Credit Risk - A Survey

Transcription

1 Credi Risk - A Survey Thorsen Schmid and Winfried Sue Absrac. This paper presens a review of he developmens in he area of credi risk. Saring in 1974, Meron developed a pricing mehod for a bond facing defaul risk, which was mainly seled in he framework of Black and Scholes (1973). Cerain aemps have been made o relax he assumpions, giving rise o a class of models called srucural models. A second class, called hazard rae models, was firs addressed in Pye (1974) and more recenly reached aenion wih he works of, e.g., Lando (1994). There are exensions in differen direcions, e.g., models which incorporae raings, models for a porfolio of bonds or marke models. The so called commercial models are readily implemened models which are widely acceped in pracice. Finally we describe cerain credi derivaives. 1. 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 ) Mahemaics Subjec Classificaion. 2 AMS Classificaion. Primary: 6-2, 91B28, 91B26, Secondary: 6H3, 6J75. Key words and phrases. Credi Risk, Srucural Models, Hazard Rae Models, Commercial Models, Marke Models, Dependen Defauls. 1

2 2 THORSTEN SCHMIDT AND WINFRIED STUTE 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). Consequenly he price of he corporae bond a ime, which we denoe by B(, T ), equals he price of he replicaing porfolio: (1.1) 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), 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. 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 1. 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. 1 The hedge consiss primarily of hedging 1 F Black-Scholes Dela-Hedging. pu and is a sraighforward consequence of he

3 CREDIT RISK - A SURVEY 3 5 x Figure 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. 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 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 7.1. The ineres raes are assumed o be consan. This assumpion is relaxed, for example, by Kim, Ramaswamy, and Sundaresan (1993), as discussed in Secion 1.4. 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.

4 4 THORSTEN SCHMIDT AND WINFRIED STUTE 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) 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 pre-specified 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 w)b(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): (1.2) Moreover, dr() = ν(θ r()) d + η dw r (). 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

5 CREDIT RISK - A SURVEY 5 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. For he price of he defaulable bond hey obain B LS (, T ) = B(, T ) IE QT [ 1 {τ>t } + (1 w)1 {τ T } F (1.3) = B(, T ) [w Q T (τ > T F ) + (1 w). Noe ha Q T (τ > T F ) is he condiional probabiliy (under he T -forward measure 2 ) ha he defaul does no happen before T. To he bes of our knowledge, a closed-form soluion for his probabiliy is no available 3. 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 (1.4) dv /V = (r λν)d + σdw V () + (Π 1)dN. 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 i ) 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 4 ha Y λν is a maringale so ha consequenly he discouned firm value is a maringale under he measure Q. i=1 2 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). 3 See discussions in Bielecki and Rukowski (22) and?). 4 See, for example, Brémaud (1981).

6 6 THORSTEN SCHMIDT AND WINFRIED STUTE 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 (24, p. 77): [ V = V exp σ V W V () + (r 1 N 2 σ2 V λν) U i. Denoe by σ 2 U he variance of ln U 1. We hen have he following Proposiion 1.1 (Zhou). Denoe ν := 1 + ν. Then he price of a defaulable bond in he above model equals B ZH (, T ) = w F V λt ν e (λνt ) j j= J= j! ( ln F Φ i=1 V (r σ2 V λν)t j(ln ν σ2 U ) ) σ 2 V T + jσu 2 +e (r+λ)t (λt ) j ( Φ ln F V (r 1 2 σ2 V λν)t j(ln ν 1 2 σ2 U ) ). j! σ 2 V T + jσu 2 Proof. The payoff of he bond equals 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 ) ( V T {τ T } w F 1)) F = e r(t )[ 1 + IE Q( ( V T 1 {VT <F } w F 1)) F = e r(t )[ 1 + w ) F IEQ( 1 {VT <F }V T F IE Q( 1 {VT <F } F ).

7 CREDIT RISK - A SURVEY 7 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 e (r 1 2 σ2 V λν)t exp[σ V W V (T ) ( = IP ln V + (r 1 2 σ2 V λν)t + σ V W V (T ) + =: IP(ξ j < ln F ), N T i=1 U i < F ) N T = j j ) ln U i < 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) = ln(1 + ν) 1 2 σ2 U, we ge ( ξ j N ln V + (r 1 2 σ2 V λν)t + j(ln ν 1 ) 2 σ2 U ), σv 2 T + jσu 2 =: N ( µ(j), σ 2 (j)). I is an easy exercise o verify ha for ξ N (µ, σ 2 V ) Conclude ha IE Q[ 1 {VT <F }V T We herefore obain B ZH (, T ) = i=1 IE ( e ξ 1 {e ξ <F }) = e µ+ 1 2 σ2 V Φ ( ln F µ σ V σ V ). = = Q(N T = j)ie Q (1 {VT <F }V T N T = j) j= j= λt (λt )j e j! = e λt V e (r λν)t (λνt ) j ( ln F Φ e rt + w F V λt (1+ν) e (λνt ) j j= Noing ha j! e (r+λ)t ( ln F Φ j= he proof is complee. (λt ) j exp( 1 ( ln F µ(j) ) 2 σ2 (j) + µ(j))φ σ(j) σ(j) j= j! V (r σ2 V λν)t j(ln(1 + ν) σ2 U ) σ 2 V T + jσu 2 V (r σ2 V λν)t j(ln(1 + ν) σ2 U ) ) σ 2 V T + jσu 2 j! ( ln F Φ ). V (r 1 2 σ2 V λν)t j(ln(1 + ν) 1 2 σ2 U ) σ 2 V T + jσu 2 e rt = e (r+λ)t (λt ) j /(j!), ).

8 8 THORSTEN SCHMIDT AND WINFRIED STUTE 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). 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 he 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 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

9 CREDIT RISK - A SURVEY 9 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 5. 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. 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 resricive so one uses Cox processes, which can be considered as Poisson processes wih random inensiies 5. A special case which suis well for our purposes is he following: Consider a sochasic process λ which is adaped o some filraion G. Poisson process N wih inensiy 1 independen of σ(λ s : s T ) se ( ) Ñ := N λ u du, T. For a 5 For a full reamen of Cox processes see Brémaud (1981) and Grandell (1997).

10 1 THORSTEN SCHMIDT AND WINFRIED STUTE 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 (1995-2). 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. 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 6. 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 w)b(, T ) + wie Q[ ( T exp r s ds )1 {τ>t }. 6 See he Longsaff and Schwarz model, Secion 1.2.

11 CREDIT RISK - A SURVEY 11 In he model of Jarrow and Turnbull we obain B(, T ) = (1 w)b(, T ) + wie Q[ exp( = (1 w)b(, T ) + wie Q[ exp[ T T = (1 w)b(, T ) + w exp( µ T v T ). r u du)q(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 T (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(). 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) = exp[a(, 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 7 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). 7 The forward rae is by definiion f(, T ) = T ln B(, T ).

12 12 THORSTEN SCHMIDT AND WINFRIED STUTE Similar o Heah, Jarrow, and Moron (1992) he auhors specify he dynamics under he objecive measure and consider an equivalen measure Q. For arbiragefreeness i is sufficien - see he work of Harrison and Pliska (1981) - 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: ( T. µ(, T ) = σ(, T ) σ(u, T ) du) On he oher hand, using he above menioned recovery of reasury value (cf. 1.2) and denoing he riskless forward rae by f(, T ), he auhors obained ( T ) v(, T ) µ(, T ) = σ(, T ) σ(u, T ) du + θ()λ() p(, T ) ( f(, T ) f(, T )). 3. 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 7. 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 greaer 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

13 CREDIT RISK - A SURVEY 13 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 (, ). probabiliies can be compued from he inensiy marix via 8 The ransiion Q() = exp(λ) := id n +Λ + 1 2! (Λ) ! (Λ)3 +..., where id n is he n n ideniy-marix. Under he recovery of reasury assumpion 9 we obain for he price of a zero coupon bond under defaul risk [ B(, T ) = 1 {τ>} IE e R τ rs ds δb(τ, T )1 {τ T } + e R T rs ds 1 {τ>t } [ = 1 {τ>} IE δ1 {τ T } e R T rs ds + 1 {τ>t } e R T rs ds [ ( = 1 {τ>} δb(, T ) + IE (1 δ)e R ) T r s ds 1 {τ>t } [ (3.1) = 1 {τ>} B(, T ) δ + (1 δ)q T (τ > T ). Q T is he T -forward measure 1. 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 8 See, for example, Israel, Rosenhal, and Wei (21). 9 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).

14 14 THORSTEN SCHMIDT AND WINFRIED STUTE {τ > } and ge (3.2) Bi (, T ) = B(, T ) ( ) δ + (1 δ)q T,i (τ > T τ > ). 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 (3.2) one obains (3.3) Q T,i (τ > T ) = B(, T )(1 δ) B i (, T ) + δb(, T ) B(, T )(1 δ) = B(, T ) B i (, T ). B(, T )(1 δ) Denoe he empirical probabiliies from he raing agency by p ij (, T ). This leads o Q T,i (τ 1) = µ i()p ik (, 1), and we obain µ i () = QT,i (τ > 1) B(, 1) = B i (, 1) p ik (, 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 o ge This leads o Q T,i (τ + 1) = QT,i (τ + 1 τ > ) QT,i (τ > ) K 1 Q T,i (τ + 1) = µ i()p i (τ + 1 τ > ) q ij (, ) µ i () = (3.3) = j=1 K 1 = µ i ()p ik (, + 1) q ij (, ). (τ + 1) K 1 j=1 q ij(, ) p ik (, + 1) Q T,i j=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.

15 CREDIT RISK - A SURVEY 15 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 11 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 ). Assume ha a risk-neural maringale measure Q is already chosen. Then he arbirage-free 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) We assume λ ij () o be adaped processes and nonnegaive for i j. Furhermore, for all s K λ i (s) = λ ij (s), i = 1,..., K 1. j i 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 12. 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. 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. 11 See also Secion 11.3 in Bielecki and Rukowski (22). 12 Cf. Chaper 15 in Caouee, Almann, and Narayanan (1998).

16 16 THORSTEN SCHMIDT AND WINFRIED STUTE Define and τ η,i := inf{ : τ := min i η τ η,i, λ η,i(x s ) ds E 1i }, η 1 := arg min i η τ η,i. i = 1,..., K 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. 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. 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. The ransiion probabiliies P (s, ) for he ime inerval (s, ) saisfy Kolmogorov s backward differenial equaion 13 P X (s, ) = Λ(s) P X (s, ). 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 3.1. Under he above assumpions he price of he defaulable bond equals ( B i (, T ) = IE exp ( T 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 arbirage-free price of he bond is he following condiional expecaion: ( B i (, T ) = IE exp ( T r s ds ) 1 {τ>t } F ). Using condiional expecaions and he independence of E 1K and (Λ(s)) one concludes ( B i (, T ) = 1 {C=i}IE exp ( T r s ds ) IP ( τ > T ) σ(λ ) s : s T ) F F ( = IE exp ( T r s ds ) (1 P X (, T ) i,k ) F ). For he calibraion o observed credi spreads explici formulas are needed and herefore furher assumpions will be necessary. Lando chooses an Eigenvaluerepresenaion of he generaor. 13 For non-commuaive Λ he soluion is in general no of he form PX (s, ) = exp R s Λ(u) du. See Gill and Johannsen (199) for soluions using produc inegrals.

17 CREDIT RISK - A SURVEY 17 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 s λ 1(u)du. C(s, ) =.... exp s λ K 1(u)du 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 3.1 simplifies considerably. Proposiion 3.2. Denoing by b ij he enries of B, he price of he defaulable bond equals B i (, T ) = K 1 j=1 b [ ij IE exp b jk ( T ) (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 K T IP X (, T ) i,k = 1 {τ>} b ij exp( λ j (u)du)b 1 jk. Wih b ik b 1 KK = 1 we obain 1 IP X (, T ) i,k = and he conclusion follows as in 3.1. K 1 j=1 j=1 b ij b jk T exp( λ j (u)du) 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 Calibraion. Assuming a Vasiček model 14 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 ). 14 see equaion (1.2).

18 18 THORSTEN SCHMIDT AND WINFRIED STUTE 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 3.1 he bond price saisfies K 1 B i (, T ) = b [ ( T ) ij IE exp γ j (1 κ j )r u du F. b jk j=1 Therefore, we obain for he bond s yield log T B i (, T ) = K 1 [ ( T ) β ij IE exp γ j (1 κ j )r u du F T = T T = j=1 K 1 [ ( T ) = β ij lim IE (γ j + (κ j 1)r T ) exp γ j + κ j r s r s ds F T j=1 K 1 = β ij (γ j + (κ j 1)r ). j=1 Hence he credi spread equals K 1 s i () = β ij (γ j + κ j r ). j=1 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: j=1 β(γ + κr ) = ŝ βκ = dŝ. 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. 4. 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.

19 CREDIT RISK - A SURVEY 19 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 secondo-defaul 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 8.6. From he second class we discuss an implemenaion based on he normal copula in Secion 4.2. 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 and assume ha for any 1,..., n T (4.1) Q(τ 1 > 1,..., τ n > n G T ) = Q(τ 1 > 1 G T ) Q(τ n > n G T ), where Q is assumed o be he unique risk neural measure. Using he represenaion via Cox processes, his yields n i (4.1) = exp( λ i (s) ds). i=1 In he recovery of reasury model, he loss of bond i upon defaul equals he prespecified 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}. Then, using he risk neural valuaion principle, he price of he bond can be compued as he expecaion w.r.. he risk-neural measure Q and equals [ T n S F = IE exp( r u du) w i 1 Ai = n w i IE [ exp( i=1 T We obain his probabiliy using he facorizaion IP(τ i T, τ k > τ i, k i G T {τ i = x}) i=1 r u du)q(a i G T ). = 1 {x T } IP(τ k > x, k i G T {τ i = x}) = 1 {x T } exp( k i x λ k (s) ds).

20 2 THORSTEN SCHMIDT AND WINFRIED STUTE We herefore obain 15 IP(τ i T, τ k > τ i, k i G T ) [ = IE 1 {τi T } exp( k i [ T = IE λ i (u) exp( = T IE [ λ i (u) exp( τi u u λ k (s) ds) G T λ i (s) ds) exp( k i n λ k (s) ds) du. k=1 We conclude for he price of he firs-o-defaul pu: n T [ T S F = δ i IE λ i (u) exp( r s ds i=1 n k=1 u This formula simplifies considerably if w i w, as in ha case [ T S F = wie n i=1 [ ( n = wie exp( u λ i (u) exp( i=1 T = (1 δ)b(, T ) [ 1 IE T ( exp( n k=1 u λ k (s) ds) du λ k (s) ds) du. T λ k (s) ds) du exp( r s ds) r s ds) λ i (u) du) ) T exp( T T n λ i (u) du) ). Using similar mehods, we deermine he swap-price, if w i is paid immediaely a defaul o he swap-holder. Se [ τ n S F = IE exp( r u du) w i 1 Ai. Cerainly, τ r u du is no G T -measurable, so ha a sligh modificaion of he previously used mehod is necessary. We obain for he facorizaion IE [ exp( x i=1 i=1 r u du)1 {x T } 1 {τk >x, k i} G T {τ i = x} = 1 {x T } exp( x r u + k i λ k (u) du) and conclude S F = n T w i IE [ λ i (u) exp( u r s + i=1 k=1 n λ k (s) ds) du. 15 See Bielecki and Rukowski (22, Proposiion ).

21 CREDIT RISK - A SURVEY 21 Similarly, he auhors provide he following price of a (firs and) second-o-defaul swap, which proecs he holder agains he firs wo defauls in he porfolio: S S = n δ i IE [ T exp( λ i (u) du) n B(, T ) δ i i=1 i=1 + T [ T n (δ i + δ j ) IE λ k (u) exp( r s ds i j j=1 n T (n 2) δ i IE [ T u λ i (u) exp( r s ds i=1 u λ j (u) du) n λ j (s) ds) j= Exended Vasiček implemenaion. Kijima and Muromachi (2) discuss a special case of he above implemenaion. The main idea is o perform a calibraion similar o he one of Hull and Whie (199) for credi risk models. Assume for he dynamics of he hazard raes (4.2) dλ i () = ( φ i () a i λ i () ) d + σ i dw i (), i = 1,..., n, where w i are sandard Brownian moions wih correlaion ρ ij, which is someimes saed as dw i dw j = ρ ij d. Furhermore, assume for he shor rae r dr = ( φ () a r ) d + σ dw (). Noe ha equaions of he ype (4.2) admi explici soluions, see Schmid (1997). From his, we ge λ i () = λ i ()e ai + φ i (s)e ai( s) ds + σ i e ai( s) dw i (s). Using he recovery of reasure assumpion he bond price equals B i (, ) = δ i B(, ) + (1 δ i )IE [ exp( (r u + λ i (u)) du). Noe ha (r u + λ i (u)) du is normally disribued and herefore he expecaion equals he Laplace ransform of a normal random variable wih mean IE [ and variance (r u + λ i (u)) du = Var [ (r u + λ i (u)) du [ u = Var σ e a(u s) dz (s) du + ( u r e au + φ (s)e a(u s) ds ) ( u λi ()e aiu + φ i (s)e ai(u s) ds ) du σ i u e ai(u s) dw i (s) du.

22 22 THORSTEN SCHMIDT AND WINFRIED STUTE To compue he variances i is sufficien o calculae he variances of all summands and he covariances. Seing ρ ii = 1, we have [ u1 u2 IE σ i σ j exp( a i (u 1 s 1 ) a j (u 2 s 2 )) dw j (s 2 ) dw i (s 1 ) du 2 du 1 [ = σ i σ j IE [ = σ i σ j IE s1 s2 = σ i σ j ρ ij exp( a i (u 1 s 1 ) a j (u 2 s 2 )) du 2 du 1 dw j (s 2 ) dw i (s 1 ) e ais1+ajs2 1 a i a j (1 e ais1 )(1 e ajs2 ) dw j (s 2 ) dw i (s 1 ) e a is+a j s 1 a i a j (1 e a is )(1 e a js ) ds = σ iσ j ρ [ ij + 1 (e ai 1) + 1 (e aj 1) + a i a j a i a j =: c ij () Therefore, [ u Var σ i e ai(u s) dw i (s) du 1 (1 e (ai+aj) ) a i + a j [ = σ2 i a (e ai 1) + 1 (1 e 2ai ) =: v 2 (). i a i 2a i Recall ha we wan o calibrae he model o he bond prices, which means calculaing φ i (s). φ (s) is compued as in he risk neural case, see Hull and Whie (199). Consider 1 B(, ) IE[ exp( (r u + λ i (u)) du) = 1 [ Bi (, ) 1 δ i B(, ) δ i =: γ i (), which can be obained from available prices, since δ i is assumed o be known. Noe ha γ i () does no involve φ (s) as [ ( u γ i () = exp λi ()e aiu + φ i (s)e ai(u s) ds ) du + 1 2( ci () + v 2 () ). As we wan o solve his expression for φ i, we consider he following derivaives: ln γ i() = λ i ()e ai + φ i (s)e ai( s) ds 1 [ c i () + v 2 () 2 Wih =: g i () g i() = a i λ i ()e ai + φ i () a i e ai we conclude φ i () = g 1 [ i() + a i g i () + a i c i () + v 2 () 2 φ i (s)e ais ds [ c i () + v 2 () [ c i () + v 2 ().

23 CREDIT RISK - A SURVEY 23 Hence and a i c i () + c i () = σ σ i ρ i [ 1 a 1 a e a 1 a e ai + 1 a e (a+ai) a i v 2 () + v 2 () = σ 2 i which finally leads o +σ σ i ρ i [ 1 a i e a + 1 a e a i a + a i a a i e (a +a i ) = σ σ i ρ i [ 1 e a a a 1 e ai + e a i [ 1 a i 2 a i e a i 1 a i e 2a i + 2 a i e a i + 2 a i e 2a i = σ2 i a i [ 1 + e 2ai φ i () = g i() + a i g i () + σ2 i 2a i (1 e 2a i ) σ σ i ρ i [ 1 e a a + e a 1 e ai a i. Using similar mehods Kijima and Muromachi (2) obain an explici formula for he firs-o-defaul swap. In Kijima (2) hese mehods are exended o pricing a credi swap on a baske, which migh incorporae a firs-o-defaul feaure Copula Models. The concep of copulas is well known in saisics and probabiliy heory, and has been applied o finance quie recenly. Modeling dependen defauls using copulas can be found, for example, in Li (2) or Frey and McNeil (21). We give an ouline of Schmid and Ward (22), who apply a special copula, he normal copula, o he pricing of baske derivaives. Fix =. The goal of he model is o presen a calibraion mehod. Consider he defaul imes τ 1,..., τ n and assume for he beginning ha =. The link beween he marginals Q i () := Q(τ i ) and he join disribuion is he so-called copula C( 1,..., n ). Assuming coninuous marginals, U i := Q i (τ i ) is uniformly disribued. The join disribuion of he ransformed random imes is he copula C(u 1,..., u n ) := Q(U 1 u 1,..., U n u n ) and defines he join disribuion of he τ i s via Q(τ 1 1,..., τ n n ) = C ( Q 1 ( 1 ),..., Q n ( n ) ). For more deailed informaion on copulas see Nelsen (1999). The choice of he copula cerainly depends on he applicaion. Schmid and Ward (22) choose he normal copula because in a Meron framework wih correlaed firm value processes such a dependence is obained, and secondary he normal copula is deermined by correlaion coefficiens which can be esimaed from daa.

24 24 THORSTEN SCHMIDT AND WINFRIED STUTE Assume ha (Y 1,..., Y n ) follows an n dimensional normal disribuion wih correlaion marix Σ = (ρ ij ), where ρ ii = 1 for all i. Denoing heir join disribuion funcion by Φ n (y 1,..., y n, Σ) yields he normal copula C(u 1,..., u n ) = Φ n ( Φ 1 (u 1 ),..., Φ 1 (u n ) ). For modeling purposes i is useful o noe ha seing τ i := Q 1 i (Φ(Y i )), resuls in {τ 1,..., τ n } having a normal copula wih correlaion marix Σ. The above mehods enable us o calculae he join disribuion of n defaul imes, and he required correlaions can be esimaed using hisorical daa. Thus, a value a risk can be deermined. For he pricing of a derivaive wih firs-o-defaul feaure, noe ha (4.3) Q(τ 1s T ) = 1 Q(τ 1 > T,..., τ n > T ) which can be calculaed from he copula and he marginals. A more involved, bu also explici formula can be obained for a kh-o-defaul opion. For example, consider a firs-o-defaul swap, which is also discussed in Secion 8.6. This is a derivaive which offers defaul proecion agains he firs defauled asse in a specified porfolio. Under he assumpion, ha all credis have he same recovery rae δ i δ, he swap pays (1 δ) a τ 1s if τ 1s T. In exchange o his, he swap holder pays he premium S a imes T 1,..., T m, bu a mos unil τ 1s. As explained in Secion 8.3, calculaing expecaions of he discouned cash flows yields he firs-o-defaul swap premium. Thus, using Equaion (8.1), we obain S 1s = (1 δ)ie[ exp( 1s τ r u du)1 {τ 1s T } m i=1 IE[ exp( T i. r u du)1 {τ 1s >T i } To calculae he expecaions, he disribuion of τ 1s under any forward measure is needed. Assuming, for simpliciy, independence of he defaul inensiy and he risk-free ineres rae, one obains IE [ exp( Ti r u du)1 {τ 1s >T i } = B(, Ti )Q(τ 1s > T i ). The bond prices are readily available and he probabiliy can be calculaed via (4.3), once he copula is deermined. For he second expecaion, use IE [ τ 1s exp( r u du)1 {τ 1s T } = T B(, s)ie [ exp( s λ 1s u du)λ 1s (s).

25 CREDIT RISK - A SURVEY 25 Noe ha his expecaion can be obained via s Q(τ 1s > s) = s IE[ exp( = IE [ exp( s s λ 1s u du) λ 1s u du)λ 1s (s). Furher on, Schmid and Ward (22) derive ineresing resuls on spread widening, once a defaul occurred. For example, if one of wo srongly relaed companies defauls, i migh be likely ha he remaining one ges ino difficulies, and herefore credi spreads increase. I seems ineresing ha raders have a good inuiion on his amoun of spread widening, which also could be used as an inpu parameer o he model, which deermines he copulas. 5. Hybrid models Hybrid models incorporae boh preceding models, for example he firm value is modeled, and a hazard rae framework is derived wihin his model. The approach of Madan and Unal (1998) mimics he behavior of he Meron model in a hazard rae framework. They assume he following srucure for he defaul inensiy: λ() = ( c ) 2. ln V () F B() Here V () denoes he firm value which as in Meron s model is assumed o follow a geomeric Brownian moion. B() is he discouning facor exp( r u du) and F is he amoun of ousanding liabiliies. If he firm value approaches F he defaul inensiy increases sharply and i is very likely ha he bond defauls. As defauls can happen a any ime his model is much more flexible han he Meron model. Unlike in Longsaff and Schwarz s model, he defaul can even happen when he firm value is far above F, hough wih low probabiliy. The auhors also consider parameer esimaion in heir model. A closed form soluion for he bond price is no available and for calculaing he prices of derivaives numerical mehods need o be used. Furher hybrid models of his ype can be found in Ammann (1999) or Bielecki and Rukowski (22). The approach of Duffie and Lando (21) accouns for he fac ha bond holders only obain imperfec informaion on he firm value. Thus, saring in a srucural framework, his leads o a hazard rae model. 6. Marke Models wih Credi Risk Schönbucher (2) discusses he framework for a defaulable marke model. The difference beween he marke models and he coninuous ime models is ha marke models rely only on a finie number of bonds, whereas coninuous ime models assume a coninuiy of bonds raded in he marke. As a maer of fac, many imporan variables are no available in hese models as, for example, he shor rae or coninuously derived forward raes, which form he basis for he seing in Heah,

26 26 THORSTEN SCHMIDT AND WINFRIED STUTE Jarrow, and Moron (1992). Inroducions o marke models wihou defaul risk can be found for example in Brace, Gaarek, and Musiela (1995), Rebonao (1996) or Brigo and Mercurio (21). Assume we are given a collecion of selemen daes T 1 < < T K, he enor srucure, which denoes he mauriies of all raded bonds. Denoe by B k () := B(, T k ) he riskless bonds raded in he marke. The discree forward rae for he inerval [T k, T k+1 is defined as F (, T k, T k+1 ) =: F k () = 1 ( Bk () ) T k+1 T k B k+1 () 1. The defaulable zero coupon bond is denoed by B(, T k ). As a saring poin for modeling, i is assumed ha his is a zero recovery bond, i.e., a defaul he value of he bond falls o zero. Pu B k () = B(, T k ) = 1 {τ>} B(, Tk ). The defaul risk facor is denoed by D k () := B k () B k (). If here exiss an equivalen maringale measure Q we have 1 Tk D k () = B k () IEQ[ exp( r u du)1 {τ>tk } F = B k() [ B k () IET k 1 {τ>tk } F = Q T ( ) k τ > Tk F where Q T k denoes he T k -forward measure 16 and IE T k he expecaion w.r.. his measure. So D k () denoes he probabiliy ha, under he forward measure, he bond survives ime T k. Define H(, T k, T k+1 ) := H k () = 1 T k+1 T k ( Dk () D k+1 () 1 ). To simplify he noaion we wrie B 1 for B 1 () (similarly for F, D, H) and T j+1 T j = δ j. 16 The Tk -forward measure is he risk neural measure which has he risk-free bond wih mauriy T k as numeraire. For deails see Björk (1997).