西南財經(jīng)大學期權(quán)期貨及其他衍生品第13章市公開課獲獎?wù)n件省名師優(yōu)質(zhì)課賽課一等獎?wù)n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企業(yè)把股票當于企業(yè)資產(chǎn)期權(quán)思想計算出風險中性世界違約距離，再利用擁有海量歷史違約數(shù)據(jù)庫，建立起風險中性違約距離與現(xiàn)實世界違約率之間對應(yīng)關(guān)系，從而得到預(yù)期違約頻率，作為違約概率預(yù)測指標。31/52貝爾斯登預(yù)期違約頻率32/52從期權(quán)價格中引出風險中性違約概率因為股票是企業(yè)資產(chǎn)期權(quán)，這么股票期權(quán)就是期權(quán)期權(quán)，其價格能夠表示為：利用最大熵方法（Capuano,)就能夠從企業(yè)同期限全部期權(quán)價格中預(yù)計出和D33/52從期權(quán)價格中能夠推導出風險中性違約概率利用上述方法，我們就可依據(jù)3月14日貝爾斯登將于3月22日到期期權(quán)價格，計算出貝爾斯登風險中性違約概率和企業(yè)價值概率分布。貝爾斯登于3月14日被摩根大通接管。下列圖顯示，市場對貝爾斯登一周后命運產(chǎn)生巨大分歧，企業(yè)價值大漲大跌概率遠遠大于小幅變動概率，這么分布與正常情況分布有天壤之別?？梢娖跈?quán)價格能夠讓我們清楚地看出市場在非常時期對未來特殊看法。34/52貝爾斯登風險中性違約概率和企業(yè)價值概率分布（3月14日）35/52風險中性違約概率風險中性違約概率即使不一樣于現(xiàn)實概率，但其改變能夠反應(yīng)現(xiàn)實世界違約概率改變。在金融危機時期，它可能比CDS價差能更敏感地反應(yīng)出違約概率改變。在貝爾斯登于3月14日被接管前后，依據(jù)上述方法計算出來風險中性概率天天改變比CDS價差更敏感。這是因為在金融危機期間，金融機構(gòu)本身信用度大幅降低，造成在OTC市場交易CDS交易量急劇萎縮，價差大幅擴大，信號失真。36/52期權(quán)隱含中性違約概率與CDS價差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