Calibration of PD Term Structures To Be Markov Or Not To Be

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CalibrationofPDTermStructures:ToBeMarkovOrNotToBe

ChristianBluhm(CreditSuisse)andLudgerOverbeck(UniversityofGiessen)February22,2007

Abstract.Termstructuresofdefaultprobabilitiesareomnipresentincreditriskmodeling:time-dynamiccreditportfoliomodels,defaulttimes,andmulti-yearpricingmodels,theyallneedthetimeevolutionofdefaultproba-bilitiesasabasicmodelinput.AlthoughpeopletendtobelievethatfromaneconomicpointofviewtheMarkovpropertyasunderlyingmodelassumptioniskindofquestionableitseemstobecommonmarketpracticetomodelPDtermstructuresviaMarkovchaintechniques.InthispaperweillustratethattheMarkovassumptioncarriesusquitefarifweallowfornonhomogeneoustimebehaviouroftheMarkovchaingeneratingthePDtermstructures.Asa‘proofofconcept’wecalibrateanonhomogeneouscontinuous-timeMarkovchain(NHCTMC)toobservedone-yearratingmigrationsandmulti-yeardefaultfrequencies,herebyachievingconvincingapproximationquality.

1MarkovChainsinCreditRiskModelingTheprobabilityofdefault(PD)foraclientisafundamentalriskparameterincreditriskman-agement.Itiscommonpracticetoassigntoeveryratinggradeinabank’smasterscaleaone-yearPDinlinewithregulatoryrequirements;see[1].Table1showsanexamplefordefaultfrequenciesassignedtoratinggradesfromStandardandPoor’s(S&P).

DAAA0.00%AA0.01%A0.04%BBB0.29%BB1.28%B6.24%CCC32.35%

Table1:One-yeardefaultfrequencies(D)assignedtoS&Pratings;see[17],Table9.

Moreover,creditriskmodelingconceptslikedependentdefaulttimes,multi-yearcreditpricing,andmulti-horizoneconomiccapitalrequiremorethanjustone-yearPDs.Formulti-yearcreditriskmodeling,banksneedawholetermstructure(p(t)R)t≥0of(cumulative)PDsforeveryratinggradeR;see,e.g.,[2]foranintroductiontoPDtermstructuresand[3]fortheirapplicationtostructuredcreditproducts.

Everybankhasitsown(proprietary)waytocalibratePDtermstructures1tobank-internalandexternaldata.AlookintotheliteraturerevealsthatforthegenerationofPDtermstructuresvariousMarkovchainapproaches,mostoftenbasedonhomogeneous2chains,dominatecurrentmarketpractice.AlandmarkingpaperinthisdirectionistheworkbyJarrow,Lando,andTurnbull[7].Furtherresearchhasbeendonebyvariousauthors,see,e.g.,Kadam[8],Lando[10],Sarfarazetal.[12],SchuermannandJafry[14,15],TrueckandOezturkmen[18],justtomentionafewexamples.AnewapproachviaMarkovmixtureshasbeenpresentedrecentlybyFrydmanandSchuermann[5].

InMarkovchaintheory(see[11])onedistinguishesbetweendiscrete-timeandcontinuous-timechains.Forinstance,adiscrete-timechaincanbespecifiedbyaone-yearmigrationortransition

1Intheliterature,PDtermstructuresaresometimescalledcreditcurves.

2AMarkovchainiscalledhomogeneousiftransitionprobabilitiesdonotdependontime.

1matrixMgeneratingmulti-yeartransitionsviapowers(Mk)k≥1ofM.Thecorresponding(yearly)discrete-timePDtermstructuresaregivenby

p(k)R=(Mk)row(R),8(k=1,2,3,...)whererow(R)denotestherowinthemigrationmatrixMcorrespondingtoratingR.Continuous-timechainsarespecifiedbyaQ-matrix3Qsuchthatexp(tQ)definesthemigrationmatrixforthetimeinterval[0,t],whereexp(·)denotesthematrixexponential.Continuous-timePDtermstructurescorrespondingtoageneratorQaregivenby

p(t)R=(exp(tQ))row(R),8(t≥0).(1)Continuous-timeMarkovchainsaresuperiortodiscrete-timechainsbecausetheyallowforacon-sistentwaytomeasuremigrationsandPDsfortimehorizonsbetweenyearlytimegridpoints.Ifforadiscrete-timechaindefinedbyaone-yearmigrationmatrixMwefindageneratorQwith

M=exp(Q),(2)onesaysthatthediscrete-timechaincanbeembeddedintoacontinuous-timechain.Ingeneral,wecanonlyexpecttofindapproximativeembeddings;seeIsrael,Rosenthal,andWei[6],Jarrow,Lando,andTurnbull[7],KreininandSidelnikova[9],and[2],Chapter6.In[3],Section2.3.1,wediscussanexampleofageneratorQalmostperfectlyfittedtoagivenone-yearmigrationmatrixfromS&P;seeAppendixII.

Theproblemisthatwefindthatawell-fittedgeneratorneverthelesscangeneratemodel-impliedPDtermstructuressignificantlydeviatingfromobservedmulti-yeardefaultfrequencies.Inthispaper,weaddressthisproblem,notbyrejectingtheMarkovassumptionbutbydroppingthehomogeneityassumptionandworkingwithnonhomogeneouscontinuous-timeMarkovchains(NHCTMC).OurresultsinFigure2showthatinthecontextofPDtermstructurecalibrationtheMarkovassumptionindeedisnotasquestionableaspeoplesometimesclaim.Infact,droppingthehomogeneityassumptionprovidessufficientflexibilitytocalibrateaMarkovprocesstoempiricalmigrationanddefaultfrequencieswithconvincingquality.Therefore,weanswerthequestionraisedinthetitleofthispaperby‘tobeMarkov’,but‘nothomogeneous’.

2CalibrationofaNHCTMCforPDtermstructuresInthesequel,weconstructaNHCTMC,whichweuseforthegenerationofPDtermstructures.InAppendixIweprovidesomecommentsonthestochasticrationaleoftheapproach.