西南财经大学期权期货及其他衍生品(00003)市公开课金奖市赛课一等奖课件

Chapter13

CreditRisk第1页第1页Whatiscreditrisk?Creditriskarisesfromthepossibilitythatborrowersandcounterpartiesinderivativestransactionsmaydefault.222第2页第2页ContentsApproachestoestimatingtheprobabilitythatacompanywilldefaultThedifferencebetweenrisk-neutralandreal-worldprobabilitiesofdefaultCreditriskofderivativeDefaultcorrelation,Gaussiancopulamodels333第3页第3页ApproachestoestimatingdefaultprobabilitiesHistoricaldefaultprobabilitiesofratingcompaniesFrombondspricesFromequitypricesFromderivativesprices第4页第4页Historicalcumulativeaveragedefaultrates(%)第5页第5页InterpretationThetableshowstheprobabilityofdefaultforcompaniesstartingwithaparticularcreditratingTheprobabilitythatabondinitiallyratedBaawilldefaultduringthesecondyearis0.506-0.181=0.325Defaultprobabilitychangewithtime第6页第6页DefaultIntensitiesvsUnconditionalDefaultProbabilitiesTheunconditionaldefaultprobabilityistheprobabilityofdefaultforacertaintimeperiodasseenattimezeroTheconditionaldefaultprobabilityistheprobabilityofdefaultforacertaintimeperiodconditionalonnoearlierdefault(say,defaultintensityorhazardrate)第7页第7页DefineV(t)ascumulativeprobabilityofthecompanysurvivingtotimet.Takinglimits,wegetDefineQ(t)astheprobabilityofdefaultbytimet.Whereistheaveragedefaultintensitybetween0andt第8页第8页RecoveryrateTherecoveryrateforabondisusuallydefinedasthepriceofthebondimmediatelyafterdefaultasapercentofitsfacevalueRecoveryratesaresignificantlynegativelycorrelatedwithdefaultrates第9页第9页Recoveryrates

(Moody’s:1982to,Table22.2,page491)第10页第10页UsingBondPricesAveragedefaultintensityoverlifeofbondisapproximatelyWheresisthespreadofthebond’syieldovertherisk-freerateandRistherecoveryrate.第11页第11页MoreExactCalculationAssumethata5yearcorporatebondpaysacouponof6%perannum(semiannually).Theyieldis7%withcontinuouscompoundingandtheyieldonasimilarrisk-freebondis5%(continuouscompounding).Priceofrisk-freebondis104.09;priceofcorporatebondis95.34;expectedlossfromdefaultsis8.75.SupposethattheprobabilityofdefaultisQperyearandthatdefaultsalwayshappenhalfwaythroughayear(immediatelybeforeacouponpayment)第12页第12页Calculations第13页第13页Calculations(Cons.)Weset288.48Q=8.75togetQ=3.03%ThisanalysiscanbeextendedtoallowdefaultstotakepacemorefrequentlyInsteadofassumingaconstantunconditionalprobabilityofdefaultwecanassumeaconstantdefaultintensityoraparticularpatternforthevariationofdefaultprobabilitieswithtime.Withseveralbondswecanusemoreparameterstodescribethetermstructureofdefaultprobability.第14页第14页TheRisk-FreeRateTherisk-freeratewhendefaultprobabilitiesareestimatedisusuallyassumedtobetheLIBOR/swapzerorate(orsometimes10bpsbelowthem)Togetdirectestimatesofthespreadofbondyieldsoverswaprateswecanlookatassetswaps第15页第15页AssetSwapsAssetswapspreadsprovideadirectestimateofthespreadofbondyieldsovertheLIBOR/swapcurve.Iftheassetswapspreadis150bpsandtheLIBOR/swapzerocurveisflatat5%.Theexpectedlossfromdefaultoverthe5-yearlifeofthebondistherefore$6.55.6.55=288.48*Q,Q=2.27%第16页第16页CreditDefaultSwapSpreads(bps)第17页第17页CreditDefaultSwapSpreads(bps)第18页第18页ComparisonhistoricalvsbondCalculationofdefaultintensitiesusinghistoricaldataarebasedonequation(22.1)andtable(22.1);Fromequation(22.1),wehaveThecalculationsusingbondpricesarebasedonequation(22.2)andbondyieldspublishedbyMerrillLynch.第19页第19页RealWorldvsRiskNeutralDefaultProbabilities,7yearaverage第20页第20页RiskPremiumsEarnedbyBondTraders第21页第21页ThedefaultprobabilityfromhistoricaldataissignificantlylowerthanthatfrombondpricesTheratiodeclineswhilethedifferenceincreasesasacompany’screditratingdeclines.第22页第22页RealWorldvs.

Risk-NeutralDefaultProbabilitiesThedefaultprobabilitiesbackedoutofbondpricesorcreditdefaultswapspreadsarerisk-neutraldefaultprobabilitiesThedefaultprobabilitiesbackedoutofhistoricaldataarereal-worlddefaultprobabilities第23页第23页PossiblereasonsfortheseresultsCorporatebondsarerelativelyilliquidThesubjectivedefaultprobabilitiesofbondtradersmaybemuchhigherthantheestimatesfromMoody’shistoricaldataBondsdonotdefaultindependentlyofeachother.Thisleadstosystematicriskthatcannotbediversifiedaway.Bondreturnsarehighlyskewedwithlimitedupside.Thenon-systematicriskisdifficulttodiversifyawayandmaybepricedbythemarket.第24页第24页Whichworldshouldweuse?Weshoulduserisk-neutralestimatesforvaluingcreditderivativesandestimatingthepresentvalueofthecostofdefaultWeshoulduserealworldestimatesforcalculatingcreditVaRandscenarioanalysis第25页第25页Merton’smodelMerton’smodelregardstheequityasanoptionontheassetsofthefirm.Inasimplesituationtheequationvalueiswhereisthevalueofthefirmandisthedebtrepaymentrequired.第26页第26页Equityvs.AssetsAnoptionpricingmodelenablesthevalueofthefirm’sequitytoday,,toberelatedtothevalueofitsassetstoday,,andthevolatilityofitsassets,Therisk-neutralprobabilitythatthecompanywilldefaultonthedebtis.第27页第27页Volatilities?第28页第28页ExampleAcompany’sequityis$3millionandthevolatilityoftheequityis80%Therisk-freerateis5%,thedebtis$10millionandtimetodebtmaturityis1yearSolvingthetwoequationsyields第29页第29页Example(Con.)TheprobabilityofdefaultisThemarketvalueofthedebtisThepresentvalueofthepromisedpaymentis9.51Theexpectedlossisabout(9.51-9.4)/9.51=1.2%Therecoveryrateis(12.7-1.2)/12.7=91%第30页第30页ImplementationofMerton’smodel(e.g.Moody’sKMV)Merton’smodelproducesagoodrankingofdefaultprobabilities(risk-neutralorreal-world)Moody公司把股票当于公司资产期权思想计算出风险中性世界违约距离，再利用拥有海量历史违约数据库，建立起风险中性违约距离与现实世界违约率之间相应关系，从而得到预期违约频率，作为违约概率预测指标。第31页第31页贝尔斯登预期违约频率第32页第32页从期权价格中引出风险中性违约概率由于股票是公司资产期权，这样股票期权就是期权期权，其价格能够表示为：利用最大熵办法（Capuano,)就能够从公司同期限所有期权价格中预计出和D第33页第33页从期权价格中能够推导出风险中性违约概率利用上述办法，我们就可依据3月14日贝尔斯登将于3月22日到期期权价格，计算出贝尔斯登风险中性违约概率和公司价值概率分布。贝尔斯登于3月14日被摩根大通接管。下图显示，市场对贝尔斯登一周后命运产生巨大分歧，公司价值大涨大跌概率远远不小于小幅变动概率，这样分布与正常情况分布有天壤之别。可见期权价格能够让我们清楚地看出市场在非常时期对未来特殊见解。第34页第34页贝尔斯登风险中性违约概率和公司价值概率分布（3月14日）第35页第35页风险中性违约概率风险中性违约概率即使不同于现实概率，但其改变能够反应现实世界违约概率改变。在金融危机时期，它可能比CDS价差能更敏感地反应出违约概率改变。在贝尔斯登于3月14日被接管前后，依据上述方法计算出来风险中性概率天天改变比CDS价差更敏感。这是因为在金融危机期间，金融机构本身信用度大幅降低，造成在OTC市场交易CDS交易量急剧萎缩，价差大幅扩大，信号失真。第36页第36页期权隐含中性违约概率与CDS价差第37页第37页CreditRiskMitigationNetting:incrementaleffectCollateralizationDowngradetriggers第38页第38页DefaultcorrelationThecreditdefaultcorrelationbetweentwocompaniesisameasureoftheirtendencytodefaultataboutthesametimeFactors

(1)macroeconomicenvironment:goodeconomy=lownumberofdefaults(2)Sameindustryorgeographicarea:companiescanbesimilarlyorinverselyaffectedbyanexternalevent(3)creditcontagion:connectionsbetweencompaniescancausearippleeffect

第39页第39页CreditderivativeCreditderivativesarecontractswherethepayoffdependsonthecreditworthinessofoneormorecompaniesorcountriesBuyers:banksorotherfinancialinstitutionsSellers:insurancecompanySinglename:creditdefaultswap,CDS第40页第40页HowdoesCDSworks?Thisisacontractthatprovidesinsuranceagainsttheriskofadefaultbyparticularcompany.Thecompanyisknownasthereferenceentityandadefaultbythecompanyisknownasacreditevent.Thebuyeroftheinsuranceobtainstherighttosellbondsissuedbythecompanyfortheirfacevaluewhenacrediteventoccurs.Thesellersoftheinsuranceagreestobuythebondsfortheirfacevaluewhenacrediteventoccur.第41页第41页ExampleA5-yearcreditdefaultswaponMarch1,.Thenotionalprincipalis$100million.

Thebuyeragreestopay90basispointsannuallyforprotectionagainstdefaultbythereferenceentity.DefaultprotectionbuyerDefaultprotectionseller90basispointsperyearPaymentifdefaultbyreferenceentity第42页第42页MechanismIfnotdefault,referenceentitypays$900,000onMarch1ofeach-Ifdefault,e.g.June1,;(1)specifiesphysicalsettlement;(2)determinethemid-marketvalueofthecheapestdeliverablebond,orsay,cashpaymentInarrearpayment,includingafinalaccrualpaymentCDSspread:thetotalamountpaidperyear,asapercentofthenotionalprincipal,tobuyprotection第43页第43页CDSandBondyieldsACDScanbeusedtohedgeapositioninacorporatebond.Then-yearCDSspreadshouldbeapproximatelyequaltotheexcessoftheparyieldonann-yearcorporatebondovertheparyieldonann-yearrisk-freebond.Howtouseit第44页第44页CDSandCheapest-to-deliverbondBondstypicallyhavethesameseniority,buttheymaynotsellforthesamepercentageoffacevalueimmediatelyafteradefault.Searchacheapest-to-deliverbond.第45页第45页ValuationofcreditdefaultswapsMid-marketCDSspreadsExample:Supposetheprobabilityduringayearconditionalonnoearlierdefaultis2%.Time(year)defaultprobabilitysurvivalprobability10.020.9820.01960.960430.01920.941240.01880.922450.01840.9039第46页第46页Valuationofcreditdefaultswaps(cons.)

(2)Defaultalwayshappenhalfwaythroughayearandthatpaymentsonthecreditdefaultswaparemadeonceayearattheendofeachyear.(3)Therisk-freeinterestrateis5%perannumwithcontinuouscompoundingandtherecoverrateis40%.第47页第47页1Default123450Default2Default3Default4Default5PayoffAccrualpayment…..…..…..…..Payment1Payment2Payment3Payment4Payment5SurvivalprobabilityDefaultprobability第48页第48页PVoftheexpectedpaymentAssumenotionalprincipalis1andpaymentatrateofsperyear.Time(year)survivalprobabilityexpectedpayment