthepre-defaultTtvalueofasurvivalclaimwiththepromisedpayo®X1. 4.3.MARTINGALEAPPROACH143Proof.Itsu±cestonotethatforY=1fT<¿gX,wehave1(Y(Y1)¡1¡K)=1X1¡KD(T;T);fT<¿gTfT<¿gwhereX1=X(Y1)¡1,andtoconsiderthepre-defaultvalues.¤TLemma4.3.5Thecredit-risk-adjustedY1-forwardpriceoftheassetY2equalsFb2;1®T¡®t®TY2jY1(t;T)=Yte=Ybte;(4.46)where®,assumedtobedeterministic,isgivenby(4.45).Proof.Itsu±cesto¯ndanFt-measurablerandomvariableKforwhich¡¯¢De(t;T)EY2(Y1)¡1¡K¯F=0:QbTTtConsequently,K=FbY2jY1(t;T);where¡¯¢Fb2;1¯F2;1®T¡®t®TY2jY1(t;T)=EQbYTt=Yte=Ybte;wherewehaveusedthefactsthatYb=Y2;1e¡®tisaQb-martingale,and®isdeterministic.¤ttWeareinapositiontoexamineavulnerableoptiontoexchangedefault-freeassetswiththepayo®Cd=1(Y1)¡1(Y2¡KY1)+=1(Y2;1¡K)+:(4.47)TfT<¿gTTTfT<¿gTThelastexpressionshowsthattheoptioncanbeinterpretedasavulnerableswaptionassociatedwiththeassetsY1andY2.Itisusefultoobservethatµ¶+Cd1Y2TfT<¿gT=¡K;Y1Y1Y1TTTsothat,whenexpressedinthenum¶eraireY1,thepayo®becomesC1;d=D1(T;T)(Y2;1¡K)+;TTwhereC1;d=Cd(Y1)¡1andD1(t;T)=D(t;T)(Y1)¡1standforthepricesrelativetoY1.ttttItisclearthatwedealherewithamodelanalogoustothemodelexaminedinprevioussubsectionsinwhich,however,allpricesarenowrelativetothenum¶eraireY1.ThisobservationallowsustodirectlyderivethevaluationformulafromProposition4.3.2.Proposition4.3.3Assumethatthevolatilitiesaredeterministic.Thecredit-risk-adjustedY1-forwardpriceofavulnerablecalloptionwrittenwiththepayo®givenby(4.47)equals¡¢¡¢FbFbCdjY1(t;T)=Y2jY1(t;T)Nd+(FbY2jY1(t;T);t;T)¡KNd¡(FbY2jY1(t;T);t;T)wherelnz¡lnK§1v2(t;T)d(z;t;T)=2§v(t;T)andZTv2(t;T)=(¾¡¾)2du:2;u1;utThereplicatingstrategyÁinthespotmarketsatis¯esforeveryt2[0;T],onthesetft<¿g,Á1Y1=¡Á2Y2;Á2=De(t;T)(Y1)¡1N(d(t;T))e®T¡®t;Á3De(t;T)=Ced;tttttt+tt¡¢Fbwhered+(t;T)=d+Y2(t;T);t;T. 144CHAPTER4.HEDGINGOFDEFAULTABLECLAIMSProof.TheproofisanalogoustothatofProposition4.3.2,andthusitisomitted.¤Itisworthnotingthatthepayo®(4.47)wasjudiciouslychosen.Supposeinsteadthattheoptionpayo®isnotde¯nedby(4.47),butitisgivenbyanapparentlysimplerexpressionCd=1(Y2¡KY1)+:(4.48)TfT<¿gTTSincethepayo®CdcanberepresentedasfollowsTCd=Gb(Y1;Y2;Y3)=Y3(Y2¡KY1)+;TTTTTTTwhereGb(y;y;y)=y(y¡Ky)+,theoptioncanbeseenanoptiontoexchangethesecondasset123321forKunitsofthe¯rstasset,butwiththepayo®expressedinunitsofthedefaultableasset.Whenexpressedinrelativeprices,thepayo®becomes1;d2;1+CT=1fT<¿g(YT¡K):where1=D1(T;T)Y1.ItisthusratherclearthatitisnotlongerpossibletoapplythesamefT<¿gTmethodasintheproofofProposition4.3.2.4.3.2DefaultableAssetwithNon-ZeroRecoveryWenowassumethatdY3=Y3(¹dt+¾dW+·dM)tt¡33t3twith·>¡1and·6=0.WeassumethatY3>0,sothatY3>0foreveryt2R.Weshall330t+brie°ydescribethesamestepsasinthecaseofadefaultableassetwithtotaldefault.Arbitrage-FreePropertyAsusual,weneed¯rsttoimposespeci¯cconstraintsonmodelcoe±cients,sothatthemodelisarbitrage-free.Indeed,anEMMQ1existsifthereexistsapair(µ;³)suchthat·i¡·1·1µt(¾i¡¾1)+³t»t=¹1¡¹i+¾1(¾i¡¾1)+»t(·i¡·1);i=2;3:1+·11+·1Toensuretheexistenceofasolution(µ;³)ontheset¿¡1;sothatthemartingalemeasureQ1existsandisunique. 4.3.MARTINGALEAPPROACH1454.3.3TwoDefaultableAssetswithTotalDefaultWeshallnowassumethatwehaveonlytwoassets,andbotharedefaultableassetswithtotaldefault.ThiscaseisalsoexaminedbyCarr[28],whostudysomeimperfecthedgingofdigitaloptions.Notethatherewepresentresultsforperfecthedging.Weshallbrie°youtlinetheanalysisofhedgingofasurvivalclaim.Underthepresentassumptions,wehave,fori=1;2;¡¢dYi=Yi¹dt+¾dW¡dM;(4.49)tt¡i;ti;tttwhereWisaone-dimensionalBrownianmotion,sothatY1=1Ye1;Y2=1Ye2;tft<¿gttft<¿gtwiththepre-defaultpricesgovernedbytheSDEs¡¢dYei=Yei(¹+°)dt+¾dW:(4.50)tti;tti;ttThewealthprocessVassociatedwiththeself-¯nancingtradingstrategy(Á1;Á2)satis¯es,foreveryt2[0;T],µZt¶V=Y1V1+Á2dYe2;1;tt0uu0whereYe2;1=Ye2=Ye1.Sincebothprimarytradedassetsaresubjecttototaldefault,itisclearthatthetttpresentmodelisincomplete,inthesense,thatnotalldefaultableclaimscanbereplicated.Weshallcheckinthefollowingsubsectionthat,undertheassumptionthatthedrivingBrownianmotionWisone-dimensional,allsurvivalclaimssatisfyingnaturaltechnicalconditionsarehedgeable,however.Inthemorerealisticcaseofatwo-dimensionalnoise,wewillstillbeabletohedgealargeclassofsurvivalclaims,includingoptionsonadefaultableassetandoptionstoexchangedefaultableassets.HedgingaSurvivalClaimForthesakeofexpositionalsimplicity,weassumeinthissectionthatthedrivingBrownianmotionWisone-dimensional.Thisisde¯nitelynottherightchoice,sincewedealherewithtworiskyassets,andthustheywillbeperfectlycorrelated.However,thisassumptionisconvenientfortheexpositionalpurposes,sinceitwillensurethemodelcompletenesswithrespecttosurvivalclaims,anditwillbelaterrelaxedanyway.Weshallarguethatinamodelwithtwodefaultableassetsgovernedby(4.49),replicationofasurvivalclaim(X;0;¿)isinfactequivalenttoreplicationofthepromisedpayo®Xusingthepre-defaultprocesses.Lemma4.3.6IfastrategyÁi;i=1;2,basedonpre-defaultvaluesYei;i=1;2,isareplicatingstrategyforanF-measurableclaimX,thatis,ifÁissuchthattheprocessVe(Á)=Á1Ye1+Á2Ye2Ttttttsatis¯es,foreveryt2[0;T],dVe(Á)=Á1dYe1+Á2dYe2;tttttVeT(Á)=X;thenfortheprocessV(Á)=Á1Y1+Á2Y2wehave,foreveryt2[0;T],tttttdV(Á)=Á1dY1+Á2dY2;tttttVT(Á)=X1fT<¿g:ThismeansthatthestrategyÁreplicatesthesurvivalclaim(X;0;¿). 146CHAPTER4.HEDGINGOFDEFAULTABLECLAIMSProof.ItisclearthatVt(Á)=1ft<¿gVt(Á)=1ft<¿gVet(Á).FromÁ1dY1+Á2dY2=¡(Á1Ye1+Á2Ye2)dH+(1¡H)(Á1dYe1+Á2dYe2);tttttttttt¡ttttitfollowsthatÁ1dY1+Á2dY2=¡Ve(Á)dH+(1¡H)dVe(Á);ttttttt¡tthatis,Á1dY1+Á2dY2=d(1Ve(Á))=dV(Á):ttttft<¿gttItisalsoobviousthatVT(Á)=X1fT<¿g.¤CombiningthelastresultwithLemma4.2.1,weseethatastrategy(Á1;Á2)replicatesasurvivalclaim(X;0;¿)wheneverwehave³ZT´Ye1x+Á2dYe2;1=XTtt0forsomeconstantxandsomeF-predictableprocessÁ2,where,inviewof(4.50),³¡¢´2;12;1dYet=Yet¹2;t¡¹1;t+¾1;t(¾1;t¡¾2;t)dt+(¾2;t¡¾1;t)dWt:WeintroduceaprobabilitymeasureQe,equivalenttoPon(•;G),andsuchthatYe2;1isanF-TmartingaleunderQe.ItiseasilyseenthattheRadon-Nikod¶ymdensity´satis¯es,fort2[0;T],µZ¶¢dQejGt=´tdPjGt=EtµsdWsdPjGt(4.51)0with¹2;t¡¹1;t+¾1;t(¾1;t¡¾2;t)µt=;¾1;t¡¾2;tprovided,ofcourse,thattheprocessµiswellde¯nedandsatis¯essuitableintegrabilityconditions.WeshallshowthatasurvivalclaimisattainableiftherandomvariableX(Ye1)¡1isQe-integrable.TIndeed,thepre-defaultvalueVetattimetofasurvivalclaimequals¡¢Ve=Ye1EX(Ye1)¡1jF;ttQeTtandfromthepredictablerepresentationtheorem,wededucethatthereexistsaprocessÁ2suchthatZt¡¢¡¢EX(Ye1)¡1jF=EX(Ye1)¡1+Á2dYe2;1:QeTtQeTuu0ThecomponentÁ1oftheself-¯nancingtradingstrategyÁ=(Á1;Á2)isthenchoseninsuchawaythatÁ1Ye1+Á2Ye2=Ve;8t2[0;T]:tttttToconclude,byfocusingonpre-defaultvalues,wehaveshownthatthereplicationofsurvivalclaimscanbereducedheretoclassicresultsonreplicationof(non-defaultable)contingentclaimsinadefault-freemarketmodel.OptiononaDefaultableAssetInordertogetacompletemodelwithrespecttosurvivalclaims,wepostulatedintheprevioussectionthatthedrivingBrownianmotionindynamics(4.49)isone-dimensional.Thisassumptionisquestionable,sinceitimpliestheperfectcorrelationofriskyassets.However,wemayrelaxthisrestriction,andworkinsteadwiththetwocorrelatedone-dimensionalBrownianmotions.Themodel 4.3.PDEAPPROACH147willnolongerbecomplete,butoptionsonadefaultableassetswillbestillattainable.Thepayo®ofa(non-vulnerable)calloptionwrittenonthedefaultableassetY2equalsC=(Y2¡K)+=1(Ye2¡K)+;TTfT<¿gTsothatitisnaturaltointerpretthiscontractasasurvivalclaimwiththepromisedpayo®X=(Ye2¡K)+.TTodealwiththisoptioninane±cientway,weconsideramodelinwhich¡¢dYi=Yi¹dt+¾dWi¡dM;(4.52)tt¡i;ti;tttwhereW1andW2aretwoone-dimensionalcorrelatedBrownianmotionswiththeinstantaneouscorrelationcoe±cient½.Morespeci¯cally,weassumethatY1=D(t;T)=1De(t;T)representsttft<¿gadefaultableZCBwithzerorecovery,andY2=1Ye2isagenericdefaultableassetwithtotaltft<¿gtdefault.Withinthepresentsetup,thepayo®canalsoberepresentedasfollowsC=G(Y1;Y2)=(Y2¡KY1)+;TTTTTwhereg(y;y)=(y¡Ky)+,andthusitcanalsobeseenasanoptiontoexchangethesecond1221assetforKunitsofthe¯rstasset.TherequirementthattheprocessYe2;1=Ye2(Ye1)¡1followsanF-martingaleunderQeimpliestttthatq¡¢dYe2;1=Ye2;1(¾½¡¾)dWf1+¾1¡½2dWf2;(4.53)tt2;tt1;tt2;tttwhereWf=(Wf1;Wf2)followsatwo-dimensionalBrownianmotionunderQe.SinceYe1=1,replica-Ttionoftheoptionreducesto¯ndingaconstantxandanF-predictableprocessÁ2satisfyingZTx+Á2dYe2;1=(Ye2¡K)+:ttT0Toobtainclosed-formexpressionsfortheoptionpriceandreplicatingstrategy,wepostulatethattheYe2¡1volatilities¾1;t;¾2;tandthecorrelationcoe±cient½taredeterministic.LetFbY2(t;T)=t(De(t;T))(Fb(t;T)=Ce(De(t;T))¡1,respectively)standforthecredit-risk-adjustedforwardpriceofthesec-Ctondasset(theoption,respectively).Theproofofthefollowingvaluationresultisfairlystandard,andthusitisomitted.Proposition4.3.4AssumethatthevolatilitiesaredeterministicandthatY1isaDZC.Thecredit-risk-adjustedforwardpriceoftheoptionwrittenonY2equals¡¢¡¢FbC(t;T)=FbY2(t;T)Nd+(FbY2(t;T);t;T)¡KNd¡(FbY2(t;T);t;T):Equivalently,thepre-defaultpriceoftheoptionequals¡¢¡¢Ce2t=YetNd+(FbY2(t;T);t;T)¡KDe(t;T)Nd¡(FbY2(t;T);t;T);wherelnzf¡lnK§1v2(t;T)d(z;t;T)=2§v(t;T)andZTv2(t;T)=(¾2+¾2¡2½¾¾)du:1;u2;uu1;u2;utMoreoverthereplicatingstrategyÁinthespotmarketsatis¯esforeveryt2[0;T],onthesetft<¿g,¡¢¡¢12Át=¡KNd¡(FbY2(t;T);t;T);Át=Nd+(FbY2(t;T);t;T): 148CHAPTER4.HEDGINGOFDEFAULTABLECLAIMS4.4PDEApproachtoValuationandHedgingIntheremainingpartofthischapter,inwhichwefollowBieleckietal.[6](seealsoRutkowskiandYousiph[81]),weshalltakeadi®erentperspective.Weassumethattradingoccursonthetimeinterval[0;T]andourgoalistoreplicateacontingentclaimoftheformY=1g(Y1;Y2;Y3)+1g(Y1;Y2;Y3)=G(Y1;Y2;Y3;H);fT¸¿g1TTTfT<¿g0TTTTTTTwhichsettlesattimeT.Wedonotneedtoassumeherethatthecoe±cientsindynamicsofprimaryassetsareF-predictable.SinceourgoalistodevelopthePDEapproach,itwillbeessential,however,topostulateaMarkoviancharacterofamodel.Forthesakeofsimplicity,weassumethatthecoe±cientsareconstant,sothat¡¢dYi=Yi¹dt+¾dW+·dM;i=1;2;3:tt¡iititTheassumptionofconstancyofcoe±cientsisrarely,ifever,satis¯edinpracticallyrelevantmodelsofcreditrisk.Itisthusimportanttonotethatitwaspostulatedheremainlyforthesakeofnotationalconvenience,andthegeneralresultsestablishedinthissectioncanbeeasilyextendedtoanon-homogeneousMarkovcaseinwhich¹=¹(t;Y1;Y2;Y3;H);¾=¾(t;Y1;Y2;Y3;H);i;tit¡t¡t¡t¡i;tit¡t¡t¡t¡etc.4.4.1DefaultableAssetwithTotalDefaultWe¯rstassumethatY1andY2aredefault-free,sothat·=·=0,andthethirdassetissubject12tototaldefault,i.e.·3=¡1;¡¢dY3=Y3¹dt+¾dW¡dM:tt¡33ttWeworkthroughoutundertheassumptionsofProposition4.3.1.ThismeansthatanyQ1-integrablecontingentclaimY=G(Y1;Y2;Y3;H)isattainable,anditsarbitragepriceequalsTTTT11¡1¼t(Y)=YtEQ1(Y(YT)jGt);8t2[0;T]:(4.54)Thefollowingauxiliaryresultisthusratherobvious.Lemma4.4.1Theprocess(Y1;Y2;Y3;H)hastheMarkovpropertywithrespecttothe¯ltrationGunderthemartingalemeasureQ1.ForanyattainableclaimY=G(Y1;Y2;Y3;H)thereexistsaTTTTfunctionv:[0;T]£R3£f0;1g!Rsuchthat¼(Y)=v(t;Y1;Y2;Y3;H).tttttWe¯nditconvenienttointroducethepre-defaultpricingfunctionv(¢;0)=v(t;y1;y2;y3;0)andthepost-defaultpricingfunctionv(¢;1)=v(t;y;y;y;1).Infact,sinceY3=0ifH=1,itsu±ces123tttostudythepost-defaultfunctionv(t;y1;y2;1)=v(t;y1;y2;0;1).Also,wewrite¹1¡¹2®i=¹i¡¾i;b=(¹3¡¹1)(¾1¡¾2)¡(¹1¡¹3)(¾1¡¾3):¾1¡¾2Let°>0betheconstantdefaultintensityunderP,andlet³>¡1begivenbyformula(4.28).Proposition4.4.1Assumethatthefunctionsv(¢;0)andv(¢;1)belongtotheclassC1;2([0;T]£R3;R).Thenv(t;y;y;y;0)satis¯esthePDE+123X21X3@tv(¢;0)+®iyi@iv(¢;0)+(®3+³)y3@3v(¢;0)+¾i¾jyiyj@ijv(¢;0)2i=1i;j=1µ¶b£¤¡®1v(¢;0)+°¡v(t;y1;y2;1)¡v(t;y1;y2;y3;0)=0¾1¡¾2 4.4.PDEAPPROACH149subjecttotheterminalconditionv(T;y1;y2;y3;0)=G(y1;y2;y3;0),andv(t;y1;y2;1)satis¯esthePDEX21X2@tv(¢;1)+®iyi@iv(¢;1)+¾i¾jyiyj@ijv(¢;1)¡®1v(¢;1)=02i=1i;j=1subjecttotheterminalconditionv(T;y1;y2;1)=G(y1;y2;0;1):Proof.Forsimplicity,wewriteCt=¼t(Y).Letusde¯ne¢v(t;y1;y2;y3)=v(t;y1;y2;1)¡v(t;y1;y2;y3;0):Thenthejump¢Ct=Ct¡Ct¡canberepresentedasfollows:¡¢¢C=1v(t;Y1;Y2;1)¡v(t;Y1;Y2;Y3;0)=1¢v(t;Y1;Y2;Y3):tf¿=tgttttt¡f¿=tgttt¡Wewrite@itodenotethepartialderivativewithrespecttothevariableyi,andwetypicallyomitthevariables(t;Y1;Y2;Y3;H)inexpressions@v;@v;¢v,etc.Weshallalsomakeuseofthet¡t¡t¡t¡tifactthatforanyBorelmeasurablefunctiongwehaveZtZtg(u;Y2;Y3)du=g(u;Y2;Y3)duuu¡uu00sinceY3andY3di®eronlyforatmostonevalueofu(foreach!).Let»=1°.Anapplicationuu¡tft<¿gofIt^o'sformulayieldsX31X3dC=@vdt+@vdYi+¾¾YiYj@vdtttit2ijt¡t¡iji=1i;j=1³´+¢v+Y3@vdHt¡3tX31X3=@vdt+@vdYi+¾¾YiYj@vdttit2ijt¡t¡iji=1i;j=1³´¡¢+¢v+Y3@vdM+»dt;t¡3ttandthisinturnimpliesthatX3¡¢1X3dC=@vdt+Yi@v¹dt+¾dW+¾¾YiYj@vdtttt¡iiit2ijt¡t¡iji=1i;j=1³´+¢vdM+¢v+Y3@v»dttt¡3t89¡1and·6=0.WeassumethatY3>0,sothatY3>0foreveryt2R.Weshall330t+brie°ydescribethesamestepsasinthecaseofadefaultableassetwithtotaldefault. 4.4.PDEAPPROACH153PricingPDEandReplicatingStrategyWeareinapositiontoderivethepricingPDEs.Forthesakeofsimplicity,weassumethatY1isthesavingsaccount,sothatProposition4.4.3isacounterpartofCorollary4.4.2.FortheproofofProposition4.4.3,theinterestedreaderisreferredtoBieleckietal.[6].Proposition4.4.3Let¾6=0andletY1;Y2;Y3satisfy2dY1=rY1dt;tt¡¢dY2=Y2¹dt+¾dW;tt22t¡¢dY3=Y3¹dt+¾dW+·dM:tt¡33t3tAssume,inaddition,that¾2(r¡¹3)=¾3(r¡¹2)and·36=0;·3>¡1.ThenthepriceofacontingentclaimY=G(Y2;Y3;H)canberepresentedas¼(Y)=v(t;Y2;Y3;H),wheretheTTTttttpricingfunctionsv(¢;0)andv(¢;1)satisfythefollowingPDEs@tv(t;y2;y3;0)+ry2@2v(t;y2;y3;0)+y3(r¡·3°)@3v(t;y2;y3;0)¡rv(t;y2;y3;0)X3¡¢1+¾i¾jyiyj@ijv(t;y2;y3;0)+°v(t;y2;y3(1+·3);1)¡v(t;y2;y3;0)=02i;j=2and@tv(t;y2;y3;1)+ry2@2v(t;y2;y3;1)+ry3@3v(t;y2;y3;1)¡rv(t;y2;y3;1)X31+¾i¾jyiyj@ijv(t;y2;y3;1)=02i;j=2subjecttotheterminalconditionsv(T;y2;y3;0)=G(y2;y3;0);v(T;y2;y3;1)=G(y2;y3;1):ThereplicatingstrategyÁequalsX32123Át=2¾iyi@iv(t;Yt;Yt¡;Ht¡)¾2Yti=2¾3¡2323¢¡2v(t;Yt;Yt¡(1+·3);1)¡v(t;Yt;Yt¡;0);¾2·3Yt1¡¢Á3=v(t;Y2;Y3(1+·);1)¡v(t;Y2;Y3;0);t·Y3tt¡3tt¡3t¡andÁ1isgivenbyÁ1Y1+Á2Y2+Á3Y3=C.ttttttttHedgingofaSurvivalClaimWeshallillustrateProposition4.4.3bymeansofexamples.First,considerasurvivalclaimoftheformY=G(Y2;Y3;H)=1g(Y3):TTTfT<¿gTThenthepost-defaultpricingfunctionvg(¢;1)vanishesidentically,andthepre-defaultpricingfunc-tionvg(¢;0)solvesthePDE@vg(¢;0)+ry@vg(¢;0)+y(r¡·°)@vg(¢;0)t22333X31gg+¾i¾jyiyj@ijv(¢;0)¡(r+°)v(¢;0)=02i;j=2 154CHAPTER4.HEDGINGOFDEFAULTABLECLAIMSwiththeterminalconditionvg(T;y;y;0)=g(y).Denote®=r¡·°and¯=°(1+·).23333Itisnotdi±culttocheckthatvg(t;y;y;0)=e¯(T¡t)v®;g;3(t;y)isasolutionoftheabove233equation,wherethefunctionw(t;y)=v®;g;3(t;y)isthesolutionofthestandardBlack-ScholesPDEequation122@tw+y®@yw+¾3y@yyw¡®w=02withtheterminalconditionw(T;y)=g(y),thatis,thepriceofthecontingentclaimg(YT)intheBlack-Scholesframeworkwiththeinterestrate®andthevolatilityparameterequalto¾3.LetCtbethecurrentvalueofthecontingentclaimY,sothatC=1e¯(T¡t)v®;g;3(t;Y3):tft<¿gtThehedgingstrategyofthesurvivalclaimis,ontheeventft<¿g,331¡¯(T¡t)®;g;331ÁtYt=¡ev(t;Yt)=¡Ct;·3·3³´22¾33¡¯(T¡t)®;g;3333ÁtYt=Yte@yv(t;Yt)¡ÁtYt:¾2HedgingofaRecoveryPayo®AsanotherillustrationofProposition4.4.3,weshallnowconsidertheclaimG(Y2;Y3;H)=TTT1g(Y2),thatis,weassumethatrecoveryispaidatmaturityandequalsg(Y2).LetvgbefT¸¿gTTthepricingfunctionofthisclaim.Thepost-defaultpricingfunctionvg(¢;1)doesnotdependony.3Indeed,theequation(wewriteherey2=y)gg122gg@tv(¢;1)+ry@yv(¢;1)+¾2y@yyv(¢;1)¡rv(¢;1)=0;2withvg(T;y;1)=g(y),admitsauniquesolutionvr;g;2,whichisthepriceofg(Y)intheBlack-TScholesmodelwiththeinterestraterandthevolatility¾2.Priortodefault,thepriceoftheclaimcanbefoundbysolvingthefollowingPDE@vg(¢;0)+ry@vg(¢;0)+y(r¡·°)@vg(¢;0)t22333X31ggg+¾i¾jyiyj@ijv(¢;0)¡(r+°)v(¢;0)=¡°v(t;y2;1)2i;j=2withvg(T;y;y;0)=0.Itisnotdi±culttocheckthat23vg(t;y;y;0)=(1¡e°(t¡T))vr;g;2(t;y):232ThereadercancomparethisresultwiththeoneofExample4.4.1.WenowassumethatdY3=Y3(¹dt+¾dW+·dM)tt¡33t3twith·>¡1and·6=0.WeassumethatY3>0,sothatY3>0foreveryt2R.Weshall330t+brie°ydescribethesamestepsasinthecaseofadefaultableassetwithtotaldefault.Arbitrage-FreePropertyAsusual,weneed¯rsttoimposespeci¯cconstraintsonmodelcoe±cients,sothatthemodelisarbitrage-free.Indeed,anEMMQ1existsifthereexistsapair(µ;³)suchthat,fori=2;3,·i¡·1·1µt(¾i¡¾1)+³t»t=¹1¡¹i+¾1(¾i¡¾1)+»t(·i¡·1):1+·11+·1 4.4.PDEAPPROACH155Toensuretheexistenceofasolution(µ;³)ontheeventft>¿g,weimposethecondition¹1¡¹2¹1¡¹3¾1¡=¾1¡;¾1¡¾2¾1¡¾3thatis,¹1(¾3¡¾2)+¹2(¾1¡¾3)+¹3(¾2¡¾1)=0:Now,ontheeventft·¿g,wehavetosolvethetwoequationsµt(¾2¡¾1)=¹1¡¹2+¾1(¾2¡¾1);µt(¾3¡¾1)+³t°·3=¹1¡¹3+¾1(¾3¡¾1):If,inaddition,(¾2¡¾1)·36=0,weobtaintheuniquesolution¹1¡¹2¹1¡¹3µ=¾1¡=¾1¡;¾1¡¾2¾1¡¾3³=0>¡1;sothatthemartingalemeasureQ1existsandisunique.4.4.3TwoDefaultableAssetswithTotalDefaultWeshallnowassumethatwehaveonlytwoassets,andbotharedefaultableassetswithtotaldefault.Weshallbrie°youtlinetheanalysisofthiscase,leavingthedetailsandthestudyofotherrelevantcasestothereader.Wepostulatethat¡¢dYi=Yi¹dt+¾dW¡dM;i=1;2;(4.56)tt¡iittsothatYi=1Yei;i=1;2,withthepre-defaultpricesgovernedbytheSDEstft<¿gt¡¢dYei=Yei(¹+°)dt+¾dW;i=1;2:ttiitInthecasewherethepromisedpayo®Xispath-independent,sothatX1=G(Y1;Y2)1=G(Ye1;Ye2)1fT<¿gTTfT<¿gTTfT<¿gforsomefunctionG,itispossibletousethePDEapproachinordertovalueandreplicatesurvivalclaimspriortodefault(needlesstosaythatthevaluationandhedgingafterdefaultaretrivialhere).WeknowalreadyfromthemartingaleapproachthathedgingofasurvivalclaimX1fT<¿gisformallyequivalenttoreplicatingthepromisedpayo®Xusingthepre-defaultvaluesoftradeableassets¡¢dYei=Yei(¹+°)dt+¾dW;i=1;2:ttiitWeneednottoworryhereaboutthebalancecondition,sinceincaseofdefaultthewealthoftheportfoliowilldroptozero,asitshouldinviewoftheequalityZ=0.Weshall¯ndthepre-defaultpricingfunctionv(t;y1;y2),whichisrequiredtosatisfytheterminalconditionv(T;y;y)=G(y;y),aswellasthehedgingstrategy(Á1;Á2).Thereplicatingstrategy1212Áissuchthatforthepre-defaultvalueCeofourclaimwehaveCe:=v(t;Ye1;Ye2)=Á1Ye1+Á2Ye2;tttttttanddCe=Á1dYe1+Á2dYe2:(4.57)ttttt 156CHAPTER4.HEDGINGOFDEFAULTABLECLAIMSProposition4.4.4Assumethat¾16=¾2.Thenthepre-defaultpricingfunctionvsatis¯esthePDEµ¶µ¶¹2¡¹1¹2¡¹1@tv+y1¹1+°¡¾1@1v+y2¹2+°¡¾2@2v¾2¡¾1¾2¡¾1³´µ¶12222¹2¡¹1+y1¾1@11v+y2¾2@22v+2y1y2¾1¾2@12v=¹1+°¡¾1v2¾2¡¾1withtheterminalconditionv(T;y1;y2)=G(y1;y2):Proof.Weshallmerelysketchtheproof.ByapplyingIt^o'sformulatov(t;Ye1;Ye2),andcomparingttthedi®usiontermsin(4.57)andintheIt^odi®erentialdv(t;Ye1;Ye2),we¯ndthattty¾@v+y¾@v=Á1y¾+Á2y¾;(4.58)1112221122whereÁi=Ái(t;y;y).SinceÁ1y=v(t;y;y)¡Á2y,wededucefrom(4.58)that121122y¾@v+y¾@v=v¾+Á2y(¾¡¾);1112221221andthus2y1¾1@1v+y2¾2@2v¡v¾1Áy2=:¾2¡¾1Ontheotherhand,byidenti¯cationofdrifttermsin(4.58),weobtain@tv+y1(¹1+°)@1v+y2(¹2+°)@2v³´1222212+y1¾1@11v+y2¾2@22v+2y1y2¾1¾2@12v=Áy1(¹1+°)+Áy2(¹2+°):2UponeliminationofÁ1andÁ2,wearriveatthestatedPDE.¤Recallthatthehistoricallyobserveddrifttermsare¹bi=¹i+°,ratherthan¹i.ThepricingPDEcanthusbesimpli¯edasfollows:µ¶µ¶¹b2¡¹b1¹b2¡¹b1@tv+y1¹b1¡¾1@1v+y2¹b2¡¾2@2v¾2¡¾1¾2¡¾1³´µ¶12222¹b2¡¹b1+y1¾1@11v+y2¾2@22v+2y1y2¾1¾2@12v=v¹b1¡¾1:2¾2¡¾1Thepre-defaultpricingfunctionvdependsonthemarketobservables(driftcoe±cients,volatilities,andpre-defaultprices),butnotonthe(deterministic)defaultintensity.Tomakeonemoresimplifyingstep,wemakeanadditionalassumptionaboutthepayo®function.Suppose,inaddition,thatthepayo®functionissuchthatG(y1;y2)=y1g(y2=y1)forsomefunctiong:R+!R(orequivalently,G(y1;y2)=y2h(y1=y2)forsomefunctionh:R+!R).Thenwemayfocusonrelativepre-defaultpricesCb=Ce(Ye1)¡1andYe2;1=Ye2(Ye1)¡1.Thecorrespondingttttt2;1pre-defaultpricingfunctionvb(t;z),suchthatCbt=vb(t;Yt)willsatisfythePDE122@tvb+(¾2¡¾1)z@zzvb=02withterminalconditionvb(T;z)=g(z).IfthepriceprocessesY1andY2in(4.49)aredrivenbythecorrelatedBrownianmotionsWandWcwiththeconstantinstantaneouscorrelationcoe±cient½,thenthePDEbecomes1222@tvb+(¾2+¾1¡2½¾1¾2)z@zzvb=0:2Consequently,thepre-defaultpriceCe=Ye1vb(t;Ye2;1)willnotdependdirectlyonthedriftcoe±cientsttt¹b1and¹b2,andthus,inprinciple,weshouldbeabletoderiveanexpressionthepriceoftheclaimintermsofmarketobservables:thepricesoftheunderlyingassets,theirvolatilitiesandthecorrelationcoe±cient.Putanotherway,neitherthedefaultintensitynorthedriftcoe±cientsoftheunderlyingassetsappearasindependentparametersinthepre-defaultpricingfunction. Chapter5DependentDefaultsandCreditMigrationsModelingofdependentdefaultsisthemostimportantandchallengingresearchareawithregardtocreditriskandcreditderivatives.Wedescribethecaseofconditionallyindependentdefaulttime,theindustrystandardcopula-basedapproach,aswellastheJarrowandYu[56]approachtothemodelingofdefaulttimeswithdependentstochasticintensities.Weconcludebysummarizingoneoftheapproachesthatwererecentlydevelopedforthepurposeofmodelingjointcreditratingsmigrationsforseveral¯rms.Itshouldbeacknowledgedthatseveralothermethodsofmodelingdependentdefaultsproposedintheliteraturearenotcoveredbythistext.Letusstartbyprovidingatentativeclassi¯cationofissuesandtechniquesrelatedtodependentdefaultsandcreditratings.Valuationofbasketcreditderivativescovers,inparticular:²defaultswapsoftypeF(Du±e[40],KijimaandMuromachi[62]){theyprovideaprotectionagainstthe¯rstdefaultinabasketofdefaultableclaims,²defaultswapsoftypeD(KijimaandMuromachi[62]){aprotectionagainstthe¯rsttwodefaultsinabasketofdefaultableclaims,²theith-to-defaultclaims(BieleckiandRutkowski[14]){aprotectionagainstthe¯rstidefaultsinabasketofdefaultableclaims.Technicalissuesarisinginthecontextofdependentdefaultsinclude:²conditionalindependenceofdefaulttimes(KijimaandMuromachi[62]),²simulationofcorrelateddefaults(Du±eandSingleton[43]),²modelingofinfectiousdefaults(DavisandLo[35]),²asymmetricdefaultintensities(JarrowandYu[56]),²copulas(LaurentandGregory[66],SchÄonbucherandSchubert[83]),²dependentcreditratings(Lando[64],BieleckiandRutkowski[13]),²simulationofdependentcreditmigrations(Kijimaetal.[61],Bielecki[2]),²simulationofcorrelateddefaultsviaMarshall-Olkincopula(Elouerkhaoui[47]).157 158CHAPTER5.DEPENDENTDEFAULTS5.1BasketCreditDerivativesBasketcreditderivativesarecreditderivativesderivingtheircash°owsvalues(andthustheirvalues)fromcreditrisksofseveralreferenceentities(orprespeci¯edcreditevents).Standingassumptions.Weassumethat:²wearegivenacollectionofdefaulttimes¿1;:::;¿nde¯nedonacommonprobabilityspace(•;G;Q);²Qf¿i=0g=0andQf¿i>tg>0foreveryiandt,²Qf¿i=¿jg=0forarbitraryi6=j(inacontinuoustimesetup).Weassociatewiththecollection¿1;:::;¿nofdefaulttimestheorderedsequence¿(1)<¿(2)<¢¢¢<¿;where¿standsfortherandomtimeoftheithdefault.Formally,(n)(i)¿(1)=minf¿1;¿2;:::;¿ngandfori=2;:::;n©ª¿(i)=min¿k:k=1;:::;n;¿k>¿(i¡1):Inparticular,¿(n)=maxf¿1;¿2;:::;¿ng:5.1.1Theith-to-DefaultContingentClaimsWesetHi=1andwedenotebyHithe¯ltrationgeneratedbytheprocessHi,thatis,bythetf¿i·tgobservationsofthedefaulttime¿i:Inaddition,wearegivenareference¯ltrationFonthespace(•;G;Q):The¯ltrationFisrelatedtosomeothermarketrisks,forinstance,totheinterestraterisk.Finally,weintroducetheenlarged¯ltrationGbysettingG=F_H1_H2_:::_Hn:The¾-¯eldGtmodelstheinformationavailableattimet:Ageneralith-to-defaultcontingentclaimwhichmaturesattimeTisspeci¯edbythefollowingcovenants:²if¿=¿·Tforsomek=1;:::;nthentheclaimpaysattime¿theamountZk,where(i)k(i)¿(i)ZkisanF-predictablerecoveryprocess,²if¿(i)>TthentheclaimpaysattimeTanFT-measurablepromisedamountX:5.1.2CaseofTwoEntitiesForthesakeofnotationalsimplicity,weshallfrequentlyconsiderthecaseoftworeferencecreditrisks.Cash°owsofthe¯rst-to-defaultcontract(FTDC):²if¿=minf¿;¿g=¿·Tfori=1;2;theclaimpaysattime¿theamountZi,(1)12ii¿i²ifminf¿1;¿2g>T;itpaysattimeTtheamountX:Cash°owsofthelast-to-defaultcontract(LTDC):²if¿=maxf¿;¿g=¿·Tfori=1;2;theclaimpaysattime¿theamountZi,(2)12ii¿i²ifmaxf¿1;¿2g>T;itpaysattimeTtheamountX: 5.2.CONDITIONALLYINDEPENDENTDEFAULTS159WerecallthatthroughouttheselecturesthesavingsaccountBequals³Zt´Bt=exprudu;0andQstandsforthemartingalemeasureforourmodelofthe¯nancialmarket(includingdefaultablesecurities,suchas:corporatebondsandcreditderivatives).Consequently,thepriceB(t;T)ofazero-coupondefault-freebondequals¡¢¡¢¡1¡1B(t;T)=BtEQBTjGt=BtEQBTjFt:ValuesofFTDCandLTDCIngeneral,thevalueattimetofadefaultableclaim(X;Z;¿)isgivenbytherisk-neutralvaluationformula³Z¯´¡1¯St=BtEQBudDu¯Gt]t;T]whereDisthedividendprocess,whichdescribesallthecash°owsoftheclaim.Consequently,thevalueattimetoftheFTDCequals:³¯´(1)¡11¯St=BtEQB¿1Z¿11f¿1<¿2;t<¿1·Tg¯Gt³¯´¡12¯+BtEQB¿2Z¿21f¿2<¿1;t<¿2·Tg¯Gt³¯´¡1¯+BtEQBTX1fT<¿(1)g¯Gt:ThevalueattimetoftheLTDCequals:³¯´(2)¡11¯St=BtEQB¿1Z¿11f¿2<¿1;t<¿1·Tg¯Gt!³¯¡12¯+BtEQB¿2Z¿21f¿1<¿2;t<¿2·Tg¯Gt³¯´¡1¯+BtEQBTX1fT<¿(2)g¯Gt:Bothexpressionsabovearemerelyspecialcasesofageneralformula.Thegoalistoderivemoreexplicitrepresentationsundervariousassumptionsabout¿1and¿2;ortoprovidewaysofe±cientcalculationofinvolvedexpectedvaluesbymeansofsimulation(usingperhapsanotherprobabilitymeasure).5.1.3RoleoftheHypothesis(H)Ifoneassumesthat(H)hypothesisholdsbetweenthe¯ltrationsFandG,then,itholdsbetweenthe¯ltrationsFandF_Hi1_¢¢¢_Hikforanyi;:::;i.However,thereisnoreasonforthehypothesis1k(H)toholdbetweenF_Hi1andG.Notethat,if(H)holdsthenonehas,fort·:::·t·T,1nQ(¿1>t1;:::;¿n>tnjFT)=Q(¿1>t1;:::;¿n>tnjF1):5.2ConditionallyIndependentDefaultsDe¯nition5.2.1Therandomtimes¿i;i=1;:::;naresaidtobeconditionallyindependentwithrespecttoFunderQifwehave,foranyT>0andanyt1;:::;tn2[0;T],YnQf¿1>t1;:::;¿n>tnjFTg=Qf¿i>tijFTg:i=1 160CHAPTER5.DEPENDENTDEFAULTSLetuscommentbrie°yonDe¯nition5.2.1.²Conditionalindependencehasthefollowingintuitiveinterpretation:thereferencecredits(creditnames)aresubjecttocommonriskfactorsthatmaytriggercredit(default)events.Inaddition,eachcreditnameissubjecttoidiosyncraticrisksthatarespeci¯cforthisname.²Conditionalindependenceofdefaulttimesmeansthatoncethecommonriskfactorsare¯xedthentheidiosyncraticriskfactorsareindependentofeachother.²Thepropertyofconditionalindependenceisnotinvariantwithrespecttoanequivalentchangeofaprobabilitymeasure.²Conditionalindependence¯tsintostaticanddynamictheoriesofdefaulttimes.²Astrongerconditionwouldbeafullconditionallyindependence,i.e.,foranyT>0andanyintervalsI1;:::;Inwehave:YnQ(¿12I1;:::;¿n2InjFT)=Q(¿i2IijFT):i=15.2.1CanonicalConstructionLet¡i;i=1;:::;nbeagivenfamilyofF-adapted,increasing,continuousprocesses,de¯nedonaprobabilityspace(•~;F;P~):Weassumethat¡i=0and¡i=1:Let(•^;F^;P^)beanauxiliaryprob-01abilityspacewithasequence»i;i=1;:::;nofmutuallyindependentrandomvariablesuniformlydistributedon[0;1]:Weset¿(~!;!^)=infft2R:¡i(~!)¸¡ln»(^!)gi+tiontheproductprobabilityspace(•;G;Q)=(•~£•^;F1•F^;P~•P^):Weendowthespace(•;G;Q)withthe¯ltrationG=F_H1_¢¢¢_Hn:Proposition5.2.1Assumethattherandomvariables»1;:::;»nareindependentandidenticallydistributed.Thentheprocess¡iistheF-hazardprocessof¿equalsi¡ii¢Qf¿>sjF_Hig=1Ee¡t¡¡sjF:ittf¿i>tgQtWehaveQf¿i=¿jg=0foreveryi6=j:Moreover,thedefaulttimes¿1;:::;¿nareconditionallyindependentwithrespecttoFunderQ:Proof.Itsu±cestonotethat,fortit;:::;¿>tjF)=Q(¡1¸¡ln»;:::;¡n¸¡ln»jF)=e¡¡ti:11nnTt11tnnTi=1Thedetailsarelefttothereader.¤RtRecallthatif¡i=°iduthen°iistheF-intensityof¿:Intuitivelyt0uiQf¿2[t;t+dt]jF_Hig¼1°idt:ittf¿i>tgt5.2.2IndependentDefaultTimesWeshall¯rstexaminethecaseofdefaulttimes¿1;:::;¿nthataremutuallyindependentunderQ:Supposethatforeveryk=1;:::;nweknowthecumulativedistributionfunctionFk(t)=Qf¿k·tgofthedefaulttimeofthekthreferenceentity.Thecumulativedistributionfunctionsof¿and¿(1)(n)are:YnF(1)(t)=Qf¿(1)·tg=1¡(1¡Fk(t))k=1 5.2.CONDITIONALLYINDEPENDENTDEFAULTS161andYnF(n)(t)=Qf¿(n)·tg=Fk(t):k=1Moregenerally,foranyi=1;:::;nwehaveXnXYYF(i)(t)=Qf¿(i)·tg=Fkj(t)(1¡Fkl(t));m=i¼2¦mj2¼l62¼where¦mdenotethefamilyofallsubsetsoff1;:::;ngconsistingofmelements.Suppose,inaddition,thatthedefaulttimes¿1;:::;¿nadmitdeterministicintensityfunctions°1(t);:::;°n(t);suchthatZt^¿iHi¡°(s)dsti0Rti¡°i(v)dvareH-martingales.RecallthatQf¿i>tg=e0.Itiseasilyseenthat,foranyt2R+,YRtQf¿>tg=Qf¿>tg=e¡0°(1)(v)dv;(1)iwhere°(1)(t)=°1(t)+:::+°n(t)henceZt^¿(1)(1)Ht¡°(1)(t)dt0isanH(1)-martingale,whereH(1)=¾(¿^t).Bydirectcalculations,itisalsopossibleto¯ndthet(1)intensityfunctionoftheithdefaulttime.Example5.2.1Weshallconsideradigitaldefaultputofbaskettype.Tobemorespeci¯c,wepostulatethatacontractpaysa¯xedamount(e.g.,oneunitofcash)attheithdefaulttime¿(i)providedthat¿(i)·T:Assumethattheinterestratesarenon-random.Thenthevalueattime0ofthecontractequalsZ¡¢S=EB¡11=B¡1dF(u):0Q¿f¿(i)·Tgu(i)]0;T]If¿1;:::;¿nadmitintensitiesthenZTZTRuS=B¡1dF(u)=B¡1°(u)e¡0°(i)(v)dvdu:0u(i)u(i)005.2.3SignedIntensitiesSomeauthors(e.g.,KijimaandMuromachi[62])examinecreditriskmodelsinwhichthenegativevaluesof"intensities"arenotprecluded.Inthatcase,theprocesschosenasthe"intensity"doesRt^¿notplaytheroleofarealintensity,inparticular,itisnottruethatHt¡0°tdtisamartingaleandnegativevaluesofthe"intensity"processclearlycontradicttheinterpretationoftheintensityastheconditionalprobabilityofsurvivaloveranin¯nitesimaltimeinterval.Moreprecisely,foragivencollection¡i;i=1;:::;nofF-adaptedcontinuousstochasticprocesses,with¡i=0;de¯ned0on(•b;F;Pb):onecande¯ne¿i;i=1;:::;n;ontheenlargedprobabilityspace(•;G;Q):¿=infft2R:¡i(^!)¸¡ln»(^!)g:i+tiLetusdenote¡^i=max¡i:Observethatiftheprocess¡iisabsolutelycontinuous,thansoittu·tutheprocess¡^i;inthiscasetheintensityof¿isobtainedasthederivativeof¡^iwithrespecttotheitimevariable.Thefollowingresultexaminesthecaseofsignedintensities. 162CHAPTER5.DEPENDENTDEFAULTSLemma5.2.1Randomtimes¿i;i=1;:::;nareconditionallyindependentwithrespecttoFunderQ:Inparticular,foreveryt1;:::;tn·T;YnP¡¡^i¡n¡^iQf¿>t;:::;¿>tjFg=eti=ei=1ti:11nnTi=15.2.4ValuationofFTDCandLTDCValuationofa¯rst-to-defaultclaim(FTDC)oralast-to-defaultclaim(LTDC)isrelativelystraight-forwardundertheassumptionofconditionalindependenceofdefaulttimes.Wehavethefollowingresultinwhich,fornotationalsimplicity,weconsideronlythecaseoftwoentities.Asusual,wedonotstateexplicitlyintegrabilityconditionsthatshouldbeimposedonrecoveryprocessesZjandtheterminalpayo®X:Proposition5.2.2Letthedefaulttimes¿j;j=1;2beF-conditionallyindependentwithF-intensitiesjiRt^¿iiii°,thatis,Ht¡0°sdsareG-martingalesand°isFadapted.AssumethattherecoveryZisanF-predictableprocess,andthattheterminalpayo®XisFT-measurable.(i)Ifthehypothesis(H)holdsbetweenFandG,thenthepriceattimet=0ofthe¯rst-to-defaultclaimequalsX2³ZT´¡¢(1)¡1j¡¡ij¡¡j¡1S0=EQBuZueu°ueudu+EQBXG;T0i;j=1;i6=jwherewedenote12G=e¡(¡T+¡T)=Q(¿>T;¿>TjF):12T(ii)Inthegeneralcase,settingFi=Q(¿·tjF)=Zi+Ai,whereZiisanFmartingale,wehavetitttthat(1)³ZT¡12¢´¡¢¡(¡+¡)1212¡1S0=EQZueuu(°u+°u)du+dhZ;Ziu+EQBXG:T0Proof.WeneedtocomputeEQ(Z¿1f¿u;¿>ujF)=Q(¿>ujF)Q(¿>ujF)=(1¡F1)(1¡F2):u12u1u2uuu²Ifweassumethatthehypothesis(H)holdsbetweenFandGi,fori=1;2,theprocessesFiareincreasing,andthus1212dF=e¡¡udF2+e¡¡udF1=e¡¡ue¡¡u(°1+°2)du:uuuuuItfollowsthat³ZT´12E(Z1)=EZe¡¡ue¡¡u(°1+°2)du:Q¿1^¿2f¿1^¿2tg=feti>»g.However,wedonotassumethatthe»arei.i.d.andwedenotebyiiikCtheircopula.Then:²ThecaseofdefaulttimesconditionallyindependentwithrespecttoFcorrespondstothechoiceoftheproductcopula¦:Inthiscase,fort1;:::;tn·TwehaveQf¿>t;:::;¿>tjFg=¦(Z1;:::;Zn);11nnTt1tniwherewesetZi=e¡¡t.t²Ingeneral,fort1;:::;tn·TweobtainQf¿>t;:::;¿>tjFg=C(Z1;:::;Zn);11nnTt1tnwhereCisthecopulausedintheconstructionof»1;:::;»n: 164CHAPTER5.DEPENDENTDEFAULTSSurvivalIntensitiesWefollowhereSchÄonbucherandSchubert[83].Proposition5.3.1Forarbitrarys·tonthesetf¿1>s;:::;¿n>sgwehaveµ1in¯¶C(Zs;:::;Zt;:::;Zs)¯Qf¿i>tjGsg=EQC(Z1;:::;Zn)¯Fs:ssProof.Theproofisratherstraightforward.WehaveQ(¿1>s;:::;¿i>t;:::;¿n>sjFs)Qf¿i>tjGsg1f¿1>s;:::;¿n>sg=1f¿1>s;:::;¿n>sg;Q(¿1>s;:::;¿i>s;:::;¿n>sjFs)whereweusedthekeylemma.¤iUndertheassumptionthatthederivatives°i=d¡texist,theithintensityofsurvivalequals,ontdtthesetf¿1>t;:::;¿n>tg,@C(Z1;:::;Zn)iii@vittii@1n¸t=°tZt1n=°tZtlnC(Zt;:::;Zt);C(Zt;:::;Zt)@viwhere¸iisunderstoodasthefollowinglimitt¸i=limh¡1Qft<¿·t+hjF;¿>t;:::;¿>tg:tit1nh#0Itappearsthat,ingeneral,theithintensityofsurvivaljumpsattimet,ifthejthentitydefaultsattimetforsomej6=i:Infact,itholdsthat2@C(Z1;:::;Zn)i;jii@vi@vjtt¸t=°tZt@1n;@vjC(Zt;:::;Zt)where¸i;j=limh¡1Qft<¿·t+hjF;¿>t;k6=j;¿=tg:titkjh#0SchÄonbucherandSchubert[83]examinealsotheintensitiesofsurvivalafterthedefaulttimesofsomeentities.Letus¯xs,andletti·sfori=1;2;:::;kTi;i=k+1;k+2;:::;njFs;¿j=tj;j=1;2;:::;k;ª¿i>s;i=k+1;k+2;:::;n³¯´@k1kk+1n¯EQ@v1:::@vkC(Zt1;:::;Ztk;ZTk+1;:::;ZTn)¯Fs=:(5.1)@kC(Z1;:::;Zk;Zk+1n)@v1:::@vkt1tks;:::;ZsRemark5.3.1Thejumpsofintensitiescannotbee±cientlycontrolled,exceptforthechoiceofC.Intheapproachdescribedabove,thedependencebetweenthedefaulttimesisimplicitlyintroducedthrough¡is,andexplicitlyintroducedbythechoiceofacopulaC.LaurentandGregoryModelLaurentandGregory[66]examineasimpli¯edversionoftheframeworkofSchÄonbucherandSchubert[83].Namely,theyassumethatthereference¯ltrationistrivial{thatis,Ft=f•;;gforeveryt2R.Themarginaldefaultintensities°^iaredeterministicfunctions,and+Rti¡°^(u)duQ(¿i>t)=1¡Fi(t)=e0:Theyobtainclosed-formexpressionsforcertainconditionalintensitiesofdefault. 5.4.JARROWANDYUMODEL165Example5.3.1Thisexampledescribestheuseoftheone-factorGaussiancopulamodel,whichistheBIS(BankofInternationalSettlements)standard.LetpXi=½V+1¡½2Vi;whereV;Vi;i=1;2;:::;n;areindependent,standardGaussianvariablesunderQand½2(¡1;1).De¯neZnto¿=inft2R:°^i(u)du>¡ln»=infft2R:1¡F(t)<»gi+i+ii0wheretherandombarriersarede¯nedas»i=1¡N(Xi).Asusual,NstandsforthecumulativedistributionfunctionofastandardGaussianrandomvariable.Thenthefollowingequalitiesholdn¡1oN(Fi(t))¡½Vf¿i·tg=f»i¸1¡Fi(t)g=Xi·p:1¡½2ijVijVijVDe¯neqt=Q(¿i>tjV)andpt=1¡qt.ThenZYnijvQf¿1·t1;:::;¿n·tng=ptin(v)dvRi=1wherenistheprobabilitydensityfunctionofV.ItiseasytoseethatÃ!N¡1(F(t))¡½VijVipt=Np1¡½2andthus,byvirtueoftheconditionalindependenceofX1;:::;XnwithrespecttoV,ZÃ!YnN¡1(F(t))¡½viiQf¿1·t1;:::;¿n·tng=Npn(v)dv:R1¡½2i=15.4JarrowandYuModelJarrowandYu[56]approachcanbeconsideredasanothersteptowardsadynamictheoryofdepen-dencebetweendefaulttimes.Foragiven¯nitefamilyofreferencecreditnames,JarrowandYu[56]proposetomakeadistinctionbetweentheprimary¯rmsandthesecondary¯rms.Attheintuitivelevel:²Theclassofprimary¯rmsencompassestheseentitieswhoseprobabilitiesofdefaultarein°u-encedbymacroeconomicconditions,butnotbythecreditriskofcounterparties.Thepricingofbondsissuedbyprimary¯rmscanbedonethroughthestandardintensity-basedmethodology.²Itsu±cestofocusonsecuritiesissuedbysecondary¯rms,thatis,¯rmsforwhichtheintensityofdefaultdependsonthestatusofsomeother¯rms.Formally,theconstructionisbasedontheassumptionofasymmetricinformation.Unilateraldependenceisnotpossibleinthecaseofcomplete(i.e.,symmetric)information.5.4.1ConstructionandPropertiesoftheModelLetf1;:::;ngrepresentthesetofall¯rms,andletFbethereference¯ltration.Wepostulatethat:²Forany¯rmfromthesetf1;:::;kgofprimary¯rms,the`defaultintensity'dependsonlyonF.²The`defaultintensity'ofeach¯rmbelongingtothesetfk+1;:::;ngofsecondary¯rmsmaydependnotonlyonthe¯ltrationF;butalsoonthestatus(defaultorno-default)oftheprimary¯rms. 166CHAPTER5.DEPENDENTDEFAULTSConstructionofDefaultTimes¿1;:::;¿nFirststep.We¯rstmodeldefaulttimesofprimary¯rms.Tothisend,weassumethatwearegivenafamilyofF-adapted`intensityprocesses'¸1;:::;¸kandweproduceacollection¿;:::;¿of1kF-conditionallyindependentrandomtimesthroughthecanonicalmethod:Zt©ª¿=inft2R:¸idu¸¡ln»i+ui0where»i;i=1;:::;karemutuallyindependentidenticallydistributedrandomvariableswithuni-formlawon[0;1]underthemartingalemeasureQ:Secondstep.Wenowconstructdefaulttimesofsecondary¯rms.Weassumethat:²Theprobabilityspace(•;G;Q)islargeenoughtosupportafamily»i;i=k+1;:::;nofmutuallyindependentrandomvariables,withuniformlawon[0;1].²Theserandomvariablesareindependentnotonlyofthe¯ltrationF;butalsoofthealreadyconstructedinthe¯rststepdefaulttimes¿1;:::;¿kofprimary¯rms.Thedefaulttimes¿i;i=k+1;:::;narealsode¯nedbymeansofthestandardformula:Zt©ª¿=inft2R:¸idu¸¡ln»:i+ui0However,the`intensityprocesses'¸ifori=k+1;:::;narenowgivenbythefollowingexpression:Xk¸i=¹i+ºi;l1;tttf¿l·tgl=1where¹iandºi;lareF-adaptedstochasticprocesses.Ifthedefaultofthejthprimary¯rmdoesnota®ectthedefaultintensityoftheithsecondary¯rm,wesetºi;j=0.MainFeaturesLetG=F_H1_:::_Hnstandfortheenlarged¯ltrationandletF^=F_Hk+1_:::_Hnbethe¯ltrationgeneratedbythereference¯ltrationFandtheobservationsofdefaultsofsecondary¯rms.Then:²Thedefaulttimes¿1;:::;¿kofprimary¯rmsareconditionallyindependentwithrespecttoF.²Thedefaulttimes¿1;:::;¿kofprimary¯rmsarenolongerconditionallyindependentwhenwereplacethe¯ltrationFbyF^.²Ingeneral,thedefaultintensityofaprimary¯rmwithrespecttothe¯ltrationF^di®ersfromtheintensity¸iwithrespecttoF:Weconcludethatdefaultsofprimary¯rmsarealso`dependent'ofdefaultsofsecondary¯rms.CaseofTwoFirmsToillustratethepresentmodel,wenowconsideronlytwo¯rms,AandBsay,andwepostulatethatAisaprimary¯rm,andBisasecondary¯rm.Lettheconstantprocess¸1=¸representthet1F-intensityofdefaultfor¯rmA,sothatZt©ª¿=inft2R:¸1du=¸t¸¡ln»;1+u110 5.4.JARROWANDYUMODEL167where»1isarandomvariableindependentofF;withtheuniformlawon[0;1]:Forthesecond¯rm,the`intensity'ofdefaultisassumedtosatisfy¸2=¸1+®1t2f¿1>tg2f¿1·tgforsomepositiveconstants¸2and®2;andthusZt©ª¿=inft2R:¸2du¸¡ln»;2+u20where»2isarandomvariablewiththeuniformprobabilitydistribution,independentofF;andsuchthat»1and»2aremutuallyindependent.Thenthefollowingpropertieshold:²¸1istheintensityof¿withrespecttoF;1²¸2istheintensityof¿withrespecttoF_H1,2²¸1isnottheintensityof¿withrespecttoF_H2:1Let¿i=infft2R+:¤i(t)¸£igZtfori=1;2where¤i(t)=¸i(s)dsand£1;Theta2areindependentrandomvariableswith0exponentialprobabilitydistributionwithparameter1.JarrowandYu[56]studythecasewhere¸1isaconstantand¸2(t)=¸2+(®2¡¸2)1f¿1·tg=¸21ft<¿1g+®21f¿1·tg:Assumeforsimplicitythatr=0andcomputethevalueofdefaultablezero-couponbondswithdefaulttime¿i,witharebate±i:Di(t;T)=EQ(1f¿i>Tg+±i1f¿is;¿2>t).²Caset·sFortsgf¿2>tg=f¿1>sgf¤2(t)<£2g=f¿1>sgf¸2t<£2g=f¸1s<£1gf¸2t<£2gleadtoQ(¿>s;¿>t)=e¡¸1se¡¸2t:12²Caset>sf¿1>sgf¿2>tg=fft>¿1>sgf¿2>tgg[ff¿1>tgf¿2>tggft>¿1>sgf¿2>tg=ft>¿1>sgf¤2(t)<£2g=ft>¿1>sgf¸2¿1+®2(t¡¿1)<£2g: 168CHAPTER5.DEPENDENTDEFAULTSTheindependencebetween£1and£2impliesthattherandomvariable¿1isindependentfrom£2(notethat¿=£(¸)¡1),andthus111³´Q(t>¿>s;¿>t)=E1e¡(¸2¿1+®2(t¡¿1))12Qft>¿1>sgZt=e¡(¸2u+®2(t¡u))¸e¡¸1udu1s³´=1¸e¡®2te¡s(¸1+¸2¡®2)¡e¡t(¸1+¸2¡®2):1¸1+¸2¡®2Setting¢=¸1+¸2¡®2,itfollowsthat1¡¢Q(¿>s;¿>t)=¸e¡®2te¡s¢¡e¡t¢+e¡¸1te¡¸2t:(5.2)121¢Inparticular,fors=0,³³´´Q(¿>t)=1¸e¡®2t¡e¡(¸1+¸2)t+¢e¡(¸1+¸2)t:21¢²ThecomputationofD1(t;T)reducestothatofQ(¿>TjG)=Q(¿>TjF_H1);1t1ttwhereF=H2.Wehavettµ¶1@2G(T;¿2)G(T;t)Q(¿1>TjGt)=1¡DZCt=1f¿1>tg1f¿2·tg+1f¿2>tg@2G(t;¿2)G(t;t)andthusD(t;T)=±+1(1¡±)e¡¸1(T¡t):11f¿1>tg1²ThecomputationofD2(t;T)followsfromthecomputationofQ(¿>TjH1)2tQ(¿2>TjGt)=1ft<¿2g1+1f¿2Tj¿2):Q(¿2>tjHt)Weobtain³D(t;T)=±+(1¡±)11e¡®2(T¡t)222f¿2>tgf¿1·tg¶+11(¸e¡®2(T¡t)+(¸¡®)e¡(¸1+¸2)(T¡t)):f¿1>tg122¢SpecialCase:ZeroRecoveryAssumethat¸1+¸2¡®26=0andthebondissubjecttothezerorecoveryscheme.Forthesakeofbrevity,wesetr=0sothatP(t;T)=1fort·T:Underthepresentassumptions,wehaveD(t;T)=Qf¿>TjH1_H2g;22ttandthegeneralformulayieldsQf¿>TjH1g2tD2(t;T)=1f¿2>tg1:Qf¿2>tjHtgRtIfweset¤2=¸2duthent0u22D(t;T)=1E(e¤t¡¤TjH1):2f¿2>tgQtFinally,wehavethefollowingexplicitresult.Corollary5.4.1If±2=0thenD2(t;T)=0onf¿2·tg:Onthesetf¿2>tgwehave³´¡®2(T¡t)1¡®2(T¡t)¡¸(T¡t)D2(t;T)=1f¿1·tge+1f¿1>tg¸1e+(¸2¡®2)e:¸¡®2 5.5.EXTENSIONOFTHEJARROWANDYUMODEL1695.5ExtensionoftheJarrowandYuModelWeshallnowarguethattheassumptionthatsome¯rmsareprimarywhileother¯rmsaresecondaryisnotrelevant.Forsimplicityofpresentation,weassumethat:²wehaven=2,thatis,weconsidertwo¯rmsonly,²theinterestrateriszero,sothatB(t;T)=1foreveryt·T,²thereference¯ltrationFistrivial,²corporatebondsaresubjecttothezerorecoveryscheme.Sincethesituationissymmetric,itsu±cestoanalyzeabondissuedbythe¯rst¯rm.Byde¯nition,thepriceofthisbondequalsD(t;T)=Qf¿>TjH1_H2g:11ttForthesakeofcomparison,weshallalsoevaluatethefollowingvalues,whicharebasedonpartialobservations,D~(t;T)=Qf¿>TjH2g11tandDb(t;T)=Qf¿>TjH1g:11t5.5.1Kusuoka'sConstructionWefollowhereKusuoka[63].UndertheoriginalprobabilitymeasureQtherandomtimes¿i;i=1;2areassumedtobemutuallyindependentrandomvariableswithexponentiallawswithparameters¸1and¸2;respectively.Girsanov'stheorem.Fora¯xedT>0;wede¯neaprobabilitymeasureQequivalenttoPon(•;G)bysettingdQ=´T;P-a.s.dPwheretheRadon-Nikod¶ymdensityprocess´t;t2[0;T];satis¯esX2Z´=1+´·idMitu¡uui=1]0;t]whereinturnZt^¿iMi=Hi¡¸du:tti0HereHi=1andprocesses·1and·2aregivenbytf¿i·tg³´³´1®12®2·t=1f¿2tg1f¿2·tg¸2=¸1+®1:t2f¿1>tg2f¿1·tgRtMainfeatures.Wefocuson¿andwedenote¤1=¸1du:Letusmakefewobservations.First,1t0utheprocess¸1isH2-predictable,andtheprocessZt^¿1M1=H1¡¸1du=H1¡¤1ttutt^¿10 170CHAPTER5.DEPENDENTDEFAULTSisaG-martingaleunderQ:Next,theprocess¸1isnotthe`true'intensityofthedefaulttime¿1withrespecttoH2underQ:Indeed,ingeneral,wehave¡11¢Qf¿>sjH1_H2g6=1Ee¤t¡¤sjH2:1ttf¿1>tgQtFinally,theprocess¸1representstheintensityofthedefaulttime¿withrespecttoH2undera1probabilitymeasureQ1equivalenttoP;wheredQ1=~´T;P-a.s.dPandtheRadon-Nikod¶ymdensityprocess~´t;t2[0;T];satis¯esZ´~=1+´~·2dM2:tu¡uu]0;t]Fors>twehave¡11¢112¤t¡¤sQf¿1>sjHt_Htg=1f¿1>tgEQ1ejFt;butalsoQf¿>sjH1_H2g=Q1f¿>sjH1_H2g:1tt1ttRecallthattheprocesses¸1and¸2havejumpsif®i6=¸i.Thenextresultshowsthattheintensities¸1and¸2are`localintensities'withrespecttotheinformationavailableattimet:Itshowsalsothatthemodelcaninfactbereformulatedasatwo-dimensionalMarkovchain(seeLando[64]).Proposition5.5.1Fori=1;2andeveryt2R+wehave¸=limh¡1Qft<¿·t+hj¿>t;¿>tg:(5.3)ii12h#0Moreover:®=limh¡1Qft<¿·t+hj¿>t;¿·tg:1112h#0and®=limh¡1Qft<¿·t+hj¿>t;¿·tg:2221h#05.5.2BondValuationProposition5.5.2ThepriceD1(t;T)onf¿1>tgequals³´¡®1(T¡t)1¡®1(T¡t)¡¸(T¡t)D1(t;T)=1f¿2·tge+1f¿2>tg¸2e+(¸1¡®1)e:¸¡®1Furthermore(¸¡®)¸e¡®1(T¡¿2)D~(t;T)=1221f¿2·tg¸®e(¸¡®2)¿2+¸(¸¡®)1222¸¡®(¸¡®)e¡¸(T¡t)+¸e¡®1(T¡t)2112+1f¿2>tg¸¡®¸e¡(¸¡®2)t+¸¡®1122and¸e¡®1T+(¸¡®)e¡¸TDb(t;T)=1211:1f¿1>tg¸e¡®1t+(¸¡®)e¡¸t211Observethat:²formulaforD1(t;T)coincideswiththeJarrowandYuformulaforthebondissuedbyasecondary¯rm,²processesD1(t;T)andDb1(t;T)representex-dividendvaluesofthebond,andthustheyvanishafterdefaulttime¿1,²thelatterremarkdoesnotapplytotheprocessD~1(t;T). 5.6.MARKOVIANMODELSOFCREDITMIGRATIONS1715.6MarkovianModelsofCreditMigrationsInthissectionwegiveabriefdescriptionofaMarkovianmarketmodelthatcanbee±cientlyusedforevaluatingandhedgingbasketcreditinstruments.Thisframework,isaspecialcaseofamoregeneralmodelintroducedinBieleckietal.[3],whichallowstoincorporateinformationrelativetothedynamicevolutionofcreditratingsandcreditmigrationprocessesinthepricingofbasketinstruments.EmpiricalstudyofthemodeliscarriedinBieleckietal.[15].Westartwithsomenotation.Lettheunderlyingprobabilityspacebedenotedby(•;G;G;Q),whereQisariskneutralmeasureinferredfromthemarket(weshalldiscussthisinfurtherdetailwhenaddressingtheissueofmodelcalibration),G=H_Fisa¯ltrationcontainingallinformationavailabletomarketagents.The¯ltrationHcarriesinformationaboutevolutionofcreditevents,suchaschangesincreditratingsordefaultsofrespectivecreditnames.The¯ltrationFisareference¯ltrationcontaininginformationpertainingtotheevolutionofrelevantmacroeconomicvariables.WeconsiderLobligors(orcreditnames)andweassumethatthecurrentcreditqualityofeachreferenceentitycanbeclassi¯edintoK:=f1;2;:::;Kgratingcategories.Byconvention,thecategoryKcorrespondstodefault.LetX`;`=1;2;:::;Lbesomeprocesseson(•;G;Q)takingvaluesinthe¯nitestatespaceK.TheprocessesX`representtheevolutionofcreditratingsofthe`threferenceentity.Wede¯nethedefaulttime¿ofthe`threferenceentitybysettingl¿=infft2R:X`=Kg(5.4)l+tWeassumethatthedefaultstateKisabsorbing,sothatforeachnamethedefaulteventcanonlyoccuronce.WedenotebyX=(X1;X2;:::;XL)thejointcreditratingprocessoftheportfolioofLcreditnames.ThestatespaceofXisX:=KLandtheelementsofXwillbedenotedbyx.Wepostulatethatthe¯ltrationHisthenatural¯ltrationoftheprocessXandthatthe¯ltrationFisgeneratedbyaRnvaluedfactorprocess,Y,representingtheevolutionofrelevanteconomicvariables,likeshortrateorequitypriceprocesses.5.6.1In¯nitesimalGeneratorWeassumethatthefactorprocessYtakesvaluesinRnsothatthestatespacefortheprocessM=(X;Y)isX£Rn.Attheintuitivelevel,wewishtomodeltheprocessM=(X;Y)asacombinationofaMarkovchainXmodulatedbytheL¶evy-likeprocessYandaL¶evy-likeprocessYmodulatedbyaMarkovchainX.Tobemorespeci¯c,wepostulatethatthein¯nitesimalgeneratorAofMisgivenasXnXnAf(x;y)=(1=2)aij(x;y)@i@jf(x;y)+bi(x;y)@if(x;y)i;j=1i=1Z¡¢X+°(x;y)f(x;y+g(x;y;y0))¡f(x;y)¦(x;y;dy0)+¸(x;x0;y)f(x0;y);Rn0x2Xwhere¸(x;x0;y)¸0foreveryx=(x1;x2;:::;xL)6=(x01;x02;:::;x0L)=x0,andX¸(x;x;y)=¡¸(x;x0;y):x02X;x06=xHere@denotesthepartialderivativewithrespecttothevariableyi.TheexistenceanduniquenessiofaMarkovprocessMwiththegeneratorAwillfollow(underappropriatetechnicalconditions)fromtherespectiveresultsregardingmartingaleproblems.We¯nditconvenienttorefertoX(Y,respectively)astheMarkovchaincomponentofM(thejump-di®usioncomponentofM,respectively).Atanytimet,theintensitymatrixoftheMarkov 172CHAPTER5.DEPENDENTDEFAULTS0chaincomponentisgivenas¤t=[¸(x;x;Yt)]x;x02X.Thejump-di®usioncomponentsatis¯estheSDE:ZdY=b(X;Y)dt+¾(X;Y)dW+g(X;Y;y0)¼(X;Y;dy0;dt);ttttttt¡t¡t¡t¡Rnwhere,fora¯xed(x;y)2X£Rn,¼(x;y;dy0;dt)isaPoissonmeasurewiththeintensitymeasure°(x;y)¦(x;y;dy0)dt;andwhere¾(x;y)satis¯estheequality¾(x;y)¾(x;y)T=a(x;y).Remarks5.6.1Ifwetakeg(x;y;y0)=y0,andwesupposethatthecoe±cients¾=[¾],b=[b],iji°,andthemeasure¦donotdependonxandythenthefactorprocessYisaPoisson-L¶evyprocesswiththecharacteristictriplet(a;b;º);wherethedi®usionmatrixisa(x;y)=¾(x;y)¾(x;y)T,thedrift"vectorisb(x;y),andtheL¶evymeasureisº(dy)=°¦(dy).Inthiscase,themigrationprocessXismodulatedbythefactorprocessY,butnotviceversa.Weshallnotstudyherethein¯niteactivity"case,thatis,thecasewhenthejumpmeasure¼isnotaPoissonmeasure,andtherelatedL¶evymeasureisanin¯nitemeasure.WeshallprovidewithmorestructuretheMarkovchainpartofthegeneratorA.Speci¯cally,wemakethefollowingstandingassumption.Asumption(M).Thein¯nitesimalgeneratoroftheprocessM=(X;Y)takesthefollowingformXnXnAf(x;y)=(1=2)aij(x;y)@i@jf(x;y)+bi(x;y)@if(x;y)i;j=1i=1Z¡¢+°(x;y)f(x;y+g(x;y;y0))¡f(x;y)¦(x;y;dy0)(5.5)RnXLX+¸l(x;x0;y)f(x0;y);lll=1xl02Kwherewewritex0=(x1;x2;:::;xl¡1;x0l;xl+1;:::;xL).lNotethatx0isthevectorx=(x1;x2;:::;:::;xL)withthelthcoordinatexlreplacedbyx0l.Inlthecaseoftwoobligors(i.e.,forL=2),thegeneratorbecomesXnXnAf(x;y)=(1=2)aij(x;y)@i@jf(x;y)+bi(x;y)@if(x;y)i;j=1i=1Z¡¢+°(x;y)f(x;y+g(x;y;y0))¡f(x;y)¦(x;y;dy0)RnXX+¸1(x;x0;y)f(x0;y)+¸2(x;x0;y)f(x0;y);1122x012Kx022Kwherex=(x1;x2);x0=(x01;x2)andx0=(x1;x02).Inthiscase,comingbacktothegeneralform,12wehaveforx=(x1;x2)andx0=(x01;x02)8<¸1(x;x0;y);ifx2=x02,1¸(x;x0;y)=¸2(x;x0;y);ifx1=x01,:20;otherwise.SimilarexpressionscanbederivedinthecaseofageneralvalueofL.Notethatthemodelspeci¯ed0by(5.5)doesnotallowforsimultaneousjumpsofthecomponentsXlandXlforl6=l0.Inotherwords,theratingsofdi®erentcreditnamesmaynotchangesimultaneously.Nevertheless,thisisnotaseriouslackofgenerality,astheratingsofbothcreditnamesmaystillchangeinanarbitrarilysmalltimeinterval.Theadvantageisthat,forthepurposeofsimulationofpathsofprocessX,ratherthandealingwithX£Xintensitymatrix[¸(x;x0;y)],weshalldealwith 5.6.MARKOVIANMODELSOFCREDITMIGRATIONS173Lintensitymatrices[¸l(x;x0;y)],eachofdimensionK£K(forany¯xedy).Thestructure(5.5)lisassumedintherestofthepaper.Letusstressthatwithinthepresentsetupthecurrentcreditratingofthecreditnameldirectlyimpactstheintensityoftransitionoftheratingofthecreditnamel0,andviceversa.Thisproperty,knownasfrailty,maycontributetodefaultcontagion.Remarks5.6.2(i)Itisclearthatwecanincorporateinthemodelthecasewhensome{possiblyall{componentsofthefactorprocessYfollowMarkovchainsthemselves.Thisfeatureisimportant,asfactorssuchaseconomiccyclesmaybemodeledasMarkovchains.Itisknownthatdefaultratesarestronglyrelatedtobusinesscycles.(ii)SomeofthefactorsY1;Y2;:::;YdmayrepresentcumulativedurationofvisitsofratingprocessesRtl1Xinrespectiveratingstates.Forexample,wemaysetYt=01fX1=1gds.Inthiscase,wehavesb1(x;y)=1fx1=1g(x),andthecorrespondingcomponentsofcoe±cients¾andgequalzero.(iii)Intheareaofstructuralarbitrage,socalledcredit{to{equity(C2E)modelsand/orequity{to{credit(E2C)modelsarestudied.Ourmarketmodelnestsbothtypesofinteractions,thatisC2EandE2C.Forexample,ifoneofthefactorsisthepriceprocessoftheequityissuedbyacreditname,andifcreditmigrationintensitiesdependonthisfactor(implicitlyorexplicitly)thenwehaveaE2Ctypeinteraction.Ontheotherhand,ifcreditratingsofagivenobligorimpacttheequitydynamics(ofthisobligorand/orsomeotherobligors),thenwedealwithaC2Etypeinteraction.Asalreadymentioned,S=(H;X;Y)isaMarkovprocessonthestatespacef0;1;:::;Lg£X£Rdwithrespecttoitsnatural¯ltration.Giventheformofthegeneratoroftheprocess(X;Y),wecaneasilydescribethegeneratoroftheprocess(H;X;Y).ItisenoughtoobservethatthetransitionintensityattimetofthecomponentHfromthestateHttothestateHt+1isequalPLl(l)tol=1¸(Xt;K;Xt;Yt),providedthatHtTS,asdescribedintheprevioussection.WeshallnowvaluethecorrespondingcreditdefaultswaptionwithexpirydateTTgfort·TTgandforarbitrarytTgBTBT·(t;T;T)N+Àt;T2Ã(1)SM!#¯!·(t;T;T)¯lnKÀt;T¯¡KN¡¯Mt;Àt;T2¯whereZTÀ2=À(t;T;TS;TM)2:=¾(s;TS;TM)2ds:t;TtProof.Wehave³¯´³¯´¡1¡(1);TS(1);TS¢+¯¡1¡(1);TS(1);TS¢+¯BtEQBTAT¡KBT¯Mt=BtEQ1f¿>TgBTAT¡KBT¯Mt³³¡´¯´¡1(1);TS(1);TS¢+X¯=BtEQ1f¿>TgBTEQAT¡KBTj¾(Mt)_FT¯Mt³³¡´¯´¡1(1);TS(1)SM¢+X¯=BtEQ1f¿>TgBTBTEQ·(T;T;T)¡Kj¾(Mt)_FT¯Mt:Inviewofourassumptions,weobtain³¡¢¯´(1)SM+¯XEQ·(T;T;T)¡K¯¾(Mt)_FTÃ(1)SM!Ã(1)SM!·(t;T;T)·(t;T;T)(1)SMlnKÀt;TlnKÀt;T=·(t;T;T)N+¡KN¡:Àt;T2Àt;T2Bycombiningtheaboveequalities,wearriveatthestatedformula.¤5.7BasketCreditDerivativesWeshallnowdiscussthecaseofcreditderivativeswithseveralunderlyingcreditnames.Feasibilityofclosed-formcalculations,suchasanalyticcomputationofrelevantconditionalexpectedvalues,dependstoagreatextentonthetypeandamountofinformationonewantstoutilize.Typically,inordertoe±cientlydealwithexactcalculationsofconditionalexpectations,onewillneedtoamendspeci¯cationsoftheunderlyingmodelsothatinformationusedincalculationsisgivenbyacoarser¯ltration,orperhapsbysomeproxy¯ltration. 178CHAPTER5.DEPENDENTDEFAULTS5.7.1kth-to-DefaultCDSWeshallnowdiscussthevaluationofagenerickth-to-defaultcreditdefaultswaprelativetoaportfolioofLreferencedefaultablebonds.ThedeterministicnotionalamountoftheithbondisdenotedasNi,andthecorrespondingdeterministicrecoveryrateequals±i.WesupposethatthematuritiesofthebondsareU1;U2;:::;UL,andthematurityoftheswapisTt.Again,ingeneral,theaboveconditionalexpectationwillneedtobeapproximatedbysimulation.Andagain,forasmallportfoliosizeL,ifeitherexactornumericalsolutionofrelevantKolmogorovequationscanbederived,thenananalyticalcomputationoftheexpectationcanbedone(atleastinprinciple).5.7.2Forwardkth-to-DefaultCDSForwardkth-to-defaultCDShasananalogousstructuretotheforwardCDS.ThenotationusedhereisconsistentwiththenotationusedpreviouslyinSections5.6.5and5.7.1.DefaultPaymentLegThecash°owassociatedwiththedefaultpaymentlegcanbeexpressedasfollowsX(1¡±i)Ni1fTS<¿(k)·TMg1¿(k)(t):i2LkConsequently,thetime-tvalueofthedefaultpaymentlegequals,foreveryt·TS,³X¯´S¯(k);T¡1At=BtEQ1fTS<¿(k)·TMgB¿(k)(1¡±i)Ni¯Mt:i2LkPremiumPaymentLegAsbefore,letT=fT;T;:::;Tgbethetenorofagenericpremiumpaymentleg,whereTS<12JT<¢¢¢0thematurityofanygivenCDSindex.CDSindicesaretypicallyissuedbyapooloflicensed¯nancialinstitutions,whichweshallcallthemarketmaker.AttimeofissuanceofaCDSindex,sayattimet=0;themarketmakerdeterminesanannualrateknownasindexspread,tobepaidouttoinvestorsonaperiodicbasis.Weshalldenotethisrateby´0.Inwhatfollows,weshallassumethat,atsometimet2[0;T],aninvestorpurchasesoneunitofCDSindexissuedattimezero.Bypurchasingtheindex,aninvestorentersintoabindingcontractwhosemainprovisionsaresummarizedbelow:²Thetimeofissuanceofthecontract0.Theinceptiontimeofthecontractistimet;thematuritytimeofthecontractisT.²Bypurchasingtheindex,theinvestorsellsprotectiontothemarketmakers.Thus,theinvestorassumestheroleofaprotectionsellerandthemarketmakersassumetheroleofprotectionbuyers.Inpractice,theinvestorsagreestoabsorballlossesduetodefaultsinthereferenceportfolio,occurringbetweenthetimeofinceptiontandthematurityT.Incaseofdefaultofareferenceentity,theprotectionsellerpaystothemarketmakerstheprotectionpaymentintheamountof(1¡±),where±2[0;1]istheagreedrecoveryrate(typically40%).Weassumethatthefacevalueofeachreferenceentityisone.ThusthetotalnotionaloftheindexisL.Thenotionalonwhichthemarketmakerpaysthespread,henceforthreferredtoasresidualprotectionisthenreducedbysomeamount.Forinstance,afterthe¯rstdefault,theresidualprotectionisupdatedasfollows(weadopthereaftertheformerconvention)L!L¡(1¡±)orL!L¡1:²Inexchange,theprotectionsellerreceivesfromthemarketmakeraperiodic¯xedpremiumontheresidualprotectionattheannualrateof´t,thatrepresentsthefairindexspread.(Wheneverareferenceentitydefaults,itsweightintheindexissettozero.Bypurchasingoneunitofindextheprotectionsellerowesprotectiononlyonthosenamesthathavenotyetdefaultedattimeofinception.)Ifthemarketindexspreadisdi®erentfromtheissuancespread,i.e.,´t6=´0,thepresentvalueofthedi®erenceissettledthroughanupfrontpayment.Wedenoteby¿therandomdefaulttimeoftheithnameintheindexandbyHitherightitcontinuousprocessde¯nedasHi=1,i=1;2;:::;L.Also,letft;j=0;1;:::;Jgwitht=ttf¿i·tgj0andtJ·Tdenotethetenorofthepremiumlegpaymentsdates.Thediscountedcumulativecash°owsassociatedwithaCDSindexareasfollows:XJB³XL´tiPremiumLeg=1¡Htj(1¡±)´tBtjj=0i=1 184CHAPTER5.DEPENDENTDEFAULTSandXLB³´ProtectionLeg=t(1¡±)(Hi¡Hi):TtB¿ii=1CollateralizedDebtObligationsCollateralizedDebtObligations(CDO)arecreditderivativesbackedbyportfoliosofassets.Iftheunderlyingportfolioismadeupofbonds,loansorothersecuritizedreceivables,suchproductsareknownascashCDOs.Alternatively,theunderlyingportfoliomayconsistofcreditderivativesreferencingapoolofdebtobligations.Inthelattercase,CDOsaresaidtobesynthetic.Becauseoftheirrecentlyacquiredpopularity,wefocusourdiscussiononstandardized(synthetic)CDOcontractsbackedbyCDSindices.Webeginwithanoverviewoftheproduct:²Thetimeofissuanceofthecontractis0:Thetimeofinceptionofthecontractist2R+anditsmaturityisT.ThenotionaloftheCDOcontractistheresidualprotectionoftheunderlyingCDSindexatthetimeofinception.²Thecreditrisk(thepotentiallossduetocreditevents)bornebythereferencepoolislayeredintodi®erentrisklevels.Therangeinbetweentwoadjacentrisklevelsiscalledatranche.Thelowerboundofatrancheisusuallyreferredtoasattachmentpointandtheupperboundasdetachmentpoint.Thecreditriskissoldinthesetranchestoprotectionsellers.Forinstance,inatypicalCDOcontractoniTraxx,thecreditriskissplitintoequity,mezzanine,andseniortranchescorrespondingto0¡3%;3¡6%;6¡9%;9¡12%,and12¡22%ofthelosses,respectively.Atinception,thenotionalvalueofeachtrancheistheCDOresidualnotionalweightedbytherespectivetranchewidth.²Thetranchebuyersellspartialprotectiontothepoolowner,byagreeingtoabsorbthepool'slossescomprisedinbetweenthetrancheattachmentanddetachmentpoint.Thisisbetterunderstoodbyanexample.Assumethat,attimet,theprotectionsellerpurchasesonecurrencyunitworthofthe6¡9%tranche.Oneyearlater,consequentlytoadefaultevent,thecumulativelossbreaksthroughtheattachmentpoint,reaching8%.Theprotectionsellerthenful¯llshisobligationbydisbursingtwothirds(=8%¡6%)ofacurrencyunit.Thetranchenotionalis9%¡6%thenreducedtoonethirdofitspre-defaulteventvalue.Werefertotheremainingtranchenotionalasresidualtrancheprotection.²Inexchange,asoftimetanduptotimeT,theCDOissuer(protectionbuyer)makesperiodicpaymentstothetranchebuyeraccordingtoapredeterminedrate(termedtranchespread)ontheresidualtrancheprotection.Wedenotethetimetspreadofthelthtrancheby·l.Returningttoourexample,afterthelossreaches8%,premiumpaymentsaremadeon1(=9%¡8%)of39%¡6%thetranchenotional,untilthenextcrediteventoccursorthecontractmatures.WedenotebyLandUthelowerandupperattachmentpointsforthelthtranche,·litstimellttspread.Itisalsoconvenienttointroducethepercentagelossprocess,PLiiti=1(Hs¡Ht)(1¡±)¡s=PLii=1(1¡Ht)whereListhenumberofreferencenamesinthebasket.(Notethatthelossiscalculatedonlyonthenameswhicharenotdefaultedatthetimeofinceptiont.)Finallyde¯nebyCl=U¡Lthellportionofcreditriskassignedtothelthtranche.Purchasingoneunitofthelthtrancheattimetgeneratesthefollowingdiscountedcash°ows:XJBXL³´PremiumLeg=t·l(1¡Hi)Cl¡min(Cl;max(¡t¡L;0))Btttjltjj=0i=1 5.7.BASKETCREDITDERIVATIVES185andXLBtiiProtectionLeg=(HT¡Ht)(1¡±)1fLk·¡t·Ukg:B¿itji=1WeremarkherethattheequitytrancheoftheCDOoniTraxxorCDXisquotedasanupfrontrate,say·0,onthetotaltranchenotional,inadditionto500basispoints(5%rate)paidannuallyonthetresidualtrancheprotection.Thepremiumlegpayment,inthiscase,isasfollows:XLXJBXL³¡¢´·0C0(1¡Hi)+t(:05)(1¡Hi)C0¡minC0;max(¡t¡L;0):ttBttj0tji=1j=0i=1First-to-DefaultSwapsThekth-to-defaultswapsarebasketcreditinstrumentsbackedbyportfoliosofsinglenameCDSs.SincethegrowthinpopularityofCDSindicesandtheassociatedderivatives,kth-to-defaultswapshavebecomeratherilliquid.Currently,suchproductsaretypicallycustomizedbanktoclientcon-tracts,andhencerelativelybespoketotheclient'screditportfolio.Forthisreason,wefocusourattentionon¯rst-to-defaultswapsissuedontheiTraxxindex,whicharetheonlyoneswithacertaindegreeofliquidity.StandardizedFTDSarenowissuedoneachoftheiTraxxsectorsub-indices.EachFTDSisbackedbyanequallyweightedportfolioof¯vesinglenameCDSsintherelativesub-index,chosenaccordingtosomeliquiditycriteria.ThemainprovisionsofaFTDScontractarethefollowing:²Thetimeofissuanceofthecontractis0:Thetimeofinceptionofthecontractist,thematurityisT.²ByinvestinginaFTDS,theprotectionselleragreestoabsorbthelossproducedbythe¯rstdefaultinthereferenceportfolio²Inexchange,theprotectionsellerispaidaperiodicpremium,knownasFTDSspread,com-putedontheresidualprotection.Wedenotethetime-tspreadby't.Recallthatftj;j=0;1;:::;Jgwitht=t0andtJ·Tdenotesthetenorofthepremiumlegpaymentsdates.Also,denoteby¿(1)the(random)timeofthe¯rstdefaultinthepool.Thediscountedcumulativecash°owsassociatedwithaFTDSonaniTraxxsub-indexcontainingNnamesareasfollows(againweassumethateachnameinthebaskethasnotionalequaltoone):XJBtPremiumLeg='t1f¿(1)¸tjgBtjj=0andBtProtectionLeg=(1¡±)1f¿(1)·Tg:B¿(1)Step-upCorporateBondsAsofnow,theseproductsarenottradedinbaskets,howevertheyareofinterestbecausetheyo®erprotectionagainstcrediteventsotherthandefaults.Inparticular,stepupbondsarecorporatecouponissuesforwhichthecouponpaymentdependsontheissuer'screditquality:thecouponpaymentincreaseswhenthecreditqualityoftheissuerdeclines.Inpractice,forsuchbonds,creditqualityisre°ectedincreditratingsassignedtotheissuerbyatleastonecreditratingsagency(Moody's-KMVorStandard&Poor's).Theprovisionslinkingthecash°owsofthestep-upbondstothecreditratingoftheissuerhavedi®erentstepamountsanddi®erentratingeventtriggers.Insomecases,astep-upofthecouponrequiresadowngradetothetriggerlevelbybothrating 186CHAPTER5.DEPENDENTDEFAULTSagencies.Inothercases,therearestep-uptriggersforactionsofeachratingagency.Here,adowngradebyoneagencywilltriggeranincreaseinthecouponregardlessoftheratingfromtheotheragency.Provisionsalsovarywithrespecttostep-downfeatureswhich,asthenamesuggests,triggeraloweringofthecouponifthecompanyregainsitsoriginalratingafteradowngrade.Ingeneral,thereisnostep-downbelowtheinitialcouponforratingsexceedingtheinitialrating.LetXtstandforsomeindicatorofcreditqualityattimet.Assumethatti;i=1;2;:::;narecouponpaymentdatesandletcn=c(Xtn¡1)bethecoupons(t0=0).Thetimetcumulativecash°owprocessassociatedtothestep-upbondequalsZBtBtDt=(1¡HT)+(1¡Hu)dCu+possiblerecoverypaymentBT(t;T]BuPwhereCt=ti·tci.5.7.5ValuationofStandardBasketCreditDerivativesWenowdiscussthepricingofthebasketinstrumentsintroducedinprevioussub-section.Inpar-ticular,computingthefairspreadsofsuchproductsinvolvesevaluatingtheconditionalexpectationunderthemartingalemeasureQofsomequantitiesrelatedtothecash°owsassociatedtoeachinstrument.InthecaseofCDSindexes,CDOsandFTDS,thefairspreadissuchthat,atincep-tion,thevalueofthecontractisexactlyzero,i.etheriskneutralexpectationsofthe¯xedlegandprotectionlegpaymentsareidentical.Thefollowingexpressionscanbeeasilyderivedfromthediscountedcumulativecash°owsgivenintheprevioussubsection.²thetimetfairspreadofasinglenameCDS:³´EXt;YtBtH`(1¡±)QB¿T´`=³`´tXt;YtPJBt`EQj=0Bt(1¡Htj)j²thetimetfairspreadofaCDSindexis:³P´EXt;YtLBt(1¡±)(Hi¡Hi)Qi=1B¿iTt´t=³P³P´´EXt;YtJBtL1¡Hi(1¡±)Qj=0Bti=1tjj²thetimetfairspreadoftheCDOequitytrancheis:³XL01Xt;YtBtii·t=PLEQ(HT¡Ht)(1¡±)1fL0·¡t·U0g0iB¿iCi=1(1¡Ht)i=1¿iXJBXL³´´¡EXt;Ytt(:05)(1¡Hi)C0¡min(C0;max(¡t¡L;0))QBttj0tjj=0i=1²thetimetfairspreadofthe`thCDOtrancheis:³P´Xt;YtLBtiiEQi=1B(HT¡Ht)(1¡±)1fL`·¡t·U`g`¿i¿i·t=³PP³´´EXt;YtJBtL(1¡Hi)Cl¡min(Cl;max(¡t¡L;0))Qj=0Btji=1ttjl 5.7.BASKETCREDITDERIVATIVES187²thetimetfairspreadofa¯rst-to-defaultswapis:Bt(1¡±)(1)B¿f¿(1)·Tg(1)'t=PJBt(1)j=0Btjf¿(1)¸tjg²thetimetfairvalueofthestepupbondis:³Z´suXt;YtBtBtB=EQ(1¡HT)+(1¡Hu)dCu+possiblerecoverypaymentBT(t;T]BuDependingonthedimensionalityoftheproblem,theaboveconditionalexpectationswillbeevaluatedeitherbymeansofMonteCarlosimulation,orbymeansofsomeothernumericalmethodand,inthelow-dimensionalcase,evenanalytically.Itisperhapsworthmentioningthatwehavealreadydonesomenumericaltestsofourmodelsotoseewhetherthemodelcanreproducesocalledmarketcorrelationskews.Thepicturebelowshowsthatthemodelperformsverywellinthisregard.3Forfurtherexamplesofmodel'simplementations,theinterestedreaderisreferredtoBieleckietal.[15].ImpliedcorrelationskewsforCDOtranchesModelv.MarketImpliedcorrelations35modelmarket302520Impliedcorrelation151050−3%3−6%6−9%9−12%12−22%TranchecutoffsTheissueofevaluatingfunctionalsassociatedwithmultiplecreditmigrations,defaultsinpartic-ular,isalsoprominentwithregardtoportfoliocreditrisk.Insomesegmentsofthecreditmarkets,onlythedeteriorationofthevalueofaportfolioofdebts(bondsorloans)duetodefaultsistypicallyconsidered.Infact,suchisthesituationregardingvarioustranchesof(eithercashorsynthetic)collateralizeddebtobligations,aswellaswithvarioustranchesofrecentlyintroducedCDSindices,suchas,DJCDXNAIGorDJiTraxxEurope.4Nevertheless,itisratherapparentthatavaluationmodelre°ectingthepossibilityofintermediatecreditmigrations,andnotonlydefaults,iscalledforinordertobetteraccountforchangesincreditworthinessofthereferencecreditnames.Likewise,forthepurposeofmanagingrisksofadebtportfolio,itisnecessarytoaccountforchangesinvalueoftheportfolioduetochangesincreditratingsofthecomponentsoftheportfolio.3WethankAndreaandLucaVidozzifromAppliedMathematicsDepartmentattheIllinoisInstituteofTechnologyfornumericalimplementationofthemodeland,inparticular,forgeneratingthepicture.4Seehttp://www.credit°ux.com/public/publications/0409CFindexGuide.pdf. 188CHAPTER5.DEPENDENTDEFAULTS Chapter6Appendix:ConditionalPoissonProcessInsome¯nancialapplications,weneedtomodelasequenceofsuccessiverandomtimes.Almostinvariably,thisisdonebymakinguseoftheso-calledF-conditionalPoissonprocess,alsoknownasthedoublystochasticPoissonprocess.Thegeneralideaisquitesimilartothecanonicalconstructionofasinglerandomtime.Westartbyassumingthatwearegivenastochasticprocess©;tobeinterpretedasthehazardprocess,andweconstructajumpprocess,withunitjumpsize,suchthattheprobabilisticfeaturesofconsecutivejumptimesaregovernedbythehazardprocess©:6.1PoissonProcesswithConstantIntensityLetus¯rstrecallthede¯nitionandthebasicpropertiesofthe(time-homogeneous)PoissonprocessNwithconstantintensity¸>0:De¯nition6.1.1AprocessNde¯nedonaprobabilityspace(•;G;P)iscalledthePoissonprocesswithintensity¸withrespecttoGifN0=0andforany0·s¿k:Nt6=N¿kg=infft>¿k:Nt¡N¿k=1g:Oneshowswithoutdi±cultiesthatPflimk!1¿k=1g=1:Itisconvenienttointroducethesequence»k;k2Nofnon-negativerandomvariables,where»k=¿k¡¿k¡1foreveryk2N:Letusquotethefollowingwellknownresult.189 190CHAPTER6.APPENDIX:CONDITIONALPOISSONPROCESSProposition6.1.1Therandomvariables»k;k2Naremutuallyindependentandidenticallydis-tributed,withtheexponentiallawwithparameter¸;thatis,foreveryk2NwehavePf»·tg=Pf¿¡¿·tg=1¡e¡¸t;8t2R:kkk+Proposition6.1.1suggestsasimpleconstructionofaprocessN;whichfollowsatime-homogeneousPoissonprocesswithrespecttoitsnatural¯ltrationFN:Supposethattheprobabilityspace(•;G;P)islargeenoughtosupportafamilyofmutuallyindependentrandomvariables»k;k2Nwiththecommonexponentiallawwithparameter¸>0:Wede¯netheprocessNon(•;G;P)bysetting:Nt=0ifft<»1gand,foranynaturalk;XkkX+1Nt=kifandonlyif»i·t<»i:i=1i=1ItcancheckedthattheprocessNde¯nedinthiswayisindeedaPoissonprocesswithparameter¸;withrespecttoitsnatural¯ltrationFN:ThejumptimesofNare,ofcourse,therandomtimesPk¿k=i=1»i;k2N:LetusrecallsomeusefulequalitiesthatarenothardtoestablishthroughelementarycalculationsinvolvingthePoissonlaw.Foranya2Rand0·s0;wehaveZXeiaNt=1+(eiaNt¡eiaNt¡)=1+(eia¡1)eiaNu¡dN;u]0;t]00;weintroduceaprobabilitymeasureQon(•;GT)bysetting¯dQ¯¯=´T;P-a.s.,(6.2)dPGTwheretheRadon-Nikod¶ymdensityprocess´t;t2[0;T];satis¯esd´t=´t¡·dNbt;´0=1;(6.3)forsomeconstant·>¡1:SinceY:=·Nbisaprocessof¯nitevariation,(6.3)admitsauniquesolution,denotedasEt(Y)orEt(·Nb);itcanbeseenasaspecialcaseoftheDol¶eans(orstochastic)exponential.Bysolving(6.3)path-by-path,weobtainYYc´=E(·Nb)=eYt(1+¢Y)e¡¢Yu=eYt(1+¢Y);ttuu0¡1:Uponsettinga=ln(1+·)inpart(iii)ofProposition6.1.2,wegetMa=´;thiscon¯rmsthattheprocess´followsaG-martingaleunderP:Wehavethusprovedthefollowingresult.Lemma6.1.1Assumethat·>¡1:Theuniquesolution´totheSDE(6.3)isanexponentialG-martingaleunderP:Speci¯cally,´=eNtln(1+·)¡·¸t=eNbtln(1+·)¡¸t(·¡ln(1+·))=Ma;(6.4)ttwherea=ln(1+·):Inparticular,therandomvariable´Tisstrictlypositive,Pa.s.andEP(´T)=1:Furthermore,theprocessMasolvesthefollowingSDE:dMa=Ma(ea¡1)dNb;Ma=1:(6.5)tt¡t0Weareinapositiontoestablishthewell-knownresult,whichstatesthatunderQtheprocessN;t2[0;T];followsaPoissonprocesswiththeconstantintensity¸¤=(1+·)¸:t 6.1.POISSONPROCESSWITHCONSTANTINTENSITY193Proposition6.1.4AssumethatunderPaprocessNisaPoissonprocesswithintensity¸withrespecttothe¯ltrationG:SupposethattheprobabilitymeasureQisde¯nedon(•;GT)through(6.2)and(6.3)forsome·>¡1:(i)TheprocessNt;t2[0;T];followsaPoissonprocessunderQwithrespecttoGwiththeconstantintensity¸¤=(1+·)¸:(ii)ThecompensatedprocessN¤;t2[0;T];de¯nedastN¤=N¡¸¤t=N¡(1+·)¸t=Nb¡·¸t;ttttfollowsaQ-martingalewithrespecttoG:Proof.Fromremark(iii)afterProposition6.1.2,weknowthatitsu±cesto¯nd¸¤suchthat,forany¯xedb2R;theprocessM~b;givenas¤bM~b:=ebNt¡¸t(e¡1);8t2[0;T];(6.6)tfollowsaG-martingaleunderQ:Bystandardarguments,theprocessM~bisaQ-martingaleifandonlyiftheproductM~b´isamartingaleundertheoriginalprobabilitymeasureP:Butinviewof(6.4),wehave³¡¢¡¢´M~b´=expNb+ln(1+·)¡t·¸+¸¤(eb¡1):tttLetuswritea=b+ln(1+·):Sincebisanarbitraryrealnumber,soisa:Then,byvirtueofpart(iii)inProposition6.1.2,wenecessarilyhave·¸+¸¤(eb¡1)=¸(ea¡1):Aftersimpli¯cations,weconcludethat,forany¯xedrealnumberb;theprocessM~bde¯nedby(6.6)isaG-martingaleunderQifandonlyif¸¤=(1+·)¸:Inotherwords,theintensity¸¤ofNunderQsatis¯es¸¤=(1+·)¸:Alsothesecondstatementisclear.¤Remark6.1.2AssumethatG=FN,i.e.,the¯ltrationGisgeneratedbysomePoissonprocessN:ThenanystrictlypositiveG-martingale´underPisknowntosatisfySDE(6.3)forsomeG-predictableprocess·:AssumethatWisaBrownianmotionandNfollowsaPoissonprocessunderPwithrespecttoG:Let´satisfy¡¢d´t=´t¡¯tdWt+·dNbt;´0=1;(6.7)forsomeG-predictablestochasticprocess¯andsomeconstant·>¡1:AsimpleapplicationoftheIt^o'sproductruleshowsthatifprocesses´1and´2satisfy:d´1=´1¯dW;d´2=´2·dNb;tt¡tttt¡tthentheproduct´:=´1´2satis¯es(6.7).TakingtheuniquenessofsolutionstothelinearSDEttt(6.7)forgranted,weconcludethattheuniquesolutiontothisSDEisgivenbytheexpression:³ZtZt´¡¢12´t=exp¯udWu¡¯uduexpNtln(1+·)¡·¸t:(6.8)020Theproofofthenextresultislefttothereaderasexercise.Proposition6.1.5LettheprobabilityQbegivenby(6.2)and(6.8)forsomeconstant·>¡1andaG-predictableprocess¯;suchthatEP(´T)=1:Rt(i)TheprocessW¤=W¡¯du;t2[0;T];followsaBrownianmotionunderQ;withrespecttott0uthe¯ltrationG:(ii)TheprocessN;t2[0;T];followsaPoissonprocesswiththeconstantintensity¸¤=(1+·)¸tunderQ;withrespecttothe¯ltrationG:(iii)ProcessesW¤andNaremutuallyindependentunderQ: 194CHAPTER6.APPENDIX:CONDITIONALPOISSONPROCESS6.2PoissonProcesswithDeterministicIntensityR1Let¸:R+!R+beanynon-negative,locallyintegrablefunctionsuchthat0¸(u)du=1:Byde¯nition,theprocessN(withN0=0)isthePoissonprocesswithintensityfunction¸ifforevery0·s0;wede¯netheprobabilitymeasureQon(•;GT)bysetting:¯dQ¯¯=´T;P-a.s.,(6.12)dPGTwheretheRadon-Nikod¶ymdensityprocess´t;t2[0;T];solvestheSDE¡¢d´t=´t¡¯tdWt+·tdNbt;´0=1;(6.13)forsomeG-predictableprocesses¯and·suchthat·>¡1andEP(´T)=1:AnapplicationofIt^o'sproductruleshowsthattheuniquesolutionto(6.13)isequaltotheproductºt³t;wheredºt=ºt¯tdWtandd³t=³t¡·tdNbt;withº0=³0=1:Thesolutionstothelasttwoequationsare³ZtZt´12ºt=exp¯udWu¡¯udu020andY³t=exp(Ut)(1+¢Uu)exp(¡¢Uu);0