期權(quán)、期貨及其他衍生產(chǎn)品 課件 hull-OFOD11-ppt-25

Options,Futures,andOtherDerivativesEleventhEditionChapter25CreditDerivativesCopyright?2022,2018,2012PearsonEducation,Inc.AllRightsReservedCreditDefaultSwapsBuyeroftheinstrumentacquiresprotectionfromtheselleragainstadefaultbyaparticularcompanyorcountry(thereferenceentity).Example:Buyerpaysapremiumof90b

p

speryearfor$100millionof5-yearprotectionagainstcompanyX.Premiumisknownasthecreditdefaultspread.Itispaidforlifeofcontractoruntildefault.Ifthereisadefault,thebuyerhastherighttosellbondswithafacevalueof$100millionissuedbycompanyXfor$100million(Severalbondsaretypicallydeliverable).C

D

SStructure(Figure25.1)Recoveryrate,R,istheratioofthevalueofthebondissuedbyreferenceentityimmediatelyafterdefaulttothefacevalueofthebond.OtherDetailsPaymentsareusuallymadequarterlyinarrears.Intheeventofdefault,thereisafinalaccrualpaymentbythebuyer.Settlementcanbespecifiedasdeliveryofthebondsor(moreusually)incash.Anauctionprocessusuallydeterminesthepayoff.Supposepaymentsaremadequarterlyintheexamplejustconsidered.Whatarethecashflowsifthereisadefaultafter3yearsand1monthandrecoveryrateis40%?AttractionsoftheC

D

SMarketAllowscreditriskstobetradedinthesamewayasmarketrisks.Canbeusedtotransfercreditriskstoathirdparty.Canbeusedtodiversifycreditrisks.UsingaC

D

StoHedgeaBondPositionPortfolioconsistingofa5-yearparyieldcorporatebondthatprovidesayieldof6%andalongpositionina5-yearC

D

Scosting100basispointsperyearis(approximately)alongpositioninarisklessinstrumentpaying5%peryear.Thisshowsthatbondyieldspreads(measuredrelativetoL

I

B

O

R)shouldbeclosetoC

D

Sspreads.C

D

SValuationHazardrateforreferenceentityis2%.Assumepaymentsaremadeannuallyinarrears,thatdefaultsalwayshappenhalfwaythroughayear,andthattheexpectedrecoveryrateis40%.SupposethatthebreakevenC

D

Srateissperdollarofnotionalprincipal.UnconditionalDefaultandSurvivalProbabilities(Table25.1)Time(years)SurvivalProbabilityDefaultProbability10.98020.019820.96080.019430.94180.019040.92310.018650.90480.0183CalculationofP

VofPayments(Table25.2,Principal=$1)Time(years)SurvivalProb.ExpectedPaymentDiscountFactorPVofExp.Pmt10.98020.9802s0.95120.9324s20.96080.9608s0.90480.8694s30.94180.9418s0.86070.8106s40.92310.9231s0.81870.7558s50.90480.9048s0.77880.7047sTotalBlankBlankBlank4.0728sPresentValueofExpectedPayoff(Table25.3,Principal=$1)Time(years)DefaultProbab.Rec.RateExpectedPayoffDiscountFactorPVofExp.Payoff0.50.01980.40.01190.97530.01161.50.01940.40.01160.92770.01082.50.01900.40.01140.88250.01013.50.01860.40.01120.83950.00944.50.01830.40.01100.79850.0088TotalBlankBlankBlankBlank0.0506P

VofAccrualPaymentMadeinEventofaDefault(Table25.4,Principal=$1)TimeDefaultProb.ExpectedAccr

PmtDisc.FactorPVofPmt0.50.01980.0099s0.97530.0097s1.50.01940.0097s0.92770.0090s2.50.01900.0095s0.88250.0084s3.50.01860.0093s0.83950.0078s4.50.01830.0091s0.79850.0073sTotalBlankBlankBlank0.0422sPuttingItAllTogetherP

VofexpectedpaymentsisThebreakevenC

D

SspreadisgivenbyThevalueofaswapnegotiatedsometimeagoC

D

Sspreadof150b

p

swouldbeperdollaroftheprincipal.ImplyingDefaultProbabilitiesfromC

D

SSpreadsSupposethatthemidmarketspreadfora5-yearnewlyissuedC

D

Sis100b

p

speryear.Wecanreverseengineerourcalculationstoconcludethatthehazardis1.63%peryear.IfprobabilitiesareimpliedfromC

D

SspreadsandthenusedtovalueanotherC

D

S,theresultisnotsensitivetotherecoveryrateprovidingthesamerecoveryrateisusedthroughout.BinaryC

D

S(Table25.5)Thepayoffintheeventofdefaultisafixedcashamount.Inourexample,theP

Voftheexpectedpayoffforabinaryswapis0.0844andthebreakevenbinaryC

D

Sspreadis205b

p

s.Table25.5Calculationofthepresentvalueofexpectedpayofffromabinarycreditdefaultswap.Principal=$1Time(years)ProbabilityofdefaultExpectedpayoff($)DiscountfactorP

Vofexpectedpayoff($)0.50.01980.01980.97530.01931.50.01940.01940.92770.01802.50.01900.01900.88250.01683.50.01860.01860.83950.01574.50.01830.01830.79850.0146TotalBlankBlankBlank0.0844CreditIndicesC

D

XN

AI

Gisaportfolioof125investmentgradecompaniesinNorthAmerica.iTraxxEuropeisaportfolioof125Europeaninvestmentgradenames.TheportfoliosareupdatedonMarch20andSeptember20eachyear.Theindexcanbethoughtofasthecostpernameofbuyingprotectionagainstall125names.TheUseofFixedCouponsIncreasinglyC

D

SsandC

D

Sindicestradelikebonds.Acouponisspecified.Ifspreadisgreaterthancoupon,thebuyerofprotectionpaysOtherwise,thesellerofprotectionpaysDurationistheamountthespreadhastobemultipliedbytogettheP

Vofspreadpayments.C

D

SForwardsandOptionsExample:Forwardcontracttobuy5-yearprotectiononFordfor280b

p

sinoneyear.IfForddefaultsduringtheone-yearlifetheforwardcontractceasestoexist.Example:Europeanoptiontobuy5-yearprotectiononFordfor280b

p

sinoneyear.IfForddefaultsduringtheone-yearlifeoftheoption,theoptionisknockedout.BasketC

D

SSimilartoaregularC

D

Sexceptthatseveralreferenceentitiesarespecified.Inafirsttodefaultswapthereisapayoffwhenthefirstentitydefaults.Second,third,andnthtodefaultdealsaredefinedsimilarly.Whydoespricingdependsondefaultcorrelation?TotalReturnSwap(Figure25.2)AgreementtoexchangetotalreturnonaportfolioofassetsforL

I

B

O

Rplusaspread.Attheendthereisapaymentreflectingthechangeinvalueoftheassets.Usuallyusedasfinancingtoolsbycompaniesthatwantexposuretoassets.Asset-BackedSecuritiesSecuritiescreatedfromaportfolioofloans,bonds,creditcardreceivables,mortgages,autoloans,aircraftleases,musicroyalties,etc.Usuallytheincomefromtheassetsistranched.A“waterfall”defineshowincomeisfirstusedtopaythepromisedreturntotheseniortranche,thentothenextmostseniortranche,andsoon.CollateralizedDebtObligations

(Section25.8)AcashC

D

OisanA

B

Swheretheunderlyingassetsaredebtobligations.AsyntheticC

D

OinvolvesformingasimilarstructurewithshortC

D

Scontracts.InasyntheticC

D

O,mostjuniortranchebearslossesfirst.Afterithasbeenwipedout,thesecondmostjuniortranchebearslosses,andsoon.SyntheticC

D

OExampleEquitytrancheisresponsibleforlossesonunderlyingC

D

S

suntiltheyreach5%oftotalnotionalprincipal(earns1000b

pspread).Mezzaninetrancheisresponsibleforlossesbetween5%and20%(earns200b

pspread).Seniortrancheisresponsibleforlossesover20%(earns10b

pspread).SyntheticC

D

ODetailsTheincomeispaidontheremainingtrancheprincipal.Example:whenlosseshavereached8%ofthetotalprincipalunderlyingtheC

D

Ss,tranche1hasbeenwipedout,tranche2earnsthepromisedspread(200basispoints)on80%ofitsprincipal.SingleTrancheTradingThisinvolvestradingtranchesofportfoliosofC

D

Sswithoutactuallyformingtheportfolios.Cashflowsarecalculatedinthesamewayastheywouldbeiftheportfolioshadbeenformed.QuotesforStandardTranchesofiTraxx(Table25.6)Quotesare30/360inbasispointsperyearexceptforthe0-3%tranchewherethequoteequalsthepercentofthetrancheprincipalthatmustbepaidupfrontinadditionto500b

p

speryear.Date0–3%3–6%6–9%9–12%12–22%IndexJanuary31,200710.34%41.5911.955.602.0023January31,200830.98%316.90212.40140.0073.6077January31,200964.28%1185.63606.69315.6397.13165ValuationofTranchesofSyntheticC

D

O

sandBasketC

D

S

s(Section25.10)Apopularapproachistouseafactor-basedGaussiancopulamodeltodefinecorrelationsbetweentimestodefault.Oftenallpairwisecorrelationsandalltheunconditionaldefaultdistributionsareassumedtobethesame.Marketlikestoimplyapairwisecorrelationsfrommarketquotes.CumulativeDefaultProbabilityConditionalonFactor(Equations25.5and25.7)Fromthebinomialdistribution,theprobabilityofkdefaultsfromnnamesbytimetconditionalonFisValuingC

D

OTranchesConsidertimesCalculatetheexpectedtrancheprincipal,ateachtime.Theexpectedpayoffbetweentimesisthereductioninexpectedprincipal.Theexpectedpaymentattimeisproportionaltotheexpectedprincipalatthattime.Valuation:voftisDiscountFactorforMaturityoft

(Equations25.9to25.11)CalculationoftheE’sDiscretizethedistributionofFsothatthereare,say,30valueswith30weights.Usebinomialdistributiontocalculatetheprobabilityof0,1,2,3…defaultsbyeachtimeontheunderlyingportfolioforeachvalueofF.ForeachvalueofFcalculateexpectedprincipaloftrancheateachtimeWeightvalueoftranchebyprobabilityofFtoobtainunconditionalexpectedprincipalsateachtimeTheF-valuesandTheirWeights(Equation25.12)CalculatedfromGaussianquadrature(orcopiedfromwww-2.rotman.utoronto.ca/~hull/

)ImpliedCorrelationsAcompound(tranche)correlationisthecorrelationthatisimpliedfromthepriceofanindividualtrancheusingtheone-factorGaussiancopulamodel.Abasecorrelationiscorrelationthatpricesthe0toX%trancheconsistentlywiththemarketwhereX%isadetachmentpoint(theendpointofastandardtranche).ProcedureforCalculatingBaseCorrelationCalculatecompoundcorrelationforeachtranche.CalculateP

Vofexpectedlossforeachtranche.SumthesetogetP

Vofexpectedlossforbasecorrelationtranches.Calculatecorrelationparameterinone-factorgaussiancopulamodelthatisconsistentwiththisexpectedloss.ImpliedCorrelationsforiTraxxonJanuary31,2007(Table25.8)Tranche0–3%3–6%6–9%9–12%12–22%CompoundCorrelation17.7%7.8%14.0%18.2%23.3%Tranche0–3%0–6%0–9%0–12%0–22%BaseCorrelation17.7%28.4%36.5%43.2%60.5%