Credit Risk Modelling and Credit Derivatives

Transcription

1 Credi Risk Modelling and Credi Derivaives Inaugural-Disseraion zur Erlangung des Grades eines Dokors der Wirschafs- und Gesellschafswissenschafen durch die Rechs- und Saaswissenschafliche Fakulä der Rheinischen Friedrich-Wilhelms-Universiä Bonn vorgeleg von Philipp J. Schönbucher aus Düsseldorf BONN 2

2 Dekan: Prof. Dr. Rüdiger Breuer Ersreferen: Prof. Dr. Dieer Sondermann Zweireferen: Prof. Dr. Klaus Schürger Tag der mündlichen Prüfung: 14. Januar 2

3 Acknowledgemens Firs and foremos I would like o hank my my supervisor Prof. Dr. Dieer Sondermann who inroduced me o he field of credi risk modelling, gave me valuable advice and suppor and who creaed he ideal environmen for producive research a his deparmen in Bonn. Amongs he many people who gave me opporuniy o discuss my ideas and provided valuable feedback I would like o hank paricularly Dr. Paul Wilmo, Dr. David Lando, Dr. Pierre Mella-Barral, Dr. Daniel Sommer, Dr. Diemar Leisen, Anje Dudenhausen, David Epsein, Chrisopher Loz, Luz Schlögl, Gabi Zimmermann, Chrisian Zühlsdorff and my fellow sudens in he EDP. I would also like o hank he paricipans a he various conferences and seminars where I presened my work for heir quesions, commens and suggesions. For financial suppor I would like o hank he Deusche Forschungsgemeinschaf who suppored me hrough he SFB 33 a he Universiy of Bonn, and he DAAD who funded my year a he London School of Economics. Las bu no leas I would like o hank my parens for being here for me a all imes during my educaion and beyond.

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5 Conens Noaion ii 1 Inroducion Firm s Value Models Modelling Approach Survey of he Lieraure Advanages and Disadvanages Inensiy Models Credi Raing Transiion Models Credi Derivaives Lieraure Srucure of he Thesis The Term Srucure of Defaulable Bonds Inroducion Seup and Noaion Pricing wih Zero Recovery Dynamics: The defaulable Forward Raes Change of Measure Absence of Arbirage Modelling he Spread beween he Forward Raes Independence of Spreads and defaul-free Raes Negaive Forward Spreads i

6 2.5 Posiive Recovery and Resrucuring The Model Seup Change of Measure Dynamics and Absence of Arbirage Senioriy Insananeous Shor Rae Modelling Jumps in he Defaulable Raes Dynamics Absence of Arbirage Raings Transiions Conclusion The Pricing of Credi Risk Derivaives Inroducion Srucures and Applicaions Terminology Asse Swap Packages Toal Rae of Reurn Swaps Defaul Swap Credi Spread Producs Credi Spread Forward and Credi Spread Swap Credi Spread Opions Opions on Defaulable Bonds Baske Srucures Credi Linked Noes Applicaions of Credi Derivaives Defaulable Bond Pricing wih Cox Processes Model Seup and Noaion The Time of Defaul ii

7 3.3.3 The Fracional Recovery Model The Equivalen Recovery Model Implied Survival Probabiliies Direc Valuaion of Credi Risk Derivaives Forms of Paymen for Defaul Proecion Defaul Digial Payoffs Payoff a Mauriy Payoff a Defaul The Defaul Digial Swap Defaul Swaps Difference o Par Difference o defaul-free Defaulable FRNs and Defaul Swaps Credi Spread Forwards Credi Spread Pu Opions Pu Opions on Defaulable Bonds Models The Mulifacor Gaussian Model Credi Derivaives in he Gaussian Model Implied Survival Probabiliies The Survival Coningen Measure Defaul Digial Payoffs The Credi Spread Pu The Pu on a Defaulable Bond The Mulifacor CIR Model Bond Prices Affine Combinaions of χ 2 Random Variables Facor Disribuions iii

8 3.9 Credi Derivaives in he CIR Model Defaul Digial Payoffs Credi Derivaives wih Opion Feaures Conclusion Appendix 112 A Calculaions o he Gaussian Model 113 A.1 Proof of Lemma A.2 Proof of Lemma A.3 Proof of Proposiion A.4 Proof of Proposiion A.5 Proof of Proposiion B Calculaions o he CIR Model 125 B.1 Proof of Lemma B.2 Proof of Proposiion B.3 Proof of Proposiion iv

9 Noaion Defaul-Free Term Srucure of Ineres Raes Bond Prices B(, T ) defaul-free zero coupon bond price µ(, T ) drif of zero coupon bond price B(, T ) η(, T ) volailiy of zero coupon bond price B(, T ) B defaul-free coupon bond price Defaul-Free Ineres Raes r defaul-free insananeous shor rae (defaul-free shor rae) β defaul-free discoun facor: β() = exp{ r(s)ds} f(, T ) defaul-free coninuously compounded insananeous forward rae (defaul-free forward rae) α(, T ) drif of he defaul-free forward rae f(, T ) σ(, T ) volailiy of he defaul-free forward rae f(, T ) a(, T ) inegral of forward rae volailiy: a(, T ) = T σ(, s)ds F (, T 1, T 2 ) simply compounded defaul-free forward rae over T 1, T 2 ] Defaulable Term Srucure of Ineres Raes Defaulable Bond Prices B(, T ) µ(, T ) η(, T ) B(, T ) defaulable quaniies usually carry an overbar defaulable zero coupon bond price drif of defaulable zero coupon bond price volailiy of defaulable zero coupon bond price defaulable zero bond price adjused for defauls before B(, T ) = B(, T )/Q() v

10 Defaulable Ineres Raes r defaulable insananeous shor rae (defaulable shor rae) β defaulable discoun facor: β() = exp{ r(s)ds} f(, T ) defaulable coninuously compounded insananeous forward rae (defaulable forward rae) α(, T ) drif of he defaulable forward rae σ(, T ) volailiy of he defaulable forward rae a(, T ) inegral of forward rae volailiy: a(, T ) = T σ(, s)ds F (, T 1, T 2 ) defaulable simply compounded forward rae over T 1, T 2 ] Spread h() h(, T ) σ h (, T ) α h (, T ) insananeous shor spread (shor spread) coninuously compounded insananeous forward rae spread (forward spread) volailiy of forward spread drif of forward spread Defaul Models Defaul Time N poin process whose firs jump riggers he defaul λ inensiy of N τ i ime of he i-h jump of N τ ime of defaul (τ = τ 1 ) Recovery Models π cash recovery model: recovery = π in cash c equivalen recovery model: recovery = c defaul -free bonds q fracional recovery model: payoff reduced by q a each defaul Q fracional recovery model: final payoff Q(T ) = τ i T (1 q τ i ) Probabiliies P (, T ) probabiliy of survival from o T P def (, T ) probabiliy of defaul in, T ] P (, T ) pseudo survival probabiliy: P (, T ) = B(, T )/B(, T ) vi

11 The CIR Model x i i-h facor (i = 1,..., n) w facor weighs of he defaul-free shor rae: r() = n w ix i () w facor weighs of he inensiy: λ() = n w ix i () η weighs of affine combinaion of noncenral chi-squared RVs vii

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13 Chaper 1 Inroducion Alhough he valuaion of defaulable securiies wih mehods of coninuous ime finance goes back o he iniial proposal of Black and Scholes (1973), his area of research has been largely ignored for a long ime. Only recenly credi risk modelling and credi risk managemen have received renewed aenion, boh by academics and praciioners alike. The praciioner s ineres has been renewed by several facors. Firs, he European moneary union and he full liberalisaion of he European capial markes removed he segmenaion of he European corporae bond marke ino naional markes, increased liquidiy and compeiion, and forced marke paricipans o compare credi risk Europe-wide. Simulaneously, since European governmen bonds are now denominaed in Euros, credi risk also became he key deerminan of differen prices in he European governmen bond markes. Secondly, apar from hese Euro-driven facors, hisorically low nominal ineres raes all over he world have driven many invesors o seek higher yields by acceping more credi risk. Wih he growing liquidiy of he governmen bond markes hese markes became very efficien and bond raders and invesors looking for mis-priced securiies had o urn owards he less liquid and less ransparen corporae and high yield bond marke. Afer he recen credi and currency crises in Asia and Russia he need arose o hedge and re-evaluae hese invesmens. Thirdly, banking supervisors have moved owards acceping inernal risk measuremen models as a basis for regulaory capial prescripions. These models are already being acceped for marke risks and have lead o a significan relaxaion of capial adequacy regulaion in hese areas. I is expeced ha in he nex years inernal risk measuremen models will also be acceped for he deerminaion of adequae capial reserves for credi 1

14 1. Inroducion 2 risks. This will generae significan advanages for hose banks ha have a credi risk model in place which is acceped by he regulaing agencies. Finally, driven by he need o hedge and manage credi risks in a flexible way, new derivaive securiies have been developed o fulfill his need. The pricing and managemen of hese credi derivaives requires more flexible and sophisicaed credi risk models. From an academic poin of view, wih he adven of he marke-based models he mahemaical modelling of he pure ineres-rae risk in he bond marke is coming closer o a generally acceped benchmark (see e.g. Sandmann and Sondermann (1997), Milersen, Sandmann and Sondermann (1997), Brace, Gaarek and Musiela (1997) or Jamshidian (1997)). Of he remaining risk componens in he bond marke, credi risk is he larges unresolved modelling problem. Furhermore, he analysis of credi risk opens he door o new fields of research which inerface wih oher areas like coninuous-ime corporae finance (see e.g. Leland (1994), Leland and Tof (1996) and Mella-Barral and Perraudin (1997)) or asymmeric informaion (see e.g. Duffie and Lando (1997)). This is par of a general move of mahemaical finance away from radiional pricing and hedging ino oher areas, where hey have no been applied before. Finally, he valuaion of credi derivaives changed he focus of many credi risk models. Insead of developing an pricing framework which yields he fair prices for defaulable bonds, now hese bonds are o be aken as inpu o derive prices for more exoic derivaive securiies. Therefore models had o be developed ha had his degree of flexibiliy. There are wo main approaches o credi risk modelling. The firs approach goes back o he iniial proposal of Black and Scholes (1973), where a defaulable securiy is regarded as a coningen claim on he value of he issuing firm s asses and is valued according o opion pricing heory. In hese models he firm s value is assumed o follow a diffusion process and defaul is modeled as he firs ime he firm s value his a pre-specified boundary. Because of he coninuiy of he processes used, he ime of defaul is a predicable sopping ime. The payoff in defaul is usually a consan cash paymen represening he proceeds from liquidaing he firm (possibly afer bankrupcy coss). The models of Meron (1974), Black and Cox (1976) Geske (1977) Longsaff and Schwarz (1993) and Das (1995) are represenaives of his approach. In a second approach, he direc reference o he firm s value is abandoned, and he ime of defaul is modeled direcly as a oally inaccessible sopping ime wih an inensiy. This

15 1.1. Firm s Value Models 3 approach is followed by Arzner and Delbaen (1992; 1994), Jarrow and Turnbull (1995), Lando (1994; 1998), Jarrow, Lando and Turnbull (1997), Madan and Unal (1998), Flesaker e.al. (1994), Duffie and Singleon (1997; 1999), Duffie, Schroder and Skiadas (1994), Duffie and Huang (1996), and Duffie (1994). The main difference beween hese models is he way in which he recovery of a defauled bond is modelled. In he following we will describe hese wo modelling approaches in more deail. 1.1 Firm s Value Models Modelling Approach In firm s value models a fundamenalis s approach o valuing defaulable deb is aken: I is assumed ha here is a fundamenal process V, usually inerpreed as he oal value of he asses of he firm ha has issued he bonds. The value of he firm V is assumed o change sochasically, usually i follows a lognormal diffusion process dv = µv d + σv dw. I is he driving force behind he dynamics of he prices of all securiies issued by he firm, all claims on he firm s value are modelled as derivaive securiies wih he firm s value as underlying. Defaul can be riggered in wo ways: Eiher V is only used o pay off he deb a he mauriy of he conrac. A defaul occurs a mauriy if V is insufficien o pay back he ousanding deb bu during he lifeime of he conrac a defaul can no be riggered. Alernaively (and more realisically) one can assume ha a defaul is already riggered as soon as he value of he collaeral V falls below a barrier S. This feaure is exacly idenical o a sandard knockou barrier in equiy opions Survey of he Lieraure The firm s value approach is hisorically he oldes o he pricing of defaulable securiies in modern coninuous-ime finance. I was firs proposed by Black and Scholes in heir aricle The Pricing of Opions and Corporae Liabiliies (1973) which already explicily refers o corporae bond pricing in is ile. Meron (1974) expands on his idea. In hese

16 1. Inroducion 4 models a defaul can only occur a mauriy of he deb, he payoff of he firm s shares is like an European opion on he firm s value. In Black and Cox (1976) his approach is exended o allow for defauls before mauriy of he deb if he firm s value his a lower boundary. Now he model has more similariy wih a barrier opion model. Black and Cox show how o value a variey of corporae bonds and bond covenans in his framework. Furher papers using his approach in a defaul-free ineres rae seup are Meron (1977) and Geske (1977). Geske models defaulable deb as a compound opion on he firm s value. Recenly Longsaff and Schwarz (1993) have managed o reach semi-closed form soluions (an infinie series) for defaulable bonds in a firm s value model wih sochasic ineres raes ha can be correlaed wih he firm s value process. They use he Vasicek (1977) model for he defaul-free ineres raes and have o assume a consan iniial defaul-free erm srucure. Anoher, much simpler soluion o his problem is given by Briys and de Varenne (1997) by modifying he approach such ha a defaul is riggered when he T -forward price of he firm s value his a lower barrier. By now here are many exensions o he original modelling approach. Schönbucher (1996b) and laer Zhou (1997) exend his approach o allow jumps in he process for V hus inroducing a jump-diffusion process for he firm s value, which solves he problem of he unrealisically low shor-erm credi spreads ha are found in all diffusion-based firm s value models. Schönbucher (1996b) also gives an efficien algorihm for he numerical soluion of he pricing problem. In Epsein e.al. (1998) he firm s value model is modified such ha no he firm s value bu he firm s cash flow is he sochasic variable. The firm s value is hen derived from accumulaed cash flow, which makes he problem similar o an opion pricing problem for Asian opions. The firm s value approach has also been implemened in a commercial model package which is markeed by KMV corporaion. The KMV model is loosely based on he original Meron (1974) and Black-Scholes (1973) approach, bu i draws is main srengh from a judicious (bu no model-consisen) use of a large daabase of hisorical defauls. Because of heir more explainaive approach firm s value models have been popular in more heoreical areas, oo. In an approach iniiaed by Leland (1994), Leland and Tof (1996) and Mella-Barral and Perraudin (1997) he firm s value framework is used o analyse sraegic ineracion beween debors and crediors. Duffie and Lando (1997) show in a seup wih asymmeric informaion ha here is a close link beween firm s value models and he inensiy models.

17 1.1. Firm s Value Models Advanages and Disadvanages The firm s value models for defaulable bonds are well suied if he relaionship beween he prices of differen securiies issued by he firm is of imporance, e.g. for converible bonds or callable bonds ha can be convered ino shares when called by he issuer. Furhermore he model allows o price defaulable bonds direcly from fundamenals, from he firm s value. Thus hese models can give a fair price of a defaulable bond as oupu. The foundaion on fundamenals makes models of his ype well-suied for he analysis of quesions from corporae finance like he relaive powers of shareholders and crediors or quesions of opimal capial srucure design. This srengh, he orienaion owards fundamenals, is also one of he model s weaknesses: Ofen i is hard o define a meaningful process for he firm s value, le alone observe i coninuously. I can be very hard o calibrae such a firm s value process o marke prices, and for some issuers, like souvereign deb, i may no exis a all. Furhermore he model may very quickly become oo complex o analyse in a real-world applicaion. If one were o model he full se of claims on he value of he asses of a medium sized corporaion one may very well have o price weny or more classes of claims: from banks, shareholders and privae crediors down o workers wages, axes and suppliers demands. This obviously becomes quickly unfeasible. On he oher hand i seems ha firm s value models are ailor-made for collaeralised loans wih raded collaeral. A second weakness of he firm s value models is he unrealisic naure of he shor erm credi spreads implied by he model. These spreads are very low and end owards zero as he mauriy of he deb approaches zero. This is immediaely obvious from he fac ha he ime of defaul is a predicable sopping ime. A predicable sopping ime has zero inensiy a all imes excep he ime of defaul, and as will be shown laer on he inensiy deermines he shor-erm credi spread. Finally, for he pricing of credi risk derivaives one would like o have a model where he prices of defaulable bonds can be aken as fundamenals and do no have o be calculaed (which hen necessarily means a calibraion process).

18 1. Inroducion Inensiy Models In he inensiy models he ime of defaul is modeled direcly as he ime of he firs jump of a Poisson process wih random inensiy (a Cox process), or more generally as a oally inaccessible sopping ime wih an inensiy. In his group of models a sriking similariy o defaul-free ineres rae modelling is found. The firs models of his ype were developed by Jarrow and Turnbull (1995), Madan and Unal (1998) and Duffie and Singleon (1997). Jarrow and Turnbull consider he simples case where he defaul is driven by a Poisson process wih consan inensiy wih known payoff a defaul. This is changed in he Madan and Unal model where he inensiy of he defaul is driven by an underlying sochasic process ha is inerpreed as firm s value process, and he payoff in defaul is a random variable drawn a defaul, i is no predicable before defaul. Madan and Unal esimae he parameers of heir process using raes for cerificaes of deposi in he Savings and Loan Indusry. Duffie and Singleon (1997) developed a similar model where he payoff in defaul is also cash, bu denoed as a fracion (1 q) of he value of he defaulable securiy jus before defaul. This model was applied o a variey of problems including swap credi risk, esimaion, and wo-sided credi risk, by a group around Duffie (Duffie and Singleon (1997), Duffie, Schroder and Skiadas (1994), Duffie and Huang (1994) and Duffie (1994)). Lando (1998) developed he Cox-process mehodology wih he ieraed condiional expecaions which will be used in he secion on he pricing of credi derivaives laer on. His model has a defaul payoff in erms a cerain number of defaul-free bonds and he applies his resuls o a Markov chain model of credi raings ransiions. In he Schönbucher (1996a; 1997a) model muliple defauls can occur and insead of liquidaion wih cash payoffs a resrucuring wih random recovery rae akes place. The model is se in a Heah- Jarrow- Moron framework and a rich variey of credi spread dynamics is allowed. For many pricing purposes he model can be reduced o a similar form of Duffie and Singleon, and in Schönbucher (1997b) i is applied o he pricing of several credi risk derivaives. There is a variey of oher models ha fall ino he class of inensiy based models, we only menion Flesaker e.al. (1994), Arzner and Delbaen (1992; 1994) and Jarrow and Turnbull (1997). On he empirical side he papers by Duffee (1995), Duffie and Singleon (1997) and Düllmann e.al. (1999) have o be menioned. In hese papers he auhors esimae he

19 1.3. Credi Raing Transiion Models 7 parameers for he sochasic process of he credi spread for he Duffie-Singleon model. The inensiy models have also been implemened in a commercial sofware package. The model is called Credi Risk+ and i was developed by Credi Suisse Financial Producs as a ool for he porfolio managemen of credi risk. In his model a defaul is riggered by he jump of a Poisson process whose inensiy is randomly drawn for each debor class. 1.3 Credi Raing Transiion Models The firs coninuous-ime model of credi-risk pricing in a raing-ransiion framework is due o Lando (1994) and Jarrow, Lando and Turnbull (1997). This model only incorporaes he Markov-chain dynamics of he raings wihou allowing for sochasic spread dynamics wihin he raing classes. Lando (1998) exends his model o have sochasic credi spreads by incorporaing a sochasic muliplier in fron of he ransiion generaor marix. This inroduces some basic sochasiciy ino he credi spreads bu sill does no allow for fully general spread dynamics in all classes, because he credi spreads of all raing classes are driven by he same facor. A differen approach o incorporaed sochasic credi spreads is aken by Das and Tufano (1994) who exend he Jarrow- Lando- Turnbull model o incorporae sochasic recovery raes. Thus hey have sochasic credi spreads wihin he individual classes alhough he defaul inensiies remain consan. Their model is se up as a discree-ime approximaion o a coninuous-ime model. Alhough his model achieves he aim of generaing sochasic credi spreads i sill has some imporan shorcomings. I does no seem plausible why he likelihood of defaul should remain consan while he expeced recovery changes (ypically he converse is he case) and he se of possible credi spreads is bounded from above by he credi spreads for zero recovery. In his hesis he credi raing ransiion models will be exended o a Heah- Jarrow- Moron model which can incorporae fully sochasic dynamics for he credi spreads of all credi classes. The erm srucures of credi spreads of all raing classes will be modelled and joined wih a defaul model in a consisen and arbirage-free way.

20 1. Inroducion Credi Derivaives Lieraure Despie a number of aricles ha have been wrien on he applicaion and uses of credi derivaives, here is very lile lieraure on he direc pricing of credi derivaives. Among he excepions here are he aricles of Duffie (1999) and Longsaff and Schwarz (1995). Das (1998) gives a simple discreisaion of he HJM- approach o credi spreads, and Pierides (1997) and Das (1995) use a firm s value approach o value credi derivaives. Baske defaul swaps are he opic of Duffie (1998) and Li (1999). Good bu no very rigorous inroducions o he applicaions and uses of credi derivaives are he books by Mahieu and d Herouville (1998), Tavakoli (1998), Das (1998) and Nelken (1999). 1.5 Srucure of he Thesis The res of he hesis is organized as follows: The nex chaper covers he erm-srucure modelling of defaulable bonds. The model proposed here is he adapaion of he Heah- Jarrow- Moron (HJM) model o he case of a erm srucure of defaulable bond prices. The arbirage-free dynamics of he defaulable bond prices are derived and a new defaul model he muliple defaul model is inroduced. The model is exended o a model wih jumps in he defaulable forward raes and hen o a raing ransiion model wih sochasic erm srucures of credi spreads in all raings classes. All hese exensions flow naurally from he original HJM forward-rae approach. In he following chaper, he pricing of credi derivaives is discussed. Afer a shor inroducion o he mos popular ypes of credi derivaives and heir applicaions, closed-form pricing formulae are given for he mos imporan credi derivaives. These pricing formulae are derived in differen recovery model frameworks (noably equivalen recovery and fracional recovery), and for a Gaussian HJM and a Cox-Ingersoll-Ross specificaion of he ineres-rae and credi spread dynamics. The applicaion of hese differen modelling approaches o concree pricing problems allows us o idenify he relaive advanages and disadvanages of hese models. The appendices conain some more echnical calculaions ha are necessary for he derivaion of he pricing formulae for he credi derivaives in he wo model specificaions.

21 1.5. Srucure of he Thesis 9 In he final secions of each chaper he main resuls of he chaper are summarised and areas for fuure research are poined ou. A more deailed descripion of he conens of he chapers can be found in he firs secion of each chaper.

22 Chaper 2 Term Srucure Modelling of Defaulable Bonds In his chaper 1 we presen a model of he developmen of he erm srucure of defaulable ineres raes ha is based on a muliple-defauls model. Insead of modelling a cash payoff in defaul we assume ha defauled deb is resrucured and coninues o be raded. We use he Heah-Jarrow-Moron (HJM) (1992) approach o represen he erm srucure of defaulable bond prices in erms of forward raes. The focus of he chaper lies on he modelling he developmen of his erm srucure of defaulable bond prices and we give condiions under which hese dynamics are arbirage-free. These condiions are a drif resricion ha is closely relaed o he HJM drif resricion for defaul-free bonds, and he resricion ha he defaulable shor rae mus always be no below he defaul-free shor rae. Similar resricions are derived for wo exensions of he model seup, he firs one is in a marked poin process framework and allows for jumps in he defaulable forward raes a imes of defaul, and he second one is a general raings ransiion framework which can incorporae sochasic dynamics for he credi spreads in all raings classes and also sochasic ransiion inensiies. 2.1 Inroducion Mos bankrupcy codes provide several alernaive procedures o deal wih defauled deb and he debors. The mos obvious opion is o liquidae he debor s remaining asses 1 This firs pars of his chaper are based upon Schönbucher (1998). 1

23 2.1. Inroducion 11 and disribue he proceeds amongs he crediors, bu a ofen more popular alernaive is o reorganize he defauled issuer and keep he issuer in operaion. The laer alernaive has he advanage of preserving he value of he debor s business as a going concern and i avoids inefficien liquidaion sales. Frequenly here is no alernaive o reorganisaion, eiher because a liquidaion of an issuer is impossible (e.g. for sovereign debors) or because i is undesirable (if a liquidaion would have a large macroeconomic effec). In heir empirical sudy Franks and Torous (1994) found he following: A defaul of a bond does no mean ha his bond becomes worhless, usually here is a posiive recovery rae beween 4 and 8 percen. This recovery rae varies significanly beween firms. The majoriy of firms in financial disress are reorganized and re-floaed, hey are no liquidaed. On average, mos of he compensaion paymens (abou wo hirds) are in erms of securiies of he reorganized firm, no in cash. If a firm is reorganized, and he debors are paid in erms of newly issued deb, hen a second defaul of his firm on is (newly issued) deb is possible. In principle here could be a sequence of any number of defauls each wih a subsequen resrucuring of he defauled firm s deb. In his chaper we presen a model of defaulable bond prices in which a defauled issuer is no liquidaed bu reorganized a defaul. Muliple defauls can occur and he magniude of he losses in defaul is no predicable. We use he Heah-Jarrow-Moron (HJM) (1992) framework o represen he erm srucure of defaulable bond prices in erms of forward raes and give condiions under which hese dynamics are arbirage-free. These condiions are a drif resricion ha is closely relaed o he HJM drif resricion for defaul-free bonds, and he resricion ha he defaulable shor rae mus always be no below he defaul-free shor rae. The model in his chaper is based on he inensiy-approach, an approach in which he ime of defaul is a oally inaccessible sopping ime which has an inensiy process. Amongs ohers his approach is followed by Arzner and Delbaen (1992; 1994), Jarrow and Turnbull (1995), Lando (1994; 1998), Jarrow, Lando and Turnbull (1997), Madan and Unal (1998), Flesaker e.al. (1994), Duffie and Singleon (1997; 1999), Duffie, Schroder and Skiadas (1994), Duffie and Huang (1996), and Duffie (1994). In all hese models a cash payoff (or a payoff in defaul-free bonds) is specified in defaul. Therefore only one defaul is allowed, and afer defaul he firm ha had issued he deb is liquidaed. This excludes

24 2. The Term Srucure of Defaulable Bonds 12 reorganisaion of defauled deb as well as muliple defauls. Usually (excep Madan and Unal (1998)) he magniude of he payoff in defaul is predicable, oo. In his model here is he possibiliy of muliple defauls and he magniude of he recovery need no be predicable. I should furhermore be poined ou ha many of he resuls (e.g. he HJM drif resricion on he defaulable bond prices or he relaionship beween he shor credi spread and he defaul inensiy) are no resriced o he inensiy-based framework, bu remain valid also for oher defaul models. We sar wih a model in which he value of a defaulable bond drops o zero upon defaul. While his case has already been exensively sudied in he lieraure i is a good inroducion o more general models. Afer deriving he inerconnecions beween he dynamics of he defaulable ineres raes and he defaulable bond prices, we derive he key relaionship ha under he maringale measure he difference beween he defaulable shor rae and he defaul-free shor rae is he inensiy of he defaul process wih an argumen using he savings accouns. This resul drives he condiions for he absence of arbirage ha are derived subsequenly, and he arbirage-free dynamics of he defaulable bond prices. We find a very srong similariy beween he defaulable and he defaul free ineres rae dynamics and drif resricions, as boh have o saisfy he HJM drif resricions. (Similar resuls have been shown by Duffie in (1994) and Lando (1998).) Nex we explore how a model of he spread of he defaulable forward raes over he defaul-free forward raes may be used o add a defaul-risk module o an exising model of defaul-free ineres raes. Surprisingly, for forward raes his spread can be negaive alhough here has o be a posiive spread for he shor raes. An example is given demonsraing ha his is only possible under srong correlaion beween spreads and defaul-free ineres raes. In he following secions we propose a model ha includes posiive recovery raes, reorganisaions of he defauled firms wih he possibiliy of muliple defauls and uncerainy abou he magniude of he defaul. Even hough i may seem ha his will make he model far more complicaed he resricions for absence of arbirage and he price dynamics remain unchanged. This model is closely relaed o he fracional recovery model proposed by Duffie and Singleon (1994; 1997; 1999). Insead of modelling he defaulable forward raes, previous inensiy models concenraed on modelling he defaul inensiy (i.e. he shor credi spread) direcly. These wo are conneced and he link is shown in he nex secion. We show ha he resuls of he classical inensiy models can be recovered from he defaulable forward model direcly. Paricularly

25 2.2. Seup and Noaion 13 he represenaion of he defaulable bond prices as expecaion of a defaulable discoun facor follows much more easily han in mos inensiy models. I is also shown ha he only resricion for absence of arbirage is o ensure a posiive spread beween defaulable and defaul-free shor rae. In he following secion he defaulable forward raes (and hus he bond prices) are allowed o change disconinously a defaul imes if here are muliple defauls. Here we use he mehods of Björk, Kabanov and Runggaldier (BKR) (1996) and give he drif- and noarbirage condiions for his more general version of he model. The chaper is concluded wih anoher exension of he model which incorporaes raings ransiions. In his secion we model he defaulable forward raes for all raing classes simulaneously and analyse he condiions ha ensure absence of arbirage for a given raings ransiion inensiy marix and a given volailiy srucure of he credi spreads wihin each raings class. Again, hese condiions urn ou o be closely relaed o he classical HJM condiions which makes he model accessible o a numerical implemenaion. Insead of inerpreing he model as a raings ransiion model i can also be viewed as a model including a credi crisis wih sochasic ransiion from he normal sae o he crisis sae which is characerised by much higher spreads, volailiy and defaul risk. A regime shif of his kind can no be replicaed in mos oher credi risk models, bu i is a very imporan feaure of real deb markes. 2.2 Seup and Noaion For ease of exposiion we firs inroduce he simples seup which will be generalised in he following secions o include posiive recovery raes, muliple defauls and jumps in he defaulable erm srucure. The model is se in a filered probabiliy space (Ω, F, (F ) ( ), P ) where P is some subjecive probabiliy measure. We assume he filraion (F ) ( ) saisfies he usual condiions 2 and he iniial filraion F is rivial. We also assume a finie ime horizon T wih F = F T, all definiions and saemens are undersood o be only valid unil his ime horizon T. The ime of defaul is defined as follows: 2 See Jacod and Shiryaev (1988).

26 2. The Term Srucure of Defaulable Bonds 14 Definiion 2.1 The ime of defaul is a sopping ime τ. We denoe wih N() := 1 {τ } he defaul indicaor funcion and A() he predicable compensaor of N(), hus M() := N() A() is a (purely disconinuous) maringale. A is nondecreasing (because N is), predicable and of finie variaion. Frequenly we will assume ha N has an inensiy λ(s) which means ha A can be represened as A() = λ(s) ds. (2.1) The filraion (F ) ( ) is generaed 3 by n independen Brownian moions W i, i = 1,..., n and he defaul indicaor N(). For he defaul-free bond markes we use he HJM seup: Definiion A any ime here are defaul-risk free zero coupon bonds of all mauriies T >. The ime- price of he bond wih mauriy T is denoed by B(, T ). 2. The coninuously compounded defaul-free forward rae over he period T 1, T 2 ] conraced a ime is defined (for T 2 > T 1 ) f(, T 1, T 2 ) = 1 T 2 T 1 (ln B(, T 1 ) ln B(, T 2 )). (2.2) 3. If he T -derivaive of B(, T ) exiss, he coninuously compounded insananeous defaul-free forward rae a ime for dae T > is defined as f(, T ) = T 4. The insananeous defaul-free shor rae r(), he defaul-free discoun facor β() and he defaul-free bank accoun b() are defined by ln B(, T ). (2.3) r() := f(, ), β() := exp{ r(s)ds}, b() := 1/β(). (2.4) 3 This assumpion will be relaxed laer on o include a marked poin process in he case of muliple defauls.

27 2.2. Seup and Noaion 15 We use similar noaion o describe he erm srucure of he defaulable bonds: Definiion A any ime here are defaulable zero coupon bonds of all mauriies T (where T > ). The ime- price of he bond wih mauriy T is denoed by B(, T ). The payoff a ime T of his bond is 1 {τ>t } = 1 N(): one uni of accoun if he defaul has no occurred unil T, and nohing oherwise. 2. The coninuously compounded defaulable forward rae over he period T 1, T 2 ] conraced a ime is defined (for T 2 > T 1 ) f(, T 1, T 2 ) = 1 T 2 T 1 ( ln B(, T1 ) ln B(, T 2 ) ). (2.5) 3. If he T -derivaive of B(, T ) exiss, he coninuously compounded insananeous defaulable forward rae a ime for dae T > is defined as f(, T ) = T 4. The insananeous defaulable shor rae r(), he defaulable discoun facor β() and he defaulable bank accoun c() are defined by ln B(, T ). (2.6) r() := f(, ), β() := exp{ r(s)ds}, c() := 1 {<τ} 1 β(). (2.7) All definiions of defaulable ineres raes are only valid for imes < τ before defaul. To shoren noaion he reference o he coninuous compounding frequency of he ineres raes or yields is ofen omied, and we only refer o defaul-free and defaulable forward raes (=coninuously compounded insananeous forward raes) and shor raes (=insananeous shor raes). We will also ofen use defaul-free in place of defaul-free o denoe non-defaulable quaniies. The defaulable forward rae f(, T 1, T 2 ) as i is defined above is no he value of a T 1 - forward conrac on a defaulable bond wih mauriy T 2, bu he promised yield of he following porfolio: shor one defaulable bond B(, T 1 ) long B(, T 1 )/B(, T 2 ) defaulable bonds B(, T 2 ).

28 2. The Term Srucure of Defaulable Bonds 16 A forward conrac on he defaulable bond T 2 would involve a shor posiion in he defaul free bond B(, T 1 ). See also secion for some consequences of his definiion. The defaulable bank accoun c() is he value of $ 1 invesed a = in a defaulable zero coupon bond of very shor mauriy and rolled over unil, given here has been no defaul unil. I will play a similar role o he defaul-free bank accoun b() in defaul-free ineres rae modelling. In he definiion of he defaulable forward raes o avoid aking logarihms of defaulable bond prices ha are zero we assume ha a fuure defaul canno be prediced wih cerainy. A any ime < τ sricly before defaul, and for every finie predicion-horizon T ( < T < ) he probabiliy of a defaul unil T is no one: P τ T F ] < 1. This can be achieved by seing he defaul ime o be he firs ime τ a which a fuure defaul can be prediced wih cerainy: τ := inf{ T < s.. P τ T F ] = 1}, effecively moving he ime of defaul forward in ime 4. We assume τ has been defined as above. This assumpion is in keeping wih he real-world legal provisions ha a bankrupcy mus be filed as soon as he fac of he bankrupcy is known. Furhermore i does no change any of he qualiaive feaures of he model. In addiion o his we assume ha all (forward) ineres raes have coninuous pahs and ha he insananeous forward raes are well-defined. 2.3 Pricing wih Zero Recovery Dynamics: The defaulable Forward Raes Given he above definiions we can sar o explore he connecions beween he dynamics of he defaulable bond prices and he defaulable forward raes. We assume he following represenaion as sochasic inegrals for he dynamics of he defaulable forward raes f(, T ) and he defaulable bonds B(, T ): Assumpion The dynamics of he defaulable forward raes are given by df(, T ) = α(, T ) d + σ i (, T ) dw i (). (2.8) 4 By definiion τ τ, bu τ = is possible.

29 2.3. Pricing wih Zero Recovery The dynamics of he defaulable bond prices are 5 db(, T ) B(, T ) = µ(, T )d + η i (, T ) dw i () dn(). (2.9) 3. The inegrands α(, T ), σ i (, T ), µ(, T ) and η i (, T ) are predicable processes ha are regular enough o allow differeniaion under he inegral sign inerchange of he order of inegraion parial derivaives wih respec o he T -variable bounded prices B(, ) for almos all ω Ω. We sar by analysing he consequences of he specificaion (2.8) of he defaulable forward raes. The dynamics of he defaulable spo rae process are 6 r() = f(, ) = f(, ) + + α(s, )ds σ i (s, )dw i (s). (2.1) From definiion (2.6) of he defaulable forward raes and definiion 2.3 of he defaulable bonds he price of a defaulable zero coupon bond is given by { T B(, T ) = (1 N()) exp } f(, s) ds. (2.11) The facor of (1 N()) follows from he defaul condiion B(, T ) = for τ. Wriing G(, T ) := T f(, s) ds his yields for τ using Iô s lemma db(, T )/B(, T ) = dg(, T ) + 1 d < G, G > dn, (2.12) 2 where we have used ha G is coninuous. For he process G(, T ) we have (see HJM (1992)) G(, T ) G(, T ) = T f(, s) f(, s) ] ds f(, s) ds 5 The noaion dy ()/Y ( ) = dx() is a shorhand for dy ()/Y ( ) = dx() for Y ( ) > and dy () = for Y ( ) =. 6 I is undersood ha dynamics of defaulable ineres raes are always he dynamics before defaul < τ.

30 2. The Term Srucure of Defaulable Bonds 18 where = = = = = T T T u α(u, s) du ds + f(, s) ds α(u, s) ds du + f(, s) ds α(u, s) ds du + f(, s) ds u γ(u, T ) du f(, s) ds s u T T T σ i (u, s) ds dw i (u) s σ i (u, s) dw i (u) ds (γ(u, T ) r(u)) du T a i (, T ) := γ(, T ) := T u α(u, s) ds du a i (u, T ) dw i (u) α(u, s) du ds σ i (u, s) dw i (u) ds σ i (u, s) ds dw i (u) σ i (u, s) ds dw i (u) a i (u, T ) dw i (u) σ i (, v) dv (2.13) α(, v) dv. (2.14) The main ool in he equaions above is Fubini s heorem and Fubini s heorem for sochasic inegrals (see e.g. HJM (1992) and Proer (199)). Wih his resul we reach he dynamics of he defaulable zero coupon bond prices db(, T ) B(, T ) = γ(, T ) + r() ] a 2 i (, T ) d

31 2.3. Pricing wih Zero Recovery 19 + a i (, T ) dw i () dn(). (2.15) The final condiion B(T, T ) = for τ < T is auomaically saisfied by he funcional specificaion of B. The above derivaion of he dynamics of G(, T ) follows he derivaion of he dynamics of he defaul-free bond prices in HJM (1992). Here he only addiion is he jump erm dn() which is inroduced by he defaul process. Summing up: Proposiion Given he dynamics of he defaulable forward raes (2.8) (i) he dynamics of he defaulable bond prices are given by db(, T ) B(, T ) = ] γ(, T ) + r() + 1 a 2 i (, T ) d 2 + a i (, T ) dw i () dn(). (2.16) where a i (, T ) and γ(, T ) are defined by (2.13) and (2.14) resp.. (ii) he dynamics of he defaulable shor rae are given by r() = f(, ) = f(, ) + + α(s, )ds σ i (s, )dw i (s). (2.17) 2. Given he dynamics (2.9) of he defaulable bond prices he dynamics of he defaulable forward raes are (for τ) given by (2.8) wih α(, T ) = η i (, T ) T η i(, T ) µ(, T ) (2.18) T σ i (, T ) = T η i(, T ). (2.19) Proof: 1.) has been derived above, 2.) follows from Iô s lemma on ln B(, T ) and aking he parial derivaive w.r.. T.

32 2. The Term Srucure of Defaulable Bonds 2 These relaionships are well-known in he case of he defaul-risk free erm srucure. Assume he following dynamics of he defaul-free forward raes f(, T ) and he defaulfree bond prices B(, T ): Assumpion The dynamics of he defaul risk free forward raes are given by df(, T ) = α(, T ) d + σ i (, T ) dw i (). (2.2) 2. The dynamics of he defaul risk free bond prices are db(, T ) B(, T ) = µ(, T )d + η i (, T ) dw i (). (2.21) 3. The inegrands α(, T ), σ i (, T ), µ(, T ) and η i (, T ) are predicable processes ha are regular enough o allow differeniaion under he inegral sign inerchange of he order of inegraion parial derivaives wih respec o he T -variable bounded prices B(, ) for almos all ω Ω. The dynamics of he defaul-free erm srucure do no conain any jumps a τ. Volailiies and drifs may change a τ bu he direc impac of he defaul is only on he defaulable bonds. Given hese dynamics he following proposiion is a well-known resul by Heah, Jarrow and Moron (1992). Proposiion Given he dynamics of he risk free forward raes (2.2) (i) he dynamics of he risk free bond prices are given by db(, T ) B(, T ) = ] γ(, T ) + r() + 1 a 2 i (, T ) d 2 + a i (, T ) dw i (). (2.22)

33 2.3. Pricing wih Zero Recovery 21 where a i (, T ) and γ(, T ) are defined by T a i (, T ) := γ(, T ) := T (ii) he dynamics of he risk free shor rae are given by r() = f(, ) = f(, ) + + σ i (, v) dv (2.23) α(, v) dv. (2.24) α(s, )ds σ i (s, )dw i (s). (2.25) 2. Given he dynamics (2.21) of he risk free bond prices he dynamics of he defaulable forward raes are given by (2.2) wih α(, T ) = η i (, T ) T η i(, T ) µ(, T ) (2.26) T σ i (, T ) = T η i(, T ). (2.27) Change of Measure Now ha he connecions beween he dynamics of he defaulable zero coupon bonds and he forward raes are clarified, we can sar analysing he condiions for absence of arbirage opporuniies in his model. We use he following sandard definiion: Definiion 2.4 There are no arbirage opporuniies if and only if here is a probabiliy measure Q equivalen o P under which he discouned securiy price processes become local maringales. This measure Q is called he maringale measure, and for any securiy price process X() he discouned price process is defined as β()x(). The main ool o classify all o P equivalen probabiliy measures is he following version of Girsanov s Theorem (see Jacod and Shiryaev (1988) III.3 and III.5 and BKR (1996)): Theorem 2.3 Assume ha he defaul process has an inensiy. Le θ be a n-dimensional predicable

34 2. The Term Srucure of Defaulable Bonds 22 processes θ 1 (),..., θ n () and φ() a sricly posiive predicable process wih θ(s) 2 ds <, φ(s) λ(s)ds < for finie. Define he process L by L() = 1 and dl() L( ) = θ i ()dw i () + (φ() 1)dM(). Assume ha E L() ] = 1 for finie. Then here is a probabiliy measure Q equivalen o P wih dq = L T dp and dq = L dp (2.28) where Q := Q F and P := P F are he resricions of Q and P on F, such ha dw () θ()d = d W () (2.29) defines W as Q-Brownian moion and λ Q () = φ()λ() (2.3) is he inensiy of he defaul indicaor process under Q. Furhermore every probabiliy measure ha is equivalen o P can be represened in he way given above. In he financial conex here he processes θ i are he marke prices of diffusion risk, and he process φ represens a marke premium on jump risk (in erms of a muliplicaive facor per uni of jump inensiy). To ensure absence of arbirage he financial requiremen of a well-defined se of marke prices of risk wih validiy for all securiies ranslaes ino he mahemaical requiremen of having a well-defined inensiy process for he change of measure. Given he defaulable bond price dynamics (2.9) he change of measure o he maringale measure leaves he volailiies of he defaulable bond prices unaffeced, he same is rue of he inegral wih respec o dn (he compensaor of his inegral has changed, hough), he only effec is a change of drif in he defaulable bond price process.

35 2.3. Pricing wih Zero Recovery 23 From now on we will assume ha he change of measure o he maringale measure has already been performed. The resuls of he preceding secion on he dynamics remain valid if he underlying measure is he maringale measure. Therefore we simplify noaion such ha all specificaions in secion are already wih respec o Q Absence of Arbirage By Iô s lemma we require under he maringale measure for absence of arbirage ha for all > T This means using (2.16) db(, T ) E B(, T ) ] = r() d. (2.31) ] db(, T ) r() d = E B(, T ) ] ] = E γ(, T ) + r() + 1 a 2 i (, T ) d 2 ] +E a i (, T ) dw i () dn() r() = γ(, T ) + r() a 2 i (, T ) λ() (2.32) Now we have o ake a closer look a he compensaor A() of he defaul indicaor process N(). A() is increasing and we assumed ha A() is also coninuous, herefore A() is differeniable almos everywhere on IR + and hus N() has an inensiy da() = λ()d. Then M() = N() + A() = N() + λ(s) ds (2.33) is a maringale by he definiion of he predicable compensaor. Now consider he value process of he defaulable bank accoun c(), i.e. he developmen of $ 1 invesed a ime a he defaulable shor rae and rolled over from hen on. By definiion is value a ime is c() = 1 {τ>} exp{ r(s)ds}. (2.34) 7 If P = Q hen θ and φ 1.

36 2. The Term Srucure of Defaulable Bonds 24 Under he maringale measure he discouned (discouning wih he defaul-free ineres rae) value process of c c() := c() b() = 1 {τ>} exp{ mus be a maringale. This is he Doleans-Dade exponenial of ˆM() := N() + τ r(s) r(s) ds} (2.35) r(s) r(s) ds, (2.36) which in urn mus also be a maringale. (The maringale propery can also be seen from ˆM() = 1 d c(s) and he uniqueness of he Doleans-Dade exponenial up o τ.) We c(s ) use he freedom we had in he specificaion of r() for τ and se r() := r() for τ. Taking he difference of (2.33) and (2.36) M() ˆM() = λ(s) r(s) + r(s) ds (2.37) one sees ha while he l.h.s. is a maringale he r.h.s. is predicable, he only predicable maringales are consan, hus we have for (almos) all s λ(s) = r(s) r(s). (2.38) The hazard rae λ(s) of he defaul is exacly he shor ineres rae spread. Noe ha his relaionship can also be invered o define he defaulable shor rae as r(s) := r(s) + λ(s). Equaion (2.38) is he key relaion ha yields, subsiued in (2.32), as necessary condiion for he absence of arbirage: γ(, T ) = 1 2 a 2 i (, T ). (2.39) Subsiuing he definiion of γ in his condiion yields he resuls of he following heorem. Theorem 2.4 The following are equivalen: 1. The measure under which he dynamics are specified is a maringale measure.