Credit Risk PDF
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CreditRisk Modellingcreditriskaccuratelyiscentraltothepracticeofmathematicalfinance. Themajorityofavailabletextsareaimedatanadvancedlevel,andaremoresuitable forPhDstudentsandresearchers.ThisvolumeoftheMasteringMathematicalFinance seriesaddressestheneedforacourseintendedformaster’sstudents,final-year undergraduates,andpractitioners.Thebookfocusesonthetwomainstreammodelling approachestocreditrisk,namelystructuralmodelsandreduced-formmodels,andon pricingselectedcreditriskderivatives.Balancingrigoroustheorywithexamples,it takesreadersthroughanaturaldevelopmentofmathematicalideasandfinancial intuition. marek capin´skiisProfessorofAppliedMathematicsatAGHUniversityof ScienceandTechnology,Krako´w.Hisresearchinterestsincludemathematicalfinance, corporatefinance,andhydrodynamics.Hehasbeenteachingforover35years,has heldvisitingfellowshipsinPolandandtheUK,andhaspublishedoverfiftyresearch papersandninebooks. tomasz zastawniakisChairinMathematicalFinanceattheUniversityof York.Hisresearchinterestsincludemathematicalfinance,stochasticanalysis, stochasticoptimisationandconvexanalysis,andmathematicalphysics.Hehas previouslytaughtatnumerousinstitutionsinPoland,theUSA,Canada,andtheUK, andhaspublishedoverfiftyresearchpublicationsandeightbooks. Mastering Mathematical Finance Mastering Mathematical Finance is a series of short books that cover all coretopicsandthemostcommonelectivesofferedinMaster’sprogrammes inmathematicalorquantitativefinance.Thebooksarecloselycoordinated and largely self-contained, and can be used efficiently in combination but alsoindividually. TheMMFbooksstartfinanciallyfromscratchandmathematicallyassume only undergraduate calculus, linear algebra, and elementary probability theory.Thenecessarymathematicsisdevelopedrigorously,withemphasis onanaturaldevelopmentofmathematicalideasandfinancialintuition,and thereadersquicklyseereal-lifefinancialapplications,bothformotivation and as the ultimate end for the theory. All books are written for both teachingandself-study,withworkedexamples,exercises,andsolutions. [DMFM] DiscreteModelsofFinancialMarkets, MarekCapin´ski,EkkehardKopp [PF] ProbabilityforFinance, EkkehardKopp,JanMalczak,TomaszZastawniak [SCF] StochasticCalculusforFinance, MarekCapin´ski,EkkehardKopp,JanuszTraple [BSM] TheBlack–ScholesModel, MarekCapin´ski,EkkehardKopp [PTRM] PortfolioTheoryandRiskManagement, MaciejJ.Capin´ski,EkkehardKopp [NMFC] NumericalMethodsinFinancewithC++, MaciejJ.Capin´ski,TomaszZastawniak [SIR] StochasticInterestRates, DaraghMcInerney,TomaszZastawniak [CR] CreditRisk, MarekCapin´ski,TomaszZastawniak [SCAF] StochasticControlAppliedtoFinance, SzymonPeszat,TomaszZastawniak Series editors Marek Capin´ski, AGH University of Science and Technol- ogy, Krako´w; Ekkehard Kopp, University of Hull; Tomasz Zastawniak, UniversityofYork Credit Risk MAREK CAPIN´ SKI AGHUniversityofScienceandTechnology,Krako´w TOMASZ ZASTAWNIAK UniversityofYork UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 4843/24,2ndFloor,AnsariRoad,Daryaganj,Delhi–110002,India 79AnsonRoad,#06–04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781107002760 10.1017/9781139047432 ©MarekCapin´skiandTomaszZastawniak2017 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2017 PrintedintheUnitedKingdombyClays,StIvesplc AcataloguerecordforthispublicationisavailablefromtheBritishLibrary ISBN978-1-107-00276-0Hardback ISBN978-0-521-17575-3Paperback Additionalresourcesforthispublicationatwww.cambridge.org/creditrisk. CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyInternetWebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchWebsitesis,orwillremain, accurateorappropriate. Contents Preface pagevii 1 Structuralmodels 1 1.1 Tradedassets 1 1.2 Mertonmodel 7 1.3 Non-tradeableassets 21 1.4 Barriermodel 24 2 Hazardfunctionmodelandnoarbitrage 39 2.1 Marketmodel 39 2.2 Defaulttimeandhazardfunction 41 2.3 Defaultindicatorprocess 48 2.4 Noarbitrageandrisk-neutralprobability 57 3 Defaultablebondpricingwithhazardfunction 71 3.1 Initialpricesandcalibration 73 3.2 Processofprices 76 3.3 Recoveryschemes 79 3.4 Martingaleproperties 81 3.5 Martingalepropertiesofintegrals 86 3.6 Bondpricedynamics 91 4 Securitypricingwithhazardfunction 94 4.1 Martingalerepresentationtheorem 94 4.2 Pricingdefaultablesecurities 96 4.3 CreditDefaultSwap 109 5 Hazardprocessmodel 114 5.1 Marketmodelandrisk-neutralprobability 114 5.2 Martingaleswithrespecttotheenlargedfiltration 116 5.3 Hazardprocess 121 5.4 Canonicalconstructionofdefaulttime 125 5.5 Conditionalexpectations 130 5.6 Martingaleproperties 136 6 Securitypricingwithhazardprocess 146 6.1 Bondpricedynamics 146 6.2 Tradingstrategies 150 v vi Contents 6.3 Zero-recoverysecurities 157 6.4 Martingalerepresentationtheorem 162 6.5 Securitieswithpositiverecovery 169 A Appendix 176 A.1 Lebesgue–Stieltjesintegral 176 A.2 PassagetimeandminimumofWienerprocess 181 A.3 Stochasticcalculuswithmartingales 185 Selectbibliography 191 Index 193 Preface Credit risk is concerned with the possibility of bankruptcy happening as a result of unpaid debt obligations. This applies to companies partially fi- nanced by debt. The key question is to find the value of debt or, equiv- alently, the amount of interest charged by the debt holders to offset their possible loss due to bankruptcy. Our goal is to explain the main issues in thesimplestpossiblecases,tokeepthetheoryfreeoftechnicalcomplica- tions. The first approach covered in this book relates bankruptcy to the com- pany’s performance. The resulting model is referred to as structural. His- torically,thiswasthefirstsystematicapproachtocreditrisk,developedby Mertonin1974. Intheothermethodstudiedhere,calledthereduced-formapproach,de- fault occurs as a result of external forces, not related directly to the func- tioning of the company, and comes as a surprise at a random time. This may be the time when some information is revealed about technological advances, court decisions, or political events, which could wipe out the company. Wedevelopanapproachsimilartotheclassicalfinancialtheories,where certainunderlyingsecuritiesarespecifiedandself-financingstrategiesare constructedfromthese.Hereitistherisk-freeanddefaultablezero-coupon bondsthatwillserveasunderlyingassets.Thismakesitpossibletopricea rangeofdefaultablesecurities,includingCreditDefaultSwaps,bymeans ofreplication.Then,afteraddingaBlack–Scholesstockasabuildingblock of the market, the range of securities broadens, and this allows us to con- clude with an example where the structural and reduced-form approaches cometogether. Readers of the book need to be familiar with material covered in some earlier volumes in this series, namely [SCF] and [BSM], that is, with the foundationsofstochasticcalculusandtheBlack–Scholesmodel.Exposure tomeasuretheorybasedprobability,coveredin[PF],wouldalsobehelpful. The text is interspersed with various examples and exercises. The nu- mericalworkunderpinningmanyoftheexamples,andsolutionstotheex- ercises,areavailableatwww.cambridge.org/creditrisk. vii 1 Structural models 1.1 Tradedassets 1.2 Mertonmodel 1.3 Non-tradeableassets 1.4 Barriermodel Consider a company launched at time 0, when some assets are purchased forV(0).Fundingcomesfromtwosources.Shareholderscontribute E(0), referred to as equity. The remaining amount D(0) = V(0)− E(0), called debt,iseitherborrowedfromabankorraisedbysellingbondsissuedby thecompany. Weconsiderthiscompanyoveratimeintervalfrom0toT,duringwhich theassetsareputtoworkinordertogeneratesomefunds,whicharethen splitbetweenthetwogroupsofinvestorsattimeT.Thedebtisfirstrepaid withinteresttothedebtholders,whohavepriorityovertheequityholders. Anyremainingamountgoestotheequityholders. The simplest way to raise money to make these payments is to sell the assetsofthecompany.Webeginouranalysiswiththiscase,bymakingthe necessaryassumptionthattheassetsaretradeable. 1.1 Tradedassets We assume that there is a liquid market for the assets, and V(t) for t ∈ [0,T]representstheirmarketvalue.Wealsoassumethattheassetsgenerate no additional cash flows. A practical example of such assets would be a portfolio of traded stocks that pay no dividends, the company being an investmentfund. 1 2 Structuralmodels Payoffs SupposethatthecompanyhastoclearthedebtattimeT,andthatthereare no intermediate cash flows to the debt holders. The interest rate applying to the loan will be quoted by the bank or implied by the bond price. We denotethisloanratebyk withcontinuouscompounding,andbyK with D D annualcompounding.TheamountdueattimeT is F = D(0)ekDT = D(0)(1+K )T. D (Throughoutthisvolumewetakeoneyearastheunitoftime.)Oneofthe goalshereistofindtheloanratethatreflectstheriskforthedebtholders. At time T we sell the assets and close down the business, at least hy- pothetically, to analyse the company’s financial position at that time. It is possible that the amount obtained by selling the assets is insufficient to settle the debt, that is, V(T) < F. In this respect, we make an important assumptionconcerningthelegalstatusofthecompany:ithaslimitedlia- bility. This means that losses cannot exceed the initial equity value E(0). If V(T) < F, the equity holders do not have to cover the loss from their personalfunds.Thecompanyisdeclaredbankrupt,andtheequityholders walkawayhavinglosttheirinitialinvestment.IfV(T)≥ F,theloancanbe paidbackwithinterest,andtheequityholderskeepthebalance.Thefinal value of equity is therefore a random amount equal to the payoff of a call option, E(T)=max{V(T)−F,0}, withthevalueoftheassetsastheunderlyingsecurityandthedebtrepay- mentamountF asthestrikeprice. Remark1.1 Acallisanoptiontobuytheunderlyingassetforaprescribedprice,which soundsparadoxicalhere.However,itisconsistentwiththegeneralpractice thatloansaresecuredonsomeassets.Theborrower’sownershiprightsin theassetsarerestricteduntiltheloanisrepaid.Thefullrights(forinstance, toselltheasset)areinasenseboughtbackwhentheloanissettled. IfV(T) ≥ F,thedebtwillbepaidinfull,butinthecaseofbankruptcy, whichwillbedeclaredifV(T)< F,thedebtholdersaregoingtotakeover the assets and sell them for V(T). This is straightforward if the assets are tradeable.TheamountreceivedattimeT willbe D(T)=min{F,V(T)}= F−max{F−V(T),0}.
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