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Pricing models and analysis of corporate coupon-bonds and credit default swaptions

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Pricing models and analysis of corporate coupon-bonds and credit default swaptions
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Shibata, Michiru
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CDS
Intensity models
Structural models
Black-scholes
Jump
Dissertations, Academic -- Mathematics and Statistics -- Doctoral -- USF   ( lcsh )
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ABSTRACT: In this work, pricing models of corporate coupon-bonds and credit default swaptions are derivedand analyzed. Corporate coupon-bonds are priced incorporating both intensity models and structural models, and also jumps introduced by seasonal effects. In deriving the models, we form portfolios to hedge the risk incurred by the instruments, then derive PDE equations using the arbitrage principle and the Ito Lemma for jump processes. The mathematical models are the parabolic-type PDE equations with terminal conditions and boundary conditions. These PDE problems are analyzed and solved by various transformations and incorporation with probabilistic properties. Either a unique solution in the exponential form is obtained, or a particular solution in the separation formis acquired. Further, the pricing model of credit default swaptions is derived using the pricing of corporate coupon-bonds in the similar manner. The main idea of deriving the price of credit default swaptions is to use the price of existing products, i.e., corporate bonds, as opposed to the existing models, which use non-existing forward credit default swap price of the reference entity. The prices of corporate coupon-bonds and credit default swaptions with unexpected default, obtained from these models, are compared to the actual market prices and analyzed.
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Dissertation (Ph.D.)--University of South Florida, 2007.
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Includes bibliographical references.
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by Michiru Shibata.
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Pricing models and analysis of corporate coupon-bonds and credit default swaptions
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ABSTRACT: In this work, pricing models of corporate coupon-bonds and credit default swaptions are derivedand analyzed. Corporate coupon-bonds are priced incorporating both intensity models and structural models, and also jumps introduced by seasonal effects. In deriving the models, we form portfolios to hedge the risk incurred by the instruments, then derive PDE equations using the arbitrage principle and the Ito Lemma for jump processes. The mathematical models are the parabolic-type PDE equations with terminal conditions and boundary conditions. These PDE problems are analyzed and solved by various transformations and incorporation with probabilistic properties. Either a unique solution in the exponential form is obtained, or a particular solution in the separation formis acquired. Further, the pricing model of credit default swaptions is derived using the pricing of corporate coupon-bonds in the similar manner. The main idea of deriving the price of credit default swaptions is to use the price of existing products, i.e., corporate bonds, as opposed to the existing models, which use non-existing forward credit default swap price of the reference entity. The prices of corporate coupon-bonds and credit default swaptions with unexpected default, obtained from these models, are compared to the actual market prices and analyzed.
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PricingModelsandAnalysisofCorporateCoupon-BondsandCreditDefaultSwaptionsbyMichiruShibataAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofDoctorofPhilosophy DepartmentofMatematicsandStatistics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:YunchengYou,Ph.D. Wen-XiuMa,Ph.D. MasahicoSaito,Ph.D. CarolWilliams,Ph.D. DateofApproval:April11,2007 Keywords:CDS,IntensityModels,StructuralModels,Black-Scholes,Jump cCopyright2007 ,MichiruShibata

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AcknowledgmentsWhenIlookbackontherstdayatthedepartmentofMathematics,IneverimaginedthatIwouldhavewhatIhavetoday.Myjourneyleadingtothisthesisalsostartedonthesamedayandisabouttoend.WhatawonderfuloneIhavehad.Itwasbothacademicandspiritual.Iowethisresulttosomanypeopleinmylife.Iwouldliketothankmyadvisor,Dr.YunchengYou.WhenIwantedtorelatemathematicstotherealworld,therstthingthatcametomymindwasnance.Hewastheverypersonwhoofferedmethebestpossibleguidanceandassistance.Iwouldalsoliketothanktheothercommitteemembers:Dr.Wen-XiuMa,Dr.MasahicoSaitoandDr.CarolWilliamswhotookeffortinreadingthisthesis.IrenaAndreevska:Wetookandstudiedforthesamenancialmathcoursestogether.ShewastheonewhoIturnedtorstwheneverIhaveaproblem.Shewasofimmensesupportformebothacademicallyandpersonally.Myfriends:Armando,Betty,David,Ed,Harry,Joji,John,Kevin,Lynda,Nathanandmysangha.AsIworkedonmydissertation,Iwentthroughamajorpersonalissueinmylife.Withoutfriendshipandsupportyouhaveprovidedme,thisthesiswouldnothaveexisted.Thankyouall.IwouldalsoliketothankBev,Denise,Francis,MaryAnn,Sarina,Nancy,AyaandBarbaraforallthehelpandsupport.YoualwayshavemademefeellikeIamhome...evenafterIresignedfromtheteachingassistantship.SpecialthankstoTony.Ithinkyouaretheonewhowantedtoseethisdaymost.Icannotthankyouenoughforallyoursupport.Lastbutnottheleast,myfamily.Mom,Dad,Satomi,Yutaka,Sho,ShunandSatoshi.Iloveyouallandthankyou.

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TableofContentsListofTables..........................................iiiListofFigures..........................................ivAbstract.............................................vChapter1Introduction....................................11.1BackgroundandMotivation.............................11.2DenitionandBasicConcepts............................31.2.1MathematicalConcepts............................31.2.2FinancialConcepts..............................51.3Black-ScholesFormulas...............................141.3.1PDEApproach................................151.3.2MartingaleApproach.............................16Chapter2DefaultableCorporateCoupon-BondPricingwithConstantInterestRate...182.1CorporateCoupon-BondwithConstantInterestRate-UnexpectedDefault.....182.1.1Formulation..................................182.1.2Derivationofthemodel...........................192.2CorporateCoupon-BondwithConstantInterestRate-ExpectedandUnexpectedDefault........................................312.2.1Formulation..................................312.2.2DerivationoftheModel...........................322.2.3ParticularSolution..............................42Chapter3CreditDerivativesPricingwithConstantInterestRate..............503.1CreditDefaultSwaptionwithConstantInterestRate-UnexpectedDefault.....503.1.1Formulation..................................513.1.2Derivationofthemodel...........................523.2CreditDerivativesPricingwithConstantInterestRate-ExpectedandUnexpectedDefault........................................553.2.1Formulation..................................553.2.2Derivationofthemodel...........................563.2.3ParticularSolution..............................58Chapter4DefaultableCorporateCoupon-BondPricingwithStochasticInterestRate...604.1CorporateCouponBondwithStochasticInterestRate-UnexpectedDefault....604.1.1Formulation..................................604.1.2Derivationofthemodel...........................61i

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4.2CorporateCouponBondwithStochasticInterestRate-ExpectedandUnexpectedDefault........................................684.2.1Formulation..................................684.2.2DerivationoftheModel...........................694.2.3ParticularSolution..............................73Chapter5CreditDerivativesPricingwithStochasticInterestRate.............795.1CreditDefaultSwaptionwithStochasticInterestRate-UnexpectedDefault....795.1.1Formulation..................................795.1.2Derivationofthemodel...........................805.2CreditDefaultSwaptionwithStochasticInterestRate-ExpectedandUnexpectedDefault........................................845.2.1Formulation..................................845.2.2DerivationoftheModel...........................855.2.3ParticularSolution..............................88Chapter6DataAnalysis...................................916.1Data..........................................916.2BondPrice......................................1006.3CreditDefaultSwaptionPrice............................103Chapter7Conclusion....................................106References............................................109AbouttheAuthor.....................................EndPageii

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ListofTablesTable1CashowoftheProtectionBuyerwithorwithoutCreditDefaultSwaption....10Table2Long-TermBondRatingSystem..........................14Table3HistoricalShort-TermRateandYieldsofCorporateBonds.............91Table4Parametersimpliedfromtheregressionestimates..................97Table5EmpiricalSurvivalRatebyOriginalCreditQuality.................99Table6FordMotorCreditBondPricewithConstantr...................101Table7Parametersimpliedfromtheregressionestimates-AAAcorporatebonds.....102Table8FordMotorCredit-CDSSwaptionPrice......................103Table9FordMotorCredit-CDSForwardPriceComparison................105iii

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ListofFiguresFigure1CreditDefaultSwap................................8Figure2HistoricalRisk-freeInterestRateandCorporate-BondYield...........92Figure3HistoricalandSamplePathsofRisk-freeInterestRate..............98Figure4SamplePathsoftheIntensityp...........................100Figure5HistoricalandSamplePathsoftheYieldofAAACorporate-Bonds.......102Figure65-Year-MaturityFORDCreditCDSQuotes....................104iv

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PricingModelsandAnalysisofCorporateCoupon-BondsandCreditDefaultSwaptionsMichiruShibataABSTRACTInthiswork,pricingmodelsofcorporatecoupon-bondsandcreditdefaultswaptionsarederivedandanalyzed.Corporatecoupon-bondsarepricedincorporatingbothintensitymodelsandstructuralmodels,andalsojumpsintroducedbyseasonaleffects.Inderivingthemodels,weformportfoliostohedgetheriskincurredbytheinstruments,thenderivePDEequationsusingthearbitrageprincipleandtheItoLemmaforjumpprocesses.Themathematicalmodelsaretheparabolic-typePDEequationswithterminalconditionsandboundaryconditions.ThesePDEproblemsareanalyzedandsolvedbyvarioustransformationsandincorporationwithprobabilisticproperties.Eitherauniquesolutionintheexponentialformisobtained,oraparticularsolutionintheseparationformisacquired.Further,thepricingmodelofcreditdefaultswaptionsisderivedusingthepricingofcorporatecoupon-bondsinthesimilarmanner.Themainideaofderivingthepriceofcreditdefaultswaptionsistousethepriceofexistingproducts,i.e.,corporatebonds,asopposedtotheexistingmodels,whichusenon-existingforwardcreditdefaultswappriceofthereferenceentity.Thepricesofcorporatecoupon-bondsandcreditdefaultswaptionswithunexpecteddefault,obtainedfromthesemodels,arecomparedtotheactualmarketpricesandanalyzed.v

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Chapter1Introduction1.1BackgroundandMotivationWhenacompanyneedsnancing,thereareonlytwowaystoraisemoney.Eithertoborrow,ortondsomeonetoco-ownthecompany.Thelatterisdonebyissuingnewstocks;theformerbasicallytakestwoforms:borrowingsfrombanksorotherinstitutions,andissuingbonds.Financialinstitutions,suchasbanks,securitieshouses,insurancecompaniesandsoon,havebeentraditionallylargestparticipantsinthebondmarket.Theyinvestinbondsnotonlybecausetheyconstituteapartoftheirinvestmentportfolios,butalsobecausetheywanttomaintaintherelation-shipwithbondissuers.Ifthatisthecase,oftenthetime,theyareexpectedtoholdthebondsuntiltheirmaturities,anditcouldbeanuisancetotheinvestinginstitutionforthefollowingreasons.First,mostnancialinstitutionshaveinternalandexternalguidelinesaboutthemaximumamountofnancialproductstheycaninvest.Soholdingbondsuntiltheirmaturitiespreventthemfromdiversifyingtheirportfolios.Second,itincursariskofunreasonablelossincaseofdefault.There-fore,itisimportantforinstitutionalinvestorstoassessthepriceofthebondsintheportfolioandalsotoavoidtheriskinherentinthebondswithoutsellingthem.Findingapricingmodelofdefaultablebondsisoneofthemainpurposesofthispaper.Therearetwofundamentalapproachestobondpricing:iintensitymodels,andiistructuralmodelsalso,knownasMerton'smodel.Eachtypeofmodelshasprosandcons;buttheemphasisofbothmodelsisonndingorestimatingthedefaultprobability.IntensitymodelsconsiderdefaultasanexogenouseventandcanbefoundinJarrow&Turnbul,Dufe&Singleton,Hughston&Turnbull,etc.Instructuralmodels,defaultisconsideredtobeanendogenous1

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event,thatis,defaultoriginatesfromwithinthermstructure.InMerton'smodel,arm'stotalassetscompriseofonezero-couponbondandonestock,whichfollowsageometricBrownianmotion.Thermdefaultsifitsassetsfallbelowthevalueofitsoutstandingbond.Black&Coxextendedthisframeworkbybringingacertainthresholdasabarrier.Mostexistingmodelsadopteitheroneapproachortheother.O,etal.combinedbothap-proachesandcameupwithanewmodeloncorporatezero-couponbonds.However,mostcorporatebondsbearcoupons,andcoupon-bondissuers,especiallysmall-sizecompanies,areexposedtotheriskofdefaultasinterestpaymentdatesapproach.InChapters2and4ofthispaper,theideaofOisextendedtocorporatecoupon-bondsandwetakeintoconsiderationthedefaultriskarisenbycouponpayment.However,ndingabetterpricingmodeldoesnotresultinavoidingthedefaultriskincurredbyholdingbonds.Newnancialderivatives,calledcreditderivatives,wereintroducedtothenan-cialmarketintheearly1990s,whichmadeitpossibleforbondholderstogetridofdefaultriskwithoutsellingbonds.Acreditderivativeisaderivativesecuritywhosepayoffisconditionedontheoccurrenceofcrediteventssuchasbankruptcyofacertainbond-issuer.Twothirdsofthecreditderivativesinthecurrentmarketarecreditdefaultswaps.ApricingmodeloncreditdefaultswapsisgivenbySchonbucher,2003b.Somepricingmodelsofcreditdefaultswaption,whichisanoptiononforwardcreditdefaultswaparegivenbySchonbucherandSchmidt.Intheirpapers,theyassumethattheyknowthedynamicsofthevalueofunderlyingforwardcreditdefaultswap,andtheyhedgetheswaptionagainsttheforwardcreditdefaultswap.However,thedynamicsofthevalueofforwardcreditdefaultswapisnoteasytoobtain.InChapters3and5ofthispaper,wewilltrytondthepricingmodeltohedgetheswaptionbythebondsissuedbytheforwarddefaultswap'sreferenceentity.Therestofthispaperisorganizedasfollows.Inthefollowingsection,basicmathematicalandnancialconceptsareintroduced.Chapter2introducesthepricingmodelsofcorporatecoupon-bonds,whichincorporatethejumptermscausedbycouponpayments.InChapter3,wewillusethedynamicsofthevalueofthecorporatecoupon-bondstondthepriceofthecreditdefaultswaption.InChapters2and3,weassumethattheshort-term,risk-freeinterestrateisconstant.InChapter4,were-examinethepricingofcorporatecoupon-bonds,however,thistimewithastochasticshort-2

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termrisk-freeinterestrate.WeremodelthepriceofacreditdefaultswaptionusingthebondpricingmodelestablishedinChapter5.Chapter6devotestothedataanalysisofthebond-priceandthecreditdefaultswaptionpricewithconstantrisk-freerate.Finally,inChapter7,wesummarizetheresultofthispaperanddiscussthedirectionsoftheresearchinthefuture.1.2DenitionandBasicConcepts1.2.1MathematicalConceptsDenition1.2.1MartingaleAstochasticprocessfXt;t0gadaptedtoaltrationFiscalledamartingaleifforanys
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Theequationhasananalyticsolution:St=S0exp)]TJ/F8 10.909 Tf 5 -8.836 Td[()]TJ/F18 10.909 Tf 10.909 0 Td[(2=2t+dWtforanarbitraryinitialvalueS0.Denition1.2.4ItoIntegralLet;F;PbeaprobabilityspaceonwhichastandardBrownianmotionWtisdened.LetXtbeasimpleprocesswhichisadaptedtothesameltrationasWtandgivenbyXt=c0I0t+n)]TJ/F16 7.97 Tf 6.587 0 Td[(1Xi=0ciIti;ti+1]t;Then,ItointegralRT0XtdWtforasimpleprocessXt,isdenedasZT0XtdWt=n)]TJ/F16 7.97 Tf 6.586 0 Td[(1Xi=0ci)]TJ/F18 10.909 Tf 5 -8.837 Td[(Wti+1)]TJ/F18 10.909 Tf 10.909 0 Td[(Wti:LetXntbeasequenceofsimpleprocessesconvergentinprobabilitytotheprocessXtsatisfyinglimn!1R10E)]TJ/F18 10.909 Tf 5 -8.836 Td[(Xt)]TJ/F18 10.909 Tf 10.909 0 Td[(Xnt2dt=0.ThenItointegralwithgeneralprocessXtisdenedasZT0XtdWt=limnZT0XntdWt:Lemma1.2.5ItoLemmaforJumpProcessesLetXtbearight-continuousstochasticprocesswithleftlimit,andithasatmostnitenumberofjumpsovernitetimeintervalsa.s.Foreverypathoftheprocess,wedene:Xt)]TJ/F8 10.909 Tf 8.485 0 Td[(:=limh!0Xt)]TJ/F18 10.909 Tf 10.909 0 Td[(h;X)]TJ/F8 10.909 Tf 8.484 0 Td[(:=X;Xt:=Xt)]TJ/F18 10.909 Tf 10.909 0 Td[(Xt)]TJ/F8 10.909 Tf 8.485 0 Td[(;Xdt:=XstXs;Xct:=Xt)]TJ/F18 10.909 Tf 10.909 0 Td[(Xdt:Alternately,letX=X1;X2;;Xnbeann-dimensionalsemi-martingalewithanitenumberofjumps,andfatwicecontinuouslydifferentiablefunctiononRd.ThenfXisalsoasemi-4

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martingale,anditfollowsthat:fXt)]TJ/F18 10.909 Tf 10.909 0 Td[(fX=nXi=1Zt0@fXs)]TJ/F8 10.909 Tf 8.485 0 Td[( @xidXc;i+1 2nXi;j=1Zt0@2fXs)]TJ/F8 10.909 Tf 8.485 0 Td[( @xi@xjds+XstfXs;wheretheintegralmeansastochasticintegralcf.Kigimaandstandsforthequadraticcovarianceoftwostochasticprocessescf.Kigima.ItsproofcanbefoundinJacodandShiryaev.1.2.2FinancialConceptsDenition1.2.6BondsThebondisadebtinstrumentissuedforaperiodoftimeinpurposeofraisingmoney.Itpromisestorepaytheprincipalamountonaspecieddaymaturitydate.Somebondsbearcoupons,whicharepromissorynotesforinterest;sotheyarecalledcouponbearingbonds.Othersdonotpayinterest;insteadtheyaresoldatadeepdiscountedprice,sotheyarecalledzero-couponbondsordeepdiscountedbond.Bondpricesuctuateinaccordancetotwofactors:changesininterestratesandchangeincreditquality.Theinterestrateconsiderediswhatisknownasarisk-free,short-terminterestrate,rt.InChapters2and3,weconsiderrttobeconstant;inchapters4and5,weconsiderrttofollowtheVasicekmodel.Hereweintroduceseveralinterestratemodels:Oneoftheearliestshort-ratemodelsintroducedbyBlackandRendlemanandBartterislognormallydistributedandgivenbydr=rdt+rdW;;:constant;whereW=fWtgt0isastandardBrownianmotion.Hereinafter,allW'susedinStochasticdifferentialequationsarestandardBrownianmotions.However,thismodeldoesnotcapturemean-revertingpropertyofinterestrate;Vasicekintroducedthefollowingnormalmean-revertingprocesswithconstantparameters,i.e.,dr=a)]TJ/F18 10.909 Tf 10.909 0 Td[(rdt+dW;;a;:constant:5

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Thedrawbackofthismodelisthattheshorttermratecanassumeanegativenumber.In1985,Cox,Ingross,andRossCIRaddedsquare-rootdiffusiontermtotheVasicekmodel,whichmakesrtchi-squaredistributedandgivenbydr=a)]TJ/F18 10.909 Tf 10.91 0 Td[(rdt+p rdW;;a;:constant:HullandWhiteextendedtheVasicekmodeltotboththecurrentstructureandvolatilitiesofinterestrates.Intheirmodel,theshort-termratefollowsanormalmean-revertingprocesswithtimedependentparametersandisgivenbydr=t)]TJ/F18 10.909 Tf 10.91 0 Td[(rdt+tdW;:constant:Ifwetaketobetime-dependent,themodelisknowasextendedVasicekmodel.Wecanwritethemodelasdr=t)]TJ/F18 10.909 Tf 10.909 0 Td[(trdt+tdW;whichcanalsobewrittenasdr=t)]TJ/F18 10.909 Tf 10.909 0 Td[(rtdt+srtdW;:constanttobetterseetheattributesoftheshorttermrate.Herethevariablesristhesameas,andisusedsothatitalignswiththerestofthepaper.InthisextendedVasicekmodel,givesthemean,giveshowfasttherateuctuates,andsrgivesthevolatilityoftheinterestrate.Thelastthreemodelsarewidelyacceptedandpopularinpracticebecauseoftheirclosedformsolutions.InChapter4and5,weshallusethisextendedVasicekmodelforrisk-free,short-termrate.Thecouponbearingbondconsistsoftheprincipalandpredeterminednumberofcouponsattachedtotheprincipal.Theprincipalispaidatthefacevalueonthematuritydateunlesstherearesomeotherconditionsstatedotherwise.Eachcouponpayspredeterminedamountofmoney,normallyexpressedasacertainpercentagecouponrateofthefacevalueoftheprincipal,orasadollaramount,onapredetermineddate.Couponsaredetachablefromtheprincipalandtransferablebythemselves.Therefore,thecouponbearingbondcanbeconsideredasaportfolioofzero-couponbearingbonds:onezero-couponbearingbondwithaprincipalbeingthesameastheoriginalbondandnzero-couponbondswithaprincipalbeingthesameamountastheoriginalbond'scoupon,eachmaturing6

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onoriginalbond'si-thcoupondate,wherenisthenumberofcouponpaymentsandi=1;2;;nLaGrandvill,2001.Soifweletthevalueofcorporatecoupon-bondattimettobeG=Gr;t;T,thevalueofzero-couponbondtobeCr;t;T,whereTisthematurityofthebond,andcitobei-thcouponrate,thenwehaveGr;t;T=Cr;t;T+XtitciCr;t;tiOftenthetime,thecouponsareseparatedfromthebodyofthebondwhichpaysonlytheprincipalamountandtheyaretradedseparately.Thusseparatedbondiscalledastrippedbond.Eachcouponentitlesthecouponholdertobepaidacertainpercentageofthefacevalueofthebond.Inmostpapersanddocumentse.g.Hanke,2003,couponsaretreatedaspayingacertainpercentageofthebondprice,whichisincorrect.Tocorrectthiskindoftreating,throughoutthispaper,thecouponbearingbondisconsideredasaportfolioofzero-couponbearingbonds.Denition1.2.7OptionTheoptionisanancialderivativewhichgivestheholdertherightbutnottheobligationtobuycalloptionortosellputoptionaparticularassetsuchasstocksorbondsataspeciedtimeortimeperiodinthefutureforpreviouslydeterminedpricestrikepriceorexerciseprice.Ifexerciseispermittedonlyatexpiry,theoptionsarecalledEuro-peanoptions,andifexerciseisallowedatanytimebeforeexpiry,theyarecalledAmericanoptions.Inthechaptersfollowed,theoptionsareassumedtobeEuropean.ThepayofffunctionsofcalloptionandputoptionaregivenbymaxST)]TJ/F18 10.909 Tf 10.909 0 Td[(E;0andmaxE)]TJ/F18 10.909 Tf 10.909 0 Td[(ST;0respectively,whereEistheexercisepriceandSTisthepriceoftheunderlyingassetattheexpirationdate.Denition1.2.8Put-CallParityIfC,PandSarethepricesofacalloption,aputoptionandtheirunderlyingassetattimet,respectively,andTistheexpirationoftheoptions,thentheysatisfythefollowingequation:C)]TJ/F18 10.909 Tf 10.909 0 Td[(P=S)]TJ/F18 10.909 Tf 10.909 0 Td[(Ee)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(t;7

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whereristheriskfreeinterestrate.Thisrelationshipiscalledput-callparity.Thisrelationshipisusefulsinceoncewendthepriceofthecalloption,itenablesustondthepriceofputoptioneasilyandviceversa.Denition1.2.9ForwardAforwardisacontractobligatingonepartytobuyandtheotherpartytosellanancialinstrument,suchasstock,bond,commodityorcurrencyataspecicfuturedate.Denition1.2.10CreditDefaultSwapTheCreditDefaultSwapCDSisabilateralnancialcontractinwhichonecounterpartytheProtectionBuyerpaysaperiodicfee,paidontheno-tionalamountandtheothercounterpartytheProtectionSellerpaysapredeterminedamountincaseacrediteventwithrespecttoareferenceentityoccurs.TheschemeisshowninFigure1. Figure1.:CreditDefaultSwapThereferenceentityistheissuerofthebonds,whosecrediteventtriggerstheprotectionseller'sobligations.Crediteventsarepreciselydenedineachcontract;theynormallyinclude:Bankruptcy,Failuretopayinterestorprincipal,Obligationdefault8

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Thisiswherethereferenceentity'sobligationbecomesdueasaresultofanycovenantbreachundertherelativeobligationcontract,ObligationaccelerationThisiswherethebond-holdersdemandimmediaterepaymentinfullasaresultofanycovenantbreachofotherobligationsofthereferenceentity,ReconstructionReconstructionincludeseventssuchasareductionintheprincipalamountorinterestpayableundertheobligation,apostponementofpayment,achangeinrankinginpriorityofpaymentoranyothercompositionofpayment.TheprotectionbuyerdoesnotnecessarilyneedtoholdthebondsissuedbyareferenceentityasinthecasewherethebuyerdealsCDSsforspeculation.Creditdefaultswapsarethemosttradedderivativeonthemarketofallcreditderivatives.Denition1.2.11CreditdefaultswaptionCreditdefaultswaptionisanoptiononcreditdefaultswap.Theunderlyingcreditdefaultswapdoesnotexistduringthelifeofcreditdefaultswaption;itisinitiatedonlyupontheexerciseoftheswaption.Example.Lett0bethetradingdateofthecalloption,T0tbethedateatwhichtheforwardcreditdefaultswapbecomeseffective,andTNT0bethematuritydateoftheforwardcreditdefaultswap.TheexpirationdateoftheoptionfallsonT0.Theswaptionholderisentitledtoenterthecreditdefaultswaptherefore,becomesaprotectionbuyerattimeT0atthepredeterminedperiodicfee,says.LetthepriceoftheswaptionbeXtor^Xt;0tT0asmodeledlater.Letstbetheperiodicfeeofaforwardcreditdefaultswapattimet,withthesamereferenceentityandthesamecontractdurationastheunderlyingcreditdefaultswapfortheswaption;sothatsT0isthepriceofthecreditdefaultswapatt=T0.Aninvestorbuystheswaptiononlyifhe/sheanticipatesthatthecreditofthereferencecompanydeterioratesandthemarketfeeatthetimeT0,i.e.,sT0exceedss.Thisway,iftheswaptionholderactuallyholdsthebondsissuedbythereferencecompany,itcanacquirethecreditdefaultswapforlesscost.Iftheswaptionholderdoesnotownthebonds,thenitcanlocktheprotsT0)]TJ/F18 10.909 Tf 11.179 0 Td[(signoringtheinitialcostbysellingtheprotectiononthebondsissuedbythereferencecompany.Table1showsthesummaryofthe9

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cashowinthreescenarios.Notethat1%=1p=100bpbasispoints.Table1:CashowoftheProtectionBuyerwithorwithoutCreditDefaultSwaption WithSwaption WithoutSwaption Case1 Case2 Case Thereferenceentity's Thereferenceentity's nancialconditiondeteriorates nancialconditionimproves Initialcost Xor^X 0 0 Periodicfee s=50bp 100bp 25bp Denition1.2.12NoArbitragePrincipleLooselystated,theprincipleassertsthatthereisnosuchthingasafreelunch.Formally,Principle1:Ifthevalueoftwoportfoliosare1tand2tattimet,then1t2tif1T2Ta.s.,t0ifandonlyifQA>0forallA2F,andthediscountedpriceprocessS=Sn0nNisamartingalewithrespectto;F;QwithltrationfFn;n=0;:::;Ng.Underariskneutralmeasurethecurrentpriceofeachsecurityintheeconomyisequaltothepresentvalueofthediscountedexpectedvalueofitsfuturepayoffsgivenarisk-freeinterestrate.Underaviablemarket,i.e.,underthemarketwherethereisnoarbitrageopportunities,theFundamental10

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TheoremofAssetPricingguaranteestheexistenceandtheuniquenessofsuchriskneutralmeasure.ItsproofcanbefoundinHarrison&Pliska.IfthepayoffpriceofanancialproductattimeTisHT,whereHTisarandomvariableontheprobabilityspacedescribingthemarket,andthediscountfactorfromtimet=0tot=TisP;T,thenthefairpriceoftheproductattimet=0isgivenbyH=P;TEQ[HT];wheretherisk-neutralmeasureisdenotedbyQ.IftherealworldprobabilitymeasureofHTisgivenbyP,thenHcanbealsogivenasH=EPdQ dPHT;wheredQ dPistheRadon-NikodymderivativeofQwithrespecttoP.Thevaluationofcreditdefaultswapsrequiresestimatingbothexpecteddefaulttimeandtheex-pectedlossofthereferenceentity.Theriskneutraldefaultprobabilitycanbeestimatedeitherfromthereferenceentity'stotalassetorfromdebtmarket.Whenweusethetotalasset,weactuallyconsiderthecompany'snancialstructure;hencedefaultisconsideredtobeanendogenousevent,andthemodelsarecalledstructuralmodels.Ontheotherhand,whenweusethedatafromdebtmarkettoestimatethedefaultprobability,wecompletelyignorethecompany'snancialstructureandconsiderdefaultisanexogenousevent.Thesemodelsarecalledintensityorreducedmod-els.Bothstructuralmodelsandintensitymodelshaveprosandcons;wewillexaminebothmodelsbrieybelow.Denition1.2.14StructuralmodelsLetthemarketvaluesofthermasset,theequity,andthedebtofacompanyattimetbeVt;StandCt,respectively.Then,wehavetherelationshipVt=St+Ct,whichisknownastheaccountingequation.Theleft-handsideofthisequationexplainshowtherm'smoneywasinvested,inanutshell.Theright-handsideofthisequationrepresentsthesourceoftherm'sassets,thatis,howtherm'smoneywasraised.ItshouldbenotedthatwhileVtandCtarenon-negative,Stcanassumenegativevalue.Inthesemodels,weassumethatVtfollowsageometricBrownianmotion.Then,thepayoffRtofthedebtatit11

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maturityTisgivenbyRT=min)]TJ/F18 10.909 Tf 5 -8.836 Td[(CT;VT:Therefore,underthesemodels,weconcludethatthecompanyhasdefaultwhenthermassetVTfallsbelowCT,orotherpredeterminedbarrieraswewillseeinSubsection2.2.1.Themeritsofusingstructuralmodelsare:Theyintuitivelymakeaneconomicsense,anddefaultisconsideredasanendogenousevent.Thetimeofdefaultisnotrandomasopposedtotheintensitymodel.Wecanassessthevalueofdefaultabledebt.However,thesemodelshavethefollowingdemerits:Theyareunwieldytoimplement.Theyareincoherentwiththehistoricaldata.Denition1.2.15IntensitymodelsInintensitymodelsthetimeofdefaultistherstjumpofanexogenouslygivenjumpprocess.Theparametergoverningthedefaultintensityareinferredfromtherelativemarket.ThedefaulttimeismodeledastherstjumpofaPoissonprocess.Letusassumethatthedefaultintensityfollowsastochasticprocessoftheformdpt=atdt+stdWtwhereWtisaBrownianmotion.Ifthedefaulttimeis,then,thesurvivalprobabilityattimetisgivenbyP[>t]=exph)]TJ/F24 10.909 Tf 10.909 14.849 Td[(Zt0psdsi:Thedefaultintensityptistheinstantaneousrateofdefault.LetPt;Tbetheconditionalproba-12

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bilityofsurvivalattimeTasseenfromtimet
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Moody's,andStandardandPoor'sS&P'sareconsideredtobeworld-widetopcredit-ratingagen-cies.Eachofthemintendstoprovidearatingsystemtohelpinvestorsdeterminetheriskassociatedwithinvestinginaspeciccompany,inaninvestinginstrumentorinamarket.Table2summarizesthedifferentratingssymbolsthatMoody'sandStandardandPoor'sissueforlong-termdebtobligationsSource:Heakal,R,WhatIsACorporateCreditRating?,Wikipedia.Table2:Long-TermBondRatingSystem Moody's StandardPoor's Risk InvestmentGrade Aaa AAA Thebestqualitycompanies,reliableandstable Aa AA Qualitycompanies,abithigherriskthanAAA A A Economicsituationcanaffectnance Baa BBB Mediumclasscompanies,whicharesatisfactoryatthemoment Non-Investment/SpeculativeGradealsoknownasjunkbonds Ba BB Morepronetochangesintheeconomy B B Financialsituationvariesnoticeably Caa CCC Currentlyvulnerableanddependentonfavorable economicconditionstomeetitscommitments Ca CC Highlyvulnerable,veryspeculativebonds C C Highlyvulnerable,perhapsinbankruptcyorinarrears butstillcontinuingtopayoutonobligations Ratingscanbeassignedtoshort-termandlong-termdebtobligationsaswellassecurities,loans,preferredstockandinsurancecompanies.Long-termcreditratingstendtobemoreindicativeofacountry'sinvestmentsurroundingsand/oracompany'sabilitytohonoritsdebtresponsibilities.1.3Black-ScholesFormulasThederivationofthecorporatecoupon-bondinthispaperhasmanysimilaritiestothederivationoftheBlack-Sholesformulas.TheBlack-ScholestheoryhasitsoriginintheseminalpaperThePric-14

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ingofOptionsandCorporateLiabilities,byBlack&Scholesandhasbeenmostwidelycitedinoptionpricing.TheBlack-ScholesFormulagivesthepriceofaEuropeancalloptionCSt;TwithexercisepriceKonastockcurrentlytradingatpriceSt,i.e.,therighttobuyashareofthestockatpriceKafterTyears.Theconstantinterestrateisr,andtheconstantstockvolatilityis.TheformulaisgivenbyCS;T=Std1)]TJ/F18 10.909 Tf 10.909 0 Td[(Ke)]TJ/F19 7.97 Tf 6.587 0 Td[(rTd2.1whered1=lnSt=K+r+2=2T p Tandd2=d1)]TJ/F18 10.909 Tf 10.909 0 Td[(p T.2Inthefollowingsubsections,wewillshowtwoapproachestoderivethisformula:iPDEapproach,andiiMartingaleapproach.Inderivation,pleasenotethatweassumethatthemarketisfrictionlessandthepriceoftheunderlyinginstrumentStfollowsthelognormalmodel,thatis,itsatisesthefollowingstochasticdifferentialequation:dSt=Stdt+StdWt;.3whereS0>0,thedriftandvolatilityareconstants,WtisaBrownianmotiondenedonalteredprobabilityspace;F;Ft;P,whereFt=Ws;st.1.3.1PDEApproachTheBlack-ScholesPDEwasrstintroducedinThePricingofOptionsandCorporateLiabilities,byBlack&Scholes.Inthisapproach,theideaofhedgingandarbitragewasused.FirstweconstructaportfolioconsistingofonederivativeCand'unitsoftheunderlyinginstrumentS.Thevalueoftheportfoliois=V+'S.Thenthechangeinpriceoftheportfoliooverasmalltimeincrementisgivenbyd=dC+'dS=@C @SdS+@C @tdt+1 2@2C @s22S2dt+'dSBylognormalmodel.3=)]TJ/F18 10.909 Tf 6.196 -1.457 Td[(@C @S+'SdW+)]TJ/F18 10.909 Tf 6.196 -1.457 Td[(@C @S+'Sdt+@C @tdt+1 22S2@2C @s2dt.415

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Bychoosing'sothatuncertaintycausedbydW-termiseliminatedcompletely,i.e.,bysetting'=)]TJ/F18 10.909 Tf 9.68 7.38 Td[(@C @S,and,applyingthearbitrageprinciple,weobtainthefollowingBlack-ScholesPDE:@C @t+1 22S2@2C @S2+rS@C @S)]TJ/F18 10.909 Tf 10.909 0 Td[(rC=0.5whereristherisk-freeinterestrateand0tT.IfthederivativeindiscussionisacalloptiononsomenancialinstrumentwithexercisepriceKandexpirationT,thenafterapplyingchangesofvariablesandFourierTransformation,wewillndtheBlack-ScholesFormulas.1and.2asasolutionfortheBlack-ScholesPDE.1.3.2MartingaleApproachHarrison&KrepsandHarrison&Pliskashowedthatanaturalmathematicalframe-workforanalysisofnancialmarketsismartingaletheoryandstochasticanalysis.LettheprocessBtbethevalueoftherisk-freeaccountsatisfyingdBt=rBtdtwithB0=1.Letfa;bgbeapairofFt)]TJ/F51 10.909 Tf 8.485 0 Td[(adaptedprocesswhichiscalledatradingstrategy,whereatandbtarenumbersoftheunitsoftheassetandtherisk-freeaccountattimet,respectively.Then,thevalueofthetradingstrategyattimetofaportfoliofat;btgisgivenbyVt=atSt+btBtLemma1.3.1Atradingstrategyisself-nancingmeaningthechangeofthevalueofthetradingstrategyisduetothechangesintheassetsprices,i.e.,dVt=atdSt+btdBtifandonlyifitsdiscountedwealthprocess~Vtsatisesd~Vt=atd~St:.6ItsproofisfoundinYan&Ju.Notethatwecanrewrite.3asd~St=~St)]TJ/F18 10.909 Tf 10.91 0 Td[(rdt+dWt.7andbyputtingdQ dPjFt=expf)]TJ/F19 7.97 Tf 15.135 4.932 Td[()]TJ/F19 7.97 Tf 6.586 0 Td[(r WT)]TJ/F16 7.97 Tf 12.299 4.296 Td[(1 2)]TJ/F19 7.97 Tf 6.586 0 Td[(r 2Tg,wehaveQ-BrownianmotionWt=Wt+)]TJ/F19 7.97 Tf 6.586 0 Td[(r tbytheGirsanov'stheorem,andd~St=~StdWt;.816

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sothat~StisaQ-martingale.Theorem1.3.2Let=fSTbeaEuropeancontingentclaimunderQ-martingale.Thenthereexistsanadmissibleself-nancingstrategyfa;bgreplicatingsuchthatitsvalueprocessVtisgivenbyVt=E[e)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(tjFt].9orequivalently,thediscountedprocessVtisaQ-martingale.ItsproofcanbefoundinYan&Ju.NotethatwecansaythatVtisthefairpriceattimetofthecontingentclaimsincethereisnoarbitrageopportunityatthisprice.Corollary1.3.3UndertheassumptionofTheorem1.3.2,wehaveVt=Ft;St,whereFt;x=e)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(tZ11f)]TJ/F18 10.909 Tf 5 -8.837 Td[(xer)]TJ/F19 7.97 Tf 6.587 0 Td[(2T)]TJ/F19 7.97 Tf 6.586 0 Td[(t+yp T)]TJ/F19 7.97 Tf 6.587 0 Td[(te)]TJ/F19 7.97 Tf 6.587 0 Td[(y2=2 p 2dy:.10IfweconsideraEuropeancalloption=ST)]TJ/F18 10.909 Tf 11.294 0 Td[(K+,whosepriceisVt=Ct;St,itspriceisgivenby.1and.2.17

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Chapter2DefaultableCorporateCoupon-BondPricingwithConstantInterestRateInthischapter,wewillndthepriceofcorporatecoupon-bond.Intherstsection,weconsiderdefaultasanexogenousevent,therefore,defaultisanunexpected.Inthesecondsection,inadditiontoexogenouscause,wetakeanendogenouseventintoconsiderationasapossiblecausefordefault.2.1CorporateCoupon-BondwithConstantInterestRate-UnexpectedDefault2.1.1FormulationInthissection,wewillassumethefollowing.Assuggestedintheintroduction,weconsidercouponbearingbondsasaportfolioofzero-couponbondsconsistingofoneprincipalportiondueonthematuritydateandcouponsdueoncouponpaymentdays.AndthroughoutChapters2and3,wewillassumethattherisk-freeshort-termraterisconstant.Assumption1:Defaultisanexogenousevent.Unexpecteddefaultprobabilityin[t;t+dt]isptdt.Ifcouponisnotdueontheinterval,thenthedefaultintensitypt=ptfollowsdp=app;tdt+spp;tdW1:whereapp;tandspp;tarethedriftandthevolatilityofprespectively.Onpredeterminedcouponpaymentdatet=j,wherejreferstoj-thinterestpaymentandj=1;2;:::;nthismeansthatn=T,whereTisthematurityofthebond,andthejumpofptisgivenbyp=pj)]TJ/F18 10.909 Tf 10.909 0 Td[(pj)]TJ/F8 10.909 Tf 10.115 1.636 Td[(=pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(Uj18

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whereUjisajumpsizeatt=j.AsequenceUj1jnisindependent,identicallydistributedrandomvariabletakingvaluesin[x;0]with)]TJ/F8 10.909 Tf 8.485 0 Td[(1
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d^C=@^C @tdt+@^C @pdp+1 2@2^C @p2dp2+f^Cpj;t)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^Cpj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g=@^C @tdt+@^C @pdp+1 2@2^C @p2apdt+spdW2+f^Cpj;t)]TJ/F8 10.909 Tf 13.379 2.757 Td[(^Cpj)]TJ/F18 10.909 Tf 7.085 1.637 Td[(;tgIfj2t;t+dt]g=@^C @tdt+@^C @pdp+1 2s2p@2^C @p2dt+f^Cpj;t)]TJ/F8 10.909 Tf 13.379 2.757 Td[(^Cpj)]TJ/F18 10.909 Tf 7.085 1.637 Td[(;tgIfj2t;t+dt]g:Incasethereisadefaultwithprobabilityptdt,thechangeinpricewillbegivenbyd^C=R)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^CwhereRisthedefaultrecovery,which,laterinthecomputation,weshalluseitsfacevalueexoge-nousrecovery.Notethatintheeventofadefault,thebondsshallnotbetradedandthebuyershallbeentitledtoreceiveitsrecoveryamount.Weconstructahedgedportfoliobyhedgingonebondwithanotherbondwithdifferentmaturity.Letusdenotethepricesofthesebondsby^Ci=^Cip;t;Ti;i=1;2.Here,Tiisthematurityofeachbond.LetRibetherecoveryrateforeach.Assumethatcouponpaymentdatesforbothbondsarethesame.Nowconstructaportfolio:=^C1)]TJ/F8 10.909 Tf 10.909 0 Td[(^C2:Thechangeofvalueinthisportfoliooverasmalltimeincrement[t;t+dt]isgivenbyd=d^C1)]TJ/F8 10.909 Tf 10.909 0 Td[(d^C2:Atthesametime,bytheArbitragePrinciple,wemusthaved=rdt.Ifthereisnodefault,wehaved=@^C1 @t+1 2s2@2^C1 @p2!dt+@^C1 @pdp+f^C1pj;t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^C1pj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(;tgIfj2t;t+dt]g)]TJ/F8 10.909 Tf 8.485 0 Td[("@^C2 @t+1 2s2@2^C2 @p2!dt+@^C2 @pdp+f^C2pj;t)]TJ/F8 10.909 Tf 13.379 2.757 Td[(^C2pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g#:Togetridofuncertaintycausedbydpterm,wechoose=@^C1 @p@^C2 @p!)]TJ/F16 7.97 Tf 6.587 0 Td[(1:20

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Then,d=24@^C1 @t+1 2s2@2^C1 @p2)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@^C1 @p@^C2 @p!)]TJ/F16 7.97 Tf 6.587 0 Td[(1@^C2 @t+1 2s2@2^C2 @p2!35dt+f^C1pj;t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^C1pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g)]TJ/F18 10.909 Tf 9.681 7.38 Td[(@^C1 @p@^C2 @p!)]TJ/F16 7.97 Tf 6.587 0 Td[(1f^C2pj;t)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^C2pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g:Ifthereisadefault,thepricechangeintheportfolioisd=R1)]TJ/F8 10.909 Tf 13.38 2.757 Td[(^C1)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@^C1 @p@^C2 @p!)]TJ/F16 7.97 Tf 6.587 0 Td[(1R2)]TJ/F8 10.909 Tf 13.38 2.757 Td[(^C2:Takingtheexpectationofd=d^C1)]TJ/F8 10.909 Tf 9.233 0 Td[(^D2=rdtandneglectingthehigherorderofinnitesimalofdt-term,wehave0@@^C1 @t+1 2s2@2^C1 @p2)]TJ/F18 10.909 Tf 12.104 7.38 Td[(@^C1 @p@^C2 @p!)]TJ/F16 7.97 Tf 6.586 0 Td[(1@^C2 @t+1 2s2@2^C2 @p2!1Adt+^C1pj;t)]TJ/F8 10.909 Tf 13.379 2.757 Td[(^C1pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tIfj2t;t+dt]g)]TJ/F18 10.909 Tf 12.104 7.38 Td[(@^C1 @p@^C2 @p!)]TJ/F16 7.97 Tf 6.586 0 Td[(1^C2pj;t)]TJ/F8 10.909 Tf 13.38 2.757 Td[(^C2pj)]TJ/F18 10.909 Tf 7.085 1.637 Td[(;tIfj2t;t+dt]g!)]TJ/F18 10.909 Tf 10.909 0 Td[(ptdt+0@R1)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^C1)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@^C1 @p@^C2 @p!)]TJ/F16 7.97 Tf 6.587 0 Td[(1R2)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^C21Aptdt=r0@^C1)]TJ/F18 10.909 Tf 12.104 7.38 Td[(@^C1 @p@^C2 @p!)]TJ/F16 7.97 Tf 6.586 0 Td[(1C21Adt;whichyields"@^C1 @t+1 2s2@2^C1 @p2)]TJ/F24 10.909 Tf 10.909 12.109 Td[(f^C1pj;t)]TJ/F8 10.909 Tf 13.379 2.757 Td[(^C1pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g)]TJ/F8 10.909 Tf 10.909 0 Td[(R1)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^C1pt)]TJ/F18 10.909 Tf 10.909 0 Td[(r^C1!dt+f^C1pj;t)]TJ/F8 10.909 Tf 13.379 2.757 Td[(^C1pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g#@^C1 @p!)]TJ/F16 7.97 Tf 6.586 0 Td[(1="@^C2 @t+1 2s2@2^C2 @p2)]TJ/F24 10.909 Tf 10.909 12.109 Td[(f^C2pj;t)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^C2pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g)]TJ/F8 10.909 Tf 10.909 0 Td[(R2)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^C2pt)]TJ/F18 10.909 Tf 10.909 0 Td[(r^C2!dt+f^C2pj;t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^C2pj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(;tgIfj2t;t+dt]g#@^C2 @p!)]TJ/F16 7.97 Tf 6.587 0 Td[(1:21

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ThelefthandsideofthisequationisafunctionofT1butnotT2,andtherighthandsideofthisequationisafunctionofT2butnotT1,sobothsidesmustbefunctionsindependentoftheirmaturitydate,say)]TJ/F18 10.909 Tf 8.484 0 Td[(app;tdt.Therefore,wehavetheequationforcorporatecouponbondwithdefaultintensitypt,takingtherecoveryassumptionintoconsideration:h@^C @t+1 2s2@2^C @p2+ap@^C @p)]TJ/F24 10.909 Tf 10.91 8.837 Td[()]TJ/F14 10.909 Tf 5 -8.837 Td[(f^Cpj;t)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^Cpj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIj2t;t+dt])]TJ/F8 10.909 Tf 10.909 0 Td[(Re)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(t)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^Cpt)]TJ/F18 10.909 Tf 10.909 0 Td[(r^Cidt+^Cpj;t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^Cpj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tIj2t;t+dt]=0:.2Hereapisariskneutraldriftofpt.Wecanwriteapintheformapp;t=ap;t)]TJ/F18 10.909 Tf 10.998 0 Td[(sp;tp;t,wherep;t=ap;t)]TJ/F19 7.97 Tf 6.586 0 Td[(app;t sp;tiscalledamarketpriceriskofptandmeasuresanextracompensationperunitofriskfortakingontheriskincurredbypt.Inthecomputationbelow,weassumethatp;t=0,sothatapp;t=ap;t.NotethattheprobabilityofthesurvivalofthebondattimeT,givenitwasnotdefaultedattimet
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Especially,ift=T)]TJ/F8 10.909 Tf 11.515 0 Td[(=n)]TJ/F51 10.909 Tf 7.085 1.636 Td[(,wehave^Cp;T)]TJ/F8 10.909 Tf 8.484 0 Td[(=1PT)]TJ/F18 10.909 Tf 8.485 0 Td[(;T+R)]TJ/F18 10.909 Tf 10.909 0 Td[(PT)]TJ/F18 10.909 Tf 8.485 0 Td[(;T=e)]TJ/F19 7.97 Tf 6.586 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.339 Td[(Un+R)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.339 Td[(UnandCp;T=1ifthereisnotdefaultuntilt=T.Inaddition,sinceasthedefaultintensityincreases,thecompanyismorelikelytogetdefaultedwehavethefollowingboundarycondition:limp!1^Cp;t=R:Ifp=0,thenitimpliesthatthebondisdefault-free.Thereforewealsohavethefollowingboundarycondition.^Cp=0;t=e)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(t:Sincetheseboundaryconditionsarenotusedinsolvingtheequation,theywillnotberepeatedbelow.Thentheequation.2oneachtimeinterval[j)]TJ/F16 7.97 Tf 6.586 0 Td[(1;jbecomesthefollowing@^C @t+1 2s2@2^C @p2+a@^C @p+Re)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^Cpt)]TJ/F18 10.909 Tf 10.909 0 Td[(r^C=0or@^C @t+1 2s2@2^C @p2+a@^C @p)]TJ/F8 10.909 Tf 10.909 0 Td[(r+pt^C+Re)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(tpt=0withtheterminalcondition:^Cp;j)]TJ/F8 10.909 Tf 7.084 1.636 Td[(=^Cp;je)]TJ/F19 7.97 Tf 6.586 0 Td[(pj)]TJ/F19 7.97 Tf 6.753 2.897 Td[(Uj+Re)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(j)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(pj)]TJ/F19 7.97 Tf 6.752 2.897 Td[(Uj:.6Thisisanonhomogeneousparabolicequationwithvariablecoefcientsandaterminalcondition.Firstwewillsolvethisonthetimeinterval[n)]TJ/F16 7.97 Tf 6.586 0 Td[(1;T,i.e.,8>><>>:@^C @t+1 2s2@2^C @p2+a@^C @p)]TJ/F8 10.909 Tf 10.909 0 Td[(r+pt^C+Re)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(tpt=0n)]TJ/F16 7.97 Tf 6.587 0 Td[(1t0;^Cp;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=^Cp;Te)]TJ/F19 7.97 Tf 6.586 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.339 Td[(Un+R)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.339 Td[(Unp>0:23

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Byletting^C=ue)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(t;.7wehave8>><>>:@u @t+1 2s2@2u @p2+a@u @p)]TJ/F18 10.909 Tf 10.909 0 Td[(ptu)]TJ/F18 10.909 Tf 10.909 0 Td[(R=0n)]TJ/F16 7.97 Tf 6.586 0 Td[(1t0;up;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=e)]TJ/F19 7.97 Tf 6.586 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.34 Td[(Un+R)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.34 Td[(Un=e)]TJ/F19 7.97 Tf 6.586 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.34 Td[(Un)]TJ/F18 10.909 Tf 10.909 0 Td[(R+Rp>0;.8since^Cp;T=1.Usingthechangeofunknownfunction^u=u)]TJ/F18 10.909 Tf 10.909 0 Td[(R;.98>><>>:@^u @t+1 2s2@2^u @p2+a@^u @p)]TJ/F18 10.909 Tf 10.909 0 Td[(pt^u=0n)]TJ/F16 7.97 Tf 6.586 0 Td[(1t0;^up;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=u)]TJ/F18 10.909 Tf 10.909 0 Td[(R=[e)]TJ/F19 7.97 Tf 6.587 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.339 Td[(Un)]TJ/F18 10.909 Tf 10.909 0 Td[(R+R])]TJ/F18 10.909 Tf 10.909 0 Td[(R=e)]TJ/F19 7.97 Tf 6.586 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.339 Td[(Un)]TJ/F18 10.909 Tf 10.909 0 Td[(Rp>0:ByafurtherchangeoftheunknownfunctiontoW=Wp;tsothat^u=We)]TJ/F19 7.97 Tf 6.586 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.339 Td[(Un)]TJ/F18 10.909 Tf 10.909 0 Td[(R;.10WeobtaintheterminalvalueproblemontheunknownW:8>><>>:@W @t+1 2s2@2W @p2+a@W @p)]TJ/F18 10.909 Tf 10.909 0 Td[(ptW=0n)]TJ/F16 7.97 Tf 6.586 0 Td[(1t0;Wp;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=1p>0:.11From.7,.9,and.10,wecanexpressthepriceofdefaultablecouponbondsforn)]TJ/F16 7.97 Tf 6.586 0 Td[(1t
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Insolving.11,wewillfollowWilmott8andOetalandrestrictap;tandsp;ttothefollowingcases.Assumption5:ap;tands2p;tarelinearinp,ap;t=bt)]TJ/F18 10.909 Tf 10.909 0 Td[(ctp;.12as2p;t=dt+etp:.12bAssumethatthesolutionof.11isgivenintheform:Wp;t=eAt;T)]TJ/F16 7.97 Tf 6.586 0 Td[()]TJ/F19 7.97 Tf 6.586 0 Td[(Bt;T)]TJ/F16 7.97 Tf 6.587 0 Td[(p:Then,since@W @t=A0)]TJ/F18 10.909 Tf 10.909 0 Td[(B0pW@W @p=)]TJ/F18 10.909 Tf 8.485 0 Td[(WB@2W @p2=WB2whereA0andB0arederivativesofAandBwithrespecttot,respectively;substitutingthesein.11givesA0+1 2dtB2)]TJ/F18 10.909 Tf 10.909 0 Td[(btB+p)]TJ/F18 10.909 Tf 8.485 0 Td[(B0+1 2etB2+ctB)]TJ/F8 10.909 Tf 10.909 0 Td[(1=0:Thisholdsforanyvalueofp,sowemusthave8><>:A0+1 2dtB2)]TJ/F18 10.909 Tf 10.909 0 Td[(btB=0;)]TJ/F18 10.909 Tf 8.485 0 Td[(B0+1 2etB2+ctB)]TJ/F8 10.909 Tf 10.909 0 Td[(1=0:.13NotethatsinceWp;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=1from.11,wehaveAT)]TJ/F18 10.909 Tf 8.485 0 Td[(;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=BT)]TJ/F18 10.909 Tf 8.485 0 Td[(;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=0.OncewesolveforBinthesecondequationin.13,wecanndAfromtherstequationasfollows:At;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=)]TJ/F24 10.909 Tf 10.303 14.849 Td[(ZT)]TJ/F19 7.97 Tf -10.955 -21.586 Td[(tbsBs;T)]TJ/F8 10.909 Tf 8.485 0 Td[()]TJ/F8 10.909 Tf 12.104 7.38 Td[(1 2dsB2s;T)]TJ/F8 10.909 Tf 8.485 0 Td[(ds:Insolving.13,wewillrestrict.12tothefollowingcases:ictcconstant;et0,iict0;etK>0constant.Casei:ctcconstant;et0.Then,wehavedp=bt)]TJ/F18 10.909 Tf 10.909 0 Td[(cpdt+p dtdW1:25

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ThiscasecoversVasicekmodelb;c;dareconstant;e=0,Ho-Leemodelc=0;e=0;disconstantandHull-Whitemodelc;dareconstante=0.Then,thesecondequationin.13becomesB0=cB)]TJ/F8 10.909 Tf 10.909 0 Td[(1;andthesolutionisBt;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=8><>:1)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(cT)]TJ/F19 7.97 Tf 6.586 0 Td[(t c;c6=0;T)]TJ/F18 10.909 Tf 10.909 0 Td[(t;c=0:Caseii:ct0;etK>0constant.Thenwehavedp=btdt+p dt+KpdW1:ThiscasecoversMertonmodelb;dareconstant;c=0;e=K=0.Thenthesecondequationin.13becomesB0)]TJ/F18 10.909 Tf 12.104 7.38 Td[(K 2B2+1=0:Lettingx=q K 2B,dx 1)]TJ/F18 10.909 Tf 10.91 0 Td[(x2=)]TJ/F24 10.909 Tf 8.484 17.354 Td[(r K 2dt:Integratingbothsides,weobtain1 2ln1+x 1)]TJ/F18 10.909 Tf 10.909 0 Td[(x+k=)]TJ/F24 10.909 Tf 8.485 17.354 Td[(r K 2t:UsingtheterminalconditionxT)]TJ/F18 10.909 Tf 8.485 0 Td[(;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=q K 2BT)]TJ/F18 10.909 Tf 8.485 0 Td[(;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=0,k=)]TJ/F24 10.909 Tf 8.485 12.863 Td[(q K 2T.BysubstitutingBbackinxandsolvingforB,weobtainthefollowing:Bt;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=)]TJ/F24 10.909 Tf 8.485 17.142 Td[(r 2 cexpp c 2T)]TJ/F18 10.909 Tf 10.909 0 Td[(t)]TJ/F18 10.909 Tf 10.909 0 Td[(exp)]TJ/F24 10.909 Tf 8.485 8.583 Td[(p c 2T)]TJ/F18 10.909 Tf 10.909 0 Td[(t expp c 2T)]TJ/F18 10.909 Tf 10.909 0 Td[(t+exp)]TJ/F24 10.909 Tf 8.485 8.582 Td[(p c 2T)]TJ/F18 10.909 Tf 10.909 0 Td[(t:Therefore,assumingthatthepriceofdefaultablecouponbondattimet=T)]TJ/F8 10.909 Tf 11.516 0 Td[(=n)]TJ/F51 10.909 Tf 9.157 1.636 Td[(is^Cp;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=^Cp;n)]TJ/F8 10.909 Tf 7.085 1.636 Td[(=e)]TJ/F19 7.97 Tf 6.586 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.339 Td[(Un+R)]TJ/F18 10.909 Tf 9.481 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.339 Td[(Un,thepriceondefaultablecouponbondforn)]TJ/F16 7.97 Tf 6.586 0 Td[(1t
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whereAt;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=)]TJ/F24 10.909 Tf 10.303 14.848 Td[(ZT)]TJ/F19 7.97 Tf -10.954 -21.585 Td[(tbsBs;T)]TJ/F8 10.909 Tf 8.484 0 Td[()]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 2dsB2s;T)]TJ/F8 10.909 Tf 8.485 0 Td[(ds.14andBt;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=8>>>>><>>>>>:1)]TJ/F19 7.97 Tf 6.586 0 Td[(e)]TJ/F20 5.978 Tf 5.756 0 Td[(cT)]TJ/F20 5.978 Tf 5.756 0 Td[(t c;dp=bt)]TJ/F18 10.909 Tf 10.909 0 Td[(cpdt+p dtdW1;c6=0n)]TJ/F18 10.909 Tf 10.909 0 Td[(t;dp=bt)]TJ/F18 10.909 Tf 10.909 0 Td[(cpdt+p dtdW1;c=0)]TJ/F24 10.909 Tf 8.485 12.708 Td[(q 2 cexpp c 2T)]TJ/F19 7.97 Tf 6.587 0 Td[(t)]TJ/F19 7.97 Tf 6.587 0 Td[(exp)]TJ/F14 10.909 Tf 6.587 6.814 Td[(p c 2T)]TJ/F19 7.97 Tf 6.587 0 Td[(t expp c 2T)]TJ/F19 7.97 Tf 6.587 0 Td[(t+exp)]TJ/F14 10.909 Tf 6.587 6.813 Td[(p c 2T)]TJ/F19 7.97 Tf 6.587 0 Td[(t;dp=btdt+p dt+KpdW1:.15Lettingt=n)]TJ/F16 7.97 Tf 6.587 0 Td[(1,wehavethevaluefor^Cp;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1,^Cp;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1=eAn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.586 0 Td[()]TJ/F19 7.97 Tf 6.586 0 Td[(Bn)]TJ/F17 5.978 Tf 5.757 0 Td[(1;T)]TJ/F16 7.97 Tf 6.587 0 Td[(pn)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F19 7.97 Tf 6.587 0 Td[(pT)]TJ/F19 7.97 Tf 6.255 1.34 Td[(Un)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(n)]TJ/F17 5.978 Tf 5.756 0 Td[(1+)]TJ/F18 10.909 Tf 10.909 0 Td[(eAn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.587 0 Td[()]TJ/F19 7.97 Tf 6.586 0 Td[(Bn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.586 0 Td[(pn)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F19 7.97 Tf 6.586 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.339 Td[(UnRe)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(n)]TJ/F17 5.978 Tf 5.757 0 Td[(1;andthenwecangettheterminalconditionontheinterval[n)]TJ/F16 7.97 Tf 6.587 0 Td[(2;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1from.6,whichis:^Cp;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F8 10.909 Tf 7.085 2.22 Td[(=^Cp;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1e)]TJ/F19 7.97 Tf 6.587 0 Td[(pn)]TJ/F17 5.978 Tf 5.757 0 Td[(1)]TJ/F19 7.97 Tf 6.752 3.42 Td[(Un)]TJ/F17 5.978 Tf 5.756 0 Td[(1+Re)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(pn)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F19 7.97 Tf 6.753 3.42 Td[(Un)]TJ/F17 5.978 Tf 5.756 0 Td[(1=e)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(n)]TJ/F17 5.978 Tf 5.756 0 Td[(10@eDn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.587 0 Td[()]TJ/F20 5.978 Tf 15.789 7.572 Td[(nPj=n)]TJ/F17 5.978 Tf 5.756 0 Td[(1pj)]TJ/F19 7.97 Tf 6.753 2.898 Td[(Uj+R)]TJ/F18 10.909 Tf 10.909 0 Td[(eDn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.586 0 Td[()]TJ/F20 5.978 Tf 15.788 7.572 Td[(nPj=n)]TJ/F17 5.978 Tf 5.756 0 Td[(1pj)]TJ/F19 7.97 Tf 6.752 2.898 Td[(Uj1AwhereDn)]TJ/F16 7.97 Tf 6.587 0 Td[(1;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=An)]TJ/F16 7.97 Tf 6.586 0 Td[(1;T)]TJ/F8 10.909 Tf 8.485 0 Td[()]TJ/F18 10.909 Tf 10.909 0 Td[(Bn)]TJ/F16 7.97 Tf 6.587 0 Td[(1;T)]TJ/F8 10.909 Tf 8.485 0 Td[(pn)]TJ/F16 7.97 Tf 6.586 0 Td[(1.From.6withj=n)]TJ/F8 10.909 Tf 10.909 0 Td[(1,wehave:8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:@^C @t+1 2s2@2^C @p2+a@^C @p)]TJ/F8 10.909 Tf 10.909 0 Td[(r+pt^C)]TJ/F18 10.909 Tf 10.909 0 Td[(Re)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(tpt=0n)]TJ/F16 7.97 Tf 6.586 0 Td[(2t0;^Cp;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F8 10.909 Tf 7.085 2.22 Td[(=e)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(n)]TJ/F17 5.978 Tf 5.756 0 Td[(10@eDn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.586 0 Td[()]TJ/F20 5.978 Tf 15.789 7.572 Td[(nPj=n)]TJ/F17 5.978 Tf 5.757 0 Td[(1pj)]TJ/F19 7.97 Tf 6.752 2.898 Td[(Uj+R)]TJ/F18 10.909 Tf 10.909 0 Td[(eDn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.587 0 Td[()]TJ/F20 5.978 Tf 15.788 7.572 Td[(nPj=n)]TJ/F17 5.978 Tf 5.756 0 Td[(1pj)]TJ/F19 7.97 Tf 6.752 2.898 Td[(Uj1Ap>0:27

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Asbefore,letting^C=ue)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(t,wehave8>>>>>><>>>>>>:@u @t+1 2s2@2u @p2+a@u @p)]TJ/F18 10.909 Tf 10.909 0 Td[(ptu)]TJ/F18 10.909 Tf 10.909 0 Td[(R=0n)]TJ/F16 7.97 Tf 6.587 0 Td[(2t0;up;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F8 10.909 Tf 7.085 2.22 Td[(=eDn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.586 0 Td[()]TJ/F20 5.978 Tf 15.789 7.572 Td[(nPj=n)]TJ/F17 5.978 Tf 5.757 0 Td[(1pj)]TJ/F19 7.97 Tf 6.752 2.897 Td[(Uj)]TJ/F18 10.909 Tf 10.909 0 Td[(R+Rp>0:.16Again,usingthechangeofunknownfunction^u=u)]TJ/F18 10.909 Tf 10.909 0 Td[(R;8>><>>:@^u @t+1 2s2@2^u @p2+a@^u @p)]TJ/F18 10.909 Tf 10.909 0 Td[(pt^u=0n)]TJ/F16 7.97 Tf 6.586 0 Td[(2t0;^up;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F8 10.909 Tf 7.085 2.22 Td[(=u)]TJ/F18 10.909 Tf 10.909 0 Td[(R=eDn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.586 0 Td[()]TJ/F20 5.978 Tf 15.788 7.572 Td[(nPj=n)]TJ/F17 5.978 Tf 5.756 0 Td[(1pj)]TJ/F19 7.97 Tf 6.752 2.898 Td[(Uj)]TJ/F18 10.909 Tf 10.909 0 Td[(Rp>0:UsingthechangeofunknownfunctionW=Wp;tsothat^u=WeDn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.586 0 Td[()]TJ/F20 5.978 Tf 15.789 7.572 Td[(nPj=n)]TJ/F17 5.978 Tf 5.756 0 Td[(1pj)]TJ/F19 7.97 Tf 6.752 2.898 Td[(Uj)]TJ/F18 10.909 Tf 10.909 0 Td[(R;8>><>>:@W @t+1 2s2@2W @p2+a@W @p)]TJ/F18 10.909 Tf 10.909 0 Td[(ptW=0n)]TJ/F16 7.97 Tf 6.586 0 Td[(2t0;Wp;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F8 10.909 Tf 7.085 2.22 Td[(=1p>0:AgainwesetWp;t=eAt;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F16 7.97 Tf 6.254 1.738 Td[()]TJ/F19 7.97 Tf 6.587 0 Td[(Bt;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F16 7.97 Tf 6.254 1.738 Td[(p.Sowecanwrite^Cp;tintermsofWas^Cp;t=WeDn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.586 0 Td[()]TJ/F20 5.978 Tf 15.788 7.572 Td[(nPj=n)]TJ/F17 5.978 Tf 5.756 0 Td[(1pj)]TJ/F19 7.97 Tf 6.752 2.897 Td[(Uje)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(t+)]TJ/F18 10.909 Tf 10.011 0 Td[(WeDn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.587 0 Td[()]TJ/F20 5.978 Tf 15.789 7.572 Td[(nPj=n)]TJ/F17 5.978 Tf 5.756 0 Td[(1pj)]TJ/F19 7.97 Tf 6.753 2.897 Td[(UjRe)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(t:AsbeforewendAt;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F8 10.909 Tf 6.753 3.294 Td[(andBt;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F8 10.909 Tf 6.752 3.294 Td[(asfollows:At;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F8 10.909 Tf 7.084 2.22 Td[(=)]TJ/F24 10.909 Tf 10.303 14.849 Td[(Zn)]TJ/F17 5.978 Tf 5.757 0 Td[(1)]TJ/F19 7.97 Tf -28.373 -19.848 Td[(tbsBs;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F8 10.909 Tf 7.084 2.22 Td[()]TJ/F8 10.909 Tf 12.104 7.38 Td[(1 2dsB2s;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F8 10.909 Tf 7.085 2.22 Td[(dsandBt;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F8 10.909 Tf 7.084 2.22 Td[(=8>>>>>>>>>><>>>>>>>>>>:1)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(cn)]TJ/F17 5.978 Tf 5.757 0 Td[(1)]TJ/F19 7.97 Tf 6.586 0 Td[(t c;dp=bt)]TJ/F18 10.909 Tf 10.909 0 Td[(cpdt+p dtdW1;c6=0n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F18 10.909 Tf 10.909 0 Td[(t;dp=bt)]TJ/F18 10.909 Tf 10.909 0 Td[(cpdt+p dtdW1;c=0)]TJ/F24 10.909 Tf 8.485 12.708 Td[(q 2 cep c 2n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F19 7.97 Tf 6.587 0 Td[(t)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F14 10.909 Tf 6.586 6.814 Td[(p c 2n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F19 7.97 Tf 6.586 0 Td[(t ep c 2n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F19 7.97 Tf 6.587 0 Td[(t+e)]TJ/F14 10.909 Tf 6.586 6.814 Td[(p c 2n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F19 7.97 Tf 6.586 0 Td[(t;dp=btdt+p dt+KpdW1:28

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Soforn)]TJ/F16 7.97 Tf 6.587 0 Td[(2t
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andBt;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:n)]TJ/F8 10.909 Tf 10.909 0 Td[(j)]TJ/F8 10.909 Tf 10.909 0 Td[(1)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(cj)]TJ/F19 7.97 Tf 6.586 0 Td[(t)]TJ/F19 7.97 Tf 21.296 10.363 Td[(nPk=j+1e)]TJ/F19 7.97 Tf 6.587 0 Td[(ck)]TJ/F19 7.97 Tf 6.587 0 Td[(k)]TJ/F17 5.978 Tf 5.756 0 Td[(1 c;ifdp=bt)]TJ/F18 10.909 Tf 10.909 0 Td[(cpdt+p dtdW1+pj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(UjIfj2t;t+dt]g;c6=0T)]TJ/F18 10.909 Tf 10.91 0 Td[(t;ifdp=bt)]TJ/F18 10.909 Tf 10.909 0 Td[(cpdt+p dtdW1+pj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(UjIfj2t;t+dt]g;c=0)]TJ/F24 10.909 Tf 8.485 12.708 Td[(q 2 cep c 2j)]TJ/F19 7.97 Tf 6.587 0 Td[(t+nPk=j+1ep c 2k)]TJ/F19 7.97 Tf 6.587 0 Td[(k)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F14 10.909 Tf 6.587 6.814 Td[(p c 2j)]TJ/F19 7.97 Tf 6.586 0 Td[(t)]TJ/F19 7.97 Tf 21.296 10.363 Td[(nPk=j+1e)]TJ/F14 10.909 Tf 6.586 6.814 Td[(p c 2k)]TJ/F19 7.97 Tf 6.586 0 Td[(k)]TJ/F17 5.978 Tf 5.756 0 Td[(1 ep c 2j)]TJ/F19 7.97 Tf 6.587 0 Td[(t+nPk=j+1ep c 2k)]TJ/F19 7.97 Tf 6.587 0 Td[(k)]TJ/F17 5.978 Tf 5.756 0 Td[(1+e)]TJ/F14 10.909 Tf 6.587 6.814 Td[(p c 2j)]TJ/F19 7.97 Tf 6.586 0 Td[(t+nPk=j+1e)]TJ/F14 10.909 Tf 6.586 6.814 Td[(p c 2k)]TJ/F19 7.97 Tf 6.586 0 Td[(k)]TJ/F17 5.978 Tf 5.756 0 Td[(1;ifdp=btdt+p dt+KpdW1+pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(UjIfj2t;t+dt]g.20with,oneachtimeinterval,j)]TJ/F16 7.97 Tf 6.587 0 Td[(1t>>>>><>>>>>>:1)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(cj)]TJ/F19 7.97 Tf 6.587 0 Td[(t c;dp=bt)]TJ/F18 10.909 Tf 10.909 0 Td[(cpdt+p dtdW1;c6=0j)]TJ/F18 10.909 Tf 10.909 0 Td[(t;dp=bt)]TJ/F18 10.909 Tf 10.909 0 Td[(cpdt+p dtdW1;c=0)]TJ/F24 10.909 Tf 8.485 12.708 Td[(q 2 cep c 2j)]TJ/F19 7.97 Tf 6.586 0 Td[(t)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F14 10.909 Tf 6.586 6.814 Td[(p c 2j)]TJ/F19 7.97 Tf 6.587 0 Td[(t ep c 2j)]TJ/F19 7.97 Tf 6.586 0 Td[(t+e)]TJ/F14 10.909 Tf 6.586 6.814 Td[(p c 2j)]TJ/F19 7.97 Tf 6.587 0 Td[(t;dp=btdt+p dt+KpdW1:.21Inthissection,weassumedthatthesolutionto.11isgivenintheexponentialform.Thisisbecausetheintensitymodelisconsideredtobeanextensionofthehazardratemodel,whichhasasolutionintheexponentialform.ItshouldbealsonotedthattheBlack-Scholesequationadmitstheexponentialsolutionformviaexponentialtransformation.WhiletheBlack-Scholesequationhasconstantdriftandconstantvolatility,weassumedthatthedriftandthevolatilityofourPDEaredependentontheintensityandtime.30

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2.2CorporateCoupon-BondwithConstantInterestRate-ExpectedandUnexpectedDe-faultUnderthissection,weconsiderdefaulteventasbothexogenousandendogenousevent.Whileexogenouscauseoccursoutsidethecompany'scontrolsowewillusedefaultintensityasintheprevioussection,incaseofendogenouscause,thecompanydecidestolebankruptcywhenitstotalassethitsthepredeterminedbarrier.2.2.1FormulationAssumption1:ThermassetsV=Vt=Vtconsistsofmsharesoftradedstock,whosepriceattimetisS=St=St,andncoupon-bondcerticates,whosepriceattimetisC=Ct=Ct:Vt=mSt+nCt:.22ThermassetsvaluealsofollowsthegeometricBrownianmotiondriftaV,volatilitysV:constantson[t;t+dt]ifcouponisnotdueontheinterval,dV=aVVtdt+sVVtdW2;.23andonpredeterminedcouponpaymentdatest=j,wherejreferstoj-thinterestpayment,j=1;:::;n,thejumpofVtisgivenbyVj=Vj)]TJ/F18 10.909 Tf 10.909 0 Td[(Vj)]TJ/F8 10.909 Tf 10.115 1.637 Td[(=ncCT=nc;wherecisthecouponrateofthebondsandTisthematurityofthebond.Hereweassumethatthebondsareredeemedattheirfacevalue;therefore,CT=1.Assumption2:Unexpecteddefaultintensityisgivenbypt=p0+Zt0ap;tdt+Zt0sp;tdW1+NtXj=1pj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(Uj;whereNtisthenumberofinterestpaymentsuptotimet.Thisisright-continuous,adaptedpro-cesswithniteandpredetermineddiscontinuities.WeassumethatEdW1dW2=0,thatis,unexpecteddefaultisnotcorrelatedwiththeassetvalueofthecompany.31

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Assumption3:ExpecteddefaultoccurswhenVVbt;Vbt=VBorVBe)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(t;whereVBisconstantandTisthematurityofthebonds.Assumption4:Thedefaultablecorporatecoupon-bondpriceisgivenbythefunctionG=GV;p;t,whichconstitutesofC=CV;p;t,thevalueattimetoftheprincipalportiononly,andci,thei-thcouponwithciCV;p;t;itobethevalueattimetofi-thcoupondueoni.Therefore,wehaveGV;p;t=CV;p;t+XitciCV;p;t;i:Assumption5:ExpectedandunexpecteddefaultrecoveryisRd=Re)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(t;0R1;constant.Problem:Undertheseassumptions,wewillndthecorporatebondpriceofdefaultablecorporatecoupon-bondwithbothexpectedandunexpecteddefault,whichisgivenasafunctionofV;pandt,thatisG=GV;p;t.2.2.2DerivationoftheModelWewillformaportfoliobybuyingonebondcerticateunderconsiderationandselling1sharesoftradedstockand2certicatesofcorporatecouponbondwithunexpecteddefaultonly,whosepricingwasconsideredintheprevioussectionandhereassumedtobetraded.Thatis,=C)]TJ/F8 10.909 Tf 10.909 0 Td[(1S)]TJ/F8 10.909 Tf 10.909 0 Td[(2^C:.24Sothepricechangeoftheportfoliooverasmallincrementoftimedtisgivenbyd=dC)]TJ/F8 10.909 Tf 10.909 0 Td[(1dS)]TJ/F8 10.909 Tf 10.909 0 Td[(2d^C:.25From.22,S=V)]TJ/F19 7.97 Tf 6.587 0 Td[(nC m.Substitutingthisin.24and.25,=+1n mC)]TJ/F8 10.909 Tf 12.104 7.38 Td[(1 mV)]TJ/F8 10.909 Tf 10.909 0 Td[(2^C.2632

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andd=+1n mdC)]TJ/F8 10.909 Tf 12.104 7.38 Td[(1 mdV)]TJ/F8 10.909 Tf 10.909 0 Td[(2d^C:Ifthereisnounexpecteddefaultintimeinterval[t;t+dt]withprobability1)]TJ/F18 10.909 Tf 10.538 0 Td[(ptdt,notingthatweassumedthatunexpecteddefaultisnotcorrelatedwiththetotalassetvalueofthecompany,byItoformula,thevaluechangeintheportfolioisgivenbyd=1+1n mdC)]TJ/F8 10.909 Tf 12.104 7.38 Td[(1 mdV)]TJ/F8 10.909 Tf 10.909 0 Td[(2d^C=1+1n m@C @t+1 2[s2p@2C @p2+s2VV2@2C @V2]dt+@C @VdV+@C @pdp+fCV;pj;t)]TJ/F18 10.909 Tf 10.909 0 Td[(CV;pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g+fCV)]TJ/F18 10.909 Tf 10.909 0 Td[(nc;pt;t)]TJ/F18 10.909 Tf 10.909 0 Td[(CV;pt;tgIfj2t;t+dt]g)]TJ/F8 10.909 Tf 12.104 7.38 Td[(1 mdV)]TJ/F8 10.909 Tf 8.485 0 Td[(2@^C dt+1 2s2p@2^C @p2dt+@^C dpdp+f^Cpj;t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^Cpj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g:.27Wewillchoose1and2sothatwecangetridofuncertaintycausedbydVanddpterms,i.e.,+1n m@C @V)]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 m=0;+1n m@C @p)]TJ/F8 10.909 Tf 10.909 0 Td[(2@^C @p=0:Solvingfor1and2,1=m@C @V1)]TJ/F18 10.909 Tf 10.909 0 Td[(n@C @V)]TJ/F16 7.97 Tf 6.587 0 Td[(1;2=@C @p@^C @p!)]TJ/F16 7.97 Tf 6.586 0 Td[(11)]TJ/F18 10.909 Tf 10.909 0 Td[(n@C @V)]TJ/F16 7.97 Tf 6.587 0 Td[(1;and1+1n m=1)]TJ/F18 10.909 Tf 10.909 0 Td[(n@C @V)]TJ/F16 7.97 Tf 6.587 0 Td[(1:Substitutingthesein.27,d=1)]TJ/F18 9.963 Tf 9.963 0 Td[(n@C @V)]TJ/F7 6.974 Tf 6.226 0 Td[(1@C @t+1 2[s2p@2C @p2+s2VV2@2C @V2]dt+fCV;pj;t)]TJ/F18 9.963 Tf 9.963 0 Td[(CV;pj)]TJ/F18 9.963 Tf 6.725 1.494 Td[(;tgIfj2t;t+dt]g+fCV)]TJ/F18 9.963 Tf 9.963 0 Td[(nc;pt;t)]TJ/F18 9.963 Tf 9.962 0 Td[(CV;pt;tgIfj2t;t+dt]g)]TJ/F18 9.963 Tf 8.945 6.74 Td[(@C @p@^C @p!)]TJ/F7 6.974 Tf 6.227 0 Td[(11)]TJ/F18 9.963 Tf 9.962 0 Td[(n@C @V)]TJ/F7 6.974 Tf 6.227 0 Td[(1[@^C dt+1 2s2p@2^C @p2]dt+f^Cpj;t)]TJ/F8 9.963 Tf 12.218 2.519 Td[(^Cpj)]TJ/F18 9.963 Tf 6.725 1.494 Td[(;tgIfj2t;t+dt]g:.2833

PAGE 41

Incaseofdefault,withprobabilityptdt,thepricechangeinCand^CaregivenbydC=Rd)]TJ/F18 10.909 Tf 10.909 0 Td[(Candd^C=^R)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^C;where^Rreferstotherecoveryrateofbondpricewithunexpecteddefaultonly.Then,thepricechangeintheportfolioisd=1+1n mRd)]TJ/F18 10.909 Tf 10.909 0 Td[(C)]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 mdV)]TJ/F8 10.909 Tf 10.909 0 Td[(2^R)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^C=1)]TJ/F18 10.909 Tf 10.909 0 Td[(n@C @V)]TJ/F16 7.97 Tf 6.587 0 Td[(1Rd)]TJ/F18 10.909 Tf 10.909 0 Td[(C)]TJ/F18 10.909 Tf 12.21 7.38 Td[(@C @VdV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@C @p@^C @p!)]TJ/F16 7.97 Tf 6.586 0 Td[(1^R)]TJ/F8 10.909 Tf 13.38 2.757 Td[(^C:.29Bythearbitrageprinciple,theexpectationofdmustbeequaltordt.Thatiswehavethefollowing::28)]TJ/F18 10.909 Tf 10.909 0 Td[(ptdt+:29ptdt=rdt=r:26dt:Therefore,@C @t+1 2s2p@2C @p2+s2VV2@2C @V2+rV@C @V)]TJ/F18 10.909 Tf 10.151 0 Td[(rC)]TJ/F24 10.909 Tf 10.151 12.109 Td[(nfCV;pj;t)]TJ/F18 10.909 Tf 10.151 0 Td[(CV;pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g+fCV)]TJ/F18 10.909 Tf 10.909 0 Td[(nc;pt;t)]TJ/F18 10.909 Tf 10.909 0 Td[(CV;pt;tgIfj2t;t+dt]gopt+Rd)]TJ/F18 10.909 Tf 10.909 0 Td[(Cpt!dt+nfCV;pj;t)]TJ/F18 10.909 Tf 10.909 0 Td[(CV;pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g+fCV)]TJ/F18 10.909 Tf 10.909 0 Td[(nc;pt;t)]TJ/F18 10.909 Tf 10.909 0 Td[(CV;pt;tgIfj2t;t+dt]go=@C @p@^C @p!)]TJ/F16 7.97 Tf 6.587 0 Td[(1"@^C @t+1 2s2p@2^C @p2+^R)]TJ/F8 10.909 Tf 13.38 2.757 Td[(^Cpt+r^C)-222(ff^Cpj;t)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^Cpj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]ggptdt+f^Cpj;t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^Cpj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(;tgIfj2t;t+dt]g#:.30From.2,thetermsinsidethebracketontherighthandsideoftheequationisequalto)]TJ/F18 10.909 Tf 8.485 0 Td[(app;t@^C @pdt,then,.30becomes@C @t+1 2s2p@2C @p2+s2VV2@2C @V2+rV@C @V+ap@C @p)]TJ/F18 10.909 Tf 10.909 0 Td[(rC+pRd)]TJ/F18 10.909 Tf 10.909 0 Td[(Cdt)]TJ/F8 10.909 Tf 10.909 0 Td[()]TJ/F18 10.909 Tf 10.909 0 Td[(ptdtnfCV;pj;t)]TJ/F18 10.909 Tf 10.909 0 Td[(CV;pj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(;tgIfj2t;t+dt]g+fCV)]TJ/F18 10.909 Tf 10.909 0 Td[(nc;pt;t)]TJ/F18 10.909 Tf 10.909 0 Td[(CV;pt;tgIfj2t;t+dt]go=0:.3134

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Fromthenancialpointofview,itisreasonabletoconsiderthataseverycoupondateapproaches,thebondholdersstarttotakeintoconsiderationthepossibilityofdefaultduetothejumpdecreaseintotalasset.Therefore,thebondpriceconvergestowhattheinvestorswouldexpectofwhatitwouldbeinthefuture,i.e.itsexpectation.So,letusassumethefollowing:Assumption6:Thedefaultablecorporatecouponbondpriceattimet=j)]TJ/F51 10.909 Tf 10.097 1.637 Td[(istheexpectationofthepriceattimet=j.Tondtheexpectation,weneedthefollowinglemma:Lemma2Theendogenoussurvivalprobabilityatt=j,giventhesurvivalattimet=j)]TJ/F51 10.909 Tf 10.283 1.636 Td[(seenfromtimet=j)]TJ/F16 7.97 Tf 6.587 0 Td[(1,denotedbyQj)]TJ/F18 10.909 Tf 6.753 2.71 Td[(;j;j)]TJ/F16 7.97 Tf 6.586 0 Td[(1orQj)]TJ/F18 10.909 Tf 6.752 2.71 Td[(;j,forshortisgivenbyQj)]TJ/F18 10.909 Tf 6.752 2.71 Td[(;j="~b p j)]TJ/F18 10.909 Tf 10.909 0 Td[(j)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F24 10.909 Tf 10.909 8.721 Td[(p j)]TJ/F18 10.909 Tf 10.909 0 Td[(j)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F18 10.909 Tf 6.196 -1.456 Td[(aV sV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(sV 2)]TJ/F8 10.909 Tf 10.909 0 Td[(exp2~maV sV)]TJ/F18 10.909 Tf 12.104 7.38 Td[(sV 2)]TJ/F8 10.909 Tf 8.098 2.879 Td[(~b p j)]TJ/F18 10.909 Tf 10.909 0 Td[(j)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F8 10.909 Tf 30.362 7.38 Td[(2~m p j)]TJ/F18 10.909 Tf 10.909 0 Td[(j)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F24 10.909 Tf 10.909 8.722 Td[(p j)]TJ/F18 10.909 Tf 10.909 0 Td[(j)]TJ/F16 7.97 Tf 6.587 0 Td[(1aV sV)]TJ/F18 10.909 Tf 12.104 7.38 Td[(sV 2#,exp2~maV sV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(sV 2)]TJ/F8 10.909 Tf 8.099 2.879 Td[(~b p j)]TJ/F18 10.909 Tf 10.909 0 Td[(j)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F8 10.909 Tf 30.362 7.38 Td[(2~m p j)]TJ/F18 10.909 Tf 10.909 0 Td[(j)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F24 10.909 Tf 10.909 8.721 Td[(p j)]TJ/F18 10.909 Tf 10.909 0 Td[(j)]TJ/F16 7.97 Tf 6.586 0 Td[(1aV sV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(sV 2;.32where~b=1 sVlnVb+nc Vj)]TJ/F17 5.978 Tf 5.756 0 Td[(1and~m=1 sVlnVb Vj)]TJ/F17 5.978 Tf 5.756 0 Td[(1.Proof.Theendogenoussurvivalprobabilityatt=j,giventhesurvivalattimet=j)]TJ/F51 10.909 Tf 10.311 1.637 Td[(seenfromtimet=j)]TJ/F16 7.97 Tf 6.586 0 Td[(1is:Qj)]TJ/F18 10.909 Tf 6.753 2.71 Td[(;j=PSurvivalatjseenfromj)]TJ/F16 7.97 Tf 6.586 0 Td[(1Survivalatj)]TJ/F51 10.909 Tf 9.811 1.636 Td[(seenfromj)]TJ/F16 7.97 Tf 6.586 0 Td[(1 PSurvivalatj)]TJ/F51 10.909 Tf 9.812 1.636 Td[(seenfromj)]TJ/F16 7.97 Tf 6.587 0 Td[(1=Pminj)]TJ/F17 5.978 Tf 5.756 0 Td[(1sj)]TJ/F18 10.909 Tf 8.571 2.852 Td[(Vs>Vb;Vj)]TJ/F8 10.909 Tf 7.085 1.636 Td[(>Vb+nc Pminj)]TJ/F17 5.978 Tf 5.756 0 Td[(1sj)]TJ/F18 10.909 Tf 8.57 2.851 Td[(Vs>Vb:.33ThereforeweneedtondthetwoprobabilitiesinEquation.33.Tondthem,wewillfollowthemethodshownbyShreve.35

PAGE 43

LetBtbeaBrownianmotionwithoutdriftanddeneMt=min0stBt:Thenbythereectionprinciple,wehave:P)]TJ/F18 10.909 Tf 5 -8.837 Td[(Mtb=P)]TJ/F18 10.909 Tf 5 -8.837 Td[(Bt<2m)]TJ/F18 10.909 Tf 10.91 0 Td[(b=1 p 2tZ2m)]TJ/F19 7.97 Tf 6.586 0 Td[(bexp)]TJ/F18 10.909 Tf 12.105 7.38 Td[(x2 2tdx;m<0;mb=Z1x=bZmy=2y)]TJ/F18 10.909 Tf 10.909 0 Td[(x tp 2texp)]TJ/F8 10.909 Tf 12.105 7.38 Td[(y)]TJ/F18 10.909 Tf 10.909 0 Td[(x2 2tdydx;m<0;m~b=Z1x=~bZ~my=2y)]TJ/F18 10.909 Tf 10.909 0 Td[(x tp 2texp)]TJ/F8 10.909 Tf 12.105 7.38 Td[(y)]TJ/F18 10.909 Tf 10.909 0 Td[(x2 2tdydx;~m<0;~m<~b:36

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ThenunderP-measure,wehaveP)]TJ/F8 10.909 Tf 9.068 -6.078 Td[(~Mt<~m;~Bt>~b=Z1x=~bZ~my=2y)]TJ/F18 10.909 Tf 10.909 0 Td[(x tp 2texp)]TJ/F8 10.909 Tf 12.105 7.38 Td[(y)]TJ/F18 10.909 Tf 10.909 0 Td[(x2 2texpfx)]TJ/F8 10.909 Tf 12.104 7.38 Td[(1 22tgdydx;~m<0;~m<~b:.34NowletusdeneVtasinAssumption1,thatis,dVt=aVVt+sVtdBt+ncIfj2[t;t+dt]g:.35Fort2;1,Thesolutionfor.35isgivenbyVt=V0expnsVBt+aV)]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 2s2Vto=V0expnsVBt+)]TJ/F18 10.909 Tf 8.014 -1.457 Td[(aV sV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(sV 2| {z }to=V0expnsV~Btowhere=aV sV)]TJ/F18 10.909 Tf 12.104 7.38 Td[(sV 2and~Bt=t+Bt:So,bysetting~Mt=min0st~Btasbefore,wehavemin0stVs=V0expfsV~Mtg;andVt>Vb+nc~Bt>1 sVlnVb+nc V0.36min0stVs
PAGE 45

So,by.36through.38,wehaveP)]TJ/F8 10.909 Tf 9.918 -8.836 Td[(min0stVsVb+nc=Pf~Mt<~m;~Bt>~bg:Byintegrating.34,wehavePf~Mt<~m;~Bt>~bg=Z1x=~bZ~my=2y)]TJ/F18 10.909 Tf 10.909 0 Td[(x tp 2texp)]TJ/F8 10.909 Tf 12.104 7.38 Td[(y)]TJ/F18 10.909 Tf 10.909 0 Td[(x2 2texpf~b)]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 22tgdydx=)]TJ/F24 10.909 Tf 10.303 14.849 Td[(Z1x=~b"1 p 2texp)]TJ/F8 10.909 Tf 12.105 7.38 Td[(y)]TJ/F18 10.909 Tf 10.909 0 Td[(x2 2t+x)]TJ/F8 10.909 Tf 12.104 7.38 Td[(1 22t#~my=dx=)]TJ/F8 10.909 Tf 19.5 7.38 Td[(1 p 2tZ1x=~bexp)]TJ/F8 10.909 Tf 12.104 7.38 Td[(~m)]TJ/F18 10.909 Tf 10.909 0 Td[(x2 2t+x)]TJ/F8 10.909 Tf 12.104 7.38 Td[(1 22tdx=1 p 2te2~mZx=)]TJ/F16 7.97 Tf 6.281 2.103 Td[(~bexp)]TJ/F24 10.909 Tf 12.104 16.909 Td[()]TJ/F18 10.909 Tf 5 -8.836 Td[(x)]TJ/F8 10.909 Tf 10.909 0 Td[(~m+t2 2tdxbylettingz=x)]TJ/F8 10.909 Tf 10.909 0 Td[(~m+t p t=1 p 2e2~mZ)]TJ/F17 5.978 Tf 5.567 1.578 Td[(~b)]TJ/F17 5.978 Tf 5.756 0 Td[(~m+t p tz=e)]TJ/F20 5.978 Tf 7.782 3.258 Td[(z2 2dz=e2~m)]TJ/F8 10.909 Tf 8.098 2.879 Td[(~b)]TJ/F8 10.909 Tf 10.909 0 Td[(2~m)]TJ/F18 10.909 Tf 10.909 0 Td[(t p tSubstituting=aV sV)]TJ/F18 10.909 Tf 12.104 7.38 Td[(sV 2back=exp2~maV sV)]TJ/F18 10.909 Tf 12.104 7.38 Td[(sV 2)]TJ/F8 10.909 Tf 8.098 2.879 Td[(~b p t)]TJ/F8 10.909 Tf 12.105 7.38 Td[(2~m p t)]TJ 10.909 9.396 Td[(p taV sV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(sV 2where~b=1 sVlnVb+nc V0and~m=1 sVlnVb V0.SinceVtislognormalontheintervalwithoutanyjump,theprobabilityVt>Vb+nc,seenfromtimet=0is:PVt>Vb+nc=lnV0 Vb+nc+)]TJ/F18 10.909 Tf 5 -8.836 Td[(aV)]TJ/F19 7.97 Tf 12.104 6.088 Td[(s2V 2t sVp t!=)]TJ/F8 10.909 Tf 15.893 10.258 Td[(~b p t)]TJ 10.909 9.396 Td[(p taV sV)]TJ/F18 10.909 Tf 12.104 7.38 Td[(sV 2!:.3938

PAGE 46

Therefore,P)]TJ/F8 10.909 Tf 15.222 -8.836 Td[(min0s1)]TJ/F18 10.909 Tf 8.072 8.254 Td[(Vs>Vb;V1)]TJ/F8 10.909 Tf 7.085 1.637 Td[(>Vb+nc=P)]TJ/F18 10.909 Tf 5 -8.837 Td[(V1)]TJ/F8 10.909 Tf 7.084 1.636 Td[(>Vb+nc)]TJ/F30 10.909 Tf 10.909 0 Td[(P)]TJ/F8 10.909 Tf 15.221 -8.837 Td[(min0s1)]TJ/F18 10.909 Tf 8.073 8.254 Td[(VsVb+nc=)]TJ/F8 10.909 Tf 18.673 10.259 Td[(~b p 1)]TJ 10.91 7.572 Td[(p 1aV sV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(sV 2)]TJ/F8 10.909 Tf 10.303 0 Td[(exp2~maV sV)]TJ/F18 10.909 Tf 12.104 7.38 Td[(sV 2)]TJ/F8 10.909 Tf 8.098 2.879 Td[(~b p 1)]TJ/F8 10.909 Tf 13.884 7.38 Td[(2~m p 1)]TJ 10.909 7.572 Td[(p 1aV sV)]TJ/F18 10.909 Tf 12.104 7.38 Td[(sV 2!where~b=1 sVlnVb+nc V0and~m=1 sVlnVb V0.Now,taking~m=~bin.34andreplacingVb+ncbyVbin.39,wehavePfmin0s1)]TJ/F18 10.909 Tf 8.072 8.254 Td[(Vs>Vb;V1)]TJ/F8 10.909 Tf 7.085 1.636 Td[(>Vbg=PfV1)]TJ/F8 10.909 Tf 7.085 1.637 Td[(>Vbg)]TJ/F30 10.909 Tf 18.788 0 Td[(Pfmin0s1)]TJ/F18 10.909 Tf 8.073 8.254 Td[(VsVbg=)]TJ/F8 10.909 Tf 18.673 7.38 Td[(~m p 1)]TJ 10.909 9.396 Td[(p taV sV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(sV 2!)]TJ/F8 10.909 Tf 10.909 0 Td[(exp2~maV sV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(sV 2)]TJ/F8 10.909 Tf 12.105 7.38 Td[(3~m p t)]TJ 10.909 9.396 Td[(p taV sV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(sV 2where~m=1 sVlnVb V0.Therefore,Q1)]TJ/F18 10.909 Tf 7.085 1.637 Td[(;1="~b p 1)]TJ 10.909 8.738 Td[(p 11)]TJ/F18 10.909 Tf 6.195 -1.456 Td[(aV sV)]TJ/F18 10.909 Tf 12.104 7.38 Td[(sV 2)]TJ/F8 10.909 Tf 10.909 0 Td[(exp2~maV sV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(sV 2)]TJ/F8 10.909 Tf 8.098 2.879 Td[(~b p 1)]TJ/F8 10.909 Tf 13.884 7.38 Td[(2~m p 1)]TJ 10.91 7.572 Td[(p 1aV sV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(sV 2#,exp2~maV sV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(sV 2)]TJ/F8 10.909 Tf 8.098 2.879 Td[(~b p 1)]TJ/F8 10.909 Tf 13.884 7.38 Td[(2~m p 1)]TJ 10.909 7.572 Td[(p 1aV sV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(sV 2;where~b=1 sVlnVb+nc V0and~m=1 sVlnVb V0.Andforj=2:::n,wehave.32.Ifwelettheendogenoussurvivalprobabilityattimet=j,giventhesurvivalattimet=j)]TJ/F51 10.909 Tf 7.085 1.637 Td[(,isQj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(;j,andtheexogenoussurvivalprobabilityattimet=j,giventhesurvivalattimet=j)]TJ/F51 10.909 Tf -424.915 -18.688 Td[(isgivenbyPj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(;j,thenAssumption6isexpressedasCj)]TJ/F8 10.909 Tf 7.085 1.637 Td[(=E[Cj]=CjQj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;jPj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(;j+Rh1)]TJ/F18 10.909 Tf 10.909 0 Td[(Qj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;jPj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;ji.4039

PAGE 47

andespeciallyCT)]TJ/F8 10.909 Tf 8.485 0 Td[(=E[CT]=1QT)]TJ/F18 10.909 Tf 8.485 0 Td[(;TPT)]TJ/F18 10.909 Tf 8.485 0 Td[(;T+Rh1)]TJ/F18 10.909 Tf 10.909 0 Td[(QT)]TJ/F18 10.909 Tf 8.485 0 Td[(;TPT)]TJ/F18 10.909 Tf 8.485 0 Td[(;Ti:Forconvenience,hereinafterwereferthesurvivalprobabilitybothendogenousandexogenouscombinedattimet=jgiventhesurvivalatt=j)]TJ/F51 10.909 Tf 9.812 1.636 Td[(asSj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(;j,thatis,Sj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;j=Qj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(;jPj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(;j.41Asintheprevioussection,ifweconsidertimeintervals[j)]TJ/F16 7.97 Tf 6.586 0 Td[(1;j,.31reducesto8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:@C @t+1 2s2p@2C @p2+s2VV2@2C @V2+ap@C @p+rV@C @V+r+pC+e)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(tRp=0j)]TJ/F16 7.97 Tf 6.587 0 Td[(1tVb;p>0CVbt;p;t=e)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(tj)]TJ/F16 7.97 Tf 6.586 0 Td[(1t0CV;p;j)]TJ/F8 10.909 Tf 7.084 1.636 Td[(=CV;p;jSj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;j+Rh1)]TJ/F18 10.909 Tf 10.909 0 Td[(Sj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;jiV>Vb;p>0limV!1CV;p;t=e)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(tj)]TJ/F16 7.97 Tf 6.587 0 Td[(1t0limp!1CV;p;t=Rj)]TJ/F16 7.97 Tf 6.586 0 Td[(1tVb:Andespeciallyfortin[n)]TJ/F16 7.97 Tf 6.587 0 Td[(1;T,wehave8>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>:@C @t+1 2s2p@2C @p2+s2VV2@2C @V2+ap@C @p+rV@C @V+r+pC+e)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(tRp=0n)]TJ/F16 7.97 Tf 6.586 0 Td[(1t0;V>Vb;p>0CVbt;p;t=e)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(tn)]TJ/F16 7.97 Tf 6.586 0 Td[(1t0CV;p;T=1V>Vb;p>0CV;p;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=1ST)]TJ/F18 10.909 Tf 8.485 0 Td[(;T+Rh1)]TJ/F18 10.909 Tf 10.909 0 Td[(ST)]TJ/F18 10.909 Tf 8.484 0 Td[(;TiV>Vb;p>0limV!1CV;p;t=e)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(tn)]TJ/F16 7.97 Tf 6.586 0 Td[(1t0limp!1CV;p;t=Rn)]TJ/F16 7.97 Tf 6.587 0 Td[(1tVb:.42Asbefore,wewilltrytosolve.42fort2[n)]TJ/F16 7.97 Tf 6.587 0 Td[(1;T,thensolveforothertbackwards.Usingthe40

PAGE 48

changeofunknownfunctionC=ue)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(t;.42becomes8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:@u @t+1 2s2p@2u @p2+s2VV2@2u @V2+ap@u @p+rV@u @V)]TJ/F18 10.909 Tf 8.485 0 Td[(pu)]TJ/F18 10.909 Tf 10.909 0 Td[(R=0n)]TJ/F16 7.97 Tf 6.587 0 Td[(1t0;V>Vb;p>0uVbt;p;t=Rn)]TJ/F16 7.97 Tf 6.587 0 Td[(1t0uV;p;T=1V>Vb;p>0uV;p;T)]TJ/F8 10.909 Tf 8.484 0 Td[(=1ST)]TJ/F18 10.909 Tf 8.485 0 Td[(;T+Rh1)]TJ/F18 10.909 Tf 10.909 0 Td[(ST)]TJ/F18 10.909 Tf 8.485 0 Td[(;Ti=R+ST)]TJ/F18 10.909 Tf 8.485 0 Td[(;T1)]TJ/F18 10.909 Tf 10.909 0 Td[(RV>Vb;p>0limV!1uV;p;t=CerT)]TJ/F19 7.97 Tf 6.586 0 Td[(tn)]TJ/F16 7.97 Tf 6.587 0 Td[(1t0limp!1uV;p;t=RerT)]TJ/F19 7.97 Tf 6.587 0 Td[(tn)]TJ/F16 7.97 Tf 6.587 0 Td[(1tVb:Usingthechangeofunknownfunctionagain,^u=u)]TJ/F18 10.909 Tf 10.909 0 Td[(R;8>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>:@^u @t+1 2s2p@2^u @p2+s2VV2@2^u @V2+ap@u @p+rV@^^u @V)]TJ/F18 10.909 Tf 10.909 0 Td[(p^u=0n)]TJ/F16 7.97 Tf 6.586 0 Td[(1t0;V>Vb;p>0^uVbt;p;t=0n)]TJ/F16 7.97 Tf 6.587 0 Td[(1t0^uV;p;T=1)]TJ/F18 10.909 Tf 10.91 0 Td[(RV>Vb;p>0^uV;p;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=)]TJ/F18 10.909 Tf 10.909 0 Td[(RST)]TJ/F18 10.909 Tf 8.485 0 Td[(;TV>Vb;p>0limV!1^uV;p;t=0n)]TJ/F16 7.97 Tf 6.587 0 Td[(1t0limp!1^uV;p;t=R)]TJ/F18 10.909 Tf 5 -8.837 Td[(erT)]TJ/F19 7.97 Tf 6.586 0 Td[(t)]TJ/F16 7.97 Tf 6.586 0 Td[(1n)]TJ/F16 7.97 Tf 6.587 0 Td[(1tVb:.43Usingthechangeofunknownfunctionagainandletting^u=WV;p;tS)]TJ/F18 10.909 Tf 10.909 0 Td[(R;.4441

PAGE 49

CV;p;tcanbeexpressedintermsWV;p;tasCV;p;t=ue)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(t=^u+Re)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(t=[WS)]TJ/F18 10.909 Tf 10.909 0 Td[(R+R]e)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(t=[WS)]TJ/F18 10.909 Tf 10.909 0 Td[(WSR+R]e)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(t=WSe)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(t+)]TJ/F18 10.909 Tf 10.909 0 Td[(WSRe)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(t:Wecaninterpretthatthepriceattimetisanexpectationofthevalueofthebondattimet.SowecanregardWSassurvivalprobability,and)]TJ/F18 10.909 Tf 9.774 0 Td[(WSasdefaultprobabilityattimet.Using.44,.43wehavethefollowingpricingmodel:Theorem3.UnderAssumptions1though6,thepriceofthedefaultablezero-couponbondwithexpectedandunexpecteddefaultattimet2[n)]TJ/F16 7.97 Tf 6.587 0 Td[(1;Tisgivenbythefollowing:CV;p;t=WSe)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(t+)]TJ/F18 10.909 Tf 10.909 0 Td[(WSRe)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(twhereSisgivenby.41andWV;p;tsatisesthefollowingPDEanddeterminingconditions:8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:@W @t+1 2s2p@2W @p2+s2VV2@2W @V2+apW @p+rV@^W @V)]TJ/F18 10.909 Tf 10.909 0 Td[(pW=0n)]TJ/F16 7.97 Tf 6.587 0 Td[(1t0;V>Vb;p>0WVbt;p;t=0n)]TJ/F16 7.97 Tf 6.587 0 Td[(1t0WV;p;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=1V>Vb;p>0limV!1WV;p;t=0n)]TJ/F16 7.97 Tf 6.586 0 Td[(1t0limp!1WV;p;t=R)]TJ/F18 10.909 Tf 5 -8.836 Td[(erT)]TJ/F19 7.97 Tf 6.586 0 Td[(t)]TJ/F16 7.97 Tf 6.586 0 Td[(1 S)]TJ/F18 10.909 Tf 10.909 0 Td[(Rn)]TJ/F16 7.97 Tf 6.586 0 Td[(1tVb:.452.2.3ParticularSolutionInadditiontoAssumptions1though6,wetrytondaparticularsolutionWV;p;tinthesepara-tiveformWV;p;t=fV;tgp;t:42

PAGE 50

Substitutingthisin.45,thePDEin.45becomes"@f @t+1 2s2vV2@2f @V2+rV@f @V#g+"@g @t+1 2s2v@2g @p2+ap@g @p)]TJ/F18 10.909 Tf 10.909 0 Td[(pg#f=0Togetthesolutionfor.45,wefurtherassumethefollowing:@f @t+1 2s2vV2@2f @V2+rV@f @V=0;@g @t+1 2s2v@2g @p2+ap@g @p)]TJ/F18 10.909 Tf 10.909 0 Td[(pg=0:Then,weneedtosolvethefollowingtwoproblems.8>>>>><>>>>>:@f @t+1 2s2vV2@2f @V2+rV@f @V=0n)]TJ/F16 7.97 Tf 6.586 0 Td[(1tVbfVbt;t=0n)]TJ/F16 7.97 Tf 6.586 0 Td[(1tVb.46and8>><>>:@g @t+1 2s2p@2g @p2+ap@g @p)]TJ/F18 10.909 Tf 10.909 0 Td[(pg=0n)]TJ/F16 7.97 Tf 6.587 0 Td[(1t0gp;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=1p>0:.47First,wewillsolve.46.Todoso,letV Vb=eyy=lnV Vb;and=T)]TJ/F8 10.909 Tf 7.084 1.636 Td[()]TJ/F18 10.909 Tf 10.909 0 Td[(ts2V 2andfV;t=Vbvy;t:Then,.46becomes8>>>>>>>><>>>>>>>>:@v @t)]TJ/F18 10.909 Tf 12.135 7.38 Td[(@2v @y2+1)]TJ/F8 10.909 Tf 14.388 7.38 Td[(2 s2vr| {z }k@v @y=00
PAGE 51

Assumingthatthesolutionvy;isinthefollowingtypev=!ey+;whereandaretobechosenlater,andlettingk=1)]TJ/F8 10.909 Tf 14.388 7.38 Td[(2 s2vr,therstequationin.48becomes!=!yy+2)]TJ/F18 10.909 Tf 10.909 0 Td[(k)]TJ/F18 10.909 Tf 10.909 0 Td[(!+)]TJ/F18 10.909 Tf 10.909 0 Td[(k!ywherethesubscriptsrefertothepartialderivativeswithrespecttothesubscript.Wechoseandsothatthecoefcientsof!and!yareequalto0,thatis=k 2=1 2)]TJ/F18 10.909 Tf 15.705 7.38 Td[(r s2V=2)]TJ/F18 10.909 Tf 10.909 0 Td[(k=)]TJ/F18 10.909 Tf 9.68 7.38 Td[(k2 4:Substitutingthesein.48,itbecomesaheatequationwiththeinitialcondition:8>><>>:!=!yy<>:hy=!y;0=1 Vbe)]TJ/F20 5.978 Tf 7.782 3.259 Td[(k 2y;y>00;otherwise:Let!1y;beasolutionoftheinitialvalueproblem8><>:!=!yy;
PAGE 52

!y;=!1y;)]TJ/F18 10.909 Tf 10.909 0 Td[(!2y;:Computing!1y;,wehave!1y;=1 2p Z10hexp)]TJ/F8 10.909 Tf 12.105 7.38 Td[(y)]TJ/F18 10.909 Tf 10.909 0 Td[(2 4dlettingz=)]TJ/F18 10.909 Tf 10.909 0 Td[(y p 2=1 Vbp 2Z1)]TJ/F20 5.978 Tf 12.486 3.693 Td[(y p 2e)]TJ/F20 5.978 Tf 7.782 3.259 Td[(k 2zp 2+ye)]TJ/F17 5.978 Tf 7.782 3.259 Td[(1 2z2dz=1 Vbp 2e)]TJ/F20 5.978 Tf 7.782 3.259 Td[(k 2y+ 4k2Z1)]TJ/F20 5.978 Tf 12.486 3.693 Td[(y p 2e)]TJ/F17 5.978 Tf 7.782 3.259 Td[(1 2z+p 2k 22dzletting=z+p 2k 2=1 Vbp 2e)]TJ/F20 5.978 Tf 7.782 3.258 Td[(k 2y+ 4k2Z1)]TJ/F20 5.978 Tf 12.486 3.693 Td[(y p 2+p 2k 2e)]TJ/F20 5.978 Tf 7.782 3.693 Td[(2 2d=1 Vbp 2e)]TJ/F20 5.978 Tf 7.782 3.258 Td[(k 2y+ 4k2d1whered1=y p 2)]TJ 12.105 15.617 Td[(p 2k 2=lnV Vb p T)]TJ/F8 10.909 Tf 7.085 1.636 Td[()]TJ/F18 10.909 Tf 10.909 0 Td[(tsV+p T)]TJ/F8 10.909 Tf 7.085 1.636 Td[()]TJ/F18 10.909 Tf 10.909 0 Td[(tsV 2)]TJ/F18 10.909 Tf 15.705 7.38 Td[(r sV:.49Similarly,!2y;=1 2p Z10hexp)]TJ/F8 10.909 Tf 12.105 7.38 Td[(y+2 4d=1 Vbp 2ek 2y+ 4k2d2whered2=)]TJ/F18 10.909 Tf 16.872 7.38 Td[(y p 2+p 2k 2=lnVb V p T)]TJ/F8 10.909 Tf 7.084 1.636 Td[()]TJ/F18 10.909 Tf 10.909 0 Td[(tsV+p T)]TJ/F8 10.909 Tf 7.085 1.636 Td[()]TJ/F18 10.909 Tf 10.909 0 Td[(tsV 2)]TJ/F18 10.909 Tf 15.705 7.38 Td[(r sV:.50Therefore,fV;t=Vbvy;t=Vbey+)]TJ/F18 10.909 Tf 5 -8.836 Td[(!1y;)]TJ/F18 10.909 Tf 10.909 0 Td[(!2y;=d1)]TJ/F24 10.909 Tf 10.909 15.381 Td[(V Vb1)]TJ/F17 5.978 Tf 9.077 3.258 Td[(2r s2Vd2whered1andd2areasdenedabove.45

PAGE 53

NoticethattheIVPofgp;tisthesameas.11.Hence,WV;p;t=eAt;T)]TJ/F16 7.97 Tf 6.587 0 Td[()]TJ/F19 7.97 Tf 6.587 0 Td[(Bt;T)]TJ/F16 7.97 Tf 6.586 0 Td[(p"d1)]TJ/F24 10.909 Tf 10.909 15.382 Td[(V Vb1)]TJ/F17 5.978 Tf 9.077 3.258 Td[(2r s2Vd2#whered1andd2aredenedby.49and.50,andAt;T)]TJ/F8 10.909 Tf 7.085 1.636 Td[(andBt;T)]TJ/F8 10.909 Tf 7.085 1.636 Td[(aredenedby.15,and.16.SoCV;p;tisgivenbyCV;p;t=WV;p;tST)]TJ/F18 10.909 Tf 8.485 0 Td[(;Te)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(t+)]TJ/F8 10.909 Tf 5 -8.836 Td[(1)]TJ/F18 10.909 Tf 10.909 0 Td[(WV;p;tST)]TJ/F18 10.909 Tf 8.485 0 Td[(;TRe)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(t=eAt;T)]TJ/F16 7.97 Tf 6.586 0 Td[()]TJ/F19 7.97 Tf 6.586 0 Td[(Bt;T)]TJ/F16 7.97 Tf 6.587 0 Td[(pd1)]TJ/F24 10.909 Tf 10.909 15.382 Td[(V Vb1)]TJ/F17 5.978 Tf 9.077 3.258 Td[(2r s2Vd2ST)]TJ/F18 10.909 Tf 8.485 0 Td[(;Te)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(t+1)]TJ/F18 10.909 Tf 10.909 0 Td[(eAt;T)]TJ/F16 7.97 Tf 6.587 0 Td[()]TJ/F19 7.97 Tf 6.587 0 Td[(Bt;T)]TJ/F16 7.97 Tf 6.586 0 Td[(pd1)]TJ/F24 10.909 Tf 10.909 15.382 Td[(V Vb1)]TJ/F17 5.978 Tf 9.078 3.258 Td[(2r s2Vd2ST)]TJ/F18 10.909 Tf 8.484 0 Td[(;TRe)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(tandwhent=n)]TJ/F16 7.97 Tf 6.587 0 Td[(1,wehaveCV;p;n)]TJ/F7 6.974 Tf 6.226 0 Td[(1=eAn)]TJ/F6 4.981 Tf 5.396 0 Td[(1;T)]TJ/F7 6.974 Tf 6.227 0 Td[()]TJ/F10 6.974 Tf 6.227 0 Td[(Bn)]TJ/F6 4.981 Tf 5.397 0 Td[(1;T)]TJ/F7 6.974 Tf 6.227 0 Td[(p~d1;n)]TJ/F24 9.963 Tf 9.962 14.048 Td[(V Vb1)]TJ/F6 4.981 Tf 8.641 2.678 Td[(2r s2V~d2;nST)]TJ/F18 9.963 Tf 7.748 0 Td[(;Te)]TJ/F10 6.974 Tf 6.227 0 Td[(rT)]TJ/F10 6.974 Tf 6.227 0 Td[(n)]TJ/F6 4.981 Tf 5.397 0 Td[(1+1)]TJ/F18 9.963 Tf 9.963 0 Td[(eAn)]TJ/F6 4.981 Tf 5.397 0 Td[(1;T)]TJ/F7 6.974 Tf 6.227 0 Td[()]TJ/F10 6.974 Tf 6.227 0 Td[(Bn)]TJ/F6 4.981 Tf 5.396 0 Td[(1;T)]TJ/F7 6.974 Tf 5.895 0.997 Td[(p~d1;n)]TJ/F24 9.963 Tf 9.963 14.048 Td[(V Vb1)]TJ/F6 4.981 Tf 8.64 2.677 Td[(2r s2V~d2;nST)]TJ/F18 9.963 Tf 7.749 0 Td[(;TRe)]TJ/F10 6.974 Tf 6.227 0 Td[(rT)]TJ/F10 6.974 Tf 6.226 0 Td[(n)]TJ/F6 4.981 Tf 5.396 0 Td[(1where,forj=1n,d1;j=lnVj)]TJET1 0 0 1 285.853 265.901 cmq[]0 d0 J0.436 w0 0.218 m18.568 0.218 lSQ1 0 0 1 -285.853 -265.901 cmBT/F19 7.97 Tf 290.785 259.63 Td[(Vb p T=nsV+p T=nsV 2)]TJ/F18 10.909 Tf 15.705 7.381 Td[(r sV.51andd2;j=lnVb Vj)]TJET1 0 0 1 266.705 178.974 cmq[]0 d0 J0.436 w0 0.218 m42.924 0.218 lSQ1 0 0 1 -266.705 -178.974 cmBT/F24 10.909 Tf 266.705 176.792 Td[(p T=nsV+p T=nsV 2)]TJ/F18 10.909 Tf 15.705 7.38 Td[(r sV:.52Wewillsolvefort
PAGE 54

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:@C @t+1 2s2p@2C @p2+s2VV2@2C @V2+ap@C @p+rV@C @V+r+pC+e)]TJ/F10 6.974 Tf 6.227 0 Td[(rT)]TJ/F10 6.974 Tf 6.227 0 Td[(tRp=0n)]TJ/F7 6.974 Tf 6.227 0 Td[(2t0;V>Vb;p>0CV;p;n)]TJ/F7 6.974 Tf 6.226 0 Td[(1)]TJ/F8 9.963 Tf 6.725 1.799 Td[(=erT)]TJ/F10 6.974 Tf 6.227 0 Td[(n)]TJ/F6 4.981 Tf 5.397 0 Td[(1"eAn)]TJ/F6 4.981 Tf 5.396 0 Td[(1;T)]TJ/F7 6.974 Tf 6.227 0 Td[()]TJ/F10 6.974 Tf 6.227 0 Td[(Bn)]TJ/F6 4.981 Tf 5.397 0 Td[(1;T)]TJ/F7 6.974 Tf 6.227 0 Td[(p~d1;n)]TJ/F24 9.963 Tf 9.962 14.048 Td[(V Vb1)]TJ/F6 4.981 Tf 8.641 2.678 Td[(2r s2V~d2;n!ST)]TJ/F18 9.963 Tf 7.748 0 Td[(;TSn)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F18 9.963 Tf 6.724 1.799 Td[(;n)]TJ/F7 6.974 Tf 6.227 0 Td[(1+R1)]TJ/F18 9.963 Tf 9.963 0 Td[(eAn)]TJ/F6 4.981 Tf 5.396 0 Td[(1;T)]TJ/F7 6.974 Tf 6.226 0 Td[()]TJ/F10 6.974 Tf 6.226 0 Td[(Bn)]TJ/F6 4.981 Tf 5.397 0 Td[(1;T)]TJ/F7 6.974 Tf 6.227 0 Td[(p~d1;n)]TJ/F24 9.963 Tf 9.963 14.048 Td[(V Vb1)]TJ/F6 4.981 Tf 8.641 2.678 Td[(2r s2V~d2;n!ST)]TJ/F18 9.963 Tf 7.749 0 Td[(;TSn)]TJ/F7 6.974 Tf 6.226 0 Td[(1)]TJ/F18 9.963 Tf 6.725 1.798 Td[(;n)]TJ/F7 6.974 Tf 6.227 0 Td[(1#V>Vb;p>0:.53Asbefore,lettingC=ue)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(t,.53becomes8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:@u @t+1 2s2p@2u @p2+s2VV2@2u @V2+ap@u @p+rV@u @V)]TJ/F18 9.963 Tf 9.963 0 Td[(pu)]TJ/F18 9.963 Tf 9.962 0 Td[(R=0n)]TJ/F7 6.974 Tf 6.227 0 Td[(2t0;V>Vb;p>0uV;p;n)]TJ/F7 6.974 Tf 6.226 0 Td[(1)]TJ/F8 9.963 Tf 6.725 1.798 Td[(=eAn)]TJ/F6 4.981 Tf 5.396 0 Td[(1;T)]TJ/F7 6.974 Tf 5.895 0.997 Td[()]TJ/F10 6.974 Tf 6.227 0 Td[(Bn)]TJ/F6 4.981 Tf 5.396 0 Td[(1;T)]TJ/F7 6.974 Tf 5.895 0.997 Td[(p~d1;n)]TJ/F24 9.963 Tf 9.963 14.048 Td[(V Vb1)]TJ/F6 4.981 Tf 8.64 2.677 Td[(2r s2V~d2;n!ST)]TJ/F18 9.963 Tf 6.725 1.494 Td[(;TSn)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F18 9.963 Tf 6.393 2.795 Td[(;n)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F18 9.963 Tf 9.962 0 Td[(R+RV>Vb;p>0:Usingthechangeofunknownfunctionagain,^u=u)]TJ/F18 10.909 Tf 10.909 0 Td[(R;8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:@^u @t+1 2s2p@2^u @p2+s2VV2@2^u @V2+ap@u @p+rV@^^u @V)]TJ/F18 10.909 Tf 10.909 0 Td[(p^u=0n)]TJ/F16 7.97 Tf 6.586 0 Td[(2t0;V>Vb;p>0^uV;p;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F8 10.909 Tf 6.752 3.294 Td[(=eAn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.254 1.074 Td[()]TJ/F19 7.97 Tf 6.587 0 Td[(Bn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.254 1.074 Td[(p~d1;n)]TJ/F24 10.909 Tf 10.909 8.837 Td[()]TJ/F19 7.97 Tf 7.141 -4.542 Td[(V Vb1)]TJ/F17 5.978 Tf 9.077 3.258 Td[(2r s2V~d2;n!ST)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;TSn)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F18 10.909 Tf 6.753 3.294 Td[(;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F18 10.909 Tf 10.909 0 Td[(RV>Vb;p>0:.54Usingthechangeofunknownfunctionagainandletting^u=WV;p;teAn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.586 0 Td[()]TJ/F19 7.97 Tf 6.586 0 Td[(Bn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.587 0 Td[(p~d1;n)]TJ/F24 10.909 Tf 10.909 15.382 Td[(V Vb1)]TJ/F17 5.978 Tf 9.077 3.259 Td[(2r s2V~d2;n!ST)]TJ/F18 10.909 Tf 8.485 0 Td[(;TSn)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F18 10.909 Tf 7.085 2.22 Td[(;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F18 10.909 Tf 10.909 0 Td[(R47

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Then.53reducestothesameIVPas.44,sothesolutionisgivenbyWV;p;t=eAt;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F16 7.97 Tf 6.254 1.738 Td[()]TJ/F19 7.97 Tf 6.587 0 Td[(Bt;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F16 7.97 Tf 6.254 1.738 Td[(p"d1)]TJ/F24 10.909 Tf 10.909 15.381 Td[(V Vb1)]TJ/F17 5.978 Tf 9.077 3.258 Td[(2r s2Vd2#.55whered1=lnV Vb q n)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F8 10.909 Tf 7.084 2.22 Td[()]TJ/F18 10.909 Tf 10.909 0 Td[(tsV+q n)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F8 10.909 Tf 7.084 2.22 Td[()]TJ/F18 10.909 Tf 10.909 0 Td[(tsV 2)]TJ/F18 10.909 Tf 15.705 7.38 Td[(r sVandd2=lnVb V q n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F8 10.909 Tf 7.085 2.22 Td[()]TJ/F18 10.909 Tf 10.909 0 Td[(tsV+q n)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F8 10.909 Tf 7.084 2.22 Td[()]TJ/F18 10.909 Tf 10.909 0 Td[(tsV 2)]TJ/F18 10.909 Tf 15.705 7.38 Td[(r sV:Fort2n)]TJ/F16 7.97 Tf 6.587 0 Td[(2;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1,CV;p;t=WV;p;teAn)]TJ/F6 4.981 Tf 5.396 0 Td[(1;T)]TJ/F7 6.974 Tf 5.895 0.996 Td[()]TJ/F10 6.974 Tf 6.227 0 Td[(Bn)]TJ/F6 4.981 Tf 5.396 0 Td[(1;T)]TJ/F7 6.974 Tf 5.895 0.996 Td[(p~d1;n)]TJ/F24 9.963 Tf 9.962 11.059 Td[(V Vb1)]TJ/F6 4.981 Tf 8.64 2.677 Td[(2r s2V~d2;nST)]TJ/F18 9.963 Tf 6.725 1.494 Td[(;TSn)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F18 9.963 Tf 6.392 2.795 Td[(;n)]TJ/F7 6.974 Tf 6.227 0 Td[(1e)]TJ/F10 6.974 Tf 6.226 0 Td[(rT)]TJ/F10 6.974 Tf 6.227 0 Td[(t+"WV;p;teAn)]TJ/F6 4.981 Tf 5.397 0 Td[(1;T)]TJ/F7 6.974 Tf 5.895 0.997 Td[()]TJ/F10 6.974 Tf 6.227 0 Td[(Bn)]TJ/F6 4.981 Tf 5.396 0 Td[(1;T)]TJ/F7 6.974 Tf 5.895 0.997 Td[(p~d1;n)]TJ/F24 9.963 Tf 9.963 11.059 Td[(V Vb1)]TJ/F6 4.981 Tf 8.641 2.678 Td[(2r s2V~d2;nST)]TJ/F18 9.963 Tf 6.725 1.494 Td[(;TSn)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F18 9.963 Tf 6.392 2.795 Td[(;n)]TJ/F7 6.974 Tf 6.227 0 Td[(1e)]TJ/F10 6.974 Tf 6.226 0 Td[(rT)]TJ/F10 6.974 Tf 6.227 0 Td[(t#Re)]TJ/F10 6.974 Tf 6.226 0 Td[(rT)]TJ/F10 6.974 Tf 6.227 0 Td[(t;whereWV;p;t;~d1;n;~d2;naregivenby.55,.51and.52respectively.Repeatingthisbackwards,foranyt2j)]TJ/F16 7.97 Tf 6.587 0 Td[(1;j;j=1;;nweobtainthefollowing:Theorem4.Undertheassumptions1through6,andadditionalassumptionsmadeinthissubsec-tion,thepriceofcorporatecoupon-bondwithunexpectedandexpecteddefaultisgivenbyGV;p;t=CV;p;t+XitciCV;p;t;iwhere48

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CV;p;t=eAt;j)]TJ/F7 6.974 Tf 5.895 0.996 Td[()]TJ/F10 6.974 Tf 6.227 0 Td[(Bt;j)]TJ/F7 6.974 Tf 5.895 0.996 Td[(p+Pn)]TJ/F6 4.981 Tf 5.396 0 Td[(1i=jAi;i+1)]TJ/F7 6.974 Tf 5.895 1.334 Td[()]TJ/F10 6.974 Tf 6.227 0 Td[(Bi;i+1)]TJ/F7 6.974 Tf 5.895 1.334 Td[(pd1)]TJ/F24 9.963 Tf 9.963 14.047 Td[(V Vb1)]TJ/F6 4.981 Tf 8.641 2.677 Td[(2r s2Vd2Sj)]TJ/F18 9.963 Tf 6.725 1.495 Td[(;jnYi=j+1~d1;n)]TJ/F24 9.963 Tf 9.963 11.058 Td[(V Vb1)]TJ/F6 4.981 Tf 8.64 2.677 Td[(2r s2V~d2;nSi)]TJ/F7 6.974 Tf 6.226 0 Td[(1)]TJ/F18 9.963 Tf 6.725 1.799 Td[(;ie)]TJ/F10 6.974 Tf 6.227 0 Td[(rT)]TJ/F10 6.974 Tf 6.226 0 Td[(t1)]TJ/F18 9.963 Tf 9.962 0 Td[(eAt;j)]TJ/F7 6.974 Tf 5.895 0.996 Td[()]TJ/F10 6.974 Tf 6.227 0 Td[(Bt;j)]TJ/F7 6.974 Tf 5.895 0.996 Td[(p+Pn)]TJ/F6 4.981 Tf 5.396 0 Td[(1i=jAi;i+1)]TJ/F7 6.974 Tf 5.894 1.334 Td[()]TJ/F10 6.974 Tf 6.226 0 Td[(Bi;i+1)]TJ/F7 6.974 Tf 5.895 1.334 Td[(pd1)]TJ/F24 9.963 Tf 9.963 14.048 Td[(V Vb1)]TJ/F6 4.981 Tf 8.641 2.678 Td[(2r s2Vd2Sj)]TJ/F18 9.963 Tf 6.725 1.494 Td[(;jnYi=j+1~d1;n)]TJ/F24 9.963 Tf 9.962 11.058 Td[(V Vb1)]TJ/F6 4.981 Tf 8.641 2.677 Td[(2r s2V~d2;nSi)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F18 9.963 Tf 6.724 1.799 Td[(;i!e)]TJ/F10 6.974 Tf 6.226 0 Td[(rT)]TJ/F10 6.974 Tf 6.227 0 Td[(t:.5649

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Chapter3CreditDerivativesPricingwithConstantInterestRate3.1CreditDefaultSwaptionwithConstantInterestRate-UnexpectedDefaultIntheexistingliteratureseeSchonbuchera,andHulletal.,thevalueofthecreditdefaultswaptionwasgivenassumingthatthepriceofunderlyingasset,i.e.,thecreditdefaultswapfollowsthegeometricBrownianmotion.However,whentheinformationregardingtheforwardcreditdefaultswapissparse,itisnotconvenienttoformapricingmodelbasedonthepriceofforwardcreditdefaultswap.InHull'spaper,thepricingformulaforforwardcreditdefaultswapisgiven;however,theformuladoesnotsuggestthattheforwardcreditdefaultswapfollowsthegeometricBrownianmotion.Thisdiscrepancywasobservedsincethearbitrageprinciplewasusedtopricetheforwardcreditdefaultswap,andthenaPDEapproachwastakentoevaluatethepriceofcreditdefaultswaption.Thisapproachwillalsobeproblematicwhenthereferenceentityhasnoexistingcreditdefaultswap,which,ifthereexists,canbeusedtoinferthedriftandvolatilityofforwardcreditdefaultswap.Inthispaper,wewilltakeadifferentapproach.First,weapplythePDEapproachtoevaluatetheforwardcreditdefaultswap,andthenusethearbitrageprincipletondthepriceofcreditdefaultswaption.Thisapproachmakessensesincetheunderlyingassetofthecreditdefaultswaptiondoesnotexistuntilthedatetheoptionexpires.Also,bytakingthisapproach,theexpectedfeefortheforwardcreditdefaultswaptakesthejumpsindefaultintensityintoconsideration.50

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3.1.1FormulationThefollowingassumptionsarevalidforChapter3only.Assumption1:Lett=0andt=T0bethetimewhenthecreditdefaultswaptionhereinafter,theswaptionstartsandexpiresrespectively.t=T0isalsowhentheforwardcreditdefaultswaptakeseffectupontheexerciseoftheswaption.Lett=TNbetheexpirationofthecreditdefaultswapand,forsimplicity,letusassumethatTj;j=1;;Nfallsontheinterestpaymentdateofthecouponbondsissuedbythereferenceentityofthecreditdefaultswapandthecreditdefaultswapterminatesonthedaythecouponbondsaretoberedeemed,thatisn=TNwherenisasdenedinsubsection2.1.1.Letzbetheexercisepriceoftheswaption.Thisisactuallythefeetheswaptionholderpaysonthenotionalamountofthecreditdefaultswaponcetheswaptionisexercised.Letzt;tT0betheannualfeeofforwardcreditdefaultswap.DeneZt:=ztE[NXi:Ti>tbTi)]TJ/F19 7.97 Tf 11.06 -1.636 Td[(iIfTig+b)]TJ/F22 7.97 Tf 11.06 4.504 Td[(IfT0TNg]wherebTi=expf)]TJ/F24 10.909 Tf 15.757 8.788 Td[(RTitrdsg,isthetimeofdefault,and)]TJ/F19 7.97 Tf 6.818 -1.637 Td[(iand)]TJ/F22 7.97 Tf 6.818 3.958 Td[(arethelengthsoftimeintervalsincethelastfeepaymenttillTiandthedefaultdate,respectively.Therefore,Ztisthepresentvalueattimetoftheforwardcreditdefaultswap.Observethatbythearbitrageprinciple,thepresentvalueofthetotalfeelegisthesameasthepresentvalueoftheprotectionleg.Thatis:Zt=E[b)]TJ/F18 10.909 Tf 10.909 0 Td[(RIfTNg]:.1Assumption2:Defaultisanexogenousevent.Unexpecteddefaultprobabilityonanyinterval[t;t+dt]isgivenby,dp=app;tdt+spp;tdW1+pj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(UjIfj2t;t+dt]g;.2whereIfj2t;t+dt]gisanindicatorfunctiontaking1whenj2t;t+dt]and0otherwise,W1isastandardBrownianmotion,andUjisdenedthesameasinAssumption1inChapter1.Assumption3:DefaultrecoveryisgivenintheformoffacevalueexogenousrecoveryRe)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.586 0 Td[(twhereRisconstantwith0R1andTisthematurityofthebondorintheformofmarketpriceexogenousrecoveryRbondpriceatdefaulttime.51

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Assumption4:Thepriceofdefaultablecorporatezero-couponbondpriceisgivenbythefunction^C=^Cp;t,whosesolutionisgivenby.18,andthepriceoftheswaptionisgivenby^X=^Xp;t.Problem:Underthissettingandaboveassumptions,weshallndthepriceofthecreditdefaultswaption^Xp;t.3.1.2DerivationofthemodelWeconstructaportfoliobyhedging^Xp;twiththereferenceentity'szero-couponbondswithexogenousdefault.Sothevalueoftheportfoliois:=^X)]TJ/F8 10.909 Tf 10.91 0 Td[(^C:Thechangeofvalueofthisportfoliooverasmalltimeincrement[t;t+dt]isgivenbyd=d^X)]TJ/F8 10.909 Tf 10.909 0 Td[(d^C:Ifthereisnodefaultover[t;t+dt]withprobability1)]TJ/F18 10.909 Tf 10.909 0 Td[(ptdt,thenwehaved=@^X @tdt+@^X @pdp+1 2s2p@2^X @p2dt)]TJ/F8 10.909 Tf 8.485 0 Td[(@^C @tdt+@^C @pdp+1 2s2p@2^C @p2dt+f^CpTj;t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^CpTj)]TJ/F18 10.909 Tf 7.084 1.689 Td[(;tgIfTj2t;t+dt]g:Togetridoftheuncertaintycausedbydpterm,wechoose=@^X @p@C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1.Then,wehaved=@^X @t+1 2s2p@2^X @p2dt)]TJ/F18 10.909 Tf 9.68 7.38 Td[(@^X @p@^C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1@^C @t+1 2s2p@2^C @p2dt+f^CpTj;t)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^CpTj)]TJ/F18 10.909 Tf 7.085 1.688 Td[(;tgIfTj2t;t+dt]g:.3Ifthereisdefaultwithprobabilityptdtbeforetheinceptionoftheforwardcreditdefaultswap,theswaptioncontractbecomesvoid;therefore,wehave:d=)]TJ/F8 10.909 Tf 11.614 2.757 Td[(^X)]TJ/F18 10.909 Tf 12.104 7.38 Td[(@^X @p@^C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1R)]TJ/F8 10.909 Tf 13.38 2.757 Td[(^C:.452

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Nowbythearbitrageprinciple,wemusthaved=rdt.Takingtheexpectationofd,bytheItoLemma,andsettingthisequaltordt,i.e.,setting:3)]TJ/F18 10.909 Tf 10.909 0 Td[(ptdt+:4ptdt=rdt,wehave@^X @t+1 2s2p@2^X @p2)]TJ/F18 10.909 Tf 10.91 0 Td[(r^X)]TJ/F8 10.909 Tf 14.038 2.758 Td[(^Xptdt)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@^X @p@^C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1@^C @t+1 2s2p@2^C @p2)]TJ/F8 10.909 Tf 10.909 0 Td[(R)]TJ/F8 10.909 Tf 13.38 2.757 Td[(^Cpt+f^CpTj;t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^CpTj)]TJ/F18 10.909 Tf 7.085 1.688 Td[(;tgIfTj2t;t+dt]gptdt)]TJ/F18 10.909 Tf 10.909 0 Td[(r^Cdt+f^CpTj;t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^CpTj)]TJ/F18 10.909 Tf 7.084 1.688 Td[(;tgIfTj2t;t+dt]g=0:By.2,@^C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1@^C @t+1 2s2p@2^C @p2)]TJ/F8 10.909 Tf 10.909 0 Td[(R)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^Cpt+f^CpTj;t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^CpTj)]TJ/F18 10.909 Tf 7.085 1.688 Td[(;tgIfTj2t;t+dt]gptdt)]TJ/F18 10.909 Tf 10.909 0 Td[(r^Cdt+f^CpTj;t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^CpTj)]TJ/F18 10.909 Tf 7.084 1.689 Td[(;tgIfTj2t;t+dt]g=)]TJ/F18 10.909 Tf 8.485 0 Td[(apdt:Becausethedurationoftheswaptionisrelativelyshorttypicallythreemonthstosixmonths,andweassumethattheexpirationdateoftheswaptionfallsononeofthecouponpaymentdateofthebondsissuedbythereferenceentity,wecanassumethatthereisnojumpindefaultintensityduringthelifetimeofoption.Therefore,@^X @t+1 2s2p@2^X @p2+ap@^X @p)]TJ/F8 10.909 Tf 10.909 0 Td[(pt+r^X=0:Notethatthepresentvalueofforwardcreditdefaultswapisgivenby.1,^XT0=[ZT0)]TJ/F18 10.909 Tf 10.909 0 Td[(Z]+=hE[b)]TJ/F18 10.909 Tf 10.909 0 Td[(RIfTNg])]TJ/F18 10.909 Tf 10.909 0 Td[(zE[bIfTNg]i+:.5Fromthenancialpointofview,wecanexpectthevalueoftheswaptionrightbeforethejumpi.e.,att=T0tobetheexpectationofthevalueatt=T0.Sowecanassumethattheterminalconditionasfollows:^XT0)]TJ/F8 10.909 Tf 7.085 1.636 Td[(=PT0)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;T0hE[b)]TJ/F18 10.909 Tf 10.909 0 Td[(RIfTNg])]TJ/F18 10.909 Tf 10.909 0 Td[(zE[bIfTNg]i+.653

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wherePT0)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;T0isgivenby.4.Letting^Xp;t=W^Xp;T0)]TJ/F8 10.909 Tf 7.084 1.636 Td[(,wehave8>><>>:@W @t+1 2s2p@2W @p2+ap@W @p)]TJ/F8 10.909 Tf 10.909 0 Td[(pt+rW=0;t0Wp;T0)]TJ/F8 10.909 Tf 7.085 1.636 Td[(=1;p>0:Asinsubsection2.1.2,wewillrestrictapp;tands2p;ttothefollowingcases:Assumption5:app;tands2p;tarelinearinp,i.e.,app;t=bt)]TJ/F18 10.909 Tf 10.909 0 Td[(ctp;s2p;t=dt+etp:WeagainwilltrytondthesolutionintheformofWp;t=eAt;T0)]TJ/F19 7.97 Tf 6.586 0 Td[(Bt;T0p:Then,wehaveA0+1 2dtB2)]TJ/F18 10.909 Tf 10.909 0 Td[(btB)]TJ/F18 10.909 Tf 10.909 0 Td[(r)]TJ/F18 10.909 Tf 10.909 0 Td[(pB0)]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 2etB2)]TJ/F18 10.909 Tf 10.909 0 Td[(ctB+1=0:Sincethisshouldholdforanyvalueofp,wehavethesystemofequations.8><>:A0+1 2dtB2)]TJ/F18 10.909 Tf 10.91 0 Td[(btB)]TJ/F18 10.909 Tf 10.909 0 Td[(r=0B0)]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 2etB2)]TJ/F18 10.909 Tf 10.909 0 Td[(ctB+1:.7NotingthatAT0)]TJ/F18 10.909 Tf 7.084 1.636 Td[(;T0)]TJ/F8 10.909 Tf 7.085 1.636 Td[(=BT0)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;T0)]TJ/F8 10.909 Tf 7.084 1.636 Td[(=0sinceWp;T0)]TJ/F8 10.909 Tf 7.084 1.636 Td[(=1andthesolutionfor.7isgivenby.16and.18byreplacingT)]TJ/F51 10.909 Tf 11.212 0 Td[(byT0)]TJ/F51 10.909 Tf 7.085 1.636 Td[(,wehavethefollowingconclusion.Theorem5.UndertheassumptionsinSubsection3.1.1,thepriceofthecreditdefaultswaptionwithunexpecteddefaultispossiblygivenby^Xt;p=^XT0;peAt;T0)]TJ/F19 7.97 Tf 6.586 0 Td[(Bt;T0p;.8whereAt;T0=)]TJ/F24 10.909 Tf 10.303 14.849 Td[(ZT0tbsBs;T0)]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 2dsB2s;T0+rds;.954

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Bt;T0=8>>>>>>>>>><>>>>>>>>>>:1)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(cT0)]TJ/F19 7.97 Tf 6.587 0 Td[(t c;dp=)]TJ/F18 10.909 Tf 5 -8.836 Td[(bt)]TJ/F18 10.909 Tf 10.909 0 Td[(cpdt+p dtdW1;c6=0T0)]TJ/F18 10.909 Tf 10.909 0 Td[(t;dp=)]TJ/F18 10.909 Tf 5 -8.836 Td[(bt)]TJ/F18 10.909 Tf 10.909 0 Td[(cpdt+p dtdW1;c=0)]TJ/F24 10.909 Tf 8.485 12.708 Td[(q 2 cexpp c 2T0)]TJ/F18 10.909 Tf 10.909 0 Td[(t)]TJ/F8 10.909 Tf 10.909 0 Td[(exp)]TJ/F24 10.909 Tf 8.485 8.582 Td[(p c 2T0)]TJ/F18 10.909 Tf 10.909 0 Td[(t expp c 2T0)]TJ/F18 10.909 Tf 10.909 0 Td[(t+exp)]TJ/F24 10.909 Tf 8.484 8.583 Td[(p c 2T0)]TJ/F18 10.909 Tf 10.909 0 Td[(t;dp=btdt+p dt+KpdW1.10and^XT0;pisgivenby.5.3.2CreditDerivativesPricingwithConstantInterestRate-ExpectedandUnexpectedDe-fault3.2.1FormulationAssumption1:WeassumethesamesettingforthecreditdefaultswaptionasinSection3.1:itsinceptionandexpiration,theonsetandfeestructureoftheunderlyingforwardcreditdefaultswap.Assumption2:Defaulteventisbothexogenousandendogenous.Unexpecteddefaultprobabilityonanyinterval[t;t+dt]isgivenbydp=app;tdt+spp;tdW1+pj)]TJ/F18 10.909 Tf 7.084 1.637 Td[(UjIfj2t;t+dt]g;.11whereIfj2t;t+dt]gisanindicatorfunctiontaking1whenj2t;t+dt]and0otherwise.ExpecteddefaultoccurswhenthermassetsV=Vt=Vtfallsbelowthebarrier,sayVbt=VB.AsinSection2.2,thermassetisthesumofitscouponbondsandstocksasin.22,andfollowsthegeometricBrownianmotiongivenby.21.Assumption3:Expectedandunexpecteddefaultrecoveryisgivenastheformoffacevalueexoge-nousrecoveryRe)]TJ/F19 7.97 Tf 6.586 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(t;0R1:constant,T:maturityofthebond.Assumption4:Thepriceofdefaultablecorporatezero-couponbondwithbothexpectedandun-expecteddefaultisgivenbythefunctionC=CV;p;tandthepriceoftheswaptionisgivenbyX=XV;p;t.55

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Problem:Underthesesettingandassumptions,wewillndthecreditdefaultswaptionX=XV;p;t.3.2.2DerivationofthemodelWeconstructaportfoliobyhedgingX=XV;p;twith1zero-couponbondsofthereferenceentitywithexogenousandendogenousdefault,and2stocksofthereferenceentity.Sothevalueoftheportfoliois:=X)]TJ/F8 10.909 Tf 10.909 0 Td[(1S)]TJ/F8 10.909 Tf 10.909 0 Td[(2C;andthechangeofvalueofthisportfoliooverasmalltimeincrement[t;t+dt]isgivenbyd=dX)]TJ/F18 10.909 Tf 10.909 0 Td[(d1S)]TJ/F18 10.909 Tf 10.909 0 Td[(d2C:SinceS=V)]TJ/F18 10.909 Tf 10.909 0 Td[(nC m,wehave=X)]TJ/F8 10.909 Tf 10.909 0 Td[(1V)]TJ/F18 10.909 Tf 10.909 0 Td[(nC m)]TJ/F8 10.909 Tf 10.909 0 Td[(2C=X)]TJ/F8 10.909 Tf 12.104 7.38 Td[(1 mV)]TJ/F24 10.909 Tf 10.909 15.382 Td[(1n m)]TJ/F8 10.909 Tf 10.909 0 Td[(2C:Bythesameargumentasbefore,sincethedurationoftheswaptionisrelativelyshort,weassumethatthereisnojumpindefaultintensityandthevalueoftotalassetsduringthetermoftheswaption,excepttheswaptionexpirationfallsonacouponpaymentdate.Therefore,ifthereisnodefaultwithprobability1)]TJ/F18 10.909 Tf 10.909 0 Td[(ptdt,d=dX)]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 mdV)]TJ/F24 10.909 Tf 10.909 15.382 Td[(1n m)]TJ/F8 10.909 Tf 10.909 0 Td[(2dC=@X @VdV+@X @t+1 2)]TJ/F18 10.909 Tf 5 -8.836 Td[(s2VV2@2X @V2+s2p@2X @p2dt+@X @pdp)]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 mdV)]TJ/F24 10.909 Tf 8.485 8.836 Td[()]TJ/F8 10.909 Tf 5 -8.836 Td[(1n m)]TJ/F8 10.909 Tf 10.909 0 Td[(2@C @VdV+@C @t+1 2)]TJ/F18 10.909 Tf 5 -8.836 Td[(s2VV2@2C @V2+s2p@2C @p2dt+@C @pdp:.12Letuschoose1and2sothatwecangetridofuncertaintycausedbydpanddVterms.Thatis,@X @V)]TJ/F8 10.909 Tf 12.104 7.38 Td[(1 m)]TJ/F24 10.909 Tf 10.909 8.837 Td[()]TJ/F8 10.909 Tf 5 -8.837 Td[(1n m)]TJ/F8 10.909 Tf 10.909 0 Td[(2@C @V=0;and@X @p)]TJ/F24 10.909 Tf 10.909 8.837 Td[()]TJ/F8 10.909 Tf 5 -8.837 Td[(1n m)]TJ/F8 10.909 Tf 10.909 0 Td[(2@C @p=0:56

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Solvingfor1and2,1=m"@X @V)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@X @p@C @V@C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1#;and2=n"@X @V)]TJ/F18 10.909 Tf 12.104 7.38 Td[(@X @p@C @V@C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1#)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@X @p@C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1;and1 m=@X @V)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@X @p@C @V@C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1;and1n m)]TJ/F8 10.909 Tf 10.909 0 Td[(2=@X @p@C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1:Puttingthesebackin.11,wehaved=@X @t+1 2)]TJ/F18 10.909 Tf 5 -8.837 Td[(s2VV2@2X @V2+s2p@2X @p2dt)]TJ/F18 10.909 Tf 11.463 7.381 Td[(@X @p@C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1@C @t+1 2)]TJ/F18 10.909 Tf 5 -8.837 Td[(s2VV2@2C @V2+s2p@2C @p2dt:Whenthereisadefaultwithprobabilityptdt,thechangeofthisportfolio'svalueisgivenbyd=)]TJ/F18 10.909 Tf 8.485 0 Td[(X)]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 mdV)]TJ/F24 10.909 Tf 10.909 8.836 Td[()]TJ/F8 10.909 Tf 5 -8.836 Td[(1n m)]TJ/F8 10.909 Tf 10.91 0 Td[(2R)]TJ/F18 10.909 Tf 10.909 0 Td[(C=)]TJ/F18 10.909 Tf 8.485 0 Td[(X)]TJ/F24 10.909 Tf 10.909 15.382 Td[(@X @V)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@X @p@C @V@C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1dV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@X @p@C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1R)]TJ/F18 10.909 Tf 10.91 0 Td[(C:Bythearbitrageprinciple,theexpectationofdisequaltordt.Ignoringthehigherorderofinnitesimaltermsofdt,wehave@X @t+1 2s2VV2@2X @V2+s2p@2X @p2)]TJ/F18 10.909 Tf 10.909 0 Td[(Xpt)]TJ/F18 10.909 Tf 10.909 0 Td[(rX+rV@X @V)]TJ/F18 10.909 Tf 12.104 7.38 Td[(@X @p@C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1@C @t+1 2s2VV2@2C @V2+s2p@2C @p2+rV@C @V+R)]TJ/F18 10.909 Tf 10.91 0 Td[(Cpt)]TJ/F18 10.909 Tf 10.909 0 Td[(rC=0:.13Butby.2,theexpressioninthebracketinthesecondlineintheequationisequalto)]TJ/F18 10.909 Tf 8.485 0 Td[(ap@C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1.So.14reducesto@X @t+1 2s2VV2@2X @V2+s2p@2X @p2+ap@X @p+rV@X @V)]TJ/F8 10.909 Tf 10.909 0 Td[(pt+rX=0:Again,theterminalconditionisgivenby.6;herethesurvivalprobabilityisgivenbyQP,whereQisthesurvivalmeasurebasedontheintensitygivenby.3,Pisthesurvivalmeasuregiven57

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by.4,andthejumpinthesurvivalprobabilityST0)]TJ/F18 10.909 Tf 7.084 1.636 Td[(;T0attimet=T0isgivenby.41.Thereforewehavethefollowingmodel.Theorem6.UnderAssumptions1though4,thepriceofthecreditdefaultswaptionwithexpectedandunexpecteddefaultprobabilityisgivenbythefollowing:8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:@X @t+1 2s2VV2@2X @V2+s2p@2X @p2+ap@X @p+rV@X @V)]TJ/F8 10.909 Tf 10.909 0 Td[(pt+rX=0;t0;V>VbXV;p;T0)]TJ/F8 10.909 Tf 7.084 1.637 Td[(=ST0)]TJ/F18 10.909 Tf 7.085 1.637 Td[(;T0hE[b)]TJ/F18 10.909 Tf 10.909 0 Td[(RIfTNg])]TJ/F18 10.909 Tf 10.909 0 Td[(zE[bIfTNg]i+;p>0;V>VblimV!1XV;p;t=0;t0limp!1XV;p;t=)]TJ/F18 10.909 Tf 10.909 0 Td[(Re)]TJ/F19 7.97 Tf 6.586 0 Td[(rTN)]TJ/F19 7.97 Tf 6.587 0 Td[(t;tVbwhereST0)]TJ/F18 10.909 Tf 7.084 1.637 Td[(;T0attimet=T0isgivenby.41.3.2.3ParticularSolutionInthissubsection,wewillndaparticularsolutiontoalimitedcase.LettingXV;p;t=WXV;p;T0,wehave8>>>>>>>>>>>>><>>>>>>>>>>>>>:@W @t+1 2s2VV2@2W @V2+s2p@2W @p2+ap@W @p+rV@W @V)]TJ/F8 10.909 Tf 10.91 0 Td[(pt+rW=0;t0;V>VbWV;p;T0)]TJ/F8 10.909 Tf 7.085 1.636 Td[(=1;p>0;V>VblimV!1WV;p;t=0;t0limp!1WV;p;t=XV;p;T0;tVb:.14WeassumethatWV;p;t=fV;tgp;t:Substitutingthisinto.14,weobtain@f @t+1 2s2VV2@2f @V2+rV@f @Vg+@g @t+1 2s2p@2g @p2+ap@g @p)]TJ/F8 10.909 Tf 10.909 0 Td[(pt+rgf=0:58

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Tondasolution,wefurtherassumethat@f @t+1 2s2vV2@2f @V2+rV@f @V=0;@g @t+1 2s2v@2g @p2+ap@g @p)]TJ/F18 10.909 Tf 10.909 0 Td[(pg=0:Thenwehavethefollowingtwosystems:8>>>>><>>>>>:@f @t+1 2s2VV2@2f @V2+rV@f @V=0;tVbfVb;t=0;tVb.15and8>><>>:@g @t+1 2s2p@2g @p2+ap@g @p)]TJ/F8 10.909 Tf 10.909 0 Td[(pt+rg=0;t0gp;T0)]TJ/F8 10.909 Tf 7.084 1.636 Td[(=1;p>0:.16Notingthat.15isthesameas.46and.16isthesameas.47,wewillhavethefollowingresult.Theorem7.UnderthesettingandassumptionsinSubsection3.2.1andfurtherassumptionsmadeinthissubsection,thepriceofthecreditdefaultswaptionwithexpectedandunexpecteddefaultisgivenbyXV;t;p=XV;T0;peAt;T0)]TJ/F16 7.97 Tf 6.254 1.107 Td[()]TJ/F19 7.97 Tf 6.587 0 Td[(Bt;T0)]TJ/F16 7.97 Tf 6.255 1.107 Td[(pd1)]TJ/F24 10.909 Tf 10.909 15.382 Td[(V Vb1)]TJ/F17 5.978 Tf 9.077 3.258 Td[(2r s2Vd2.17whereXV;T0;pisdenedin.5,At;T0)]TJ/F8 10.909 Tf 7.085 1.637 Td[(andBt;T0)]TJ/F8 10.909 Tf 7.084 1.637 Td[(aregivenby.9and.10respec-tively,andd1andd2aredenesasfollows:d1=lnV Vb p T0)]TJ/F18 10.909 Tf 10.909 0 Td[(tsV+p T0)]TJ/F18 10.909 Tf 10.909 0 Td[(tsV 2)]TJ/F18 10.909 Tf 15.704 7.38 Td[(r sVd2=lnVb V p T0)]TJ/F18 10.909 Tf 10.909 0 Td[(tsV+p T0)]TJ/F18 10.909 Tf 10.909 0 Td[(tsV 2)]TJ/F18 10.909 Tf 15.704 7.38 Td[(r sV:.1859

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Chapter4DefaultableCorporateCoupon-BondPricingwithStochasticInterestRate4.1CorporateCouponBondwithStochasticInterestRate-UnexpectedDefault4.1.1FormulationUnderthissection,wewillassumethefollowing.Assumption1:Theriskfreeshortterminterestratert=rtfollowsVasicekmodel:drt=t)]TJ/F18 10.909 Tf 10.909 0 Td[(rtdt+srtdW1t:whereisaconstant,tandsrtaredeterministicfunctionsoft,andW1tisastandardBrownianmotion.Underthisassumption,thepriceZtofdefault-freezero-couponbondsatisesthefollowingPDE:8><>:@Z @t+1 2s2r@2Z @r2+t)]TJ/F18 10.909 Tf 10.909 0 Td[(rt@Z @r)]TJ/F18 10.909 Tf 10.909 0 Td[(rZ=0;Zr;T=1:.1andZr;t;T=eAt;T)]TJ/F16 7.97 Tf 8.565 2.015 Td[(Bt;TrBt;T=1)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(T)]TJ/F19 7.97 Tf 6.587 0 Td[(t At;T=)]TJ/F24 10.909 Tf 10.303 14.849 Td[(ZTtsBs;T)]TJ/F8 10.909 Tf 12.105 7.381 Td[(1 2s2rB2s;Tds:.2Assumption2:Unexpecteddefaultprobabilityin[t;t+dt]isptdt,thedefaultintensityptsatisesdp=apr;p;tdt+spr;p;tdW2+pj)]TJ/F18 10.909 Tf 7.25 4.611 Td[(UjIfj2t;t+dt]g60

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withapr;p;t=t+tr+tp;s2p=t+tr+tp;whereW2tisastandardBrownianmotion,andUnexpecteddefaultrecoverisgivenbyRd=RZ,whereR;0R1,isaconstant,andZisthepriceatthedefaulttimeofdefault-freezero-couponbond.Assumption3:CovarianceCovdW1;dW2=.Assumption4:Thedefaultablecorporatecoupon-bondpriceisgivenbythefunction^G=^Gr;p;t,whichconstitutesof^C=^Cr;p;t,thevalueattimetoftheprincipalportiononly,andci,thei-thcouponwithci^Cr;p;t;itobethevalueattimetofi-thcoupondueoni.Therefore,wehave^Gr;p;t=^Cr;p;t+Xitci^Cr;p;t;i:Problem:Undertheseassumptions,weshallndthepriceofdefaultablecorporatecoupon-bond^G=^Gr;p;t.4.1.2DerivationofthemodelAsinChapter2,tohedgetheriskofptweconstructaportfoliobyhedgingonebondwithanotherbondwithdifferentmaturity.Wewillincludedefault-freezero-couponbondheretohedgetheriskcausedbyrt.LetusdenotethepricedofabondwithmaturityTianddefaultrecoveryRibyCir;p;t:Ti;i=1;2.Then,theportfoliois:=^C1)]TJ/F8 10.909 Tf 10.909 0 Td[(1Z)]TJ/F8 10.909 Tf 10.909 0 Td[(2^C2:61

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Thechangeofvaluesintheportfoliooverasmalltimeincrement[t;t+dt],ifthereisnodefaultwithprobability1)]TJ/F18 10.909 Tf 10.909 0 Td[(ptdt,isd=@^C1 @t+1 2s2r@2^C1 @r2+2srsp@2^C1 @r@p+s2p@2^C1 @p2!dt+@^C1 @rdr+@^C1 @pdp+f^C1r;pj;t)]TJ/F8 10.909 Tf 13.379 2.757 Td[(^C1r;pj)]TJ/F18 10.909 Tf 7.085 1.637 Td[(;tgIfj2t;t+dt]g)]TJ/F8 10.909 Tf 10.909 0 Td[(1"@Z @t+1 2s2r@2Z @r2dt+@Z @rdr#)]TJ/F8 10.909 Tf 8.485 0 Td[(2"@^C2 @t+1 2s2r@2^C2 @r2+2srsp@2^C2 @r@p+s2p@2^C2 @p2dt+@^C2 @rdr+@^C2 @pdp+f^C2r;pj;t)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^C2r;pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g#=@^C1 @t+1 2s2r@2^C1 @r2+2srsp@2^C1 @r@p+s2p@2^C1 @p2!dt)]TJ/F8 10.909 Tf 10.909 0 Td[(1@Z @t+1 2s2r@2Z @r2dt)]TJ/F8 10.909 Tf 8.485 0 Td[(2@^C2 @t+1 2s2r@2^C2 @r2+2srsp@2^C2 @r@p+s2p@2^C2 @p2!dt+@^C1 @r)]TJ/F8 10.909 Tf 10.909 0 Td[(1@Z @r)]TJ/F8 10.909 Tf 10.909 0 Td[(2@^C2 @rdr+@^C1 @p)]TJ/F8 10.909 Tf 10.909 0 Td[(2@^C2 @pdp+f^C1r;pj;t)]TJ/F8 10.909 Tf 13.379 2.757 Td[(^C1r;pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g+f^C2r;pj;t)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^C2r;pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g:.3Wewillchoose1and2sothatwecangetridofuncertaintycausedbydranddpterms.Thatis,1=@^C1 @r)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@^C1 @p@^C1 @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1@^C2 @r@^Z @r)]TJ/F16 7.97 Tf 6.586 0 Td[(1and2=@^C1 @p@^C1 @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1:Then.3becomesd=@^C1 @t+1 2s2r@2^C1 @r2+2srsp@2^C1 @r@p+s2p@2^C1 @p2!dt)]TJ/F8 10.909 Tf 10.909 0 Td[(1@Z @t+1 2s2r@2Z @r2dt)]TJ/F8 10.909 Tf 8.485 0 Td[(2@^C2 @t+1 2s2r@2^C2 @r2+2srsp@2^C2 @r@p+s2p@2^C2 @p2!dt+f^C1r;pj;t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^C1r;pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;tgIfj2t;t+dt]g+f^C2r;pj;t)]TJ/F8 10.909 Tf 13.38 2.757 Td[(^C2r;pj)]TJ/F18 10.909 Tf 7.085 1.637 Td[(;tgIfj2t;t+dt]g:.4Ifthereisadefault,withprobabilityptdt,thenthepricechangeintheportfolioisd=R1)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^C1)]TJ/F8 10.909 Tf 10.909 0 Td[(1dZ)]TJ/F8 10.909 Tf 10.909 0 Td[(2R2)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^C2:.562

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BytheArbitragePrinciple,weshouldhaveE[d]=rtdt,i.e.,:4)]TJ/F18 10.909 Tf 9.572 0 Td[(ptdt+:5ptdt=rtdt.BytheItoLemma,andconsideringthetimeperiodwithnojumpi.e.,j62[t;t+dt],wehave@^C1 @t+1 2s2r@2^C1 @r2+2srsp@2^C1 @r@p+s2p@2^C1 @p2)]TJ/F18 10.909 Tf 10.91 0 Td[(rC1+ptR1)]TJ/F18 10.909 Tf 10.909 0 Td[(C1)]TJ/F8 10.909 Tf 8.485 0 Td[(2@^C2 @t+1 2s2r@2^C2 @r2+2srsp@2^C2 @r@p+s2p@2^C2 @p2)]TJ/F18 10.909 Tf 10.909 0 Td[(rC1+ptR2)]TJ/F18 10.909 Tf 10.909 0 Td[(C2)]TJ/F8 10.909 Tf 8.485 0 Td[(1@Z @t+1 2s2r@2Z @r2)]TJ/F18 10.909 Tf 10.909 0 Td[(rZ=0:Notethatby.1@Z @t+1 2s2r@2Z @r2)]TJ/F18 10.909 Tf 10.909 0 Td[(rZ=)]TJ/F18 10.909 Tf 8.485 0 Td[(t)]TJ/F18 10.909 Tf 10.909 0 Td[(rt@Z @r:.6Substituting.6andsimplifying,weobtain@^C1 @t+1 2s2r@2^C1 @r2+2srsp@2^C1 @r@p+s2p@2^C1 @p2+t)]TJ/F18 9.963 Tf 9.963 0 Td[(rt@^C1 @r)]TJ/F18 9.963 Tf 9.962 0 Td[(rC1+ptR1)]TJ/F18 9.963 Tf 9.963 0 Td[(C1@^C1 @p)]TJ/F7 6.974 Tf 6.226 0 Td[(1=@^C2 @t+1 2s2r@2^C2 @r2+2srsp@2^C2 @r@p+s2p@2^C2 @p2+t)]TJ/F18 9.963 Tf 9.963 0 Td[(rt@^C2 @r)]TJ/F18 9.963 Tf 9.963 0 Td[(rC2+ptR2)]TJ/F18 9.963 Tf 9.963 0 Td[(C2@^C2 @p)]TJ/F7 6.974 Tf 6.226 0 Td[(1:ThelefthandsideoftheequationisafunctionofT1butnotT2andtherighthandsideisafunctionofT2butnotT1sothatbothsidesmustbeafunctionindependentofT1andT2,say)]TJ/F18 10.909 Tf 8.485 0 Td[(r;p;t.Sowehave@^C @t+1 2s2r@2^C @r2+2srsp@2^C @r@p+s2p@2^C @p2+t)]TJ/F18 10.909 Tf 8.919 0 Td[(rt@^C @r)]TJ/F18 10.909 Tf 8.919 0 Td[(r^C+ptRd)]TJ/F8 10.909 Tf 11.389 2.757 Td[(^C=)]TJ/F18 10.909 Tf 8.485 0 Td[(r;p;t@^C @p:Herer;p;tisariskneutraldriftofpt.Wecanwriter;p;tintheformofr;p;t=apr;p;t)]TJ/F18 10.909 Tf 11.397 0 Td[(spr;p;tr;p;torr;p;t=apr;p;t)]TJ/F19 7.97 Tf 6.587 0 Td[(r;p;t spr;p;t.Thefunctionr;p;tiscalledamarketpriceofriskptandmeasuresanextracompensationperunitofriskfortakingontheriskincurredbypt.Inamodelwherewecandynamicallyhedgetheportfolio,wecanelimi-natesuchrisktotally,thereforeinthecomputationbelow,weassumer;p;t=0.Thatis,r;p;t=apr;p;t.Asbefore,inadditiontothepreviousassumptions,letusalsoassumethefollowing.63

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Assumption5:Thedefaultablecouponbondpriceattimet=j)]TJ/F51 10.909 Tf 9.687 1.636 Td[(istheexpectationofthepriceattimet=j.Thatis,^Cr;p;j)]TJ/F8 10.909 Tf 7.084 1.637 Td[(=E^Ct;p;j;.7whereexpectationistakenwithrespecttothesurvivalprobabilityP;.Especially,ift=T)]TJ/F8 10.909 Tf 11.745 0 Td[(=n)]TJ/F51 10.909 Tf 7.085 1.636 Td[(,wehave^Cr;p;T)]TJ/F8 10.909 Tf 8.484 0 Td[(=1PT)]TJ/F18 10.909 Tf 8.485 0 Td[(;T+R)]TJ/F18 10.909 Tf 10.909 0 Td[(PT)]TJ/F18 10.909 Tf 8.485 0 Td[(;Tand^Cr;p;T=1ifthereisnotdefaultuntilt=T.Then,usingRd=RZ,onthetimeinterval[n)]TJ/F16 7.97 Tf 6.586 0 Td[(1;TwehaveaPDEwiththeterminalcondition.8>>>>>>>>><>>>>>>>>>:@^C @t+1 2s2r@2^C @r2+2srsp@2^C @r@p+s2p@2^C @p2+t)]TJ/F18 10.909 Tf 10.909 0 Td[(rt@^C @r+apr;p;t@^C @p)]TJ/F8 10.909 Tf 8.485 0 Td[(r+pt^C+pRZ=0;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1t0;p>0^Cr;p;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=1PT)]TJ/F18 10.909 Tf 8.485 0 Td[(;T+R)]TJ/F18 10.909 Tf 10.909 0 Td[(PT)]TJ/F18 10.909 Tf 8.485 0 Td[(;T=e)]TJ/F19 7.97 Tf 6.587 0 Td[(pT)]TJ/F19 7.97 Tf 6.255 1.34 Td[(Un+R)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.34 Td[(Unr>0;p>0:.8Inordertondanexplicitsolution,assumethatthedriftandvolatilityofptisnotcorrelatedtotheshorttermratert.Thatis,ap=t+tp;s2p=t+tp:.9Usingthechangeofunknownfunctionandvariableas^Cr;p;t=Zr;tup;t;andusing.1,.8reducesto8>><>>:@u @t+1 2s2p@2u @p2+ap@u @p)]TJ/F18 10.909 Tf 10.909 0 Td[(pu)]TJ/F18 10.909 Tf 10.909 0 Td[(R=0;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1t0up;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=e)]TJ/F19 7.97 Tf 6.587 0 Td[(pT)]TJ/F19 7.97 Tf 6.255 1.339 Td[(Un+R)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(pT)]TJ/F19 7.97 Tf 6.254 1.339 Td[(Un;r>0;p>0:.1064

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Notingthatthissystemisthesameas.8,wehavethesolutionfort2[n)]TJ/F16 7.97 Tf 6.586 0 Td[(1;T,^Cr;p;t=Zr;teAt;T)]TJ/F16 7.97 Tf 6.587 0 Td[()]TJ/F19 7.97 Tf 6.587 0 Td[(Bt;T)]TJ/F16 7.97 Tf 6.586 0 Td[(pt+)]TJ/F18 10.909 Tf 10.91 0 Td[(eAt;T)]TJ/F16 7.97 Tf 6.586 0 Td[()]TJ/F19 7.97 Tf 6.586 0 Td[(Bt;T)]TJ/F16 7.97 Tf 6.587 0 Td[(ptRwhereAt;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=)]TJ/F24 10.909 Tf 10.303 14.849 Td[(ZT)]TJ/F19 7.97 Tf -10.955 -21.586 Td[(tsBs;T)]TJ/F8 10.909 Tf 8.485 0 Td[()]TJ/F8 10.909 Tf 12.104 7.38 Td[(1 2sB2s;T)]TJ/F8 10.909 Tf 8.485 0 Td[(dsandBt;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=8>>>>>>>>>>>>><>>>>>>>>>>>>>:1)]TJ/F19 7.97 Tf 6.587 0 Td[(e)]TJ/F20 5.978 Tf 5.756 0 Td[(T)]TJ/F20 5.978 Tf 5.756 0 Td[(t ;dp=t)]TJ/F18 10.909 Tf 10.909 0 Td[(pdt+p tdW2;6=0n)]TJ/F18 10.909 Tf 10.909 0 Td[(t;dp=t)]TJ/F18 10.909 Tf 10.909 0 Td[(pdt+p tdW2;=0)]TJ/F24 10.909 Tf 8.485 11.933 Td[(q 2 expp 2T)]TJ/F19 7.97 Tf 6.587 0 Td[(t)]TJ/F19 7.97 Tf 6.587 0 Td[(exp)]TJ/F14 10.909 Tf 6.587 7.031 Td[(p 2T)]TJ/F19 7.97 Tf 6.587 0 Td[(t expp 2T)]TJ/F19 7.97 Tf 6.587 0 Td[(t+exp)]TJ/F14 10.909 Tf 6.587 7.031 Td[(p 2T)]TJ/F19 7.97 Tf 6.587 0 Td[(t;dp=tdt+p t+kpdW2:Now,using.7,wecanobtaintheterminalconditionfortheperiod[n)]TJ/F16 7.97 Tf 6.586 0 Td[(2;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1,andwehavethefollowingPDEwiththeterminalcondition:8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:@^C @t+1 2s2r@2^C @r2+2srsp@2^C @r@p+s2p@2^C @p2+t)]TJ/F18 10.909 Tf 10.909 0 Td[(rt@^C @r+apr;p;t@^C @p)]TJ/F8 10.909 Tf 10.909 0 Td[(r+pt^C+pRZ=0;n)]TJ/F16 7.97 Tf 6.586 0 Td[(2t0;p>0^Cr;p;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F8 10.909 Tf 7.085 2.22 Td[(=Zr;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1E[Cr;p;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1]=Zr;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1eDn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.587 0 Td[()]TJ/F20 5.978 Tf 15.789 7.572 Td[(nPj=n)]TJ/F17 5.978 Tf 5.756 0 Td[(1pj)]TJ/F19 7.97 Tf 6.753 2.897 Td[(Uj+R)]TJ/F18 10.909 Tf 10.909 0 Td[(eDn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.586 0 Td[()]TJ/F20 5.978 Tf 15.789 7.572 Td[(nPj=n)]TJ/F17 5.978 Tf 5.756 0 Td[(1pj)]TJ/F19 7.97 Tf 6.752 2.897 Td[(Uj=Zr;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1eDn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.586 0 Td[()]TJ/F20 5.978 Tf 15.789 7.572 Td[(nPj=n)]TJ/F17 5.978 Tf 5.756 0 Td[(1pj)]TJ/F19 7.97 Tf 6.753 2.898 Td[(Uj)]TJ/F18 10.909 Tf 10.909 0 Td[(R+Rr>0;p>0whereDn)]TJ/F16 7.97 Tf 6.586 0 Td[(1;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=An)]TJ/F16 7.97 Tf 6.587 0 Td[(1;T)]TJ/F8 10.909 Tf 8.485 0 Td[()]TJ/F18 10.909 Tf 10.655 0 Td[(Bn)]TJ/F16 7.97 Tf 6.587 0 Td[(1;T)]TJ/F8 10.909 Tf 8.485 0 Td[(pn)]TJ/F16 7.97 Tf 6.587 0 Td[(1.Again,letting^C=Zr;tup;t,andusing.1,wehavethefollowingPDEwiththeterminalcondition:8>><>>:@u @t+1 2s2p@2u @p2+ap@u @p)]TJ/F18 10.909 Tf 10.909 0 Td[(pu)]TJ/F18 10.909 Tf 10.909 0 Td[(R=0;n)]TJ/F16 7.97 Tf 6.586 0 Td[(2t0up;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F8 10.909 Tf 7.085 2.22 Td[(=eDn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F16 7.97 Tf 6.586 0 Td[()]TJ/F20 5.978 Tf 15.788 7.572 Td[(nPj=n)]TJ/F17 5.978 Tf 5.756 0 Td[(1pj)]TJ/F19 7.97 Tf 6.752 2.897 Td[(Uj)]TJ/F18 10.909 Tf 10.909 0 Td[(R+R;r>0;p>0:Notingthatthisisthesamesystemas.16,fort2[n)]TJ/F16 7.97 Tf 6.586 0 Td[(2;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1,wehavethefollowingsolution.^Cr;p;t=Zr;teAt;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F16 7.97 Tf 6.254 1.738 Td[()]TJ/F19 7.97 Tf 6.586 0 Td[(Bt;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F16 7.97 Tf 6.254 1.738 Td[(pt+)]TJ/F18 10.909 Tf 10.91 0 Td[(eAt;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F16 7.97 Tf 6.254 1.738 Td[()]TJ/F19 7.97 Tf 6.587 0 Td[(Bt;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F16 7.97 Tf 6.254 1.738 Td[(ptR65

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whereAt;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F8 10.909 Tf 7.085 2.22 Td[(=)]TJ/F24 10.909 Tf 10.303 14.848 Td[(Zn)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F19 7.97 Tf -28.373 -19.848 Td[(tsBs;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F8 10.909 Tf 7.085 2.22 Td[()]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 2sB2s;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F8 10.909 Tf 7.084 2.22 Td[(dsandBt;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F8 10.909 Tf 7.084 2.22 Td[(=8>>>>>>>>><>>>>>>>>>:1)]TJ/F19 7.97 Tf 6.586 0 Td[(e)]TJ/F20 5.978 Tf 5.756 0 Td[(n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ 6.254 2.424 Td[()]TJ/F20 5.978 Tf 5.756 0 Td[(t ;dp=t)]TJ/F18 10.909 Tf 10.909 0 Td[(pdt+p tdW2;6=0n)]TJ/F16 7.97 Tf 6.587 0 Td[(1)]TJ/F18 10.909 Tf 10.909 0 Td[(t;dp=t)]TJ/F18 10.909 Tf 10.909 0 Td[(pdt+p tdW2;=0)]TJ/F24 10.909 Tf 8.485 11.933 Td[(q 2 expp 2n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F19 7.97 Tf 6.586 0 Td[(t)]TJ/F19 7.97 Tf 6.586 0 Td[(exp)]TJ/F14 10.909 Tf 6.586 7.031 Td[(p 2n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F19 7.97 Tf 6.587 0 Td[(t expp 2n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F19 7.97 Tf 6.586 0 Td[(t+exp)]TJ/F14 10.909 Tf 6.586 7.031 Td[(p 2n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F19 7.97 Tf 6.587 0 Td[(t;dp=tdt+p t+KpdW2:Byextendingthisbackwards,wehavethefollowingresult.Theorem8.UnderAssumptions1through5,thepriceofthecorporatecoupon-bondwithunex-pecteddefaultandstochasticshorttermrate,forany0t
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andBt;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:n)]TJ/F8 10.909 Tf 10.909 0 Td[(j)]TJ/F8 10.909 Tf 10.909 0 Td[(1)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(j)]TJ/F19 7.97 Tf 6.587 0 Td[(t)]TJ/F19 7.97 Tf 21.297 10.364 Td[(nPk=j+1e)]TJ/F19 7.97 Tf 6.586 0 Td[(k)]TJ/F19 7.97 Tf 6.586 0 Td[(k)]TJ/F17 5.978 Tf 5.756 0 Td[(1 ;ifdp=t)]TJ/F18 10.909 Tf 10.909 0 Td[(pdt+p tdW2+pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(UjIfj2t;t+dt]g;6=0T)]TJ/F18 10.909 Tf 10.909 0 Td[(t;ifdp=t)]TJ/F18 10.909 Tf 10.909 0 Td[(pdt+p tdW2+pj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(UjIfj2t;t+dt]g;=0)]TJ/F24 10.909 Tf 8.485 11.933 Td[(q 2 ep 2j)]TJ/F19 7.97 Tf 6.586 0 Td[(t+nPk=j+1ep 2k)]TJ/F19 7.97 Tf 6.586 0 Td[(k)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F14 10.909 Tf 6.586 7.031 Td[(p 2j)]TJ/F19 7.97 Tf 6.587 0 Td[(t)]TJ/F19 7.97 Tf 21.297 10.364 Td[(nPk=j+1e)]TJ/F14 10.909 Tf 6.586 7.031 Td[(p 2k)]TJ/F19 7.97 Tf 6.586 0 Td[(k)]TJ/F17 5.978 Tf 5.756 0 Td[(1 ep 2j)]TJ/F19 7.97 Tf 6.586 0 Td[(t+nPk=j+1ep 2k)]TJ/F19 7.97 Tf 6.586 0 Td[(k)]TJ/F17 5.978 Tf 5.756 0 Td[(1+e)]TJ/F14 10.909 Tf 6.586 7.031 Td[(p 2j)]TJ/F19 7.97 Tf 6.587 0 Td[(t+nPk=j+1e)]TJ/F14 10.909 Tf 6.586 7.031 Td[(p 2k)]TJ/F19 7.97 Tf 6.586 0 Td[(k)]TJ/F17 5.978 Tf 5.756 0 Td[(1;ifdp=tdt+p t+KpdW2+pj)]TJ/F18 10.909 Tf 7.084 1.637 Td[(UjIfj2t;t+dt]g.13with,oneachtimeinterval,j)]TJ/F16 7.97 Tf 6.587 0 Td[(1t>>>>><>>>>>>:1)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(j)]TJ/F19 7.97 Tf 6.586 0 Td[(t ;dp=t)]TJ/F18 10.909 Tf 10.91 0 Td[(pdt+p tdW2;6=0j)]TJ/F18 10.909 Tf 10.909 0 Td[(t;dp=t)]TJ/F18 10.909 Tf 10.91 0 Td[(pdt+p tdW2;=0)]TJ/F24 10.909 Tf 8.485 11.933 Td[(q 2 ep 2j)]TJ/F19 7.97 Tf 6.586 0 Td[(t)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F14 10.909 Tf 6.587 7.031 Td[(p 2j)]TJ/F19 7.97 Tf 6.586 0 Td[(t ep 2j)]TJ/F19 7.97 Tf 6.586 0 Td[(t+e)]TJ/F14 10.909 Tf 6.587 7.031 Td[(p 2j)]TJ/F19 7.97 Tf 6.586 0 Td[(t;dp=tdt+p t+KpdW2:.1467

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4.2CorporateCouponBondwithStochasticInterestRate-ExpectedandUnexpectedDe-fault4.2.1FormulationUnderthissection,wewillassumethefollowing.Assumption1:Asintheprevioussection,riskfreeshortterminterestratert=rtfollowsVasicekmodel:drt=t)]TJ/F18 10.909 Tf 10.909 0 Td[(rtdt+srdW1t:whereisaconstant,tandsraredeterministicfunctionsoft.Again,asintheprevioussection,thepriceZtsatisesthePDEgivenby.1anditssolutionisgivenby.2.Assumption2:Unexpecteddefaultprobabilityin[t;t+dt]isptdt,thedefaultintensityptsatisesdp=apr;p;tdt+spr;p;tdW2+pjUjIfj2t;t+dt]gwithapr;p;t=t+tr+tp;s2p=t+tr+tp;andunexpecteddefaultrecoveryisgivenbyRd=RZ,whereR:0r1,constant,andZisthepriceatthedefaulttimeofdefaultfreezerocouponbond.Assumption3:ThermassetsVt=Vtconsistsofmsharesoftradedstock,whosepriceattimetisSt=St,andncoupon-bondcerticates,whosepriceattimetisCt=Ct:Vt=mSt+nCt:ThermassetsvaluefollowsageometricBrownianmotion:dV=aVVtdt+sVVtdW3;68

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aVandSVareconstant,andonpredeterminedcouponpaymentdatest=j,wherejreferstoj-thinterestpayment,j=1;;n,thejumpofVtisgivenbyVj=Vj)]TJ/F18 10.909 Tf 10.909 0 Td[(Vj)]TJ/F8 10.909 Tf 10.28 4.611 Td[(=ncCT=nc;wherecisthecouponrateofthebondsandTisthematurityofthebond.Here,weassumethatthebondsareredeemedattheirfacevalue;therefore,CT=1.ExpecteddefaultoccurswhenVVbt;Vbt=VBorVBZandthedefaultrecoveryisalsogivenbyRd=RZ;R:0r1,whereZisasdenedabove.Assumption4:dWidWj=ijdt;i=1;2;3:Howeverweassumethatunexpecteddefaultandexpecteddefaultarenotcorrelated,i.e.,23=0.Assumption5:Thedefaultablecorporatecoupon-bondpriceisgivenbythefunctionG=Gr;V;p;t,whichconstitutesofC=Cr;V;p;t,thevalueattimetoftheprincipalportiononly,andci,thei-thcouponwithciCr;V;p;t;itobethevalueattimetofi-thcoupondueoni.Therefore,wehaveGr;V;p;t=Cr;V;p;t+XitciCr;V;p;t;i:Problem:Undertheseassumptions,weshallndthepriceofdefaultablecoupon-bondwithbothexpectedandunexpecteddefault,whichisgivenasafunctionofr;V;pandt,thatisG=Gr;V;p;t.4.2.2DerivationoftheModelWewillformtheportfoliobybuyingonebondcerticateunderconcernandhedgetheriskincurredbyV;pandrbyselling1sharesofthetradedstock,2coupon-bondcerticateswithunexpecteddefaultonly,whosepriceisgivenby^Cr;p;tand3defaultfreezerocouponbond.Wedenotethedefaultrecoveryfor^Cr;p;tby^R:=C)]TJ/F8 10.909 Tf 10.909 0 Td[(1S)]TJ/F8 10.909 Tf 10.909 0 Td[(2^C)]TJ/F8 10.909 Tf 10.909 0 Td[(3Z:69

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BytheArbitragePrinciple,thepricechangeoftheportfoliooverasmallincrementoftimedtisequaltordt,sothat,aftertakingAssumption3intoconsideration,wehavethefollowing:d=dC)]TJ/F8 10.909 Tf 10.91 0 Td[(1dS)]TJ/F8 10.909 Tf 10.909 0 Td[(2d^C)]TJ/F8 10.909 Tf 10.909 0 Td[(3dZ=1+1n mdC)]TJ/F8 10.909 Tf 12.104 7.38 Td[(1 mdV)]TJ/F8 10.909 Tf 10.909 0 Td[(2d^C)]TJ/F8 10.909 Tf 10.909 0 Td[(3dZ=rdt:Ifthereisnodefaultover[t;t+dt]withprobability1)]TJ/F18 10.909 Tf 11.251 0 Td[(ptdt,thenthechangeinthevalueoftheportfoliosisgivenbyd=1+1n mn@C @t+1 2s2r@2C @r2+213srsVV@2C @r@V+s2VV2@2C @V2+212srsp@2C @r@p+s2p@2C @p2odt+@C @rdr+@C @VdV+@C @pdp+Cr;Vj;pj;t)]TJ/F18 10.909 Tf 10.909 0 Td[(Cr;Vj)]TJ/F18 10.909 Tf 6.752 2.852 Td[(;pj)]TJ/F18 10.909 Tf 6.753 2.852 Td[(;tIfj2t;t+dt]g)]TJ/F8 10.909 Tf 9.68 7.38 Td[(1 mdV)]TJ/F8 10.909 Tf 10.909 0 Td[(2n@^C @t+1 2s2r@2^C @r2+212srsp@2^C @r@p+s2p@2^C @p2odt+@^C @rdr+@^C @pdp+^Cr;pj;t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^Cr;pj)]TJ/F18 10.909 Tf 6.752 2.851 Td[(;tIfj2t;t+dt]g)]TJ/F8 10.909 Tf 8.485 0 Td[(3n@Z @t+1 2s2r@2Z @r2odt+@Z @rdr:Wewillchoose1;2and3sothatwecangetridofuncertaintycausedbydr;dpanddVterms,thatis,1=m@C @V1)]TJ/F18 10.909 Tf 10.909 0 Td[(n@C @V)]TJ/F16 7.97 Tf 6.586 0 Td[(1;and1+1n m=1)]TJ/F18 10.909 Tf 10.909 0 Td[(n@C @V)]TJ/F16 7.97 Tf 6.587 0 Td[(12=@C @p@^C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(11)]TJ/F18 10.909 Tf 10.909 0 Td[(n@C @V)]TJ/F16 7.97 Tf 6.587 0 Td[(1;and3=1)]TJ/F18 10.909 Tf 10.909 0 Td[(n@C @V)]TJ/F16 7.97 Tf 6.586 0 Td[(1@C @r)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@C @p@^C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1@^C @r@Z @r)]TJ/F16 7.97 Tf 6.587 0 Td[(1:70

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Onthetimeinterval[t;t+dt,wehaved=1)]TJ/F18 10.909 Tf 12.209 7.38 Td[(@C @V)]TJ/F16 7.97 Tf 6.587 0 Td[(1n@C @t+1 2s2r@2C @r2+213srsVV@2C @r@V+s2VV2@2C @V2+212srsp@2C @r@p+s2p@2C @p2odt+Cr;Vj;pj;t)]TJ/F18 10.909 Tf 10.909 0 Td[(Cr;Vj)]TJ/F18 10.909 Tf 6.752 2.851 Td[(;pj)]TJ/F18 10.909 Tf 6.752 2.851 Td[(;tIfj2t;t+dt]g)]TJ/F24 10.909 Tf 8.484 12.109 Td[(1)]TJ/F18 10.909 Tf 12.21 7.38 Td[(@C @V)]TJ/F16 7.97 Tf 6.587 0 Td[(1@C @p@^C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1n@^C @t+1 2s2r@2^C @r2+212srsp@2^C @r@p+s2p@2^C @p2odt+^Cr;pj;t)]TJ/F8 10.909 Tf 13.38 2.757 Td[(^Cr;pj)]TJ/F18 10.909 Tf 6.752 2.852 Td[(;tIfj2t;t+dt]g)]TJ/F24 10.909 Tf 8.485 12.109 Td[(1)]TJ/F18 10.909 Tf 12.21 7.38 Td[(@C @V)]TJ/F16 7.97 Tf 6.587 0 Td[(1@Z @r@C @r)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@C @p@C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1@^C @rn@Z @t+1 2s2r@2Z @r2odt:.15Incaseofdefaultwithprobabilityptdt,sincedC=Rd)]TJ/F18 10.909 Tf 10.909 0 Td[(Candd^C=^R)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^C;thechangeinthevalueisgivenbyd=1+@1n mRd)]TJ/F18 10.909 Tf 10.909 0 Td[(C)]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 mdV)]TJ/F8 10.909 Tf 10.909 0 Td[(2^R)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^C)]TJ/F8 10.909 Tf 10.91 0 Td[(3dZ=1)]TJ/F18 10.909 Tf 12.21 7.38 Td[(@C @V)]TJ/F16 7.97 Tf 6.586 0 Td[(1Rd)]TJ/F18 10.909 Tf 10.909 0 Td[(C)]TJ/F18 10.909 Tf 12.21 7.38 Td[(@C @VdV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@C @p@^C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1^R)]TJ/F8 10.909 Tf 13.38 2.757 Td[(^C)]TJ/F24 10.909 Tf 8.485 12.11 Td[(@Z @r)]TJ/F16 7.97 Tf 6.587 0 Td[(1@C @r@^C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1@^C @rdZ:.16Wetaketheexpectationofdandequateitwithrdt,thatis:15)]TJ/F18 10.909 Tf 10.529 0 Td[(ptdt+4:16ptdt.71

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Then,multiplyingby1)]TJ/F19 7.97 Tf 12.212 4.295 Td[(@C @V)]TJ/F16 7.97 Tf 6.587 0 Td[(1,weobtainn@C @t+1 2s2r@2C @r2+213srsVV@2C @r@V+s2VV2@2C @V2+212srsp@2C @r@p+s2p@2C @p2odt+)]TJ/F18 10.909 Tf 10.909 0 Td[(ptdtCr;Vj;pj;t)]TJ/F18 10.909 Tf 10.909 0 Td[(Cr;Vj)]TJ/F18 10.909 Tf 6.752 2.851 Td[(;pj)]TJ/F18 10.909 Tf 6.752 2.851 Td[(;tIfj2t;t+dt]g)]TJ/F18 10.909 Tf 9.68 7.381 Td[(@C @p@^C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1n@^C @t+1 2s2r@2^C @r2+212srsp@2^C @r@p+s2p@2^C @p2odt)]TJ/F8 10.909 Tf 8.484 0 Td[()]TJ/F18 10.909 Tf 10.909 0 Td[(ptdt@C @p@^C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1^Cr;pj;t)]TJ/F8 10.909 Tf 13.38 2.757 Td[(^Cr;pj)]TJ/F18 10.909 Tf 6.752 2.852 Td[(;tIfj2t;t+dt]g)]TJ/F24 10.909 Tf 8.485 12.109 Td[(@Z @r)]TJ/F16 7.97 Tf 6.586 0 Td[(1@C @r)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@C @p@^C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1@^C @rn@Z @t+1 2s2r@2Z @r2odt+Rd)]TJ/F18 10.909 Tf 10.909 0 Td[(C)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@C @p@^C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1^R)]TJ/F8 10.909 Tf 13.379 2.757 Td[(^Cptdt=r"C)]TJ/F18 10.909 Tf 12.21 7.38 Td[(@C @V)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@C @p@^C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1^C)]TJ/F24 10.909 Tf 10.909 15.382 Td[(@C @rV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@C @p@^C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1@^C @r@Z @r)]TJ/F16 7.97 Tf 6.587 0 Td[(1Z#ptdt:.17Now,on[j;j+1,.17becomes@C @t+1 2s2r@2C @r2+213srsVV@2C @r@V+s2VV2@2C @V2+212srsp@2C @r@p+s2p@2C @p2+rV@C @V)]TJ/F18 10.909 Tf 10.909 0 Td[(rC+pRd)]TJ/F18 10.909 Tf 10.909 0 Td[(C)]TJ/F18 10.909 Tf 9.68 7.38 Td[(@C @p@^C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1@^C @t+1 2s2r@2^C @r2+212srsp@2^C @r@p+s2p@2^C @p2)]TJ/F18 10.909 Tf 10.909 0 Td[(r^C+p^R)]TJ/F8 10.909 Tf 13.38 2.757 Td[(^C)]TJ/F24 10.909 Tf 8.484 12.11 Td[(@Z @r)]TJ/F16 7.97 Tf 6.586 0 Td[(1@C @r)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@C @p@^C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1@^C @rn@Z @t+1 2s2r@2Z @r2)]TJ/F18 10.909 Tf 10.909 0 Td[(rZo=0:.18By.6and.1,.18becomes@C @t+1 2s2r@2C @r2+213srsVV@2C @r@V+s2VV2@2C @V2+212srsp@2C @r@p+s2p@2C @p2+rV@C @V)]TJ/F18 10.909 Tf 10.909 0 Td[(rC+pRd)]TJ/F18 10.909 Tf 10.909 0 Td[(C+ap@C @p+t)]TJ/F18 10.909 Tf 10.909 0 Td[(r@C @r=0:.19Asintheprevioussections,weassumethefollowing:Assumption6:Thedefaultablecorporatecouponbondpriceattimet=j)]TJ/F51 10.909 Tf 10.097 1.636 Td[(istheexpectationofthepriceattimet=j.72

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SoifweletSj)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;jbethesurvivalprobabilityofthebondattimet=jgiventhesurvivalattimet=j)]TJ/F51 10.909 Tf 7.085 1.637 Td[(,thenthepriceofthecorporatebondatt=n)]TJ/F8 10.909 Tf 10.115 1.637 Td[(=T)]TJ/F51 10.909 Tf 7.085 1.637 Td[(,whichistheterminalconditionforthePDE.18,isgivenbyCr;V;p;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=1ST)]TJ/F18 10.909 Tf 8.485 0 Td[(;T+R[1)]TJ/F18 10.909 Tf 10.909 0 Td[(ST)]TJ/F18 10.909 Tf 8.485 0 Td[(;T]:.20Thereforewehavethefollowingmodel:Theorem9.UnderAssumptions1through6,thepriceofthedefaultablecorporatecouponbondpriceCon[n)]TJ/F16 7.97 Tf 6.587 0 Td[(1;T)]TJ/F8 10.909 Tf 7.084 1.636 Td[(ismodeledby8>>>>>><>>>>>>:@C @t+1 2s2r@2C @r2+213srsVV@2C @r@V+s2VV2@2C @V2+212srsp@2C @r@p+s2p@2C @p2+rV@C @V)]TJ/F18 10.909 Tf 10.909 0 Td[(rC+pRd)]TJ/F18 10.909 Tf 10.909 0 Td[(C+ap@C @p+t)]TJ/F18 10.909 Tf 10.909 0 Td[(r@C @r=0Cr;V;p;T)]TJ/F8 10.909 Tf 8.485 0 Td[(=1ST)]TJ/F18 10.909 Tf 8.485 0 Td[(;T+R[1)]TJ/F18 10.909 Tf 10.909 0 Td[(ST)]TJ/F18 10.909 Tf 8.485 0 Td[(;T]:4.2.3ParticularSolutionTosolve.19with.20astheterminalcondition,weusethechangeofunknownfunctionandvariableasfollows:x=V Zandux;p;t=Cr;V;p;t Z:Then,.19and.20become8>>>>>>>>><>>>>>>>>>:@u @t+1 2[s2rB2t+s2V+213srsVBt]x2@2u @x2+s2pp;r;t@2u @p2+212srspBtx@2u @p@x+[app;r;t)]TJ/F8 10.909 Tf 10.909 0 Td[(212srspBt]@u @p)]TJ/F18 10.909 Tf 10.909 0 Td[(pu)]TJ/F18 10.909 Tf 10.909 0 Td[(R=0;x>VB;p>0;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1tVB;p>0uVB;p;t=Rd;p>0;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1t
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andalsoletting^u=u)]TJ/F18 10.909 Tf 10.909 0 Td[(R;wehave8>>>>>>>>><>>>>>>>>>:@^u @t+1 2s2tx2@2^u @x2+s2pp;t@2^u @p2+2x@2^u @p@x+app;t@^u @p)]TJ/F18 10.909 Tf 10.909 0 Td[(p^u=0;x>VB;p>0;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1tVB;p>0^uVB;p;t=0;p>0;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1t>>>>>>>><>>>>>>>>>:@W @t+1 2s2tx2@2W @x2+s2pp;t@2W @p2+2x@2W @p@x+app;t@W @p)]TJ/F18 10.909 Tf 10.909 0 Td[(pW=0;x>VB;p>0;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1tVB;p>0WVB;p;t=0;p>0;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1t>>>>>>>><>>>>>>>>>:@W @t+1 2s2tx2@2W @x2+s2pp;t@2W @p2+app;t@W @p)]TJ/F18 10.909 Tf 10.909 0 Td[(pW=0;x>VB;p>0;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1tVB;p>0WVB;p;t=0;p>0;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1t
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Asinbefore,assumingthecoefcienttermsofgandfarezero,wehavethefollowingtwoPDEproblems:8>>>>><>>>>>:@f @t+1 2s2tx2@2f @x2=0;x>VB;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1tVB.228>><>>:@g @t+1 2s2pp;t@2g @p2+app;t@g @p)]TJ/F18 10.909 Tf 10.909 0 Td[(pg=0;p>0;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1t0:.23Noticethat.23isthesameas.47.Now,tosolve.22,usingthefollowingtimescaletrans-formation,s=Ztn)]TJ/F17 5.978 Tf 5.756 0 Td[(1s2uduandT=ZTn)]TJ/F17 5.978 Tf 5.756 0 Td[(1s2udu;andlettingfx;s=fx;t;.22becomes8>>>>><>>>>>:@f @s+1 2x2@2f @x2=0;x>VB;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1sVB:.24Tosolvethis,letx VB=ey;andsoy=lnx VB=T)]TJ/F18 10.909 Tf 10.909 0 Td[(s 2;fx;s=VBvy;:Then,.24becomes8>>>>><>>>>>:@v @)]TJ/F18 10.909 Tf 12.134 7.38 Td[(@2v @y2+@v @y=0;
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Usingthechangeofunknownfunctionv=wey+,thePDEin.25becomesw=wyy+2)]TJ/F18 10.909 Tf 10.909 0 Td[(w+)]TJ/F8 10.909 Tf 10.909 0 Td[(1wy:Wechooseandsothatthecoefcientsofwandwyarezero.Thatis,=1 2and=)]TJ/F16 7.97 Tf 9.68 4.295 Td[(1 4.Then,wehavethefollowingheatequation:8><>:w=wyy;<>:hy=wy;0=1 VBe)]TJ/F17 5.978 Tf 7.782 3.259 Td[(1 2yy>00;otherwiseletw1y;beasolutionoftheIVP,8><>:w=wyy;
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whered1=lnV VBZ q RTts2udu)]TJ/F24 10.909 Tf 11.769 23.455 Td[(q RTts2udu 2andd2=lnVBZ V q RTts2udu)]TJ/F24 10.909 Tf 11.769 23.455 Td[(q RTts2udu 2.Sothepriceofthecorporatecouponbondwithexpectedandunexpecteddefaultfort2[n)]TJ/F16 7.97 Tf 6.586 0 Td[(1;TisgivenbyCr;V;p;t=Zux;p;t=Z^u+R=Z)]TJ/F18 10.909 Tf 5 -8.836 Td[(W)]TJ/F18 10.909 Tf 10.909 0 Td[(RST;T)]TJ/F8 10.909 Tf 8.485 0 Td[(+R=Z)]TJ/F18 10.909 Tf 5 -8.837 Td[(fg)]TJ/F18 10.909 Tf 10.909 0 Td[(RST;T)]TJ/F8 10.909 Tf 8.485 0 Td[(+R=RZt;T+eAt;T)]TJ/F19 7.97 Tf 6.587 0 Td[(Bt;Tp)]TJ/F18 10.909 Tf 10.909 0 Td[(RZd1)]TJ/F18 10.909 Tf 14.532 7.38 Td[(V VBd2Now,usingthis,Cr;V;p;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1isgivenbyCr;V;p;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1=RZn)]TJ/F16 7.97 Tf 6.586 0 Td[(1;T+eAn)]TJ/F17 5.978 Tf 5.757 0 Td[(1;T)]TJ/F19 7.97 Tf 6.586 0 Td[(Bn)]TJ/F17 5.978 Tf 5.757 0 Td[(1;Tp)]TJ/F18 10.909 Tf 10.909 0 Td[(RhZn)]TJ/F16 7.97 Tf 6.586 0 Td[(1;Td1;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F18 10.909 Tf 14.532 7.38 Td[(V VBd2;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1whereAt;andBt;aregivenby.19through.21andd1;jandd2;jaregivenbyd1;j=lnV VBZj;j+1 q Rj+1js2udu)]TJ/F24 10.909 Tf 12.105 23.454 Td[(q Rj+1js2udu 2;andd2;j=lnVBZj;j+1 V q Rj+1js2udu)]TJ/F24 10.909 Tf 12.105 23.455 Td[(q Rj+1js2udu 2Then,onthetimeinterval[n)]TJ/F16 7.97 Tf 6.586 0 Td[(2;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1,bythearbitrageprinciple,theterminalconditionbecomesCr;V;p;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F8 10.909 Tf 6.752 3.294 Td[(=RZn)]TJ/F16 7.97 Tf 6.587 0 Td[(1;T+eAn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F19 7.97 Tf 6.587 0 Td[(Bn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;Tp)]TJ/F18 10.909 Tf 10.909 0 Td[(R[Zn)]TJ/F16 7.97 Tf 6.587 0 Td[(1;Td1;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F18 10.909 Tf 12.108 7.38 Td[(V VBd2;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1]Sn)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F18 10.909 Tf 6.753 3.294 Td[(;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1+RZn)]TJ/F16 7.97 Tf 6.586 0 Td[(1;T)]TJ/F8 10.909 Tf 5 -8.836 Td[(1)]TJ/F18 10.909 Tf 10.91 0 Td[(Sn)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F18 10.909 Tf 6.753 3.294 Td[(;n)]TJ/F16 7.97 Tf 6.587 0 Td[(1=Sn)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F18 10.909 Tf 6.752 3.293 Td[(;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F18 10.909 Tf 10.909 0 Td[(ReAn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F19 7.97 Tf 6.586 0 Td[(Bn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;Tp[Zn)]TJ/F16 7.97 Tf 6.586 0 Td[(1;Td1;n)]TJ/F17 5.978 Tf 5.757 0 Td[(1)]TJ/F18 10.909 Tf 12.107 7.38 Td[(V VBd2;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1]+RZn)]TJ/F16 7.97 Tf 6.586 0 Td[(1;T:Solving.19withthisterminalconditiongivesthesolutionfort2[n)]TJ/F16 7.97 Tf 6.587 0 Td[(2;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1asfollows:Cr;V;p;t=RZt;T+eAt;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1)]TJ/F19 7.97 Tf 6.587 0 Td[(Bt;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1p)]TJ/F18 10.909 Tf 10.909 0 Td[(R[Zt;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1d1)]TJ/F18 10.909 Tf 12.105 7.38 Td[(Vt VBd2]Sn)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F18 10.909 Tf 6.752 3.294 Td[(;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1)]TJ/F18 10.909 Tf 10.909 0 Td[(ReAn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;T)]TJ/F19 7.97 Tf 6.586 0 Td[(Bn)]TJ/F17 5.978 Tf 5.756 0 Td[(1;Tp[Zn)]TJ/F16 7.97 Tf 6.586 0 Td[(1;Td1;n)]TJ/F17 5.978 Tf 5.757 0 Td[(1)]TJ/F18 10.909 Tf 9.68 7.38 Td[(Vn)]TJ/F16 7.97 Tf 6.586 0 Td[(1 VBd2;n)]TJ/F17 5.978 Tf 5.756 0 Td[(1]77

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whereAt;,Bt;,d1;jandd2;jarethesameasabove;andd1andd2aregiven,foranyt2[j;j+1,asd1=lnV VBZt;j+1 q Rj+1ts2udu)]TJ/F24 10.909 Tf 12.104 23.455 Td[(q Rj+1ts2udu 2;andd2=lnVBZt;j+1 V q Rj+1ts2udu)]TJ/F24 10.909 Tf 12.104 23.454 Td[(q Rj+1ts2udu 2:Byiteratingthisprocess,weshallndthesolutionforanyt2[0;Tasfollows:Theorem10.UnderAssumptions1through5,foranyt2[0;Twitht2[j)]TJ/F16 7.97 Tf 6.587 0 Td[(1;j;j=1;:::;n)]TJ/F8 10.909 Tf -423.515 -20.323 Td[(1,thepriceofcorporatecouponbondwithexpectedandunexpecteddefaultisgivenbyGr;V;p;t=Cr;V;p;t+XitciCr;V;p;t;iwhereCr;V;p;t=RZt;T+eAt;j)]TJ/F19 7.97 Tf 6.587 0 Td[(Bt;jp)]TJ/F18 10.909 Tf 10.91 0 Td[(R[Zt;jd1)]TJ/F18 10.909 Tf 12.105 7.38 Td[(Vt VBd2]n)]TJ/F16 7.97 Tf 6.587 0 Td[(1Yk=jSk)]TJ/F18 10.909 Tf 6.752 2.851 Td[(;k)]TJ/F18 10.909 Tf 10.909 0 Td[(ReAk;k+1)]TJ/F19 7.97 Tf 6.586 0 Td[(Bk;k+1p[Zk;k+1d1;n)]TJ/F17 5.978 Tf 5.757 0 Td[(1)]TJ/F18 10.909 Tf 9.68 7.38 Td[(Vk VBd2;k].27andCr;V;p;t;iisthepriceofzero-couponbondattimetofthesamecompanywithmaturityi.78

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Chapter5CreditDerivativesPricingwithStochasticInterestRateInthischapter,weshallndthepriceofcreditdefaultswaptionusingthebondpricingmodelsformulatedinthepreviouschapter.5.1CreditDefaultSwaptionwithStochasticInterestRate-UnexpectedDefault5.1.1FormulationWemakethefollowingassumptionsinthissection.Assumption1:Weassumethatthetimestructureofthecreditdefaultswaptionandthebondsunderlyingtheswaptionarethesameasinsubsection3.1.1.Thatis:Lett=0andt=T0bethetimewhenthecreditdefaultswaptionhereinafter,theswaptionstartsandexpiresrespectively.t=T0isalsowhentheforwardcreditdefaultswaptakeseffectupontheexerciseoftheswaption.Lett=TNbetheexpirationofthecreditdefaultswapandforsimplicity,letusassumethatTj;j=1;;Nfallsontheinterestpaymentdateofthecouponbondsissuedbythereferenceentityofthecreditdefaultswapandthecreditdefaultswapterminatesonthedaythecouponbondsaretoberedeemed,thatisn=TNwherenisasdenedinsubsection2.1.1.Letzbetheexercisepriceoftheswaption.Thisisactuallythefeetheswaptionholderpaysonthenotionalamountofthecreditdefaultswaponcetheswaptionisexercised.Letzt;tT0bethefeelegofforwardcreditdefaultswap.DeneZt:=ztE[NXi;Ti>tbTiiIfTig+bIfT0TNg]79

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wherebTi=expfRTitrdsg,isthetimeofdefault,andiandarethelengthsoftimeintervalsincethelastfeepaymenttillTiandthedefaultdate,respectively.Therefore,Stisthepresentvalueattimetoftheforwardcreditdefaultswap.Observethatbythearbitrageprinciple,thepresentvalueofthetotalfeelegisthesameasthepresentvalueoftheprotectionleg.Thatis:Zt=E[b)]TJ/F18 10.909 Tf 10.91 0 Td[(RIfTNg]:.1Assumption2:Theriskfreeshortterminterestratert=rtfollowstheVasicekmodel:dr=t)]TJ/F18 10.909 Tf 10.909 0 Td[(rtdt+srtdW1twhere,t,srtandWtaredenedasinsection4.1.Assumption3:Defaultisanexogenousevent.Unexpecteddefaultprobabilityonanyinterval[t;t+dt]isgivenby,dp=app;tdt+spp;tdW2+pj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(UjIfj2t;t+dt]g;whereIfj2t;t+dt]gisanindicatorfunction.dW1dW2=.DefaultrecoveryisgivenastheformoffacevalueexogenousrecoveryRe)]TJ/F19 7.97 Tf 6.587 0 Td[(rT)]TJ/F19 7.97 Tf 6.587 0 Td[(t;0R1:constant,T:maturityofthebondorastheformofmarketpriceexogenousrecoveryRbondpriceatdefaulttime.Thepriceofdefaultablecorporatezero-couponbondwithexogenousdefaultisgivenbythefunction^C=^Cr;p;t.Problem:Underthesesettingandassumptions,wewillndthepriceofthecreditdefaultswaption^Xr;p;t.5.1.2DerivationofthemodelWeconstructaportfolioconsistingofionecreditdefaultswaption^Xr;p;t,ii1unitsofreferenceentity'scoupon-bondswithexogenousdefault^Ctogetridoftheriskarisenbyp,andiii2unitsofdefault-freezero-couponbondZtogetridoftheriskarisenbyr.Then,valueoftheportfoliois:=^X)]TJ/F8 10.909 Tf 10.909 0 Td[(1^C)]TJ/F8 10.909 Tf 10.909 0 Td[(2Z:80

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Sothechangeofvalueofthisportfoliooverasmalltimeincrement[t;t+dt]isgivenbyd=d^X)]TJ/F18 10.909 Tf 10.909 0 Td[(d1^C)]TJ/F18 10.909 Tf 10.909 0 Td[(d2Z:Ifthereisnodefaultoverasmalltimeincrement[t;t+dt]withprobability1)]TJ/F18 10.909 Tf 10.99 0 Td[(ptdt,thenbyItoLemma,thechangeofvalueintheportfoliooverthisperiodisgivenbyd=@^X @tdt+@^X @pdp+@^X @rdr+1 2s2p@2^X @p2+s2r@2^X @r2+2spsr@2^X @r@pdt)]TJ/F8 10.909 Tf 8.485 0 Td[(1@^C @tdt+@^C @pdp+@^C @rdr+1 2s2p@2^C @p2+s2r@2^C @r2+2spsr@2^C @r@pdtf^Cr;pj;t)]TJ/F8 10.909 Tf 13.38 2.758 Td[(^Cr;pj)]TJ/F18 10.909 Tf 7.251 4.611 Td[(;tgIfj2t;t+dt]g)]TJ/F8 10.909 Tf 8.485 0 Td[(2@Z @tdt+1 2s2r@2Z @r2dt+@Z @rdr:Togetridoftheuncertaintycausedbydpanddrterms,wechoose1and2asfollows:1=@^X @p@^C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1;2=@Z @r)]TJ/F16 7.97 Tf 6.587 0 Td[(1@^X @r)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@^X @p@^C @r@^C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1:Then,wehaved=@^X @t+1 2s2p@2^X @p2+s2r@2^X @r2+2spsr@2^X @r@pdt)]TJ/F18 10.909 Tf 9.681 7.38 Td[(@^X @p@^C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1@^C @t+1 2s2p@2^C @p2+s2r@2^C @r2+2spsr@2^C @r@pdtf^Cr;pj;t)]TJ/F8 10.909 Tf 13.379 2.758 Td[(^Cr;pj)]TJ/F18 10.909 Tf 7.251 4.611 Td[(;tgIfj2t;t+dt]g)]TJ/F24 10.909 Tf 8.485 15.382 Td[(@Z @r)]TJ/F16 7.97 Tf 6.586 0 Td[(1@^X @r)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@^X @p@^C @r@^C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1@Z @tdt+1 2s2r@2Z @r2dt:.2Ifthereisdefaultwithprobabilityptdtbeforetheinceptionoftheforwardcreditdefaultswap,theswaptioncontractbecomesvoid;therefore,wehave:d=)]TJ/F8 10.909 Tf 11.614 2.757 Td[(^X)]TJ/F8 10.909 Tf 10.909 0 Td[(1R)]TJ/F8 10.909 Tf 13.379 2.757 Td[(^C)]TJ/F8 10.909 Tf 10.91 0 Td[(2dZ=)]TJ/F8 10.909 Tf 11.614 2.757 Td[(^X)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@^X @p@^C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1R)]TJ/F8 10.909 Tf 13.379 2.757 Td[(^C)]TJ/F24 10.909 Tf 10.909 15.382 Td[(@Z @r)]TJ/F16 7.97 Tf 6.587 0 Td[(1@^X @r)]TJ/F18 10.909 Tf 12.104 7.38 Td[(@^X @p@^C @r@^C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1dZ:.381

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Then,bythearbitrageprinciple,wemusthave:2)]TJ/F18 10.909 Tf 11.119 0 Td[(ptdt+5:3ptdt=rdt.Sowehave@^X @t+1 2s2p@2^X @p2+s2r@2^X @r2+2spsr@2^X @r@p)]TJ/F8 9.963 Tf 9.963 0 Td[(pt+rXdt)]TJ/F18 9.963 Tf 8.944 6.74 Td[(@^X @p@^C @p)]TJ/F7 6.974 Tf 6.227 0 Td[(1@^C @t+1 2s2p@2^C @p2+s2r@2^C @r2+2spsr@2^C @r@p+R)]TJ/F18 9.963 Tf 9.962 0 Td[(Cpt)]TJ/F18 9.963 Tf 9.963 0 Td[(r^Cdtf^Cr;pj;t)]TJ/F8 9.963 Tf 12.218 2.518 Td[(^Cr;pj)]TJ/F18 9.963 Tf 6.891 4.096 Td[(;tgIfj2t;t+dt]g+ptdt)]TJ/F24 9.963 Tf 7.749 14.047 Td[(@Z @r)]TJ/F7 6.974 Tf 6.227 0 Td[(1@^X @r)]TJ/F18 9.963 Tf 11.159 6.74 Td[(@^X @p@^C @r@^C @p)]TJ/F7 6.974 Tf 6.227 0 Td[(1@Z @t+1 2s2r@2Z @r2)]TJ/F18 9.963 Tf 9.963 0 Td[(rZdt=0:.4Notingthat,by.8and.6,forthetimeintervalsuchthatj662[t;t+dt],@^C @t+1 2s2p@2^C @p2+s2r@2^C @r2+2spsr@2^C @r@p+R)]TJ/F18 9.963 Tf 9.963 0 Td[(Cpt)]TJ/F18 9.963 Tf 9.963 0 Td[(r^C=)]TJ/F18 9.963 Tf 7.749 0 Td[(t)]TJ/F18 9.963 Tf 9.962 0 Td[(r@^C @r)]TJ/F18 9.963 Tf 9.962 0 Td[(ap@^C @pand@Z @t+1 2s2r@2Z @r2)]TJ/F18 9.963 Tf 9.962 0 Td[(rZ=)]TJ/F18 9.963 Tf 7.748 0 Td[(t)]TJ/F18 9.963 Tf 9.963 0 Td[(r@^Z @r;.4becomes@^X @t+t)]TJ/F18 10.909 Tf 10.909 0 Td[(r@^X @r+1 2s2p@2^X @p2+s2r@2^X @r2+2spsr@2^X @r@p)]TJ/F8 10.909 Tf 10.909 0 Td[(pt+rX=0:Sincethepresentvalueoftheforwardcreditdefaultswapisgivenby.1,thevalueoftheswaptionatt=T0isgivenby^XT0=[ZT0)-222(Z]+=E[b)]TJ/F18 9.963 Tf 9.963 0 Td[(RIfTNg])]TJ/F18 9.963 Tf 9.963 0 Td[(zE[NXi;Ti>tbTiiIfTig+bIfT0TNg]+:Fromthenancialpointofview,wecanexpectthevalueoftheswaptionrightbeforethejumpi.e.,att=T0)]TJ/F51 10.909 Tf 7.084 1.637 Td[(,tobetheexpectationofthevalueatt=T0.Sowecanassumethattheterminalconditionasfollows:^XT0)]TJ/F8 10.909 Tf 7.085 1.636 Td[(=PT0)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;T0EE[b)]TJ/F18 10.909 Tf 10.909 0 Td[(RIfTNg])]TJ/F18 10.909 Tf 8.484 0 Td[(zE[NXi;Ti>tbTiiIfTig+bIfT0TNg]+.5whereP;isdenedby.4.Usingthechangeofunknownfunction^Xr;p;t=^XT0)]TJ/F8 10.909 Tf 7.085 1.637 Td[(Wr;p;t;82

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wehavethefollowingPDEproblem:8>>>>><>>>>>:@W @t+t)]TJ/F18 9.963 Tf 9.963 0 Td[(r@W @r+ap@W @p+1 2s2p@2W @p2+s2r@2W @r2+2spsr@2W @r@p)]TJ/F8 9.963 Tf 9.962 0 Td[(pt+rW=0;t0;p>0Wr;p;T0)]TJ/F8 9.963 Tf 6.725 1.495 Td[(=1;r>0;p>0:.6Tosolvethis,rstweassumethatthedriftandvolatilityofptarenotcorrelatedtotheshorttermratert,thatis,ap=t+tp;s2p=t+t;and=0:.7Usingthechangeofunknownfunctionagain,weconsiderthefollowingcase.Wr;p;t=up;tZr;t;whereZr;tisgivenby.2,wehaveu@Z @t+1 2s2ru@2Z @r2+@Z @r)]TJ/F18 9.963 Tf 4.567 -8.07 Td[(t)]TJ/F18 9.963 Tf 11.862 0 Td[(r)]TJ/F18 9.963 Tf 11.862 0 Td[(rZ+Z@u @t+1 2s2p@2u @p2+ap@u @p)]TJ/F18 9.963 Tf 11.861 0 Td[(ptu=0:By.1,.6reducesto8>><>>:@u @t+1 2s2p@2u @p2+ap@u @p)]TJ/F18 10.909 Tf 10.909 0 Td[(ptu=0;t0;p>0ur;p;T0)]TJ/F8 10.909 Tf 7.085 1.637 Td[(=1;r>0;p>0:Wewillseekthesolutionintheformofu=eAt;T0)]TJ/F19 7.97 Tf 6.587 0 Td[(Bt;T0psothat^Xr;p;t=^Xr;p;T0)]TJ/F8 10.909 Tf 7.084 1.636 Td[(Zt;T0eAt;T0)]TJ/F19 7.97 Tf 6.586 0 Td[(Bt;T0p.Notingthesimilarityoftheequationwith.9,wehavethefollowingsolution.Theorem11.UnderAssumptions1though4,thepriceofthecreditdefaultswaptionisgivenby^Xr;p;t=^Xr;p;T0)]TJ/F8 10.909 Tf 7.085 1.636 Td[(Zt;T0eAt;T0)]TJ/F19 7.97 Tf 6.587 0 Td[(Bt;T0p.883

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where^Xr;p;T0)]TJ/F8 10.909 Tf 7.085 1.636 Td[(andZt;T0aregivenby.5and.2respectively,andAt;T0andBt;T0aregivenbyAt;T0=)]TJ/F24 10.909 Tf 10.303 14.849 Td[(ZT0t)]TJ/F18 10.909 Tf 5 -8.837 Td[(bsBs;T0)]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 2sB2s;T0dsBt;T0=8>>>>>>>>>><>>>>>>>>>>:1)]TJ/F18 10.909 Tf 10.909 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(T0)]TJ/F19 7.97 Tf 6.586 0 Td[(t ;dp=)]TJ/F18 10.909 Tf 5 -8.836 Td[(t)]TJ/F18 10.909 Tf 10.909 0 Td[(pdt+p tdW2;6=0T0)]TJ/F18 10.909 Tf 10.909 0 Td[(t;dp=)]TJ/F18 10.909 Tf 5 -8.837 Td[(t)]TJ/F18 10.909 Tf 10.909 0 Td[(pdt+p tdW2;=0r 2 exp)]TJ/F14 10.909 Tf 5 -1.805 Td[(p 2T0)]TJ/F19 7.97 Tf 6.587 0 Td[(t)]TJ/F16 7.97 Tf 6.587 0 Td[(exp)]TJ/F22 7.97 Tf 5 -8.836 Td[()]TJ/F14 10.909 Tf 6.586 7.031 Td[(p 2T0)]TJ/F19 7.97 Tf 6.586 0 Td[(t exp)]TJ/F14 10.909 Tf 5 -1.805 Td[(p 2T0)]TJ/F19 7.97 Tf 6.587 0 Td[(t+exp)]TJ/F22 7.97 Tf 5 -8.836 Td[()]TJ/F14 10.909 Tf 6.586 7.031 Td[(p 2T0)]TJ/F19 7.97 Tf 6.586 0 Td[(t;dp=tdt+p t+KpdW2;K:constant5.2CreditDefaultSwaptionwithStochasticInterestRate-ExpectedandUnexpectedDe-fault5.2.1FormulationAssumption1:WeassumethatthesamesettingforthecreditdefaultswaptionasinSection5.1:itsinceptionandexpiration,theonsetandfeestructureoftheunderlyingforwardcreditdefaultswap.Assumption2:Theriskfreeshortterminterestratert=rtfollowstheVasicekmodel:dr=t)]TJ/F18 10.909 Tf 10.91 0 Td[(rtdt+srtdW1twhere,t,andsrtaredenedasinsection4.1.Assumption3:Defaulteventisbothexogenousandendogenous.Unexpecteddefaultprobabilityonanyinterval[t;t+dt]isgivenby,dp=app;tdt+spp;tdW2+pj)]TJ/F18 10.909 Tf 7.084 1.636 Td[(UjIfj2t;t+dt]g;whereIfj2t;t+dt]gisanindicatorfunction.ExpecteddefaultoccurswhenthermassetsV=Vt=Vtfallsbelowthebarrier,sayVbt.AsinSection4.2,thermassetsVtisthesumofitscouponbondswhosepriceisCr;p;V;tandstocks,andfollowsthegeometricBrownianmotion,givenbydV=aVVtdt+sVVtdW3:84

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DefaultrecoveryisgivenastheformoffacevalueexogenousrecoveryRd=RZ;R:0R1:constant,whereZisthepriceofriskfreezerocouponbond,givenby.2.Assumption4:dWidWj=ijdt;i=1;2;3:However,weassumethattheunexpecteddefaultandexpecteddefaultarenotcorrelated,i.e.,23=0.Problem:Underthesesettingandassumptions,wewillndthepriceofthecreditdefaultswaptionX=Xr;p;V;t.5.2.2DerivationoftheModelWeconstructaportfoliobyhedgingXr;p;V;twiththereferenceentity'scoupon-bondswithexpectedandunexpecteddefault,thereferenceentity'sstock,andzero-coupondefault-freebond,togetridoftheriskarisenbyp,V,andr.Sothevalueoftheportfoliois:=X)]TJ/F8 10.909 Tf 10.909 0 Td[(1S)]TJ/F8 10.909 Tf 10.909 0 Td[(2C)]TJ/F8 10.909 Tf 10.909 0 Td[(3Z=X)]TJ/F8 10.909 Tf 10.909 0 Td[(1V)]TJ/F18 10.909 Tf 10.909 0 Td[(nC m)]TJ/F8 10.909 Tf 10.909 0 Td[(2C)]TJ/F8 10.909 Tf 10.909 0 Td[(3Z=X)]TJ/F8 10.909 Tf 12.104 7.38 Td[(1 mV)]TJ/F24 10.909 Tf 10.909 12.109 Td[(n m1)]TJ/F8 10.909 Tf 10.909 0 Td[(2C)]TJ/F8 10.909 Tf 10.909 0 Td[(3Z:Sothechangeofvalueofthisportfoliooverasmalltimeincrement[t;t+dt]isgivenbyd=dX)]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 mdV)]TJ/F24 10.909 Tf 10.909 12.109 Td[(n m1)]TJ/F8 10.909 Tf 10.909 0 Td[(2dC)]TJ/F8 10.909 Tf 10.909 0 Td[(3dZ:85

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Ifthereisnodefaultoverasmalltimeincrement[t;t+dt]withprobability1)]TJ/F18 10.909 Tf 10.99 0 Td[(ptdt,thenbyItoLemma,thechangeofvalueintheportfoliooverthisperiodisgivenbyd=@X @tdt+@X @pdp+@X @rdr+@X @VdV+1 2s2p@2X @p2+s2r@2X @r2+s2VV2@2X @V2+212srsp@2X @r@p+213srsVV@2X @r@Vdt)]TJ/F8 9.963 Tf 8.945 6.74 Td[(1 mV)]TJ/F24 9.963 Tf 9.963 11.059 Td[(n m1)]TJ/F8 9.963 Tf 9.963 0 Td[(2@C @tdt+@C @pdp+@C @rdr+@C @VdV+1 2s2p@2C @p2+s2r@2C @r2+s2r@2C @r2+s2VV2@2C @V2+212srsp@2^C @r@p+213srsVV@2^C @r@Vdt+f^Cr;pj;Vj;t)]TJ/F8 9.963 Tf 12.219 2.519 Td[(^Cr;pj)]TJ/F18 9.963 Tf 6.891 4.095 Td[(;Vj)]TJ/F18 9.963 Tf 6.891 4.095 Td[(;tgIfj2t;t+dt]g)]TJ/F8 9.963 Tf 7.749 0 Td[(3@Z @tdt+1 2s2r@2Z @r2dt+@Z @rdr:Togetridoftheuncertaintycausedbydp,dranddVterms,wechoose1,2and3asfollows:1=m@^X @V)]TJ/F18 10.909 Tf 12.104 7.38 Td[(@X @p@C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1@C @V;2=n@X @V)]TJ/F18 10.909 Tf 12.104 7.381 Td[(@X @p@C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(11+n@C @V;3=@Z @r)]TJ/F16 7.97 Tf 6.587 0 Td[(1@X @r)]TJ/F18 10.909 Tf 12.104 7.38 Td[(@X @p@C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1@C @r:Then,wehaved=@X @t+1 2s2p@2X @p2+s2r@2X @r2+s2VV2@2X @V2+212srsp@2X @r@p+213srsVV@2X @r@Vdt)]TJ/F18 9.963 Tf 8.944 6.739 Td[(@X @p@C @p)]TJ/F7 6.974 Tf 6.227 0 Td[(1@C @t+1 2s2p@2C @p2+s2r@2C @r2+s2VV2@2C @V2+212srsp@2C @r@p+213srsVV@2C @r@Vdt+fCr;pj;Vj;t)]TJ/F18 9.963 Tf 9.963 0 Td[(Cr;pj)]TJ/F18 9.963 Tf 6.89 4.096 Td[(;Vj)]TJ/F18 9.963 Tf 6.891 4.096 Td[(;tgIfj2t;t+dt]g)]TJ/F24 9.963 Tf 7.749 14.047 Td[(@Z @r)]TJ/F7 6.974 Tf 6.227 0 Td[(1@^X @r)]TJ/F18 9.963 Tf 11.159 6.739 Td[(@X @p@^C @p)]TJ/F7 6.974 Tf 6.227 0 Td[(1@C @r@Z @tdt+1 2s2r@2Z @r2dt:.9Ifthereisdefaultwithprobabilityptdtbeforetheinceptionoftheforwardcreditdefaultswap,the86

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swaptioncontractbecomesvoid;therefore,wehave:d=)]TJ/F18 10.909 Tf 8.485 0 Td[(X)]TJ/F8 10.909 Tf 12.105 7.38 Td[(1 mdV)]TJ/F24 10.909 Tf 10.909 15.382 Td[(n m1)]TJ/F8 10.909 Tf 10.909 0 Td[(2Rd)]TJ/F18 10.909 Tf 10.909 0 Td[(C)]TJ/F8 10.909 Tf 10.909 0 Td[(3dZ=)]TJ/F18 10.909 Tf 8.485 0 Td[(X)]TJ/F24 10.909 Tf 10.909 15.382 Td[(@X @V)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@X @p@^C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1@C @VdV)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@X @p@^C @p)]TJ/F16 7.97 Tf 6.586 0 Td[(1Rd)]TJ/F18 10.909 Tf 10.909 0 Td[(C)]TJ/F24 10.909 Tf 8.485 15.382 Td[(@Z @r)]TJ/F16 7.97 Tf 6.587 0 Td[(1@^X @r)]TJ/F18 10.909 Tf 12.105 7.38 Td[(@^X @p@^C @p)]TJ/F16 7.97 Tf 6.587 0 Td[(1@^C @rdZ:.10Then,bythearbitrageprinciple,wemusthave:9)]TJ/F18 10.909 Tf 10.004 0 Td[(ptdt+:10ptdt=rdt.Wehave@X @t+1 2s2p@2X @p2+s2r@2X @r2+s2VV2@2X @V2+212srsp@2X @r@p+213srsV@2X @r@V)]TJ/F8 9.963 Tf 9.963 0 Td[(pt+rXdt)]TJ/F18 9.963 Tf 8.944 6.739 Td[(@X @p@C @p)]TJ/F7 6.974 Tf 6.226 0 Td[(1@C @t)]TJ/F18 9.963 Tf 9.962 0 Td[(rV@C @V+1 2s2p@2C @p2+s2r@2C @r2+s2VV2@2C @V2+212srsp@2C @r@p+213srsV@2C @r@V+Rd)]TJ/F18 9.963 Tf 9.963 0 Td[(Cpt)]TJ/F18 9.963 Tf 9.963 0 Td[(rCdt+fCr;pj;Vj;t)]TJ/F18 9.963 Tf 9.962 0 Td[(Cr;pj)]TJ/F18 9.963 Tf 6.891 4.096 Td[(;Vj)]TJ/F18 9.963 Tf 6.891 4.096 Td[(;tgIfj2t;t+dt]g+ptdt)]TJ/F24 9.963 Tf 7.748 14.047 Td[(@Z @r)]TJ/F7 6.974 Tf 6.227 0 Td[(1@X @r)]TJ/F18 9.963 Tf 11.158 6.74 Td[(@X @p@C @p)]TJ/F7 6.974 Tf 6.227 0 Td[(1@C @r@Z @t+1 2s2r@2Z @r2)]TJ/F18 9.963 Tf 9.963 0 Td[(rZdt=0:Notingthat,by.18and.6,forthetimeintervalsuchthatj662[t;t+dt],@C @t)]TJ/F18 10.909 Tf 10.909 0 Td[(rV@C @V+1 2s2p@2C @p2+s2r@2C @r2+s2VV2@2C @V2+212srsp@2C @r@p+213srsV@2C @r@V+Rd)]TJ/F18 10.909 Tf 10.909 0 Td[(Cpt)]TJ/F18 10.909 Tf 10.909 0 Td[(rC=)]TJ/F18 10.909 Tf 8.485 0 Td[(t)]TJ/F18 10.909 Tf 10.909 0 Td[(r@C @r)]TJ/F18 10.909 Tf 10.909 0 Td[(ap@C @p;and@Z @t+1 2s2r@2Z @r2)]TJ/F18 10.909 Tf 10.909 0 Td[(rZ=)]TJ/F18 10.909 Tf 8.485 0 Td[(t)]TJ/F18 10.909 Tf 10.909 0 Td[(r@^Z @r:.11.11becomes@X @t+ap@X @p+t)]TJ/F18 10.909 Tf 10.909 0 Td[(r@X @r+rV@X @V+1 2s2p@2X @p2+s2r@2X @r2+s2VV2@2X @V2+212srsp@2X @r@p+213srsVV@2X @r@V)]TJ/F8 10.909 Tf 8.484 0 Td[(pt+rX=0:.12Sincethepresentvalueoftheforwardcreditdefaultswapisgivenby.1,thevalueoftheswaptionatt=T0isgivenbyXT0=[ZT0)-222(Z]+=E[b)]TJ/F18 10.909 Tf 10.909 0 Td[(RIfTNg])]TJ/F18 10.909 Tf 8.485 0 Td[(zE[NXi;Ti>tbTiiIfTig+bIfT0TNg]+:87

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Asbefore,fromthenancialpointofview,wecanexpectthevalueoftheswaptionrightbeforethejumpi.e.,att=T0)]TJ/F51 10.909 Tf 7.084 1.637 Td[(,tobetheexpectationofthevalueatt=T0.Sowecanassumethattheterminalconditionasfollows:XT0)]TJ/F8 10.909 Tf 7.085 1.637 Td[(=ST0)]TJ/F18 10.909 Tf 7.085 1.637 Td[(;T0EE[b)]TJ/F18 10.909 Tf 10.909 0 Td[(RIfTNg])]TJ/F18 10.909 Tf 8.485 0 Td[(zE[NXi;Ti>tbTiiIfTig+bIfT0TNg]+whereP;isdenedby.4.Therefore,wehavethefollowingpricingmodel:Theorem12.UnderAssumptions1through4,thepriceofcreditdefaultswaptionwithexpectedandunexpecteddefaultprobabilityismodeledasfollows:8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:@X @t+ap@X @p+t)]TJ/F18 10.909 Tf 10.91 0 Td[(r@X @r+rV@X @V+1 2s2p@2X @p2+s2r@2X @r2+s2VV2@2X @V2+212srsp@2X @r@p+213srsVV@2X @r@V)]TJ/F8 10.909 Tf 8.485 0 Td[(pt+rX=0XT0)]TJ/F8 10.909 Tf 7.085 1.637 Td[(=ST0)]TJ/F18 10.909 Tf 7.084 1.637 Td[(;T0EE[b)]TJ/F18 10.909 Tf 10.909 0 Td[(RIfTNg])]TJ/F18 10.909 Tf 8.485 0 Td[(zE[PNi;Ti>tbTiiIfTig+bIfT0TNg]+:5.2.3ParticularSolutionUnderthissubsection,wewillndaparticularsolutionwithadditionalcondition.Usingthechangeofunknownfunction^Xr;p;V;t=XT0)]TJ/F8 10.909 Tf 7.084 1.637 Td[(Wr;p;V;t;88

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.12becomesthefollowingPDEproblemwiththeterminalcondition.8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:@W @t+ap@W @p+t)]TJ/F18 10.909 Tf 10.909 0 Td[(r@W @r++rV@W @V+1 2s2p@2W @p2+s2r@2X @r2+s2VV2@2W @V2+212srsp@2W @r@p+213srsVV@2W @r@V)]TJ/F8 10.909 Tf 10.909 0 Td[(pt+rW=0;t0;r>0;V>VBWr;p;V;T0)]TJ/F8 10.909 Tf 7.085 1.637 Td[(=1;p>0;r>0;V>VBWr;p;VB;t=0;t0;r>0:.13Tosolve.13astheterminalcondition,weusethechangeofunknownfunctionandvariableasfollows:x=V Zandux;p;t=Wr;V;p;t Z:Then,.14reducesto8>>>>>>>>><>>>>>>>>>:@u @t+1 2[s2rB2t+s2V+213srsVBt]x2@2u @x2+s2pp;r;t@2u @p2+212srspBtx@2u @p@x+[ap)]TJ/F8 10.909 Tf 10.909 0 Td[(212srspBt]@u @p)]TJ/F18 10.909 Tf 10.909 0 Td[(pu=0;x>VB;p>0;0tVB;p>0uVB;p;t=0;p>0;0t>>>>>>>><>>>>>>>>>:@u @t+1 2s2tx2@2u @x2+s2pp;t@2u @p2+2x@2^u @p@x+app;t@u @p)]TJ/F18 10.909 Tf 10.909 0 Td[(pu=0;x>VB;p>0;0tVB;p>0uVB;p;t=0;p>0;n)]TJ/F16 7.97 Tf 6.586 0 Td[(1t
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NotingthattheabovePDEisthesameas.20wehavethepriceoftheswaptionasfollows:Theorem13.UndertheAssumptions1through4,thepriceofthecreditdefaultswaptionforthecorporatecoupon-bondwithexpectedandunexpecteddefaultisgivenbyXr;p;V;t=Zt;T0ST0)]TJ/F18 10.909 Tf 7.085 1.636 Td[(;T0EE[b)]TJ/F18 10.909 Tf 10.909 0 Td[(RIfTNg])]TJ/F18 10.909 Tf 8.485 0 Td[(zE[NXi;Ti>tbTiiIfTig+bIfT0TNg]+d1)]TJ/F18 10.909 Tf 15.808 7.38 Td[(x VBd2eAt;T0)]TJ/F19 7.97 Tf 6.586 0 Td[(Bt;T0p.14whered1=lnV VBZt;T0 q RT0ts2udu)]TJ/F24 10.909 Tf 12.104 23.454 Td[(q RT0ts2udu 2;andd2=lnVBZt;T0 V q RT0ts2udu)]TJ/F24 10.909 Tf 12.104 23.455 Td[(q RT0ts2udu 2:.1590

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Chapter6DataAnalysisInthischapter,weshalltrytondthepriceofBBandBBBratedcorporatecouponbondwithunexpecteddefaultprobability,andwiththeriskfreeinterestratertobestochastic.6.1DataTable3andFigure2showsthehistoricalmonthlydataof3-monthtreasurybill,andthehistor-icalyieldsofAAA,AA,AandBBBratedcorporate-bondsfromOctober1991throughNovem-ber/December2000Source:Moody'sInvestorsService.Weshallusetheratefor3-monthtreasurybillastherisk-freeinterestrateinpercent.Forexample,intherstrow,thedailyaverageoftheyield-to-maturityof3-monthtreasurybillinJanuary1991was6.41%,whilethedailyaveragesoftheyield-to-maturityofAAA,AA,A,andBBBratedcorporatecouponbondsinthesameperiodwere9.04%,9.37%,9.61%,and10.45%respectively.Table3:HistoricalShort-TermRateandYieldsofCorporateBonds Months 3MT-BillAAAAAABBB Jan-91 6.419.049.379.6110.45Feb-91 6.128.839.169.3810.07Mar-91 6.098.939.219.510.09 Continuedonnextpage.91

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Figure2.:HistoricalRisk-freeInterestRateandCorporate-BondYieldContinuedfrompreviouspage Months 3MT-BillAAAAAABBB Apr-91 5.838.869.129.399.94May-91 5.638.869.159.419.86Jun-91 5.759.019.289.559.96Jul-91 5.7599.259.519.89Aug-91 5.508.758.999.269.65Sep-91 5.378.618.869.119.51Oct-91 5.148.558.839.089.49Nov-91 4.698.488.789.019.45Dec-91 4.188.318.618.829.26 Continuedonnextpage.92

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Continuedfrompreviouspage Months 3MT-BillAAAAAABBB Jan-92 3.918.28.518.729.13Feb-92 3.958.298.678.839.23Mar-92 4.148.358.738.899.25Apr-92 3.848.338.698.879.21May-92 3.728.288.638.819.13Jun-92 3.758.228.568.79.05Jul-92 3.288.078.378.498.84Aug-92 3.207.958.218.348.65Sep-92 2.977.928.178.318.62Oct-92 2.937.998.328.498.84Nov-92 3.218.18.48.588.96Dec-92 3.297.988.248.378.81Jan-93 3.077.918.118.268.67Feb-93 2.997.717.98.038.39Mar-93 3.017.587.727.868.15Apr-93 2.937.467.627.88.14May-93 3.037.437.617.858.21Jun-93 3.147.337.517.748.07Jul-93 3.117.177.357.537.93Aug-93 3.096.857.067.257.6Sep-93 3.016.666.857.057.34Oct-93 3.096.676.877.047.31Nov-93 3.186.937.127.297.66Dec-93 3.136.937.127.317.69Jan-94 3.046.937.127.37.65Feb-94 3.337.087.297.447.76 Continuedonnextpage.93

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Continuedfrompreviouspage Months 3MT-BillAAAAAABBB Mar-94 3.597.487.697.828.13Apr-94 3.787.888.088.228.52May-94 4.277.998.198.328.62Jun-94 4.257.978.178.318.65Jul-94 4.468.118.318.448.8Aug-94 4.618.078.258.388.74Sep-94 4.758.348.498.618.98Oct-94 5.108.578.718.829.2Nov-94 5.458.688.838.949.32Dec-94 5.768.468.628.739.11Jan-95 5.908.468.68.79.08Feb-95 5.948.268.398.488.85Mar-95 5.918.128.248.338.7Apr-95 5.848.038.128.238.6May-95 5.857.657.747.868.2Jun-95 5.647.37.437.537.9Jul-95 5.597.417.547.658.04Aug-95 5.577.577.697.798.19Sep-95 5.427.327.457.567.93Oct-95 5.447.127.277.397.75Nov-95 5.527.027.187.327.68Dec-95 5.296.826.997.137.49Jan-96 5.156.816.997.127.47Feb-96 4.966.997.167.317.63Mar-96 5.107.357.527.688.03Apr-96 5.097.57.687.838.19 Continuedonnextpage.94

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Continuedfrompreviouspage Months 3MT-BillAAAAAABBB May-96 5.157.627.777.948.3Jun-96 5.237.717.878.028.4Jul-96 5.307.657.827.978.35Aug-96 5.197.467.637.778.18Sep-96 5.257.667.827.958.35Oct-96 5.127.397.587.78.07Nov-96 5.177.17.317.417.79Dec-96 5.047.27.417.517.89Jan-97 5.177.427.637.718.09Feb-97 5.147.317.547.597.94Mar-97 5.287.557.777.828.18Apr-97 5.307.737.937.988.34May-97 5.207.587.87.868.2Jun-97 5.077.417.627.688.02Jul-97 5.197.147.367.427.75Aug-97 5.287.227.47.467.82Sep-97 5.087.157.347.397.7Oct-97 5.1177.27.277.57Nov-97 5.286.877.077.157.42Dec-97 5.306.766.997.057.32Jan-98 5.186.616.826.937.19Feb-98 5.236.676.887.017.25Mar-98 5.166.726.937.057.32Apr-98 5.086.696.97.037.33May-98 5.146.696.917.037.3Jun-98 5.126.536.787.037.13 Continuedonnextpage.95

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Continuedfrompreviouspage Months 3MT-BillAAAAAABBB Jul-98 5.096.556.786.887.15Aug-98 5.046.526.776.897.14Sep-98 4.746.416.696.897.09Oct-98 4.076.376.696.827.18Nov-98 4.536.416.796.857.34Dec-98 4.506.226.656.957.23Jan-99 4.456.246.686.87.29Feb-99 4.566.46.796.847.39Mar-99 4.576.626.986.977.53Apr-99 4.416.646.967.147.48May-99 4.636.937.237.137.72Jun-99 4.727.237.527.48.02Jul-99 4.697.197.487.697.95Aug-99 4.877.47.687.658.15Sep-99 4.827.397.687.848.2Oct-99 5.027.557.797.848.38Nov-99 5.237.367.627.998.15Dec-99 5.367.557.787.798.19Jan-00 5.507.787.967.968.33Feb-00 5.737.687.828.158.29Mar-00 5.867.687.838.068.37Apr-00 5.827.647.828.078.4May-00 5.997.998.248.078.9Jun-00 5.867.677.878.498.48Jul-00 6.147.657.818.188.35Aug-00 6.287.657.78.118.26 Continuedonnextpage.96

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1 -0.084292805 Meanreversionrate 1:01151 0 0.003526218 Revertingmean 4.1833% Standarderror 0.00570581 Volatility sr 0.571% Table4:ParametersimpliedfromtheregressionestimatesContinuedfrompreviouspage Months 3MT-BillAAAAAABBB Sep-00 6.187.627.838.028.35Oct-00 6.297.557.818.138.34Nov-00 6.367.457.758.098.28Dec-00 5.947.21 AsweintroducedinChapter4,weshalladoptVasicekmodelforrisk-freeshortterminterestrate,i.e.,drt=t)]TJ/F18 10.909 Tf 10.909 0 Td[(rtdt+srdW1t:.1whereandsrisaconstant,andtisadeterministicfunctionoft.WefollowthemethodintroducedbyChangtoobtaintheimpliedparameters.UsingthedatafromTable3,werunthefollowingregression:drt=0+1rt+:.2Table4summarizestheimpliedparametersbythislinearregressionestimates.Themeanreversionrateiscalculatedasanegativeslopefrom1=)]TJ/F18 10.909 Tf 8.485 0 Td[(dtandtherevertingmeaniscalculatedfrom0=dt.Theinverseofthemeanreversionrate=canbeinterpretedasthenumberofperiodselapsedbetweenreversion,orspeedofreversion.Therevertingmeanrepresentsthelevelwhichtherisk-freeinterestraterevertstoafterwanderingoff.Thevolatilityhereisannualizedstandarddeviationoftherisk-freeinterestrate.Inthiscase,wecanexpectthatontheaverage0:9886=1=1:01151yearelapsesbetweenreversions,theleveloftherisk-freeinterestraterevertstois4:1833%,andtheannualizedstandarddeviationis0:571%.Therefore,weestimatetheshort-termratebythefollowingequationandFigure3belowshowsasimulationofthismeanrevertingmodel.97

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dr=1:01151:04133)]TJ/F18 10.909 Tf 10.909 0 Td[(rtdt+0:00570581dW.3wheretisinyears. Figure3.:HistoricalandSamplePathsofRisk-freeInterestRateNowTable5showsempiricalsurvivalrateofthecorporatebondsbytheoriginalcreditqualityconstructedfromhistoricalbonddefaultdatafor1991-2000.EachnumberinTable5givestheprobabilitythatthebondsurvivestillthetimeinyearselapsed.WeusethedataforBBBandBB.First,usingthecurve-ttingsoftwareDataFitavailableathttp://www.curvetting.com/index.html,thefunctionptwithoutjumpwasmodeled.Datawasregressedtoseveraltwicedifferentiablefunctions,i.e.,polynomialsofdegreetwothroughfour,ex-ponentialfunctions,reciprocalfunctionsofpolynomialsofdegreeoneandtwo,andthefollowingresultwasobtainedforBBB-ratedcompanieswithR2=:9971:98

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Table5:EmpiricalSurvivalRatebyOriginalCreditQuality Years AAAAAABBBBBBCCC 1 1.00001.00001.00000.99880.99040.98400.95652 1.00001.00001.00000.99400.97410.93540.82973 1.00000.99650.99980.98860.93500.87970.69004 1.00000.99460.99910.98270.92880.82150.63385 0.99970.99460.99880.97720.90880.77270.61476 0.99970.99460.99800.97150.90020.74060.55857 0.99970.99460.99750.96450.88530.71750.53308 0.99970.99460.99660.96300.88130.70240.51569 0.99970.99430.99600.96250.86590.69080.515610 0.99970.99410.99600.96020.83340.68490.4942 pt=p0+Zt0app;tdt+Zt0spp;tdW2.4withp0=0;app;t=0:0000674957t2)]TJ/F8 10.909 Tf 10.909 0 Td[(0:001170141t+0:003576423spp;t=0:001506268:.5Figure4showssomesamplespathswithjumpsoftheintensitypt,eachrepresentedbythesequenceofsquares,pluses,anddiamonds,computedfrom.4and.5.Here,fortheseasonaljumpsUjoncouponpaymentdatesattheendofmultiplesof6months,weassumethatitisconstantUj=)]TJ/F18 10.909 Tf 8.485 0 Td[(:1.Inthesamemanner,theparametersofthedefaultintensityforBB-ratedcompanieswasfoundtobep0=0;app;t=)]TJ/F8 10.909 Tf 8.485 0 Td[(0:004114407675t2)]TJ/F8 10.909 Tf 10.909 0 Td[(0:03025322658tspp;t=0:174409213.699

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Figure4.:SamplePathsoftheIntensitypwithR2=:9207.6.2BondPriceInthissection,usingthedataobtainedintheprevioussectionandthemodelsderivedinsubsec-tions2.1.2and4.1.2,thepriceofnonsecuredcouponbondsofFordMotorCreditCompanywithunexpecteddefaultonlywascomputed.ThetermsandconditionsoftherelativebondsareIssuedate:November21,1999Issueprice:99.812%100

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Coupon:7.375%s.a.dueonFebruaryandOctoberMaturity:October28,2009TheactualmarketpricerangeofthesebondsinYear2002was88.45%and104.28%,andthe3-monthT-billwas1.76%duringFebruary2002.Table6showsthebondpricecomputedwithrtobeconstant.Table6:FordMotorCreditBondPricewithConstantr r% Price 1.76 133:89% 4.82 112:29% Asitcanbeeasilyseen,thepricecomputedusingthemodelwithr=1:76%,whichistheshortterminterestrateasofFebruary2002,isoverpricedcomparedtotheactualmarketprice.Evenwhentheaverageshorttermrateover1991through2001,whichisr=4:82%,itisstilloverpriced.Atleastacoupleofreasonsforthiscanbeconsidered.First,eventhoughtheshorttermratefromFebruary2002wasused,theprospectoftheshort-termratewasunseenatthispoint.Actually,theshorttermratedrasticallydroppedatthebeginningofYear2002,afterwhichitgraduallycameback.Therefore,thestochasticmodelispreferred.Second,eventheaverageyieldofAAAcorporatebondsduringthesameperiodwas200)]TJ/F8 10.909 Tf 11.202 0 Td[(300pbhigherthantheshorttermsrate.Itshouldbealsomentionedthatthelongtermrateismorestablethanshorttermrate,whichimpliesthatthelongtermbondpriceislesssensitivetothechangeinshorttermrate.Sincethedefaultintensitygivesthedefaultprobabilityoverthedefaultfreebonds,ifwecomputethebondpriceusingtheyieldofAAAbondsasrisk-freeinterestrate,insteadofshorttermrate.Forr=7:55%,whichistheaverageover1991-2000,weobtainedthepriceof96:32%;andforr=6:51%,whichisthevalueasofFebruary2002,thepricewasestimatedtobe102:08%,bothofwhicharemorerealisticthanusingtherateof3-monthT-bill.Next,thepriceofthesamecorporatebondswascomputedusingstochasticprocessforr.However,asseeninthepricingwithconstantr,usingshorttermrateoverpricesthebonds,giving124:99%.WeranthelinearregressionontheyieldsofAAAcorporatebondsfromJanuary1991toDecember101

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2000andobtainedthefollowingresult.Table7:Parametersimpliedfromtheregressionestimates-AAAcorporatebonds 1 -0.048039321 Meanreversionrate 0:576471853 0 0.003470349 Revertingmean 7.22% Standarderror 0.001648158 Volatility sr 0.165% Therefore,weestimatetheyieldofAAAcorporatebondsbythefollowingequation,andFigure5showsthehistoricalandsamplepathoftheyieldofAAAcorporatebonds.drAAA=0:576471853:07224)]TJ/F18 10.909 Tf 10.909 0 Td[(rdt+0:001648158dW:.7 Figure5.:HistoricalandSamplePathsoftheYieldofAAACorporate-BondsHowever,usingthisequationresultsinthebondpriceof120:33%.Consideringusingtheconstantrresultedinamorerealisticvalue,itcanbeassessedthattheproblemliesintheanalysisofthe102

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stochasticprocessoftheyieldoftheAAAcorporatebonds.Itisobviousthattheyielddoesnotriseforeverorfalluntilitgetszero.Soitisplausibletoconsiderthemeanrevertingmodelandusingthelinearmodelforthevariable.However,thelongtermaverageoftheyieldinthisstochasticprocesswas7.22%comparedto7.54%,thearithmeticmeanforthesameperiod,reectingthehistoricalpath.However,wedonotknowthesentimentofthemarketjustfromthisdata,whichmightpredicttheriseinthelong-termyield.6.3CreditDefaultSwaptionPriceInthissubsection,weshallcomputethepriceofthecreditdefaultswaptionofFordMotorCreditCompanyonthefollowingterms,basedonthepricingofthebondscomputedintheprevioussubsection.Sinceusingthe3-monthT-billfortheriskfreeinterestratesgivesoverpricingofthebonds,weshalluseAAA-bondyieldastheriskfreeinterestrate.Referenceentity:FordMotorCreditCompanyOnsetoftheswaption:March,2004Expirationoftheswaption:September,2004DurationoftheunderlyingCDS:1year,3years,5yearsTable8givesthesummaryofthepricing:Table8:FordMotorCredit-CDSSwaptionPrice Riskfreeinterestrate 1Year 3Years 5Years % z=0bp z=50bp z=0bp z=50bp z=0bp z=50bp 7.22 AAAyieldaverage 30.22bp 15.11bp 110.63bp 55.32bp 163.88bp 81.94bp on1991-2000 5.33 AAAyield 31.26bp 15.63bp 116.78bp 58.38bp 175.43bp 87.72 asofMarch2004 Stochasticmodel 30.28bp 15.14bp 113.68bp 56.84bp 172.53bp 86.27bp Forexample,therightcolumnunder1yeargivesthepriceofCDSswaption,whichentitlesits103

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holdertoenteraCDSagreementforpayingz=50bpevery6months.Basedonthestochasticmodel,suchpriceis15:14bp.Notethatifz=0bp,thepriceoftheswaptionistheoreticallythesameasthatoftheforwardswaption.SincethereisnomarketforforwardCDSoptionorCDSswaption,weshallcomparethepriceoftheswaptionwithz=0bp,andthepriceofCDSforwardoptioncomputedusingtheactualCDSpricemidprice;seeFigure6,discountedusingthediscountratebasedonthecomputationintheprevioussection.Table9belowshowsthecomparisonbetweentheforwardCDSoptionpricebasedonourmodelandthatbasedontheactualCDSprice. Figure6.:5-Year-MaturityFORDCreditCDSQuotesInusingourmodel,thecouponpaymentdateofthebondsofthereferenceentityshouldtechnicallymatchtheonsetandmaturityoftheswaption,andmaturityoftheunderlyingforwardCDS.How-ever,incomputingtheabove,sincethematchingCDSratewasnotavailable,wecouldnotmatchthemthereisonemonthgapbetweenthem.104

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Table9:FordMotorCredit-CDSForwardPriceComparison Riskfree 1Year 3Years 5Years interestrate OurModel Estimated OurModel Estimated OurModel Estimated from from from % CDSprice CDSprice CDSprice 7.22 AAAyieldaverage 30.22bp 36.00bp 110.63bp 126.39bp 163.88bp 163.59bp on1991-2000 5.33 AAAyield 31.26bp 36.30bp 116.78bp 127.43bp 175.43bp 164.94 asofMarch2004 Stochasticmodel 30.28bp 36.73bp 113.68bp 128.94bp 172.53bp 166.89bp WecanseethatthattheforwardCDSoptionpriceobtainedfromourmodelisrelativelyclosetothatobtainedfromthemarketrate.105

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Chapter7ConclusionInthispaper,wecombinedtheintensitymodelandthestructuralmodeltondthepriceofthecorporatecoupon-bearingbonds.Weformedtheportfoliosothatwecouldhedgetheriskcausedbydefaultintensityand/oructuationoftheassetvalue.WeusedthearbitrageprincipleandtheItoLemmatoderivethePDEwithterminalandboundaryconditionsforpricing.Weassumedthatthesolutionwouldbeintheexponentialform.Thisisbecausetheintensitymodelisbasicallyderivedfromthehazardratemodel,whichhasthesolutionintheexponentialform.TheBlack-Scholesequation,whichisahomogeneousparabolicequationwithvariablecoefcientscombinedwiththeterminalconditionandasymptoticboundarycondition,admitsthesolutionintheexponentialformviaexponentialtransformation.Furtheranalysiswillbeneededtoseewhetheranonhomogeneousparabolicequationwithvariablecoefcients,terminalcondition,andasymptoticboundarycondi-tionyieldsauniquesolution.Indataanalysis,forunpredictabledefaultoccurrence,wemerelydependedonthehistoricaldataofdefaultprobability/intensity.Thehistoricaldataweusedwasovertheperiod1991-2000andacrossalltheindustrialsectors.Inactualpricing,thedefaultprobability/intensityneedstobecomputedbytheindustrialsector.Toincreasetheaccuracyoftheestimateofthedefaultprobability,weneedtotakeintotheconsiderationotherelementssuchaseconomicuctuationgrowth,recession,ordepression,thesizeofthecompany,monetarypolicyandsoon.Also,thesizeofthejumpinthedefaultintensitywasarbitraryassignedinSection6.2.Eventhoughthisistheoreticallyplausible,weneedtocollectempiricalevidenceandincorporateitinmeasuringthevariableUj.InChapters4and5,weassumedthattherisk-freeinterestratefollowstheVasicekmodelsincethismodelismostusedinthemarket.Applyingthismodel,weusedlinearregressiontoestimate106

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theparameters.Theshortcomingofusingthelinearregressionisthatthelong-termaverageisdeterminedbythetrendoftheperiodfromwhichthedatacomefrom.Weneedtotakeintotheconsiderationthefuturetrendoftherelativeinterestratetoestimatetheparameters.InSections2.2,3.2,4.2and5.2,topredictexpecteddefault,weassumedthatthermassetsvaluefollowsageometricBrownianmotion,withjumpsateachcouponpaymentdate,andthatexpecteddefaultoccurswhentheassetsvaluehitsthepredeterminedbarrier.InSection2.2,weappliedthereectionprincipletondtheexpecteddefaultprobability,theprobabilitythatthetotalassetsvaluehitsthepredeterminedbarrier.Asmentionedintheintroduction,eventhoughthismodelseemstomakesensetheoretically,therearestillsomeshortcomings.Theamountoftotalassetsvaluedoesnotnecessarilydeterminethenancialhealthofthecompany.Ahugecompanywithlargeassetscanbeunhealthynancially.Also,twocompanieswiththesameassetsvaluecanbeconsiderablydifferentintheirnancialconditions,whichwillleadtothedifferentlevelofthebarrierfordefault.Weshallneedtofurtherinvestigatethequalityoftheirassets.Inadditiontotheseshortcomings,thismodelisdifculttoimplementsincetherequiredquantitiesarenotreadilyobservable;wecanhaveanaccesstotherms'nancialstatementsonlyquarterlyforthebestinmostcases.Wealsoneedtoincorporatetheunforeseenfactorsinthefuture,suchasmarkettrendIsthemarketgrowingornot?,overalleconomytrendIsitingrowthperiod,inrecession,ordepression?andsoon.SchonbucherbsuggestsstockpriceandKMVforalternateparametersforpredictingex-pecteddefault.Stockprice,eventhoughweincorporatedstockpriceaspartofrmassetsvalue,istoospeculative,andtherefore,doesnotreectthecompany'snancialconditioninanybetterway.TheKMVmodel,marketedbyMoody's,setsthedefaultbarriersomewherebetweenthefacevalueoftotalliabilitiesandthefacevalueofshort-termliabilities.Theideabehindthisisthatthecompanyneedstorenanceitsshort-termliabilitiescontinuouslytocontinueitsdailyoperationswhilethelong-termliabilitiesdonotrequirerenancinguntiltheirmaturities.Thisideamakesabettersensethanusingthermassetsvalue;howeveritwillalsorequireasmuchefforttocollectthenecessarydata.107

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InSections2.2,3.2,4.2and5.2,insolvingtheproblem,weassumedthatendogenousdefaulteventsandexogenousdefaulteventsareuncorrelated,thatis,theintensityrateisuncorrelatedwiththevalueofthermassets,whichisnotrealistic.InSections4.2and5.2,wealsoassumedthatthereisnocorrelationbetweentherisk-freeinterestrateandtheintensityrate,orbetweentherisk-freerateandthermassetsvalue.Inmostexistingpaper,thepricingmodelofcreditdefaultswaptionisbasedonapplyingBlack-ScholesFormulatothepriceofforwardCreditDefaultSwap.However,thereisnoobservableforwardCDSmarket,whichmakestheexistingapproachlessattractive.Inthispaper,thepriceofcreditdefaultswaptionwascomputeddirectlyfromthebondprice.108

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References[1]Achdou,Y.andTchou,N.,2002,VariationalAnalysisfortheBlackandScholesEquationwithStochasticVolatility,MathematicalModellingandNumericalAnalysis,Vol.36,pp.373-395.[2]Ameur,B.,etal.,2006,ADynamicProgrammingApproachforPricingCDSandCDSOptions,Workingpaper,DepartmentofManagementSciences,HECMontreal.[3]Black,F.,andCox,J.C.,1976,ValuingCorporateSecurities:SomeEffectsofBondIndentureProvi-sions,JournalofFinance,Vol.31,No.2,pp.351-367.[4]Black,F.,andScholes,M.S.,1973,ThePricingofOptionsandCorporateLiabilities,JournalofPoliticalEconomy,Vol.81,No.3,pp.637-54.[5]Chang,Y.,2006,AnApplicationof'EarningatRisk'toAssessCorporateFinancialRisk,Proceed-ingsofthe2006CrystalBallUserConference,Gyeonggi,Korea.[6]Dufe,D.,andSingleton,K.J.,1999,ModelingTermStructuresofDefaultableBonds,WorkingPaper,GraduateSchoolofBusiness,StanfordUniversity.[7]Heakal,R.,2003,WhatIsACorporateCreditRating?,availablefromhttp://www.investopedia.com/articles/03/102203.asp[8]Hanke,Michael,2003,CreditRisk,CapitalStructure,andthePricingofEquityOptions,Springer,NewYork.[9]Harrison,J.M.,andStanleyR.P.,1981,MartingalesandStochasticIntegralsintheTheoryofCon-tinuousTrading,StochasticProcessesandtheirApplications,Vol.11,pp.215-260.[10]Hughston,L.P.,andTurnbull,S.,2001,CreditRisk:ConstructingtheBasicBuildingBlocks,Eco-nomicNotes,Vol.30,No.2,pp.281-292.[11]Hull,J.,andWhite,A.,2003,TheValuationofCreditDefaultSwapOptions,JournalofDerivatives,Vol.10,No.3,pp.40-50.109

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AbouttheAuthorMichiruShibata,borninChiba,Japan,workedinsecuritiesbusinessinTokyobeforeshemovedtoTampa,Florida.ShereceivedaB.A.inpsychologyinAugust,2001andaM.A.inmathematicsinMay,2003fromtheUniversityofSouthFlorida.CurrentlysheisanactuarialanalystatWakelyConsultingGroup,Inc.,Clearwater.