西南财经大学期权期货及其他衍生品第13章市公开课获奖课件省名师优质课赛课一等奖课件

Chapter13

CreditRisk1/52Whatiscreditrisk?Creditriskarisesfromthepossibilitythatborrowersandcounterpartiesinderivativestransactionsmaydefault.222/522ContentsApproachestoestimatingtheprobabilitythatacompanywilldefaultThedifferencebetweenrisk-neutralandreal-worldprobabilitiesofdefaultCreditriskofderivativeDefaultcorrelation,Gaussiancopulamodels3333/52ApproachestoestimatingdefaultprobabilitiesHistoricaldefaultprobabilitiesofratingcompaniesFrombondspricesFromequitypricesFromderivativesprices4/52Historicalcumulativeaveragedefaultrates(%)5/52InterpretationThetableshowstheprobabilityofdefaultforcompaniesstartingwithaparticularcreditratingTheprobabilitythatabondinitiallyratedBaawilldefaultduringthesecondyearis0.506-0.181=0.325Defaultprobabilitychangewithtime6/52DefaultIntensitiesvsUnconditionalDefaultProbabilitiesTheunconditionaldefaultprobabilityistheprobabilityofdefaultforacertaintimeperiodasseenattimezeroTheconditionaldefaultprobabilityistheprobabilityofdefaultforacertaintimeperiodconditionalonnoearlierdefault(say,defaultintensityorhazardrate)7/52DefineV(t)ascumulativeprobabilityofthecompanysurvivingtotimet.Takinglimits,wegetDefineQ(t)astheprobabilityofdefaultbytimet.Whereistheaveragedefaultintensitybetween0andt8/52RecoveryrateTherecoveryrateforabondisusuallydefinedasthepriceofthebondimmediatelyafterdefaultasapercentofitsfacevalueRecoveryratesaresignificantlynegativelycorrelatedwithdefaultrates9/52Recoveryrates

(Moody’s:1982to,Table22.2,page491)10/52UsingBondPricesAveragedefaultintensityoverlifeofbondisapproximatelyWheresisthespreadofthebond’syieldovertherisk-freerateandRistherecoveryrate.11/52MoreExactCalculationAssumethata5yearcorporatebondpaysacouponof6%perannum(semiannually).Theyieldis7%withcontinuouscompoundingandtheyieldonasimilarrisk-freebondis5%(continuouscompounding).Priceofrisk-freebondis104.09;priceofcorporatebondis95.34;expectedlossfromdefaultsis8.75.SupposethattheprobabilityofdefaultisQperyearandthatdefaultsalwayshappenhalfwaythroughayear(immediatelybeforeacouponpayment)12/52Calculations13/52Calculations(Cons.)Weset288.48Q=8.75togetQ=3.03%ThisanalysiscanbeextendedtoallowdefaultstotakepacemorefrequentlyInsteadofassumingaconstantunconditionalprobabilityofdefaultwecanassumeaconstantdefaultintensityoraparticularpatternforthevariationofdefaultprobabilitieswithtime.Withseveralbondswecanusemoreparameterstodescribethetermstructureofdefaultprobability.14/52TheRisk-FreeRateTherisk-freeratewhendefaultprobabilitiesareestimatedisusuallyassumedtobetheLIBOR/swapzerorate(orsometimes10bpsbelowthem)Togetdirectestimatesofthespreadofbondyieldsoverswaprateswecanlookatassetswaps15/52AssetSwapsAssetswapspreadsprovideadirectestimateofthespreadofbondyieldsovertheLIBOR/swapcurve.Iftheassetswapspreadis150bpsandtheLIBOR/swapzerocurveisflatat5%.Theexpectedlossfromdefaultoverthe5-yearlifeofthebondistherefore$6.55.6.55=288.48*Q,Q=2.27%16/52CreditDefaultSwapSpreads(bps)17/52CreditDefaultSwapSpreads(bps)18/52ComparisonhistoricalvsbondCalculationofdefaultintensitiesusinghistoricaldataarebasedonequation(22.1)andtable(22.1);Fromequation(22.1),wehaveThecalculationsusingbondpricesarebasedonequation(22.2)andbondyieldspublishedbyMerrillLynch.19/52RealWorldvsRiskNeutralDefaultProbabilities,7yearaverage20/52RiskPremiumsEarnedbyBondTraders21/52ThedefaultprobabilityfromhistoricaldataissignificantlylowerthanthatfrombondpricesTheratiodeclineswhilethedifferenceincreasesasacompany’screditratingdeclines.22/52RealWorldvs.

Risk-NeutralDefaultProbabilitiesThedefaultprobabilitiesbackedoutofbondpricesorcreditdefaultswapspreadsarerisk-neutraldefaultprobabilitiesThedefaultprobabilitiesbackedoutofhistoricaldataarereal-worlddefaultprobabilities23/52PossiblereasonsfortheseresultsCorporatebondsarerelativelyilliquidThesubjectivedefaultprobabilitiesofbondtradersmaybemuchhigherthantheestimatesfromMoody’shistoricaldataBondsdonotdefaultindependentlyofeachother.Thisleadstosystematicriskthatcannotbediversifiedaway.Bondreturnsarehighlyskewedwithlimitedupside.Thenon-systematicriskisdifficulttodiversifyawayandmaybepricedbythemarket.24/52Whichworldshouldweuse?Weshoulduserisk-neutralestimatesforvaluingcreditderivativesandestimatingthepresentvalueofthecostofdefaultWeshoulduserealworldestimatesforcalculatingcreditVaRandscenarioanalysis25/52Merton’smodelMerton’smodelregardstheequityasanoptionontheassetsofthefirm.Inasimplesituationtheequationvalueiswhereisthevalueofthefirmandisthedebtrepaymentrequired.26/52Equityvs.AssetsAnoptionpricingmodelenablesthevalueofthefirm’sequitytoday,,toberelatedtothevalueofitsassetstoday,,andthevolatilityofitsassets,Therisk-neutralprobabilitythatthecompanywilldefaultonthedebtis.27/52Volatilities?28/52ExampleAcompany’sequityis$3millionandthevolatilityoftheequityis80%Therisk-freerateis5%,thedebtis$10millionandtimetodebtmaturityis1yearSolvingthetwoequationsyields29/52Example(Con.)TheprobabilityofdefaultisThemarketvalueofthedebtisThepresentvalueofthepromisedpaymentis9.51Theexpectedlossisabout(9.51-9.4)/9.51=1.2%Therecoveryrateis(12.7-1.2)/12.7=91%30/52ImplementationofMerton’smodel(e.g.Moody’sKMV)Merton’smodelproducesagoodrankingofdefaultprobabilities(risk-neutralorreal-world)Moody企业把股票当于企业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对应关系，从而得到预期违约频率，作为违约概率预测指标。31/52贝尔斯登预期违约频率32/52从期权价格中引出风险中性违约概率因为股票是企业资产期权，这么股票期权就是期权期权，其价格能够表示为：利用最大熵方法（Capuano,)就能够从企业同期限全部期权价格中预计出和D33/52从期权价格中能够推导出风险中性违约概率利用上述方法，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企业价值概率分布。贝尔斯登于3月14日被摩根大通接管。下列图显示，市场对贝尔斯登一周后命运产生巨大分歧，企业价值大涨大跌概率远远大于小幅变动概率，这么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看法。34/52贝尔斯登风险中性违约概率和企业价值概率分布（3月14日）35/52风险中性违约概率风险中性违约概率即使不一样于现实概率，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。36/52期权隐含中性违约概率与CDS价差37/52CreditRiskMitigationNetting:incrementaleffectCollateralizationDowngradetriggers38/52DefaultcorrelationThecreditdefaultcorrelationbetweentwocompaniesisameasureoftheirtendencytodefaultataboutthesametimeFactors

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

39/52CreditderivativeCreditderivativesarecontractswherethepayoffdependsonthecreditworthinessofoneormorecompaniesorcountriesBuyers:banksorotherfinancialinstitutionsSellers:insurancecompanySinglename:creditdefaultswap,CDS40/52HowdoesCDSworks?Thisisacontractthatprovidesinsuranceagainsttheriskofadefaultbyparticularcompany.Thecompanyisknownasthereferenceentityandadefaultbythecompanyisknownasacreditevent.Thebuyeroftheinsuranceobtainstherighttosellbondsissuedbythecompanyfortheirfacevaluewhenacrediteventoccurs.Thesellersoftheinsuranceagreestobuythebondsfortheirfacevaluewhenacrediteventoccur.41/52ExampleA5-yearcreditdefaultswaponMarch1,.Thenotionalprincipalis$100million.

Thebuyeragreestopay90basispointsannuallyforprotectionagainstdefaultbythereferenceentity.DefaultprotectionbuyerDefaultprotectionseller90basispointsperyearPaymentifdefaultbyreferenceentity42/52MechanismIfnotdefault,referenceentitypays$900,000onMarch1ofeach-Ifdefault,e.g.June1,;(1)specifiesphysicalsettlement;(2)determinethemid-marketvalueofthecheapestdeliverablebond,orsay,cashpaymentInarrearpayment,includingafinalaccrualpaymentCDSspread:thetotalamountpaidperyear,asapercentofthenotionalprincipal,tobuyprotection43/52CDSandBondyieldsACDScanbeusedtohedgeapositioninacorporatebond.Then-yearCDSspreadshouldbeapproximatelyequaltotheexcessoftheparyieldonann-yearcorporatebondovertheparyieldonann-yearrisk-freebond.Howtouseit44/52CDSandCheapest-to-deliverbondBondstypicallyhavethesameseniority,buttheymaynotsellforthesamepercentageoffacevalueimmediatelyafteradefault.Searchacheapest-to-deliverbond.45/52ValuationofcreditdefaultswapsMid-marketCDSspreadsExample:Supposetheprobabilityduringayearconditionalonnoearlierdefaultis2%.Time(year)defaultprobabilitysurvivalprobability10.020.9820.01960.960430.01920.941240.01880.922450.01840.903946/52Valuationofcreditdefaultswaps(cons.)

(2)Defaultalwayshappenhalfwaythroughayearandthatpaymentsonthecreditdefaultswaparemadeonceayearattheendofeachyear.(3)Therisk-freeinterestrateis5%perannumwithcontinuouscompoundingandtherecoverrateis40%.47/521Default123450Default2Default3Default4Default5PayoffAccrualpayment…..…..…..…..Payment1Payment2Payment3Payment4Payment5SurvivalprobabilityDefaultprobability48/52PVoftheexpectedpaymentAssumenotionalprincipalis1andpaymentatrateofsperyear.Time(year)survivalprobabilityexpectedpaymentdisc