Recent Advances in Credit Risk Modeling

Transcription

1 WP/09/162 Recen Advances in Credi Risk Modeling Chrisian Capuano, Jorge Chan-Lau, Giancarlo Gasha, Carlos Medeiros, Andre Sanos, and Marcos Souo

2 2009 Inernaional Moneary Fund WP/09/162 IMF Working Paper Moneary and Capial Markes Deparmen Recen Advances in Credi Risk Modeling Prepared by: Chrisian Capuano, Jorge Chan-Lau, Giancarlo Gasha, Carlos Medeiros, Andre Sanos, and Marcos Souo 1 Augus 2009 Absrac This Working Paper should no be repored as represening he views of he IMF. The views expressed in his Working Paper are hose of he auhor(s) and do no necessarily represen hose of he IMF or IMF policy. Working Papers describe research in progress by he auhor(s) and are published o elici commens and o furher debae. As is well known, mos models of credi risk have failed o measure he credi risks in he conex of he global financial crisis. In his conex, financial indusry represenaives, regulaors and academics worldwide have given new impeus o effors o improve credi risk modeling for counries, corporaions, financial insiuions, and financial insrumens. The paper summarizes some of he recen advances in his regard. I considers modificaions of srucural models, including of he classical Meron model, and effors o reconcile he srucural and he reduced-form models. I also discusses he reassessmen of he defaul correlaions using copulas, he pricing of credi index opions, and he deerminaion of he prices of disressed deb and esimaion of recovery values. JEL Classificaion Number: G000 Keyword: credi risk Auhors Addresses: ccapuano@imf.org; jchanlau@imf.org; ggasha@imf.org; cmedeiros@imf.org; asanos2@imf.org; and msouo@imf.org 1 The auhors would like o hank Chrisopher Morris and oher colleagues in he Fund for many useful discussions on credi risk modeling during he preparaion of his working paper over he course of he las year.

3 2 Conens Page I. Inroducion...3 II. Srucural Models...4 A. Single-Issuer Defaul Risk...4 B. Disance-o-Defaul: Variaions on a Theme...6 C. Porfolio Credi Risk Models...7 III. Reduced-Form Models...9 A. Srucural and Reduced-Form Models: Reconciliaion Aemps...9 B. Some Models...12 C. Nonlinear Filering...14 IV. Oher Innovaions in he Modeling of Credi Risk...17 A. Defaul Correlaion Using Copulas and Oher Recen Approaches...17 B. Pricing of Credi Index Opions...20 C. Disressed Deb Prices and Recovery Rae Esimaion...21 V. Conclusions...22 Figure 1. Dah-Sing Bank: Disance-o-Defaul...6 Boxes 1. Compensaors and Pricing Trends: Some Definiions Elizalde (2006) The Modeling Sraegy of Frey, Schmid, Gabih (2007)...16 Appendix Filraion and he Pricing of Credi Index Opions...24 References...27

4 3 I. INTRODUCTION As is well known, mos models of credi risk have failed o measure he credi risks in he conex of he global financial crisis. The failure of hese models has made i difficul, if no impossible in some cases, for invesors o manage he credi risk associaed wih counries, corporaions, financial insiuions, and even some financial insrumens. This failure reflecs parly he fac ha he criical assumpions or elemens ha underlie hese models have lacked he needed flexibiliy o ake ino accoun he recen changes in economic and financial circumsances or occurrence of exreme evens. 2 Such a failure also demonsraes ha he correlaions of he facors ha drive credi risk in hese models have broken down in he conex of he crisis. No surprisingly, mos models of credi risk have hen provided lile guidance for he managemen of credi risk. In his conex, financial indusry represenaives, regulaors and academics worldwide, among ohers, have given new impeus o effors o improve he modeling of credi risk. This reflecs he criical need o boh measure and manage credi risk in he conex of wha has become he wors global financial crisis in recen memory. To his end, hese ineresed observers have begun o change or innovae key assumpions and elemens of he modeling of credi risk. They have aken seps o look a changes or innovaions o boh srucural models, which consider ha a defaul occurs whenever he value of he asses underlying he liabiliies falls below some hreshold, and reduced-form models, which depend on a random defaul ime whose disribuion depends on economic variables. They have sressed he need o include jump erms based on Poisson disribuions in he modeling of credi risk, while incorporaing a robus assumpion of filraion, or he observed developmens of he facors ha drive credi risk. Emphasis also has been given o he benefis of including variable recovery raes in he modeling of credi risks. In addiion, hey have highlighed he need o develop simple bu robus models o assess he credi risk of srucured financial insrumens. The paper summarizes some of he recen advances in he modeling of credi risk, including by Fund saff. This paper considers he modificaions of he srucural models, and effors o reconcile he srucural and he reduced-form models. I also mulls over he reassessmen of he defaul correlaions using copulas, he pricing of credi index opions and he imporance of filraion in his regard, he use of nonlinear filering, and he deerminaion of he prices of disressed deb and esimaion of recovery values. Even hough he paper aemps o simplify he exposiion of hese innovaions, i falls shor in many respecs no leas because he innovaions make use of ever more complex echniques and mehodologies of a number of disciplines, including mahemaical saisics and financial economics. 2 Anoher imporan facor ha conribued o he global financial crisis was he misuse of credi risk models in invesmen banks and credi raing agencies, a problem associaed wih governance issues and conflics of ineres beween revenue producing unis, such as underwriers and proprieary rading desks and risk managemen unis.

5 4 The paper is divided as follows. Secion II describes he recen modificaions of srucural models of defaul risk, including boh of single issues and muliple issuers. Secion III explains he recen aemps o reconcile he srucural and reduced-form models. Secion IV summarizes oher innovaions in he modeling of credi risk. This secion includes a summary of he recen work on defaul correlaions using Copulas, he pricing of credi index opions, and he laes work on disressed deb prices and recovery rae esimaion. Secion V provides a conclusion. II. STRUCTURAL MODELS A. Single-Issuer Defaul Risk Defaul-a-mauriy and firs-passage ime models As is well known, wo main approaches are in use for modeling he defaul risk of a single issuer: he inensiy-based or reduced-form and srucural approaches. The reduced-form approach assumes ha he iming of defaul depends on an exogenous sochasic process, and he defaul even is no linked o any observable characerisic of he firm. In conras, he srucural approach, which races is roos o Black and Scholes (1973) and Meron (1974), sars wih he observaion ha defaul occurs when a firm is unable o coninue servicing is deb, say, because of economic reasons relaed o he business cycle. 3 Under absolue prioriy rules, equiy shareholders are residual claimans on he asses of he firm since bondholders are paid firs in case of defaul. Equiy shareholders, in effec, hold a call opion on he asses of he firm, wih a srike price equal o he deb owed o bondholders. Similarly, he value of he deb owed by he firm is equivalen o a defaul-free bond plus a shor posiion on a pu opion on he asses of he firm. Srucural models rely on he concepual insigh ha defaul occurs when he asse value of he firm is less han wha he firm owes o is debors. However, hese models differ wih respec o heir assumpions regarding he iming of defaul. In he model of Meron (1974) as in oher srucural models, for a firm ha issues a zero-coupon bond, 4 defaul occurs a mauriy since his is he only period in which crediors can verify he asse value of he firm. These are examples of defaul-a-mauriy models. In oher srucural models, defaul occurs he he asse value of he firm, V, falls below he value of he liabiliies of he firm, L, a some defaul ime τ. The problem of defaul, in mahemaical language, is equivalen o a firs passage ime problem, also known as a firs sopping or exi ime problem. 5 Firs passage ime models include, among ohers, hose of Kim, Ramaswamy, and Sundaresan (1993), Nielsen, Saá-Requejo, and Sana-Clara (1993), Longsaff and Schwarz (1995) and Saá- Requejo and Sana Clara (1999). 3 See Duffie and Singleon (2003), among ohers, for a exbook reamen of he srucural and reduced-form approaches o credi risk. 4 See Geske (1977) for an exension o coupon bonds. 5 For a comprehensive discussion of sopping imes, see Karazas and Shreve (1991) or Proer (1992).

6 5 More recenly, Capuano (2008) has proposed a non-parameric srucural model o esimae he probabiliy of defaul. This model esimaes he probabiliy of defaul implied by equiyopions by calibraing he probabiliy densiy funcion of he value of he asses using he marke prices of opion conracs. Such a model makes i possible o esimae he defaul barrier wihin he model, while capuring deviaions from log-normaliy. The model has performed well in he conex of he global financial crisis, providing early warning signals of disress for some key financial insiuions. (IMF, 2009). Disance-o-defaul Srucural models rely on he concep of disance-o-defaul. This concep is a sandardized measure of he difference beween he firms asse and liabiliy values, which, heoreacally, depends on he opion-like feaures of he equiy value of a firm. Such feaures are derived from an elemenary accouning ideniy whereby he value of he firm, V (or he value of is asses), is equal o he sum of he values of is deb, D, and equiy, E. Because deb is senior o equiy, shareholders are residual claimans on he firm: he firm s asses are firs used o pay deb holders in case of defaul, and whaever is lef is disribued o shareholders. Concisely, he value of equiy can be wrien as (II.1) E = max(0, V D) The payoff o equiy holders is equivalen o a call opion on he value of he firm wih a srike price equal o he face value of deb. The srike price is also known as he defaul barrier. Given an opion pricing formula, knowledge of any wo of he following hree variables he value of he firm, he deb owed by he firm, and he marke value of equiy is sufficien for esimaing he remaining unknown variable. The Black-Scholes-Meron opion pricing formula for European call opions is he basis for mos pracical applicaions. The srike price is se equal o he level of he firm s shor-erm liabiliies and half is long-erm liabiliies. For he Meron (1974) model, he disance-odefaul T periods ahead, DD, is given by T (II.2) DD T V 1 2 ln + μ σ T D 2 =, σ T where μ is he growh rae of he asse value of he firm and σ is he asse volailiy. Equaion (II.2) simply saes ha he disance-o-defaul is he expeced difference beween he asse value of he firm relaive o he defaul barrier, afer correcing and normalizing for he volailiy of asses. The disance-o-defaul measure has become a useful measure o assess he credi risk of nonfinancial corporaions. 6 Empirical resuls by Moody s KMV have shown ha he disance-o- 6 Crosbie and Bohn (2003), and Vassalou and Xing (2004).

7 6 defaul predics well corporae defauls. Furhermore, work by Gropp, Vesala and Vulpes (2002), and Chan-Lau, Jober, and Kong (2004) shows ha he disance-o-defaul predics banks downgrades in developed and emerging marke counries. For insance, he figure below shows he evoluion of he disance-o-defaul of Dah-Sing Bank, a Hong Kong SARbased bank. Clearly, he disance-o-defaul of he bank poins owards a subsanial credi qualiy deerioraion in he second half of 1997, while showing a recovery o pre-crisis levels in mid Figure 1. Dah-Sing Bank: Disance-o-Defaul B. Disance-o-Defaul: Variaions on a Theme Differen variaions of he disance-o-defaul arise from he use of differen opion pricing formulas and/or differen calibraion procedures. The appropriaeness of he assumpions underlying specific disance-o-defaul of a paricular insiuion and he qualiy of he daa used for calibraion are criical in his regard. For insance, illiquid sock markes may yield lile informaion abou he profiabiliy and, herefore, he defaul risk of a firm. As described below, i is possible o adap he basic disance-o-defaul o paricular siuaions. Currency mismaches in he balance shee A currency mismach exiss when a borrower funds is operaions in one currency, while he earnings derived from hese operaions accrue in anoher currency. In emerging marke counries, and especially Lain America, currency mismaches in he corporae secor arise from balance shees heavily iled owards foreign-currency-denominaed deb and localcurrency-denominaed asses and/or earnings. Modeling he impac of changes in he exchange rae ino he sandard disance-o-defaul based in Meron (1974) is a raher involved process ha requires giving up he assumpion ha he defaul barrier is fixed and allowing i o change i sochasically. Neverheless, borrowing on advanced opion pricing heory, i is possible o derive racable formulas for differen assumpions regarding he behavior of he exchange rae. Furhermore, he disanceo-defaul can be esimaed using simple maximum likelihood echniques as shown in Chan- Lau and Sanos (2006). Promp correcive acion frameworks in he banking sysem Despie he empirical suppor for using he disance-o-defaul for assessing disress in financial insiuions, he definiion of defaul embedded in his measure may no capure he regulaory and supervisory complexiies associaed wih bank inervenions and closures. The disance-o-defaul may well undersae he likelihood ha a bank may be required o

8 7 underake correcive acions by regulaors. The disance-o-defaul may, in effec, represen a bridge oo far for regulaory purposes. On a firs pass, he problem may appear inracable. However, as shown by Chan-Lau and Sy (2006), he defaul barrier can be adaped in a relaively simple way o accoun for he rigger hresholds prescribed by promp correcive acion, which leads o a similar measure beer defined as disance-o-regulaory capial. The choice of opion pricing models and calibraion mehodologies o esimae his measure involves he same selecion as he mehodologies o approximae he disance-o-defaul. Sovereign risk The disance-o-defaul is now in use o measure credi risk of sovereign counries. 7 The main cavea arises from he fac ha he mapping of he conceps of corporae equiy and asse value o a sovereign counry is no sraighforward. Also, here is an imporan implici assumpion ha equiy holders are subordinae o deb holders, which is no he case for a sovereign counry. Ulimaely, he proof of he usefulness of he use of disance-o-defaul depends on he empirical evidence: Is he disance-o-defaul a good empirical predicor of defaul? Is i highly correlaed wih oher defaul indicaors such as credi defaul swaps? So far, he answers have been posiive. C. Porfolio Credi Risk Models Knowledge of he probabiliy of defaul of individual firms opens he way o use porfolio credi risk models o assess he probabiliy ha a subse of he firms in a sample defaul during a pre-specified period of ime. Pu differenly, if here is informaion abou losses given defaul associaed wih securiies issued by each single issuing firms, i is possible o esimae he loss disribuion of a porfolio ha holds hese securiies. Assessing he probabiliy of defaul among a subse of firms requires compuing he disribuion of he number of defauls. The muli-facor normal Gaussian copula, which was inroduced by Vacisek (1987) and exended by Li (2000), is he workhorse srucural model for such a compuaion,. 8 In he Gaussian copula, he normalized asse value of firm i, x i, depends on a single common facor, M, and an idiosyncraic shock, Z i : (II.3) x am a Z 2 i = i + 1 i i, where x i, M, and Z i are sandard normally disribued variables. The coefficien a i, or facor loading, is resriced o values beween 0 and 1 and measures he dependence of he asse 7 See Gapen, Gray, Lim, and Xiao (2004). 8 Secion IV.A discusses he use of Gaussian copula in he conex of he credi risk of srucured securiies.

9 8 value on he common facor. For insance, a common facor could be he exchange rae or some economy-wide index. Firm i defauls when he asse value x i falls below a hreshold value x i. The hreshold value can be deermined if he probabiliy of defaul q() i for firm i in period is known since x 1 i =Φ ( qi), where Φ is he cumulaive sandard normal disribuion funcion. Once he hreshold value is known, i follows ha he condiional defaul probabiliy is equal o: x am Prob < = ( ) =Φ. 1 a i i i (II.4) { xi xi M} qi M 2 The disribuion of he number of defauls can be obained using he recursive procedure K proposed by Andersen, Sidenius, and Basu (2003). As in Gibson (2004), p ( l, M) is he probabiliy of experiencing l defauls during a ime horizon condiional on he common facor M for a se of K firms. If he defaul disribuion is known for K firms, he defaul disribuion if an addiional firm is added o he se can be obained from he following recursion: K+ 1 K (II.5) p (0, M) = p (0, M) ( 1 q ( M) ) K (II.6) ( ) p ( l, M) = p ( l, M) 1 q ( M) + p ( l 1, M) q ( M), l=1,...,k K K K K+ 1 K+ 1 (II.7) K+ 1 K p K + M = p K M qk + 1 M ( 1, ) (, ) ( ) Recursion in hese equaions sars wih he degenerae defaul disribuion p 0 (0, M ) = 1 for K=0 o deermine he defaul disribuion for a se of N firms, N p (, l M ), l = 0,..., N. The uncondiional defaul disribuion p(l,) is obained by inegraion: N (II.8) p( l, ) = p ( l, M) φ( M) dm where φ is he sandard normal disribuion funcion. Calibraion of he one-facor Vacisek model requires firs esimaing he correlaion of each firm s asse value wih he common shock or facor. This correlaion can be obained using principal componen analysis. Such a mehod assumes ha a limied number of unobserved variables (or facors) explain he oal variaion of he larger se of variables. Tha is, he higher is he degree of co-movemen across all individual firm defaul probabiliy ime series, he fewer is he number of principal componens (facors) needed o explain a large porion of he variance of he original series.

10 9 In he case where he original variables are idenical (perfecly collinear), he firs principal componen would explain 100 percen of he variaion in he original series. Alernaively, if he series are orhogonal o one anoher (i.e., uncorrelaed), i would ake as many principal componens as here are series o explain all he variance in he original series. In ha case, no advanage would be gained by looking a common facors, as none exis. Resuls obained by Chan-Lau and Gravelle (2005) sugges ha he firs principal componen accouns for around 70 o 80 percen of he variance. A number of defaul-a-mauriy and firs passage ime models accommodae differen assumpions abou he behavior of he exchange rae. The firs model is a firs passage ime model ha assumes ha he exchange rae follows a diffusion process. The use of a diffusion process is indirecly validaed by he empirical success of simple implemenaions of he Meron model in capuring defaul risk in boh he corporae and banking secors. 9 The model is easy o calibrae since i yields simple closed form soluions. The second model is a defaul-a-mauriy model, which assumes implicily ha he exchange rae follows a jumpdiffusion process. Empirical sudies underaken by Jorion (1988), Dumas, Jennergren, and Naslund (1995) and Baes (1996) find ha jump-diffusion processes capure he behavior of exchange raes beer han alernaive models such as diffusion processes and sochasic volailiy models. The hird model is a firs passage ime model based on a double exponenial jump-diffusion process (Kou, 2002). In conras o jump-diffusion processes, he double exponenial jump-diffusion process capures he sylized fac ha he disribuion of reurns is asymmeric by specifying differen probabiliy disribuions for posiive and negaive jumps. This model is well suied for analyzing siuaions under which he exchange rae is prone o move in only one direcion, i.e. undervalued or overvalued exchange rae pegs. Preliminaries III. REDUCED-FORM MODELS A. Srucural and Reduced-Form Models: Reconciliaion Aemps As observed in he previous secion, conrary o srucural models, reduced-form models assume ha he defaul even does no depend on he characerisics of he firm, which has promped aemps by many auhors o reconcile his difference. To his end, hese auhors have focused on he role ha informaion srucure plays in deermining he predicabiliy of he defaul even in srucural and reduced-form models. Under cerain condiions relaed o how informaion is revealed o he marke paricipans, i is possible o show he equivalence beween srucural and reduced-form models. This secion reviews work along hese lines by, among ohers, Duffie and Lando (2001), Giesecke (2004, 2005), Giesecke and Goldberg 9 See Crosbie and Bohn (2003) for corporaes, and Gropp, Vessala, and Vulpes (2006) and Chan-Lau, Jober, and Kong (2004) for banks in maure and emerging marke counries, respecively.

11 10 (2004), Çein, Jarrow, Proer, and Yildirim (2004), and Guo, Jarrow, and Zen (2005a) o reconcile he differences beween srucural and reduced-form models. The basic insigh relaes o wha informaion is available o a modeler as explained by Jarrow and Proer (2004). They argue ha he difference beween he srucural and reduced-form models reflecs he informaion available o he modeler. While srucural models assume ha he modeler has he same informaion ha he firm s manager has complee knowledge of he processes of all he firm s asses and liabiliies, which, in mos siuaions, leads o a predicable defaul ime reduced-form models assume ha he modeler has he same informaion se ha he marke has incomplee knowledge of he firm s financial condiion, which, in mos cases, resuls in impossible o predic or inaccessible defaul ime. As he informaion available o he modeler declines or shrinks, i becomes possible o ransform a srucural model in which defaul is a predicable sopping ime ino a reduced-form model in which defaul is an inaccessible sopping ime. A simple way o undersand he argumen offered by Jarrow and Proer (2004) is o focus on he simple srucural model of Meron (1974). If he asse value is observed coninuously, he defaul even is predicable in he sense ha i is possible o observe if he asse value is moving owards he defaul barrier (or he face value of he firm s liabiliies). However, if he asse value were o be observed only in relaively long, discree inervals of ime, i would no be possible o know wheher he firm is close o defaul in beween inervals. In he laer case, he defaul is unpredicable. 10 In mahemaical erms, he main difference beween he srucural and reduced-form models is one of he appropriae filraion process, or how informaion is conveyed o he marke and how sopping imes, e.g. defaul evens, behave under differen filraion. The srucural models clearly make i possible o deermine he defaul ime. However, as he informaion available o he invesor is reduced, here is a need o consider a smaller filraion, which, depending upon he circumsances, could even make he defaul ime compleely inaccessible. As noed by Jarrow and Proer (2004), he difference beween srucural and reduced form models depends on wheher he defaul ime is par of a filraion ha is observed by invesors. Compensaors and pricing measures The reconciliaion beween models requires he specificaion of srucural models wih incomplee informaion abou he processes of asses, defaul hreshold or boh. The use of compensaors o deermine he defaul processes opens he way o deermine he srucural models cumulaive defaul raes or he so-called pricing rends (Box 1). 10 Asse values are obained from equiy prices in mos empirical implemenaion of srucural models. Since equiy prices are available a high frequencies, e.g. inra-day and inra-minue in some insances, i can be argued ha equiy-based srucural models do no need o be reduced o an equivalen reduced-form model. However, his argumen requires weak marke efficiency, e.g. ha curren and pas prices of equiy and equiyrelaed securiies conain all relevan informaion. If his is no he case, hen he defaul even is no be predicable.

12 11 Box 1. Compensaors and Pricing Trends: Some Definiions Elizalde (2006) Definiion 1. A sopping ime τ is predicable if here exis a sequence of sopping imes which announce τ such ha: τ τ... < τ 1 2 lim n τ = τ And τ is considered a oally inaccessible sopping ime if here does no exis any predicable sopping ime which can give informaion abou τ, ha is: Pr[ τ = ~ τ < ] = 0 for any predicable sopping imeτ ~. The defaul indicaor process N generaed by τ is given by: N =1 {τ }. Definiion 2. A process C is called he (F )-compensaor of he process N if and only if: C is a (F )-predicable increasing process, wih C 0 =0. The process N C, called he compensaed process, follow a (F ) maringale. Definiion 3. A process Γ is said o be a pricing rend associaed wih he (F )-compensaor C such ha: C = Γ min{,τ} The condiional defaul probabiliy of his pricing rend can be expressed as: Γ ΓT P[ τ T / F ] = E[ e / F ] Moreover, if a defaulable securiy which pays X unis a ime T if defaul has no occurred before T and zero oherwise is considered, he price of he securiy a ime T can be expressed as: E T Γ ΓT rs ds [ Xe n / F ] where r is he (F )-adaped ineres rae process. The wo previous expressions for he condiional defaul probabiliy and he price of he defaulable securiies are similar o hose observed in reduced models where, if λ is he inensiy process, Γ would be he cumulaive defaul process λ d S s. The pricing rend Γ only admis an inensiy represenaion when i is differeniable. In he cases where Γ is differeniable, here exiss a process λ such ha: Γ = 0 λ d s s 0, which represens he inensiy of he couning process N, i.e., he inensiy of arrival of he sopping ime τ. Therefore, he pricing rend is he cumulaive defaul rae. The pricing rend Γ is characerized by a compensaor process C such ha he difference beween he defaul process N and he compensaor follows an (F )-maringale. If he filraion (F ) represens informaion ha invesors receive over ime, differen specificaions of (F ) imply differen compensaor processes and, herefore, differen pricing measures. The pricing rends are deermined by he specificaion of a sopping ime τ, and an informaion framework (F ). This is he link beween srucural and reduced-form models. A srucural model is a way of specifying a defaul ime τ based on he economic ime of he firm, or: (III.1) τ = inf{ 0 / V K }

13 12 where V and K are he firm s asses and defaul hreshold, respecively. Equipped wih a specificaion for he defaul ime, each specificaion of he informaion (F ) available o invesors wih respec o he asse value and he defaul hreshold processed yields a differen pricing rend and, herefore, a differen reduced-form model. Duffie and Lando (2001) B. Some Models 11 Duffie and Lando (2001) consider a model in which he defaul ime is fixed by he firm s managers so as o maximize he value of equiy. Invesors canno observe he asses direcly, and receive only periodic and imperfec accouning repors. Assuming a given Markov process, A = ( A ) 0, where A represens he firms value a ime, Duffie and Lando obscure he process A so ha i can be observed only a discree ime inervals, and add independen noise. A discree ime process Z = A + Y is obained, where Y is he added noise, and which is observed a imes i for i = 1,...,. The auhors derive he disribuion of he firm s asse value condiional on invesors informaion, and, from his disribuion, he inensiy of defaul in erms of he condiional asse disribuion and he defaul hreshold. The paricular specificaion of he defaul ime τ and he filraion (F ) herefore make i possible o derive an inensiy for he defaul ime. The sopping ime, τ, is herefore inaccessible. The defaul ime τ is ransformed from a predicable sopping ime ino an inaccessible sopping ime since i is unclear how he asse value evolves beween he ime of he observaions of he asse value. Defaul could occur unexpecedly prior o he nex observaion. Under hese circumsances, he srucural model becomes a reduced-form model by obscuring and reducing he informaion. Giesecke (2005) Giesecke (2005) deals wih he case of a srucural model in which invesors have complee informaion abou he asse value bu incomplee informaion abou he defaul hreshold. Alhough consan, he defaul hreshold is no known by he invesors, who are forced o work under a disribuion funcion for he defaul hreshold. The impossibiliy of observing he defaul hreshold makes he defaul ime an unpredicable even. In his case, invesors calculae he pricing rend in erms of he disribuion funcion for he hreshold and he observable hisorical asse value. Giesecke also sudies he cases of incomplee informaion for boh he asse value and he defaul hreshold. In conras wih he previous case in which invesors have incomplee informaion abou he defaul hreshold bu complee informaion abou he asse value process, his case wih imperfec informaion abou he pricing rend calculaed in erms of he hreshold disribuion and he disribuion of he minimum hisorical asse level he pricing rend, calculaed in erms of he hreshold disribuion and he disribuion for he minimum hisorical asse level admis an inensiy represenaion. 11 For a comprehensive survey of some of hese models, see Jarrow and Proer (2004).

14 13 Giesecke and Goldberg (2004) Giesecke and Goldberg (2004a) consider he case in which he defaul barrier is random and unobserved modeled as an horizonal line of he form y = L, where L iself is unknown and random. Since his random curve is independen of he underlying srucural model, he defaul ime τ is inaccessible. Given ha he rue level of liabiliies is no disclosed o he public, invesors use a priory disribuion for he defaul hreshold. Giesecke (2004) akes he incomplee informaion assumpion in srucural model one sep furher o model he defaul correlaion. He provides a srucural model in which he firms defaul probabiliies are linked via a join disribuion o heir defaul hresholds. Invesors do no have perfec informaion abou eiher such hresholds or heir join disribuion. However, hey form a prior disribuion which is updaed when one such hresholds is revealed, which only happens when one of he firms defauls. In Giesecke (2004), invesors have incomplee informaion abou he firms defaul hresholds bu complee informaion abou heir asse processes. Giesecke and Goldberg (2004b) exend his framework o one in which invesors do no have informaion abou eiher he firms asse values or heir defaul hresholds. In his case, defaul correlaion is inroduced hrough correlaed asse processes, and, again, invesors receive informaion abou he firms asse and defaul barrier only when hey defaul. Such informaion is used o updae heir priors abou he disribuion of he remaining firms asse values. Çein, Jarrow, Proer, and Yildirim (2004) Çein, Jarrow, Proer, and Yildirim (2004) depar from a srucural model as in Duffie and Lando where he modeler s filraion, (F ), is a sric subfilraion of ha variable o he firm s managers invesors receive only a reduced version of he informaion ha firm s managers have. The auhors claim ha he defaul ime is a predicable even for firm s managers, since hey have enough informaion abou he firm s fundamenals. Bu invesors do no have access o such informaion. Insead, invesors observe a reduced version of his informaion. In he model, he firm s cash flow (L) is he variable ha riggers defaul, afer reaching some minimum levels during a given period of ime. Firm s managers can see L levels, bu invesors only receive informaion abou he sign of he L, making he defaul ime an unpredicable even from heir perspecive. In his seing, invesors derive he defaul inensiy as seen by he marke. The relevan barrier is now L = 0, for all 0, he cash flows. Invesors only observe wheher he cash flow is posiive, zero or negaive, and assume ha he defaul ime is he firs ime ha he cash flows fall below zero, or when he cash flow boh remains below zero for a cerain period of ime, and hen doubles in absolue magniude. The defaul ime is also inaccessible in his case Guo, Jarrow and Zeng (2005) argue ha he way in which he previous papers inroduce incomplee informaion abou he variables generaing defaul are illusraive bu oo simple o be applied in pracice. Their paper represens a generalizaion o formalize he heory linking srucural and reduced form models.

15 14 Jarrow, Turnbull, and ohers A class of reduced-form models ha separaes bankrupcy and he firm s underlying asses has araced a lo of aenion. These models rely on an approach suggesed by Jarrow and Turnbull (1995, 1992) o price derivaives. The basic idea of his approach is o assume he presence of wo exogenous sochasic erm-srucures one risk-free and he oher one ha would be a credi spread over he firs one and bankrupcy ha is an exogenous process, independen of he firm s underlying asses. The combined erm-srucure is hen used o price insrumens under he absence of arbirage opporuniies and using maringale echnology (Harrison and Kreps (1979) and Harrison and Pliska (1981)). This approach does no require esimaes for he parameers of he firm s unobservable asse value, a common problem in he srucural models, or a payoff prioriy srucure of he firm s liabiliies. By way of exension of his approach, Jarrow, Lando and Turnbull (1997) specify he bankrupcy process as a discree Markov chain, whose parameers are easily esimaed using observable daa. Duffie and Singleon (1999) parameerize he losses a defaul as a reducion of he marke value of defaulable securiies observed a defaul, and show ha hese securiies can be priced using a defaul-adjused, risk-free rae process. They show ha he price of he securiies using heir framework accouns for boh he probabiliy and iming of defaul, as well as he effec of losses on defaul. Guo, Jarrow and Zeng (2005b) model he recovery rae process wihin a reduced-form model using he firm s balance shee srucure. Wong and Wong (2007) develop a regime-swiching model over he enire yield curve o examine he changes in defaul probabiliies across differen credi raings. C. Nonlinear Filering Nonlinear filering problems usually arise in a naural way in srucural models of credi risk wih incomplee informaion abou he value of asses or liabiliies as in Duffie and Lando (2001), Jarrow and Proer (2004), and Frey and Runggaldier (2007). Frey and Runggaldier (2007) sudy he pricing of credi derivaives in he conex of incomplee informaion. They assume ha he sae variable process is no direcly observable, and invesors have informaion only on he defaul hisory of he porfolio and noisy price observaions of raded credi derivaives. They address he filering problems using a Markovian model where he unobservable facor or sae variable process X may jump a defaul imes. In his conex, hey derive a finie-dimensional filer for he case where X follows a finie-sae Markov chain. Frey and Runggaldier propose wo seps for modeling he pricing of derivaives wih incomplee informaion. In he firs sep, hey propose a model where he sae process X driving he defaul inensiies is observable. This so-called full-informaion model makes use of Markov-process echniques. In he second sep, hey sudy he pricing of derivaives in a more realisic seup of incomplee-informaion, which is buil by projecing he defaul inensiies and price dynamics from he full-informaion model ono he informaion (F I ) available o invesors. (F I ) conains he defaul hisory (H ) of he porfolio under consideraion and noisy price observaions for raded credi derivaives, which are modeled using Z observaions of nonlinear funcions of he sae process X in addiive Gaussian noise.

16 15 The second sep leads o a nonlinear filering problem. This reflecs he fac ha he esimaion of he condiional disribuion of X depends on he informaion available o he invesors, or F I. Frey and Runggaldier consider ha he sae process X and he defaul indicaor process Y have a common jump, which makes he filering problem paricularly complex: X and Y canno be made independen by a change-of-measure or viewing he same process under a differen se of likelihoods. 13 They hen derive he filer by exploiing he recursive srucure of he defaul hisory (H ). In so doing, hey employ recursive soluions using a finie-dimensional filer in he case when X is a finie-sae, coninuous-ime Markov chain. The filer for such a Markov chain can be a useful ool for evaluaing he filer for general sae variable processes of jump-diffusion ype. Frey and Runggaldier (2008) use nonlinear filers in boh ineres rae and credi risk models wih incomplee informaion. In paricular, hey employ he filers o price credi derivaives. Box 2 provides a general descripion of he use of nonlinear filers for modeling credi risk. 13 Analysis of common jumps may be found also in Ceci and Gerardi (2001).

17 16 Box 2. The Modeling Sraegy of Frey, Schmid, Gabih (2007) Frey, Schmid, Gabih (2007) presen a modeling sraegy for he pricing and hedging of credi derivaives based on hree layers of informaion. They consider defaulable securiies issued by m firms, while noing ha he random ime τ i denoes he defaul ime of firm I; Y,i =1 {τi } is he corresponding defaul indicaor; and Y =(Y,1,..., Y,m ) gives he curren sae of he porfolio. They also assume ha he defaul inensiies (he inensiies of he mulivariae poin process Y) depend on some facor process X. They also consider hree layers of informaion: full informaion; informaion of informed marke paricipans (marke informaion); and informaion of secondary-marke-invesors (invesor-informaion). Full Informaion. The auhors assume a filered probabiliy space (Ω, Π, F, Q) wih Q being he risk neural measure, and F he full-informaion filraion. They also assume ha τ i is condiionally independen doubly sochasic random imes wih (Q, F)-defaul inensiy λ,i = λ i (X ); X follows a finiesae Markov chain; and he risk-free rae equal o zero. They define he full-informaion value of a Y - T measurable claim P (such as a ypical credi derivaive) by E Q (P/Π ) and denoe he naural filraion of process Y as F Y. By he Markov propery of (X, Y), he full-informaion value is given by p (X ) for some Y -measurable funcion p. Marke Informaion. The auhors assume ha he prices of raded credi derivaives are deermined by informed marke paricipans. These paricipans have access o so-called marke informaion, given by he filraion F M :=F Y VF Z. The sochasic process Z represens noisy observaions of X and can be viewed as an absrac form of insider informaion. Z is given by Z = a( X ) ds + db, where B is a sandard F-Brownian moion independen of X and Y. The marke price of a raded securiy wih payoff P is defined as: pˆ : = E Q ( P / M ) = E Q 0 ( p ( X s ) / Since Y is known, o compue he marke price pˆ i is necessary o deermine he condiional M disribuion of X given, given he probabiliy vecor π =(π 1,..., π k ) wih M ) π k = Q( X k / Π M ), 1 k K which is a nonlinear filering problem. The auhors solved he problem using maringale represenaion resuls, and he innovaion approach o nonlinear filering. Invesor Informaion. Since he process Z is no direcly relaed o observable economic quaniies, he pricing and hedging of credi derivaives need o be analyzed from he view poin of secondary marke paricipans wih informaion se F I F M. I is assumed ha F I conains he defaul hisory, F Y, and he noisy price observaions of raded credi derivaives. The auhors show ha, under his seup, he compuaion of prices and risk-minimizing hedging sraegies lead o a second filering problem in which he condiional disribuion of he probabiliy vecor π needs o be deermined given invesor informaion I Π. In his se up: (i) prices are weighed averages of full-informaion values, pˆ, and, herefore, compuaions are done mosly in he conex of a full-informaion model, which are relaively simple o handle; (ii) he fac ha prices of raded securiies are given by he projecion of heir full-informaion value on he marke filraion F M leads o rich credi spread dynamics: spread risk (as credi spreads flucuaed in response o flucuaions in Z), and defaul conagion (as defauls of firms in he porfolio lead o an updae of he condiional disribuion of X given F M and, herefore, a jump in he (Q, F M )- defaul inensiies) are allowed; and (iii) he se up has a naural facor srucure wih facors given by he k condiional probabiliies π, 1 k K.

18 17 IV. OTHER INNOVATIONS IN THE MODELING OF CREDIT RISK A. Defaul Correlaion Using Copulas and Oher Recen Approaches 14 The naure of defaul correlaion is relaed o he analysis of muliple defauls, which are ypically observed in a porfolio of loans or a baske of securiies, for example in collaeralized deb obligaions (CDOs). In his conex, when modeling muliple random variables, i is necessary o analyze heir mulivariae disribuion. This is where he dependence srucure and correlaion beween he random variables become relevan. The same applies o muliple defauls. Since defauls are hisorically rare, hey represen a aileven of he disribuion. As a consequence, defaul correlaion relaes o he modeling, in a mulivariae framework, of he ail-dependence. Following he radiional porfolio analysis, he mulivariae normal disribuion and he relaed Gaussian copula have been adoped for a generaion of muliple defauls models. As described above, in is simples (bi-variae) form, he Gaussian copula saes ha if we consider wo sandard normal random variables x 1 and x 2, wih correlaion ρ 12, he join disribuion of x 1 and x 2 will be bi-variae normal, and he degree of dependence beween x 1 and x 2 will be enirely described by ρ 12. In his se-up, he x i ' s would be percenile-opercenile ransformaion of he random variables describing he ime o defaul of each firm or loan of ineres. One of he advanages of he copula framework, and he Gaussian copula in paricular, is ha he correlaion can be modeled separaely from he marginal densiy of each random variable. 15 Wih more han wo random variables, such as in a CDO, one common way o model he correlaion srucure has been hrough facor models, wih he mos common being he one-facor model. The one-facor model ypically implies ha he defaul of each variable x is linearly affeced by a common facor, V and a specific idiosyncraic facor, Z, i generally endowed wih independen sandard normal disribuions. This srucure ensures x, remains linear. ha he correlaion beween each pair of defauls ( ) i x j The Gaussian copula, discussed in secion II.C, generally relies on a op-down approach. In paricular, given a choice of marginal probabiliy disribuions for he random variables, he researcher specifies a one facor (or muli-facor) Gaussian copula ha summarizes he dependence among he variables. The advanages and, hence, he populariy of his approach relae o he inuiive specificaion, he relaively simple srucure, and he exisence of closed form soluions, which make his approach easily implemenable. i 14 This secion only ouches on inensiy defaul modeling, and does no cover defaul correlaion based on he binomial and Poisson disribuions, inensiy correlaion, and correlaion in raing changes. Lando (2004) provides an exensive reamen of hese opics. 15 This is a salien feaure of he reduced form models.

19 18 A a regulaory level, he Basel II framework developed by he Bank for Inernaional Selemens (BIS) is also based on assumpion of op-down saic correlaions beween differen asse classes, BIS (2005). More recenly, his assumpion has been severely quesioned given he rapid increase in correlaions experienced by differen classes in he face of he global financial crisis (Fich Raings, 2008, 2004). The main drawbacks of he op-down approach relae o he problem of choosing he correc copula, and he opimal incorporaion of new informaion ino he framework, i.e. he framework is saic. Jarrow and Yu (2001) have made some effors o overcome hese drawbacks. In paricular, hey propose an asymmeric informaion framework in which companies are divided ino primary, whose defaul is influenced only by macroeconomic variables, and secondary, whose defaul depends on he probabiliy of defaul of oher companies. This framework aemps o beer incorporae new informaion concerning acual defauls. Boh he academic lieraure and financial indusry represenaives, however, have devoed considerable amoun of effor o design boom-up frameworks. Crouhy, Galai, and Mark (2000) presen a deailed comparaive analysis of he mos popular credi risk models in he finance indusry. Fund saff have also developed credi risk models ha ake ino accoun defaul correlaion. Avesani, Liu, Miresean and Salvai (2006) show how o exend he popular Credi Risk+ framework o accoun for correlaed defauls. Specifically, hey allow for a fixed degree of correlaion in he risk facors ha drive muliple defauls in a porfolio of loans. Avesani, Garcia-Pascual, and Li (2006) presen a marke-based defaul indicaor based on a baske of CDS spreads. In his case, he correlaion srucure is specified by allowing he defaul probabiliy of he CDS baske o be linked wih he covariance of he sock reurns of he underlying corporaions. More recenly, Huang, Zhou, and Zhu (2008) propose a framework o esimae defaul correlaions from high-frequency reurn daa, while IMF (2009) presens a new se of models aimed a esimaing direcly he join disribuion of defauls, hereby delivering a dependence srucure (copula) in he porfolio ha is no exogenously specified. In he face of he U.S. financial crisis, hese models obain ineresing resuls from a financial sabiliy purposes. In his ligh, why has he lieraure recenly moved away from he Gaussian copula in modeling defaul correlaions? The answer is mosly an empirical one. While he Gaussian copula is racable and inuiive, is empirical performance has been weak, paricularly in he curren crisis. I is possible o find many reasons ha explain he failure of he Gaussian copula, bu probably he mos relevan one relaes o he non-lineariy of defaul dependence. 16 While correlaion is a linear concep he covariance is a linear operaor 16 The seemingly failure of he Gaussian copula, however, should no be oversaed. Marke praciioners mosly used he Gaussian copula as a pricing convenion o quoe prices for credi derivaives. The exisence of a correlaion smile is evidence ha he marke did no believe in he Guassian copula as he righ pricing model. A more appropriae asserion perhaps is ha he Gaussian copula was similar o he use of he Black- (coninued )

20 19 defauls end o cluser during sress evens, i.e. muliple defauls represen non-linear evens. As a consequence, correlaion is ill-suied o describe muliple defauls. More pracically, he absence of any dynamics in he general Gaussian framework makes i also difficul o adop i for hedging purposes. The recen lieraure ries o address he empirical drawbacks of earlier models. An ineresing approach o he modeling of muliple defauls, or an alernaive o he choice of copula, is hrough nework dependence as described by Eisenberger and Noe (2001). In his framework, firms are linked hrough cross-liabiliies, which generae a marix, i.e. a nework, of iner-linkages. Wih sufficien deailed informaion, i is possible o design an opimal algorihm ha generaes conagion in he nework and, herefore, correlaion and muliple defauls, following he failure of one or more firms o honor is liabiliies. To accoun for defaul correlaion in a baske of CDSs or an index such as he CDX, Couderc (2007) suggess firs o use he enire informaion from he underlying defaul spreads and hen o granularly model he discree loss disribuion of he index by compounding he (calibraed) correlaions of he differen ranches. I is also possible o inroduce oher ypes of srucure along he line of he volailiy smile in he Black-Scholes opion pricing lieraure. Overall, i is sill oo early o deermine he empirical success of hese modificaions, bu he addiional compuaional coss appear o represen a limiing facor. Schönbucher (2008) proposes innovaions o exising mulivariae inensiy models of credi risk ha appear promising in capuring observed defaul correlaions in ime of disress. The main idea is o divide he informaion srucure of he model ino wo pars: a pre imechange and a pos ime-change srucure. In a pre ime-change seing, defauls are condiionally independen; however, in a pos ime-change seing muliple defauls become condiionally dependen. In oher words, an individual defaul is more likely when muliple defauls have already occurred. This observaion suggess ha perhaps wha maers is no calendar ime bu business or defaul ime, which is measured by he realizaion of defauls. The key o model business ime is o accuraely model he ime-change sochasic process, while calibraing dependence o mach any observed correlaion. In his conex, he problem becomes o separae he modeling of he ime-change process, which inuiively accoun for he jump in he defaul dependence. Along hese lines, Joshi and Sacey (2005) propose he use of Levy processes, in paricular Variance Gamma processes, o accoun for ime-change, which appears o capure he recen surge in defaul correlaions. Duffie, Horel and Saia (2008) presen an empirical analysis of porfolio defaul losses on U.S. corporae deb during Their resuls indicae ha he probabiliy of exreme porfolio losses can no be explained solely by observable risk facors. Even afer accouning for boh macro-and-micro observable risk facors, he auhors find srong evidence for he presence of unobservable (laen) common facors. These resuls sugges ha, in order o capure he observable ime-variaion in exreme defaul losses (ail evens) in he U.S. Scholes model for quoing equiy opions premia, or he Garman-Kohlhagen model for quoing he price of FXopions. The exisence of a correlaion smile is akin o he volailiy smile.

21 20 corporae secor, i is of key imporance o allow for laen facors o drive he porfolio loss disribuion. B. Pricing of Credi Index Opions Morini and Brigo (2007) offer a mehodology o price credi swap opions. The mos common of hese opions is he credi index opion, which is an opion on he spread of a credi index ha consiss of a sandardized porfolio of credi defaul swaps (CDS). The credi index opion allows an invesor o ener a forward credi index a a pre-specified spread, and o receive upon exercise of his opion a fron-end proecion corresponding o index losses from opion incepion o opion expiry. Concepually, a payer credi index opion a incepion or ime 0, wih a srike price K and an exercise dae Ta, and wrien on an index wih mauriy offers a buyer seeking proecion he righ (bu no he obligaion) o ener ino an index a Ta wih final paymen a. The buyer pays a fixed K, which gives him he righ o receive proecion from losses in he period beween Ta and. In addiion, however, he buyer receives, upon exercise, he so-called fron-end proecion, which covers he losses from he opion incepion 0 o he exercise dae Ta. The fron-end proecion, herefore, provides he buyer wih proecion from losses in he period beween 0 and Ta. The credi index opion includes he fron-end proecion as a way o arac more invesors. Examples of his opion are he itraxx Europe ha includes he CDS of equally-weighed European names, he itraxx Crossover index ha includes he mos liquid subinvesmen grade eniies, and itraxx HiVol ha comprises a subse of he main index wih he riskies names. According o Morini and Brigo, he pricing of he credi index opions has undergone significan modificaions over ime. The iniial marke sandard for pricing of credi index opions cenered on he use of a Black formula o price he opion as a call on he spread, while adding he fron-end proecion. However, his marke sandard negleced he fac ha i is no possible o separae he price of he opion from he fron-end proecion since he laer is an inegral par of he invesor s decision o exercise he opion. This led o a change in he marke sandard for he pricing of he credi index opions ha involved a redefiniion of he underlying index spread. Morini and Brigo, however, argue ha he pricing of credi index opions sill suffers from shorcomings. They noe ha he index spread does no ake ino accoun he fron-end proecion in momens of sress in financial markes. They also indicae ha he marke pracice o compue he index spread does no consider all saes of he world. In addiion, hey noe ha i is no possible o jusify heoreically he use of he Black formula in his conex. They argue ha, according o he fundamenal heorem of asse pricing, a rigorous derivaion of he Black formula for he pricing of a credi index opion requires he definiion of an appropriae numeraire for change of he pricing measure under which he underlying spread is a maringale. Since he index includes many defaulable names, he quaniy ha appears o be he naural choice of a numeraire is no sricly posiive. To address his difficuly, i is possible o use a echnique for single name producs based on a numeraire ha is no sricly posiive. However, his measure would no be equivalen o he sandard risk neural, forward and swap measures used in mahemaical finance.