Credit Risk Modeling with Random Fields

Transcription

1 Credi Risk Modeling wih Random Fields Inaugural-Disseraion zur Erlangung des Dokorgrades an den Naurwissenschaflichen Fachbereichen (Mahemaik der Jusus-Liebig-Universiä Gießen vorgeleg von Thorsen Schmid Gießen 23

2 D - 26 Dekan: Prof. Dr. Albrech Beuelspacher Guacher: Prof. Dr. Winfried Sue (Gießen Prof. Dr. Ludger Overbeck (Gießen Daum der Dispuaion:

3 Preface The demand for invesmens wih higher reurns in areas oher han he sock marke has increased enormously due o he sock marke crash in he las wo years. In exchange for an aracive yield he invesors ake a credi risk, and as a resul mehodologies for pricing and hedging credi derivaives as well as for risk managemen of credi risky asses became very imporan. The effors of he Basel Commiee is jus one of many examples which subsaniae his. In he las years he credi markes developed a a remendous speed while a he same ime he number of corporae defauls increased dramaically. I is herefore no surprising ha he demand for credi derivaives is growing rapidly. In view of his, he goal of his work is wofold. In he firs par, a survey of he credi risk lieraure is given, which offers a quick inroducion ino he area and presens he mahemaical mehods in a unifying way. Second, we propose wo new models of credi risk, focusing on differen needs. The firs model generalizes exising models using random fields in Hilber spaces. The second model uses Gaussian random fields leading o explici formulas for a number of derivaives, for which we propose wo calibraion procedures. This work is organized as follows. In Chaper 1, a survey of he credi risk lieraure is given. This includes srucural models, hazard rae models, mehods incorporaing credi raings, models for baskes of credi risky bonds, hybrid models, marke models and commercial models. In he las secion we illusrae several credi derivaives. Generally he mahemaical framework for he models is provided and some models are discussed in greaer deail. Addiionally, an explici formula for he defaul inensiy in he imperfec informaion model of Duffie and Lando (21 is derived. Chapers 2 and 3 focus on credi risk modeling using sochasic differenial equaions (SDEs in infinie dimensions. Alhough known in ineres rae heory, he applicaion of hese mehods is new o credi risk. Chaper 2 conains an inroducion o SDEs in Hilber spaces providing an Iô formula which is adequae for our purposes. In Chaper 3 a Heah- Jarrow-Moron formulaion of credi risk in infinie dimensions is given. The work of Duffie and Singleon (1999 and Bielecki and Rukowski (2 was enhanced wih alernaive recovery models and exended o infinie dimensions. These new models comprise mos of he known credi risk models and sill offer frameworks which are racable. Recen research in Özkan and Schmid (23 exends his furher o Lévy processes in infinie dimensions. I

4 II In Chaper 4, a credi risk model is presened which uses Gaussian random fields and ransfers he framework of Kennedy (1994 o credi risk. In conras o he funcional analyic approach in he previous wo chapers, he mehods used in his secion concenrae on deriving formulas for pricing and hedging. Explici expressions for he prices of several credi defaul opions are obained and an example for hedging credi derivaives is presened. Based on hese pricing formulas, wo calibraion mehodologies are provided. The firs calibraion procedure fis he model o prices of derivaives using a leas squares approach. As he daa for derivaives like credi defaul swapions is sill scarce, he second approach akes his ino accoun and in addiion uses hisorical daa. This new approach allows o calibrae perfecly o marke prices and is applicable using only a small amoun of credi derivaives daa. I am mos graeful o my supervisor, Prof. Dr. Winfried Sue, for his vial suppor. His fascinaing lecures and his way of inspiring mahemaics were a highly valuable encouragemen. Always having ime for fruiful discussions is jus one example of his coninual suppor hroughou he making of his hesis. I also warmly hank my friends and colleagues from he Sochasik-AG. Special hanks go o Sue, Charlie and Oli for spending hours and hours reading crypic noes. I wish, especially, o hank my family for heir educaion which encouraged he search for answers and helping me whenever I needed hem. Finally, I hank my deares Kirsen as she brighens my life wih her love.

5 Conens 1 Credi Risk - A Survey Inroducion Srucural Models Meron ( Longsaff and Schwarz ( Jump Models - Zhou ( Furher Srucural Models Hazard Rae Models Mahemaical Preliminaries Jarrow and Turnbull ( Duffie and Singleon ( Credi Raings Based Mehods Jarrow, Lando and Turnbull ( Lando ( Baske Models Kijima and Muromachi ( Copula Models Hybrid models Madan and Unal ( Duffie and Lando ( Marke Models wih Credi Risk Commercial Models The KMV Model ( CrediMonior Moody s CrediMerics Credi Derivaives Digial Opions Defaul Opions and Credi Defaul Swap Defaul Swapions Credi Spread Opions kh-o-defaul Opions SDEs on Hilber Spaces Preliminaries The Sochasic Inegral Covariances Iô s formula The Fubini Theorem Girsanov s Theorem III

6 CONTENTS IV 3 An Infinie Facor Model for Credi Risk An Infinie Facor HJM Exension Change of Measure Models wih Credi Risk Recovery of Marke Value Recovery of Treasury Models Using Raings Raing Based Recovery of Marke Value Raing Based Recovery of Treasury Pricing Credi Risk Modeling wih Gaussian Random Fields Preliminaries A Model wihou Credi Risk Models wih Credi Risk Zero Recovery Recovery of Treasury Value Fracional Recovery of Treasury Value Explici Pricing Formulas Defaul Digials Defaul Pu Credi Spread Opions Credi Defaul Swap and Swapion Hedging - an Example Calibraion Calibraion Using Gaussian Random Fields Calibraion Using he Karhunen-Loève Expansion A Basic Seup for Hazard Rae Models 136 B Auxiliary Calculaions 141 B.1 Normal Random Variables B.2 Boundary Crossing Probabiliies B.3 Some Inegrals B.4 Tools for Gaussian Models

7 Chaper 1 Credi Risk - A Survey 1.1 Inroducion The firs regulaions of lending and ineres were menioned in Hammurabi s Code of Laws. Hammurabi was a famous Babylonian king, who lived circa 18 BC. The mos remarkable source for his legal code is a sone slab discovered in 191 which is preserved in he Louvre, Paris. Oher cuneiform ables record a number of exbook-like ineres rae problems. For example, he cuneiform able VAT 8528 poses he following problem 1 : If I len one mina of silver a he rae of 12 shekels (1/6 of a mina per year, and I received in repaymen, one alen (6 minas and 4 minas. For how long was he money len? As long as lending is subjec o a person s employmen, here is risk of losing par of he loan, which in modern financial language would be called credi risk. A common definiion of credi risk is he following: Credi risk refers o he possibiliy ha a conracual counerpar may no be able o mee his obligaions so ha he lender faces a financial loss. The financial objec, which is subjec o credi risk, is a so-called bond. In oday s financial markes here is a vas variey of bonds raded, from Treasuries issued by differen counries or saes o bonds issued by corporaes. Generally speaking, a bond is a cerificae confirming ha is owner, he credior, has len a cerain amoun of money o a specified issuer. The len sum is called he principal or face value of he bond and has o be repaid a a fixed dae, called mauriy of he bond. Addiionally he bond offers a fixed rae of ineres and appears as an example of a fixed-income insrumen. Even if he credior has no kind of ownership righs, i is imporan o noe ha in he even of business liquidaion, bond holders have prioriy over shareholders in erms of abiliy o reclaim capial. 1 See Neugebauer (1969. Furher hisorical informaion on ineres raes in hisory may be found in chaper wo of James and Webber (2. 1

8 1.1 Inroducion 2 The risk of he bond holder o lose a cerain porion of his invesmen is he above menioned credi risk. Accordingly, he crediworhiness of he issuer is an imporan kind of informaion. Agencies like Moody s and Sandard & Poor s classify he crediworhiness of he issuers by he so-called raing. As a consequence, marke paricipans demand higher yields for lower raed bonds as a compensaion for he aken risk. The excess reurn of he corporae bond over a Treasury bond, i.e., a bond which is assumed o be free of credi risk, is called he credi spread; see Bielecki and Rukowski (22. A defaul occurs if he issuer is no able o mee his obligaions. The precise definiion of a defaul is complicaed, because i is iself negoiable; see Tavakoli (1998. Cerainly, an amoun of money is los, and he pos-defaul value of he bond, which is called recovery, significanly differs from he pre-defaul value. For his reason, spread-widening risk or changes in credi qualiy are also implied when alking abou credi risk. The occurrence of credi risk raises he demand for possibiliies o manage hem. This is when credi derivaives come ino play. They enable proecion agains differen ypes of credi risk o he effec ha cerain risk profiles are achieved. For example, credi derivaives can be used, if an invesor wans o hedge himself agains a credi risk, bu no agains ineres risk. As boh are enangled in a bond, credi derivaives provide he ailor-made possibiliy o rade his specific risk. I is imporan o disinguish beween reference risk and counerpary risk. The former refers o a conrac of wo defaul free paries, where he conrac relaes o he credi risk of some reference eniy. If, on he oher hand, over-he-couner derivaives are raded, which are in conras o exchange-raded conracs no backed by a clearinghouse or an exchange, hen each pary faces he defaul risk of is counerpary. We inroduce several classes of models of credi risk, which serve differen needs. Some ry o deermine he magniude of credi risk in a cerain produc while ohers are more suiable for he managemen of whole porfolios or for pricing derivaives. Srucural models dae back o he Nobel Prize paper of Meron (1974. They make a specific assumpion abou he capial srucure of a company, which leads o a precise specificaion when obligaions canno be fulfilled. Therefore, he probabiliy of a defaul can be deermined and furher calculaions done. A commercial implemenaion of his model is presened in Secion Conversely, hazard rae models focus on modeling he ime, a which he defaul even occurs, while he capial srucure of he company is no modeled a all. The defaul even is specified in erms of an exogenous jump process, which iself migh depend on ineres raes, credi raings, firms asses or ohers. Ofen also called reduced-form or inensiy based models, hey were firs menioned in Pye (1974. An imporan class of hazard rae models incorporae credi raings, readily available informaion on he crediworhiness of he bonds issuer. So-called hybrid models ry o combine hese ideas and incorporae boh hazard raes and he capial srucure of he company. From his perspecive hese ineresing models are relaively new in he financial lieraure and a lo of research is going on in his field.

9 1.2 Srucural Models 3 In he secion on baske models we presen wo mehods of modeling a porfolio of credi risky securiies. Baske models are mainly used o value credi derivaives wih a firs-odefaul feaure. Marke models represen he ransfer of a very successful class of ineres rae models o credi risk. They mainly cover he fac, ha yields (or bonds, respecively in he marke are available wih respec o a finie number (less han 2 of mauriy imes, and no for any mauriy as assumed by mos oher models. Quie differen are he commercial models which represen readily available sofware packages. These models show he implemenaion of several mehods handling credi risk and applicaions o large porfolios. Finally we presen cerain credi derivaives in a precise specificaion. These include credi defaul swaps and swapions, credi defaul opions, credi spread opions and opions wih a firs-o defaul feaure, and provide he basis for deriving prices in differen models. 1.2 Srucural Models The firs class of models ries o measure he credi risk of a corporae bond by relaing he firm value of he issuing company o is liabiliies. If he firm value a mauriy T is below a cerain level, he company is no able o pay back he full amoun of money, so ha a defaul even occurs Meron (1974 In his landmark paper Meron (1974 applied he framework of Black and Scholes (1973 o he pricing of a corporae bond. A corporae bond promises he repaymen F a mauriy T. Since he issuing company migh no be able o pay he full amoun of money back, he payoff is subjec o defaul risk. Le V denoe he firm s value a ime. If, a ime T, he firm s value V T is below F, he company is no able o make he promised repaymen so ha a defaul even occurs. In Meron s model i is assumed ha here are no bankrupcy coss and ha he bond holder receives he remaining V T, hus facing a financial loss. If we consider he payoff of he corporae bond in his model, we see ha i is equal o F in he case of no defaul (V T F and V T oherwise, i.e., 1 {VT >F }F + 1 {VT F }V T = F (F V T +. If we spli he single liabiliy ino smaller bonds wih face value 1, hen we can replicae he payoff of his bond by a porfolio of a riskless bond B(, T wih face value 1 (long and 1/F pus wih srike F (shor.

10 1.2 Srucural Models 4 Consequenly he price of he corporae bond a ime, which we denoe by B(, T, equals he price of he replicaing porfolio: B(, T = B(, T 1/F P (F, V,, T, σ V = e r(t 1 ( F e r(t Φ( d 2 V Φ( d 1 F = e r(t Φ(d 2 + V F Φ( d 1, (1.1 where Φ( is he cumulaive disribuion funcion of a sandard normal random variable. Furhermore, P (F, V,, T, σ V denoes he price of a European pu on he underlying V wih srike F, evaluaed a ime, when mauriy is T and he volailiy of he underlying is σ V. This price is calculaed using he Black and Scholes opion pricing formula. The consans d 1 and d 2 are d 1 = ln V F e r(t σ2 (T σ T d 2 = d 1 σ T. If he curren firm value V is far above F he pu is worh almos nohing and he price of he corporae bond equals he price of he riskless bond. If, oherwise, V approaches F he pu becomes more valuable and he price of he corporae bond reduces significanly. This is he premium he buyer receives as a compensaion for he credi risk included in he conrac. Price reducion implies a higher yield for he bond. The excess yield over he risk-free rae is direcly conneced o he crediworhiness of he bond and is called he credi spread. In his model he credi spread a ime equals see Figure 1.1. s(, T = 1 T ln B(, T e r(t ( Φ(d 2 + = 1 T ln V F e Φ( d 1, r(t The quesion of hedging he corporae bond is easily solved in his conex, as hedging formulas for he pu are readily available. To replicae he bond he hedger has o rade he risk-free bond and he firm s share simulaneously 2. This reveals he fac ha in Meron s model he corporae bond is a derivaive on he risk-free bond and he firm s share. We face he following problems wihin his model: The credi spreads for shor mauriy are close o zero if he firm value is far above F. This is in conras o observaions in he credi markes, where hese shor mauriy spreads are no negligible because even close o mauriy he bond holder 2 The hedge consiss primarily of hedging 1 F Scholes Dela-Hedging. pu and is a sraighforward consequence of he Black-

11 1.2 Srucural Models 5 5 x Figure 1.1: This plo shows he credi spread versus ime o mauriy in he range from zero o wo years. The upper line is he price of a bond issued by a company whose firm value equals wice he liabiliies while for he second he liabiliies are hree imes as high. Noe ha if mauriy is below.3 years he credi spreads approach zero. is uncerain wheher he full amoun of money will be paid back or no; cf. Wei and Guo (1991 and Jones, Mason and Rosenfeld (1984. The reason for his are he assumpions of he model, in paricular coninuiy and log-normaliy of he firm value process. On he oher hand, he inrinsic modeling of he defaul even may also be quesionable. In realiy here can be many reasons for a defaul which are no covered by his model. The model is no designed for differen bonds wih differen mauriies. Also i can happen ha no all bonds defaul a he same ime (senioriy. In pracice no all liabiliies of a firm have o be paid back a he same ime. One disinguishes beween shor-erm and long-erm liabiliies. To deermine he criical level where he company migh defaul Vasiček (1984 inroduced he defaul poin as a mixure of he level of ousandings. This concep is discussed in Secion The ineres raes are assumed o be consan. This assumpion is relaxed, for example, by Kim, Ramaswamy and Sundaresan (1993, as discussed in Secion As here are only few parameers which deermine he price of he bond, his model canno be calibraed o all raded bonds on he marke, which reveals arbirage possibiliies. Geske and Johnson (1984 exended he Meron model o coupon-bearing bonds while Shimko, Tejima and van Devener (1993 considered sochasic ineres raes using he ineres rae model proposed in Vasiček (1977. The second exension is essenially equivalen o pricing a European pu opion wih Vasiček ineres raes, where closed-form soluions are available. Of course, any oher ineres rae model can be used in his framework, like Cox, Ingersoll and Ross (1985 or Heah, Jarrow and Moron (1992.

12 1.2 Srucural Models Longsaff and Schwarz (1995 As already menioned defauls in he Meron model are resriced o happen only a mauriy, if a all. In pracice defauls may happen a any ime. Also, when a company offers more han one bond wih differen mauriies or senioriies, inconsisencies in he Meron model show up which can be solved by he following approach. Black and Cox (1976 firs used firs passage ime models in he conex of credi risk. This means ha a defaul happens a he firs ime, when he firm value falls below a prespecified level. They used a ime dependen boundary, F ( = ke γ(t, which resuled in a random defaul ime τ. Unforunaely, his framework proves o be unsaisfacory. Longsaff and Schwarz (1995 exended he Meron, respecively Black and Cox, framework wih respec o he following issues: Defaul may happen a he firs ime, denoed by τ, when he firm value V drops below a cerain level F. Ineres raes are sochasic and assumed o follow he Vasiček model. As a consequence, he firm value a defaul equals F. In he Meron model he value of he defauled bond was assumed o be V T /F which equals 1 in his conex. The recovery value of he bond is herefore assumed o be a pre-specified consan (1 w. This is he fracion of he principal he bond holder receives a mauriy. Since furher defauls are excluded in his model, he bond value a defaul equals B(τ, T = (1 wb(τ, T, where B(, T is he value of a risk-free bond mauring a T. This assumpion is ofen referred o as recovery of reasury value. In he following, we presen he model of Longsaff and Schwarz (1995 in greaer deail. The firm value is assumed o follow he sochasic differenial equaion dv ( V ( = µ( d + σ dw V (, and he spo rae is modeled according o he model of Vasiček (1977: dr( = ν(θ r( d + η dw r (. (1.2 Moreover, IE(W V (s W r ( = ρ (s for all and s. The las equaion reveals a possible correlaion beween he wo Brownian Moions W V and W r. The Vasiček model exhibis a mean-reversion behavior a level θ and easily allows for an explici represenaion of r. I is a classical model used in ineres rae heory and ofen aken as a saring poin for more sophisicaed models. A drawback of his model is he fac ha i may exhibi negaive ineres raes wih posiive probabiliy. See, for example, Brigo and Mercurio (21 and he discussions herein.

13 1.2 Srucural Models 7 For he price of he defaulable bond hey obain B LS (, T = B(, T IE QT 1 {τ>t } + (1 w1 {τ T } F = B(, T w Q T (τ > T F + (1 w. (1.3 Noe ha Q T (τ > T F is he condiional probabiliy (under he T -forward measure 3 ha he defaul does no happen before T. To he bes of our knowledge, a closed-form soluion for his probabiliy is no available 4. Neverheless here are cerain quasi-explici resuls provided by Longsaff and Schwarz (1995. See also Lehrbass (1997 for an implemenaion of he model. In he empirical invesigaion of Wei and Guo (1991, he Longsaff and Schwarz model reveals a performance worse han he Meron model. According o hese auhors his is mainly due o he exogenous characer of he recovery rae Jump Models - Zhou (1997 Anoher approach o solve he problem of shor mauriy spreads is o exend he firm value process o allow for jumps. Mason and Bhaacharya (1981 exended he Black and Cox (1976 model o a pure jump process for he firm value. The size of he jumps has a binomial disribuion. In his model here is some considerable probabiliy for he defaul o happen even jus before mauriy. Alernaively, Zhou (1997 exended he Meron model by assuming he firm value o follow a jump-diffusion process. The immediae consequence is ha defauls are no predicable. The model is formulaed direcly under an equivalen maringale measure Q, and he firm value is assumed o follow dv /V = (r λνd + σdw V ( + (Π 1dN. (1.4 (N is a Poisson process wih consan inensiy λ. The jumps are Π := U N, where U 1, U 2,... are i.i.d. and assumed o be independen of (N, (r and (W V (. Denoe ν := IE(U 1 1. Noe ha he inegral of (Π 1 dn is shorhand for Y s := s N s (Π 1 dn = (U i 1, so ha (Y is a marked poin process. I can be proved 5 ha (Y λν is a maringale so ha consequenly he discouned firm value is a maringale under he measure Q. 3 The T -forward measure is he risk neural measure which has he risk-free bond wih mauriy T as numeraire. For deails see Björk ( See discussions in Bielecki and Rukowski (22 and Goldsein ( See, for example, Brémaud (1981. i=1

14 1.2 Srucural Models 8 The ineres rae is assumed o be sochasic and follow he Vasiček model; see (1.2. The recovery rae is deermined by a deerminisic funcion w, so ha he bond holder receives ( 1 w(vτ /F a defaul. The funcion w represens he loss of he bond s value due o he reorganizaion of he firm. For w = 1 we have he zero recovery case. Zhou considers wo models. The firs, more general model, assumes ha defaul happens a he firs ime when he firm value falls below a cerain hreshold. See he previous chaper for more examples of his class of models. Since in his case no closed-form soluions are available, he auhor proposes an implemenaion via Mone-Carlo echniques. In he second, more resricive model, he auhor obains closed form soluions. For his a consan ineres rae and log-normaliy of he U i s is assumed and defaul happens only a mauriy T, when V T < F. Furhermore w is assumed o be linear, i.e., w(x = 1 w x. For w = 1 we obain he recovery srucure of he Meron model. Equaion (1.4 akes he form of a Doleans-Dade exponenial and can be explicily solved under hese assumpions, cf. Proer (1992, p. 77: We hen have he following V = V exp σ V W V ( + (r 1 N 2 σ2v λν Proposiion (Zhou. Denoe σ 2 U := Var(ln U 1 and ν := 1 + ν. Then he price of a defaulable bond in he above model equals B ZH (, T = w F V e λt ν (λ νt j e (r+λt j= j= Proof. The payoff of he bond equals j! ( ln F Φ i=1 U i. V (r σ2 V λνt j(ln ν σ2 U σ 2 V T + jσu 2 (λt j ( Φ ln F V (r 1 2 σ2 V λνt j(ln ν 1 2 σ2 U j! σ 2 V T + jσu 2 B ZH (, T = 1 {τ>t } + 1 {τ T } ( 1 w(vt /F = 1 {τ>t } + 1 {τ T } w V T F = ( V T {τ T } w F 1. To compue he presen value of he bond we consider he expecaion of he discouned payoff B ZH (, T = IE Q e r(t ( {τ T } ( w V T = e r(t 1 + IE Q (1 {VT <F } = e r(t 1 + w F IEQ ( 1 {VT <F }V T F F 1 F ( V T w F 1 F IE (1 Q {VT <F } F..

15 1.2 Srucural Models 9 Noe ha condiionally on {N T = j} we obain a log-normal disribuion for V T : IP(V T < F N T = j = IP (V exp (r 1 2 σ2v λνt + σ V W V (T N T ( = IP ln V + (r 1 2 σ2 V λνt + σ V W V (T + i=1 U i < F NT = j j ln U i < ln F =: IP(ξ j < ln F, where σ V W (T + j i=1 ln U i as a sum of independen normally disribued random variables is again normally disribued. Recall σu 2, he variance of ln U 1. As IE(ln U i = ln(1 + ν 1 2 σ2 U, we ge ( ξ j N ln V + (r 1 2 σ2 V λνt + j(ln ν 1 2 σ2 U, σ2 V T + jσ2 U =: N ( µ(j, σ 2 (j. I is an easy exercise o verify ha for ξ N (µ, σ 2 Conclude ha IE Q 1 {VT <F }V T = = i=1 IE ( ( e ξ 1 {e ξ <F } = e µ+ 1 ln F µ 2 σ2 Φ σ. σ Q(N T = jie Q (1 {VT <F }V T N T = j j= j= λt (λt j e j! = e λt V e (r λνt (λ νt j ( ln F Φ exp( µ(j + 1 ( ln F µ(j 2 σ2 (jφ σ(j σ(j j= j! V (r σ2 V λνt j(ln ν σ2 U σ 2 V T + jσu 2. We herefore obain B ZH (, T = e rt + w F V λt (1+ν e (λ νt j j= j! e (r+λt ( ln F Φ j= (λt j j! V (r σ2 V λνt j(ln ν σ2 U σ 2 V T + jσu 2 ( ln F Φ V (r 1 2 σ2 V λνt j(ln ν 1 2 σ2 U σ 2 V T + jσu 2. Noing ha he proof is complee. e rt = e (r+λt (λt j /(j!,

16 1.2 Srucural Models 1 In he case where no jumps are presen, i.e., λ =, he sum reduces o he summand wih j = so ha he bond price formula of Meron (1.1 is obained as a special case. This model feaures some properies which are also found in empirical invesigaions on credi risk: The erm srucure of he credi spreads can be upward-sloping, fla, humped or downward-sloping. The shor mauriy spreads can be significanly higher han in he Meron model. As he firm value a defaul is random, especially no equal o F as in he Longsaff and Schwarz (1995 model, he recovery is more realisic. The recovery rae is correlaed wih he firm value also jus before defaul Furher Srucural Models Kim, Ramaswamy and Sundaresan (1993 exended he firs passage ime models o also incorporae sochasic ineres raes following he model of Cox, Ingersoll and Ross (1985. In heir model here is an addiional possibiliy for a defaul o happen a mauriy. The payoff hey considered equals min(f, V. Possibly he company is no able o mee is liabiliies a mauriy bu did no face a defaul up o his ime. Nielsen, Saà-Requejo and Sana-Clara (1993 exended hese models o incorporae a sochasic defaul boundary. For he ineres rae hey used he model of Hull and Whie (199 bu were only able o obain explici formulas in he special case of he Vasiček model, cf. formula (1.2. Denoe σ 2 U := Var(ln U 1 and ν := 1 + ν. In he work of Ammann (1999 vulnerable claims are considered. These are possibly sochasic payoffs which face a counerpary risk. Counerpary risk plays a role if he buyer of a claim considers he defaul probabiliy of he seller as significan. He herefore will ask for a risk premium which compensaes for a possible loss in case of a defaul. The defaul is assumed o happen if V T < F, similar o Meron s model. In ha case he buyer of he claim X receives he fracion V T F X. Explici prices are derived for he Heah, Jarrow and Moron (1992 forward rae srucure and Meron-like firm dynamics. This secion on srucural models heavily relies on he assumpion ha he firm s value is observable or even radeable. From a pracical poin of view his seems no jusifiable as he firm s value is no radeable and even difficul o observe. This difficuly is discussed by Buffe (22 and also solved in he KMV-model; see Secion

17 1.3 Hazard Rae Models Hazard Rae Models In comparison o srucural models, inensiy based models or hazard rae models use a oally differen approach for modeling he defaul. In he srucural approach defaul occurs when he firm value falls below a cerain boundary. The hazard rae approach akes he defaul ime as an exogenous random variable and ries o model or fi is probabiliy o defaul. The main ool for his is a Poisson process wih possibly random inensiy λ, and jumps denoing he defaul evens. As in he firs passage ime models recovery is no inrinsic o his model and is ofen assumed o be a somehow deermined consan. The reason for his new approach lies in he very differen causes for defaul. Precise deerminaion as done in srucural models seems o be very difficul. Furhermore, in srucural models he calibraion o marke prices ofen causes difficulies, while inensiy based models allow for a beer fi o available marke daa. In some approaches basic ideas of hese model classes are combined, for example by Madan and Unal (1998 and Ammann (1999 where he defaul inensiy explicily depends on he firm value. These models are called hybrid models and will be discussed in Secion 1.6. As he firm value approaches a cerain boundary, inensiy increases sharply and defaul becomes very likely. So basic feaures of he srucural models are mimicked. A more involved hybrid model is presened by Duffie and Lando (21 where a firm value model wih incomplee accouning daa is considered. Basically we may disinguish hree ypes of hazard rae models. In he firs approach he defaul process is assumed o be independen of mos economic facors, someimes i is even modeled independenly from he underlying. The raing based approach incorporaes he firm s raing as his consiues readily available informaion on he company s crediworhiness. In principle one ries o model he company s way hrough differen raing classes up o a possible fall o he lowes raing class which deermines he defaul. A hird and very recen class is in he line of he famous marke models of Jamshidian (1997 and Brace, Gaarek and Musiela (1995, see Chaper Mahemaical Preliminaries In his secion we consider he modeling of he defaul process in greaer deail. The approach is mainly based on Lando (1994 and also discussed in many aricles and books like Jeanblanc (22 and Bielecki and Rukowski (22. We firs presen a brief inroducion o Cox processes. More deails can be found in Appendix A. As already menioned differen sopping imes denoing he defaul evens need o be modeled. The Poisson process is aken as a saring poin. Consan inensiy seems oo

18 1.3 Hazard Rae Models 12 resricive so one uses Cox processes, which can be considered as Poisson processes wih random inensiies 6. A special case which suis well for our purposes is he following: Consider a sochasic process λ which is adaped o some filraion G. For a Poisson process (N wih inensiy 1 independen of σ(λ s : s T se ( Ñ := N λ u du, T. (Ñ is a Cox process. Observe ha for posiive λ he process λ u du is sricly increasing and so Ñ can be viewed as a Poisson process under a random change of ime. This reveals a very powerful concep for he problems considered in credi risk. If jus one defaul ime τ is considered, his will be equal o he firs jump τ 1 of Ñ. If more defaul evens are considered, for example, ransiion o oher raing classes, furher jumps τ i are aken ino accoun. The bigger λ is, he sooner he nex jump may be expeced o occur. We obain, for any < T, IP(τ > = IE IP (τ > (λ s s = IE exp ( λ u du. Conclude ha condiionally on σ(λ s : s T he jumps are exponenially disribued wih parameer λ u du. I may be recalled ha a fundamenal assumpion o obain his is he independence of λ and N Jarrow and Turnbull ( In he work of Jarrow and Turnbull (1995 a binomial model is considered. In exension of he classical Cox., Ross and Rubinsein (1979 approach he auhors also modeled he non-defaul and he defaul sae. So for every ime period four possible saes may be aained: {up,down} {non-defaul,defaul}. They discovered an analogy o he foreignexchange markes. As he inensiy of he model is assumed o be consan we do no discuss i in greaer deail. In Jarrow and Turnbull (2 a Vasiček model for he spo rae is used and he hazard rae is explicily modeled. Correlaion of he hazard rae and spo raes are allowed. Denoe by Z and W Brownian moions under he risk neural measure Q, wih consan correlaion ρ. Z can be some economic facor, like an index or he logarihm of he firm value. 6 For a full reamen of Cox processes see Brémaud (1981 and Grandell (1997.

19 1.3 Hazard Rae Models 13 Assume he following dynamics dr = κ(θ r d + σdw, λ = a ( + a 1 (r + a 2 (Z. Noe ha λ may ake on negaive values wih posiive probabiliy. Recovery mus be modeled exogenously and he auhors use he already menioned recovery of reasury value 7. This means if defaul happens prior o mauriy of he bond, he bond holder receives a fracion (1 w of he principal a mauriy. For he value of he bond we calculae he expecaion of he discouned payoff under he risk-neural measure Q. For ease of noaion we consider =. By equaion (1.3, ( B(, T = (1 wb(, T + wie Q exp r u du 1 {τ>t }. In he model of Jarrow and Turnbull we obain B(, T = (1 wb(, T + wie Q exp( = (1 wb(, T + wie Q exp = (1 wb(, T + w exp( µ T v T. r u duq(τ T λ s : s T (r u + λ u du In he las equaion µ T and v T denoe expecaion and variance of (r u + λ u du. Under he saed assumpions his inegral is normally disribued and µ and v can be easily calculaed. The flexibiliy of he model leads o a good fi o marke daa, which is no obained by mos srucural models. Also he model incorporaes economic facors (Z Duffie and Singleon (1999 The paper by Duffie and Singleon (1999 combines wo very successful model classes in ineres rae modeling o access Credi Risk: exponenial affine models and he Heah, Jarrow and Moron (1992 mehodology. For he exponenial affine model he auhors model a vecor of hidden facors which underlie he erm srucure of ineres raes. This vecor is assumed o follow a mulidimensional Cox-Ingersoll-Ross model: dy( = K(Θ y(d + Σ diag(y( 1/2 dw(. 7 See he Longsaff and Schwarz model, Secion

20 1.3 Hazard Rae Models 14 Consequenly he componens of y are nonnegaive random numbers. Spo and hazard rae are assumed o be linear in y(: r( = δ + δ y(, λ((1 θ( = γ + γ y(. A main feaure of he exponenial affine models is ha he soluion of he above SDE can be explicily expressed in an exponenial affine form. Hence we obain deerminisic funcions a(, b( such ha ( IE exp iξ y(u du = expa(, ξ + b(, ξ y(. Thus he price of he defaulable bond can be calculaed in closed form as he value of he characerisic funcion a a proper poin. The second approach uses he well known Heah-Jarrow-Moron model of forward raes. Denoe by f(, T he forward raes deermined by he erm srucure of he defaulable bond prior o defaul 8 and by W(, T a d-dimensional sandard Brownian moion. Assume he dynamics of he forward rae o be f(, T = f(, T + µ(u, T du + σ(u, T dw(u. Similar o Heah, Jarrow and Moron (1992 he auhors specify he dynamics under he objecive measure and consider an equivalen measure Q. For arbirage-freeness i is sufficien - see he work of Harrison and Pliska ( ha all discouned price processes are maringales. Naurally his heavily relies on he recovery assumpion. Duffie and Singleon (1999 inroduced he recovery of marke value which means ha immediaely a defaul he bond loses a fracion of is value. This seup is paricularly well suied for working wih SDEs. The loss rae w is assumed o be an adaped process. Hence B(τ, T = (1 w B(τ, T. Under hese assumpions he auhors derived he following drif condiion for µ and σ: ( µ(, T = σ(, T σ(u, T du. On he oher hand, using he above menioned recovery of reasury value (cf and denoing he riskless forward rae by f(, T, he auhors obained ( µ(, T = σ(, T 8 The forward rae is by definiion f(, T = T ln B(, T. v(, T σ(u, T du + θ(λ( p(, T ( f(, T f(, T.

21 1.4 Credi Raings Based Mehods Credi Raings Based Mehods Simple hazard rae models are ofen criicized because hey do no incorporae available economic fundamenal informaion like firm value or credi raings. This secion reveals some models which incorporae hese daa. This is also a basic feaure of commercial models; see Secion 1.8. Credi raings consiue a published ranking of he credior s abiliy o mee his obligaions. Such raings are provided by independen agencies, for example Sandard & Poor s or Moody s and mosly financed by he gauged companies. The firms are raed even if hey are no willing o pay, bu for a fee hey ge deailed insigh in he resuls of he examinaions and migh reain fundamenal insighs in heir inernal divisions o idenify weaknesses. Each raing company uses a differen sysem of leers o classify he crediworhiness of he raed agencies. Sandard & Poor s, for example, describes he highes raed deb (riple-a=aaa wih he words Capaciy o pay ineres and repay principal is exremely srong. An obligaion wih he lowes raing, D, is in sae of defaul or is no believed o make paymens in ime or even during a grace period. The lower he raing, he higher is he risk ha ineres or principal paymens will no be made Jarrow, Lando and Turnbull (1997 The model proposed by Jarrow, Lando and Turnbull (1997 circumvens some disadvanages of he hihero inroduced models. Especially he use of credi raings is an aracive feaure. The movemens beween he single raing classes is modeled by a ime homogenous Markov chain, he enry ino he lowes raing class yielding a defaul. For example, if a bond is raed AAA, i is a member of he highes raing class (= class 1. If here exis K 1 raing classes, denoe by K he class of defaul. Defaul is assumed o be an absorbing sae, resrucuring afer defaul is no considered in his model. The generaor of he Markov chain is defined as λ 1 λ 12 λ 13 λ 1K λ 21 λ 2 λ 23 λ 2K Λ = λ K 1,1 λ K 1,2 λ K 1 λ K 1,K The ransiion raes for he firs raing class are in he firs row. So λ 1 = j 1 λ 1j is he rae for leaving his class, while λ 12 is he rae for downgrading o class 2 and so on. The rae for a defaul direcly from class one is λ 1K. We denoe q ij (, := IP(Raing is in class i a and in class j a, and by Q( he marix of he ransiion probabiliies q ij (,.

22 1.4 Credi Raings Based Mehods 16 The ransiion probabiliies can be compued from he inensiy marix via 9 Q( = exp(λ := id n +Λ + 1 2! (Λ ! (Λ3 +..., where id n is he n n ideniy-marix. Under he recovery of reasury assumpion 1 we obain for he price of a zero coupon bond under defaul risk r s ds 1 {τ>t } B(, T = 1 {τ>} IE e R τ rs ds δb(τ, T 1 {τ T } + e R T = 1 {τ>} IE δ1 {τ T } e R T r s ds + 1 {τ>t } e R T r s ds ( = 1 {τ>} δb(, T + IE (1 δe R T r s ds 1 {τ>t } = 1 {τ>} B(, T δ + (1 δq T (τ > T. (1.5 Q T is he T -forward measure 11. I is herefore crucial o have a model which deermines he ransiion probabiliies under his measure. While raing agencies esimae he ransiion probabiliies using hisorical observaions, i.e., under he objecive measure P, Jarrow, Lando and Turnbull (1997 propose a mehod which uses he defaulable bond prices and calculaes ransiion probabiliies under he he risk-neural measure Q. Consider he bond wih raing i and se Q T,i (τ > T he probabiliy ha he bond will no defaul unil T given i is raed i a. As i makes no sense o alk abou bond prices afer defaul, we furher on jus consider he bond price on {τ > } and ge ( B i (, T = B(, T δ + (1 δq T,i (τ > T τ >. (1.6 Jarrow, Lando and Turnbull (1997 spli he inensiy marices ino an empirical par (under P and a risk adjusmen like a marke price of risk: They assume ha he inensiies under Q T have he form UΛ and U denoes a diagonal marix where he enries are he risk adjusing facors µ i. For he ransiion probabiliies his yields ha q ij (, T is he ij h enry of he marix exp(uλ. Time homogeneiy of µ would enail exac calibraion being impossible. For he discree ime approximaion,, T is divided ino seps of lengh 1. Saring wih (1.6 one obains Q T,i (τ T τ > = 1 B i(, T δb(, T (1 δb(, T = B(, T B i (, T. (1.7 B(, T (1 δ 9 See, for example, Israel, Rosenhal and Wei (21. 1 The bond holder receives δ equivalen and riskless bonds in case of defaul. See Secion The T -forward measure is he risk neural measure which has he risk-free bond wih mauriy T as numeraire. For deails see Björk (1997.

23 1.4 Credi Raings Based Mehods 17 Denoe he empirical probabiliies from he raing agency by p ij (, T. Q T,i (τ 1 = µ i (p ik (, 1, and we obain This leads o µ i ( = QT,i (τ 1 = p ik (, 1 B(, 1 B i (, 1 p ik (, 1 B(, 1(1 δ. By his one obains (µ 1,..., µ K 1 and consequenly q ij (, 1. For he sep from o + 1 use Q T,i (τ + 1 = Q T,i (τ + 1 τ > Q T,i (τ > o ge This leads o K 1 Q T,i (τ + 1 = µ i(p i (τ + 1 τ > q ij (, = µ i (p ik (, + 1 K 1 j=1 j=1 q ij (,. µ i ( = (1.7 = Q T,i (τ + 1 K 1 j=1 q ij(, p ik (, + 1 B(, + 1 B i (, + 1 ( K 1, B(, + 1(1 δ j=1 q ij(, p ik (, + 1 and, via q ij (, + 1 = µ i (p ij (, + 1, he required probabiliies are obained. This model exends Jarrow and Turnbull (1995 using ime dependen inensiies bu sill working wih consan recovery raes. Das and Tufano (1996 propose a model which also allows for correlaion beween ineres raes and defaul inensiies. I seems problemaic ha all bonds wih he same raing auomaically have he same defaul probabiliy. In realiy his is definiely no he case. Naurally differen credi spreads occur for bonds wih he same raing. A furher resricive assumpion is he ime independence of he inensiies. The yield of a bond in his model may only change if he raing changes. Usually he marke price precedes he raings wih informaions on a possible raing change which is an imporan insigh of he KMV model; see Secion Lando (1998 The work of Lando (1998 uses a condiional Markov chain 12 o describe he raing ransiions of he bond under consideraion. All available marke informaion like ineres raes, asse values or oher company specific informaion is modeled as a sochasic process (X. This is analogous o he case wihou raings, where Lando used λ = λ(x. 12 See also Secion 11.3 in Bielecki and Rukowski (22.

24 1.4 Credi Raings Based Mehods 18 Assume ha a risk-neural maringale measure Q is already chosen. Then he arbiragefree price of a coningen claim is he condiional expecaion under his measure Q. The auhor lays ou he framework for raing ransiions where all probabiliies are already under he risk-neural measure and calibraes hem o available marke prices. As no hisorical informaion is used he probabiliy disribuion under he objecive measure is no needed. If one wans o consider risk-measures like Value-a-Risk, noe ha he objecive measure is sill required. We denoe he generaor of he condiional Markov chain (C by λ 1 (s λ 12 (s λ 13 (s λ 1K (s λ 21 (s λ 2 (s λ 23 (s λ 2K (s Λ(s = λ K 1,1 (s λ K 1,2 (s λ K 1 (s λ K 1,K (s, where for all s λ i (s = K j=1, j i λ ij (s, i = 1,..., K 1. We assume (λ ij ( o be adaped and nonnegaive processes. I is imporan for he inensiies o depend on boh ime and ineres raes. Especially for low raed companies he defaul raes vary considerably over ime 13. I was observed by Duffee (1999, e.g., ha defaul raes significanly depend on he erm srucure of ineres raes. I is cerainly bad news for companies wih high deb when ineres raes increase whereas for oher companies i migh be good news. The consrucion of (C can be done as follows. Consider a series of independen exponenial(1-disribued random variables E 11,..., E 1K, E 21,..., E 2K,... which are also independen of σ(λ(s : s and denoe he raing class of he company a he beginning of he observaion by η. Define τ η,i := inf{ : λ η,i(s ds E 1i }, i = 1,..., K and τ := min i η τ η,i, η 1 := arg min i η τ η,i. The τ η,i model he possible ransiions o oher raing classes saring from raing η. The firs ransiion o happen deermines he ransiion ha really akes place, compare Figure 1.2. The reached raing class is denoed by η 1 while τ denoes he ime a which his occurs. Analogously, he nex change in raing saring in η 1 is defined, and similarly for η i and τ i. Then, for τ i 1 < τ i, C is defined by C := η i. Defaul is assumed o be an absorbing sae of he Markov chain and we denoe he overall-ime o defaul by τ. This is he firs ime when η i = K. 13 Cf. Chaper 15 in Caouee, Almann and Narayanan (1998.

25 1.4 Credi Raings Based Mehods 19 Raing η η 1 η 2 τ τ 1 Figure 1.2: A possible realizaion of raing ransiions. The raing sars in η and drops o η 1 a τ. The nex change is a τ 1, o raing class η 2. The ransiion probabiliies P (s, for he ime inerval (s, saisfy Kolmogorov s backward differenial equaion 14 PX(s, s = Λ(s P X (s,. Consider he price of a defaulable zero recovery bond a ime, Bi (, T, which has mauriy T and is raed in class i a ime. Then we obain he following Theorem. Theorem Under he above assumpions he price of he defaulable bond equals ( B i (, T = 1 {C=i}IE exp ( r s ds (1 P X (, T i,k F. Here P X (, T i,k is he (i,k-h elemen of he marix of ransiion probabiliies for he ime inerval (, T, P X (, T. Proof. As already menioned he Markov chain is modeled under Q so ha he arbiragefree price of he bond is he following condiional expecaion: ( B i (, T = IE exp ( r s ds 1 {τ>t,c=i} F. Using condiional expecaions and he independence of E 1K and (Λ(s one concludes ( B i (, T = 1 {C=i}IE exp ( ( = 1 {C=i}E exp ( r s ds IP ( τ > T σ(λs : s T F F r s ds (1 P X (, T i,k F. 14 For non-commuaive Λ he soluion is in general no of he form P X (s, = exp Λ(u du. See Gill s and Johannsen (199 for soluions using produc inegrals.

26 1.4 Credi Raings Based Mehods 2 For he calibraion o observed credi spreads explici formulas are needed and herefore furher assumpions will be necessary. Lando chooses an Eigenvalue-represenaion of he generaor. Denoe wih A(s he marix wih enries λ 1 (s,..., λ K 1 (s, on he diagonal and zero oherwise. Assume ha Λ(s admis he represenaion Λ(s = B A(s B 1, where B is he K K-marix of he Eigenvecors of Λ(s. We conclude P X (s, = B C(s, B 1 wih exp C(s, = λ s 1(udu..... exp s λ K 1(udu 1 I is easy o see ha P X (s, saisfies he Kolmogorov-backward differenial equaion. For uniqueness, see Gill and Johannsen (199. Under hese addiional assumpions he price of he defaulable bond in Theorem simplifies considerably. Proposiion Denoing by β ij := B ij B 1 jk, he price of he defaulable bond equals B i (, T = K 1 j=1 β ij IE exp ( (λ j (u r u du F. Proof. In his seup he condiional probabiliy for a defaul when he bond is in raing class i equals Wih B ik B 1 KK = 1 we obain IP X (, T i,k = 1 {τ>} K 1 IP X (, T i,k = K 1 j=1 j=1 B ij exp( T B ij B 1 jk exp( λ j (udub 1 jk. λ j (udu. and he conclusion follows as in Using he readily available ools for hazard rae models i is now easy o consider opions which explicily depend on he credi raing or credi derivaives wih a credi rigger.

27 1.4 Credi Raings Based Mehods 21 Calibraion Assuming a Vasiček model 15 for he ineres rae we are in he posiion o use he model laid ou above for calibraion o observed credi spreads. There are no economic facors considered oher han he ineres rae and, as a consequence, λ mus be adaped o G = σ(r s : s. Furhermore, we assume wih consans γ j, κ j. λ j (s = γ j + κ j r s, j = 1,..., K 1, The dynamics of he generaor marix is Λ(s = B A(s B 1 and B has o be esimaed from hisorical daa while γ j, κ j are calibraed. The credi spread is he difference of he offered yield o he spo rae. By Theorem he bond price saisfies K 1 B i (, T = j=1 β ij IE exp Therefore, we obain for he bond s yield T T = log B i (, T = T K 1 = j=1 T = K 1 j=1 ( β ij IE exp (γ j (1 κ j r u du F. ( β ij lim IE (γ j + (κ j 1r T exp T K 1 = β ij (γ j + (κ j 1r. j=1 Hence he credi spread equals K 1 s i ( = β ij (γ j + κ j r. j=1 (γ j (1 κ j r u du F ( (γ j + κ j r u r u du F For calibraion a second relaion is needed. Lando uses he sensiiviy of he credi spreads w.r.. he spo rae: K 1 s i ( = β ij κ j. r Denoe by ŝ, dŝ he observed credi spreads and heir esimaed sensiiviies. One finally has o solve he following equaion o calibrae he model: 15 see equaion (1.2. j=1 β(γ + κr = ŝ βκ = dŝ.

28 1.5 Baske Models 22 I urns ou o be problemaic ha observed credi spreads are no always monoone wih respec o he raings. The auhor argues ha in pracice his would occur raher seldom. 1.5 Baske Models Usually here is a whole porfolio under consideraion insead of jus one single asse. Therefore he so far presened models were exended o models which may handle he behavior of a larger number of individual asses wih defaul risk, a so-called porfolio or baske. There are several approaches in he lieraure and hey can be grouped ino models which use a condiional independence concep and ohers which are based on copulas. From he firs class we presen he mehods of Kijima and Muromachi (2, which provide a pricing formula for a credi derivaive on baskes wih a firs- or second-odefaul feaure. An example is he firs-o-defaul pu, which covers he loss of he firs defauled asse in he considered porfolio, see also Secion From he second class we discuss an implemenaion based on he normal copula in Secion Besides ha, Jarrow and Yu (21 model a kind of direc ineracion beween defaul inensiies of differen companies. In heir model he defaul of a primary company has some impac on he hazard rae of a secondary company, whose income significanly depends on he primary company Kijima and Muromachi (2 Consider a porfolio of n defaulable bonds and denoe by τ i he defaul ime of he i-h bond. Le (G represen he general marke informaion (see Appendix A. Furhermore assume ha for any 1,..., n T Q(τ 1 > 1,..., τ n > n G T = Q(τ 1 > 1 G T Q(τ n > n G T, (1.8 where Q is assumed o be he unique risk neural measure. Using he represenaion via Cox processes, his yields (1.8 = exp( n i=1 i λ i (s ds. In he recovery of reasury model, he loss of bond i upon defaul equals he pre-specified consan w i := (1 δ i. So he firs-o-defaul pu is he opion which pays w i if he ih asse is he firs one o defaul before T and zero if here is no defaul. Denoe he even ha he firs defauled bond is number i by D i := {τ i T, τ j > τ i, j i}.