Double Exponential Jump Diffusion Process

DoubleExponentialJumpDiffusionProcess:AStructuralModelofEndogenousDefaultBarrierwithRoll-overDebtStructureBinhDAO∗MoniqueJEANBLANC†March9,2006AbstractInthispaper,weextendtheframeworkofLeland(1994b)whoproposedastructuralmodelofroll-overdebtstructureinaBlack-Scholesframeworktothecaseofadoubleexponentialjumpdiffusionprocess.Weconsideratrade-offmodelwithfirm’sparametersasfirmrisk,riskfreeinterestrate,payoutrateaswellastaxbenefitofcouponpayments,defaultcosts,violationoftheabsolutepriorityruleandtaxrebate.Weobtaintheequity,debt,firmandcreditspreadsvaluesinclosedformformulae.Weanalyzethesevaluesasfunctionsofcoupon,leverageandmaturity.Keywords:DoubleExponentialJumps,StructuralApproach,CreditSpreads,“Roll-over”DebtStructure.JELClassification:G12,G13,G331.IntroductionTheproblemofthefirm’soptimalcapitalstructureanditsendogenousdefaultbarrierhasbeenconsideredgreatlyinaseriesofpapersbyLeland(1994a)[14],Leland(1994b)[15]andLelandandToft(1996)[16].Leland(1994b)[15]hasstudiedaroll-overdebtstructurewhichallowssimultaneousconsiderationofthecoupon-payingbondwitharbitrarymaturity,while∗Universit´eParisDauphine.CentrederechercheCEREG.1,PlaceduMar´echaldeLattredeTassigny.75016Paris.Email:[e-mail]:[phone].†Universit´ed’´Evry.Laboratoired’AnalyseetProbabilit´e.D´epartementdeMath´ematiques.RueduP`ereJarlan.91025´EvryCedex.Email:[e-mail][phone]maintaininginatime-homogeneousenvironment.LelandandToft(1996)[16]derivedclosed-formformulaeforriskycouponpayingbondswithafixedmaturity.DuetothechoiceofageometricBrownianmotionprocessforthefirmvaluedynamics,thecreditspreadsgeneratedbytheirmodelandespeciallythecreditspreadsofshorttermdebtweretoolowanddidnotmatchempiricaldata.Furthermore,thepresenceofjumpsintheassetpriceprocesseswasfurtherendorsedandsupportedbyBates(1996)[1]andJorion(1988)[10]publicationsamongothers.Inconcordancewiththeseempiricalworks,ajumpdiffusionprocessisparticularlyhelpfulincorrectingthetwomainempiricalfailuresofthenormaldistributionprocesssuchasfirstly,thelargerandomfluctuations(discontinuouspaths)ascrashesandsec-ondly,thenon-normalfeatures,asthenegativeskewnessandtheleptokurticfeatureofthereturndistribution(seePapapantoleonandSenge(2002)[17]andRamezaniandZeng(2004)[18],forexample).Asweshallsee,thejumpdiffusionprocessalsohelpstoincreasethecreditspreads.HilberinkandRogers(2002)[8]introducedL´evyprocesses1inthemodelofLelandreferringtoanendogenousdefaultbarrierframework.LeCourtoisandQuittard-Pinon(2004)[13]haveextendedtheLeland(1994b)frameworkbyusingaL´evystableprocesshavingtwo-sidedjumpswithoutdiffusionpart,astheassetpriceprocess.Inthispaper,weexaminethedebtvaluesubjectedtodefaultriskinthestructuralapproach.Weproposeacontinuoustimeframeworkwithadoubleexponentialjumpdiffusionprocesswhichoffersbothnegativeandpositivejumps.AsinLeland(1994b)[15],weconsideraroll-overdebtstructurewithregularrepaymentsandrenewalsofprincipalandofcoupon.Thankstothisspecialdebtstructure,weareabletoexaminecouponpayingbondswitharbitrarymaturity,whileremaininginatimehomogeneousenvironment.Thisflexibleroll-overdebtstructureproposesawideclassofpossibledebtstructuresandoffersananalysisofdebtvalueandofyieldspreadswitharbitrarymaturity.Thedoubleexponentialjumpdiffusion,firstproposedbyKou(2002)[11]2fortheassetprocessisaspecialcaseoftheL´evyprocesseswhichhasthetwointerestingpropertiesoftheexponentialdistribution.Thefirstpropertyisthatthedoubleexponentialdistribution,whichperformstwo-sidedjumps,hastheleptokurticfeatureofthejumpsizethatprovidesthepeakandtailsofthereturndistributionfoundinreality.Thesecondpropertyisthatthedoubleexponentialdistributionhasamemorylessfeaturewhichmakesiteasiertocalculateexpectedmeansandvarianceterms.Thismemorylesspropertyhelpsalsotosolvetheproblemofovershoots:thisproblemoccurswhenajumpdiffusionprocessVcrossesabarrierlevelLattimeτassometimesithitsthebarrierlevelexactly,i.e.Vτ=L,andsometimesitincursan“overshoot”overthebarrier,i.e.Vτ6=L.Asaconsequence,wecancomputetheLaplacetransformofthefirstpassagetimes.ChenandKou(2005)[3]alsoindependentlyproposedadoubleexponentialjumpdiffusionmodelveryclosetoourmodel,emphasizingonimpliedvolatilityanddifferentformsofcreditspreads.Wefurtherconsiderinourmodelthetaxbenefitofcouponpaymentsandthereorganizationcostsatdefaultwhilestudyingtwoadditionalfactorsusuallyobservedinfinancialmarkets,inordertoobtainamorerealisticmodel.ThefirstfactoristheviolationoftheAbsolutePriorityRule(APR),whichmeansatdefaulttime,theshareholderstakeapartofthefirm’svalueanddonotrespectthebondholdersfirstpriority.EmpiricalstudiesconductedbyFranksandTorous1SeethebookofContandTankov(2003)[4]forastudyofL´evyProcesses2ThedoubleexponentialdistributionwasfirstintroducedbyLaplace,andthesymmetricdoubleexponentialdistributionwascalledLaplacedistribution.2(1989)[7]andEberhart,Moore,andRoenfeldt(1990)[6]showedthatAPRisviolatedinabout50%ofthecases.Wesupposethattheshareholderswilltakeγofthefirm’sexpectedvalue,afterthereorganizationcostsatdefault,andthebondholderswillreceiveonly(1−γ)oftheresidualexpectedvalueatdefault.Thesecondfactoris