Information Reduction in Credit Risk Models

Transcription

1 Information Reduction in Credit Risk Models Xin Guo Robert A. Jarrow Yan Zeng (First Version March 9, 2005 April 25, 2006 Abstract This paper builds the mathematical foundation for credit risk models with incomplete information. We provide rigorous mathematical definitions for the continuously and discretely delayed filtrations. Our definitions unify two well-known types of incomplete information: the noisy information in Duffie and Lando (200 and partial information in Collin-Dufresne et al. (2004. Under this framework, we first show how delayed information sets, such as those available to market participants, transform structural models into reduced-form intensity-based models, via obtaining eplicit analytic epressions for default intensities under various models. We then establish relations of risk debt pricings under complete and delayed filtrations. Corresponding author. 224 Rhodes Hall, School of Operations Research and Industrial Engineering. Tel: Fa: Cornell University, Ithaca, NY Sage Hall, Johnson School of Management. Tel: Cornell University, Ithaca, NY 4853 Department of Mathematics. Cornell University, Ithaca, NY 4853

2 Introduction Given the size and rapid growth of the credit risk market in recent years (see Creditflu, 2004, it is not surprising that the credit risk literature has eperienced similar growth, as evidenced by the number of books and articles being published on this topic. Two types of models are studied in the credit risk literature: structural and reduced-form. Structural models view a firm s liabilities as comple put options on the firm s assets. Therefore, modelled in this approach are the firm s liability structure and the firm s asset value process. This methodoy originated with Black and Scholes (973 and Merton (974. In these models, the default time is usually characterized as the first hitting time of a firm s asset value to a given boundary, the boundary being determined by the firm s liabilities. As such, if the firm s asset value process follows a diffusion, then the default time is usually a predictable stopping time. The difficulty of using the structural approach is twofold: first, the firm s asset value process is not directly observable, making empirical implementation difficult; and second, a predictable default time implies credit spreads should be near zero on short maturity corporate debt. This second implication is well known to be inconsistent with historical market credit spread data. In contrast, the reduced-form approach was developed precisely to avoid directly modelling the firm s unobservable asset value process. This was instead accomplished by modelling the price process of the firm s liabilities, for eample, a zero-coupon bond issued by the firm. This approach was originated by Jarrow and Turnbull (992, 995, Artzner and Delbaen (995, and Duffie and Singleton (999. Typically, reduced-form models characterize default as the first jump time of a point process, often assumed to follow a Co process (i.e., a doubly stochastic Poisson process. As such, the default time is usually a totally inaccessible stopping time, implying non-zero credit spreads for short maturity corporate debt. A review of the credit risk literature can be found in many good books, including Ammann (200, Bielecki and Rutkowski (2002, Duffie and Singleton (2003, and Lando (2004. A systematic study of the mathematical tools in reduced-form models is available in Elliott, Jeanblanc, and Yor (2000 and Jeanblanc and Rutkowski (2002. To summarize, structural and reduced-form models are viewed as competing paradigms. However, recent work by Duffie and Lando (200, Collin-Dufresne, Goldstein, and Helwege (2003, Çetin et al. (2004 and Jarrow and Protter (2004 point out an intrinsic connection between these two approaches. Reduced-form models can be viewed as structural models that are analyzed under different filtrations. Structural models are based on the information set available to the firm s management, which includes continuous-time observations of both asset values and liabilities. Reduced-form models are based on the information set available to the market, typically including only partial observations of both the firm s asset values and liabilities. As shown in eamples by Duffie and Lando (200, Collin-Dufresne, Goldstein, and Helwege (2003, Çetin et al. (2004, and Jarrow and Protter (2004, it is possible to transform a structural model with a predictable default time into a reduced-form model, with a totally inaccessible default time, by formulating the so called incomplete information. For instance, Duffie and Lando (200 used noisy and discretely observed asset value in a continuous-time model, Collin-Dufresne, Goldstein, and Helwege (2003 used a simple form of delayed information in a Brownian motion type model. However, up to date, the notion of incomplete information has not been mathematically and systematically studied. Indeed, it has not been even well-defined mathematically, despite the fast growing literature of information-based credit risk studies. Furthermore, it is unclear if incomplete information such as the noisy information and the delayed information can be unified under a proper mathematical framework. 2

3 Our paper is to address this issue. We define rigorously the notion of delayed information, for both discrete and continuous type. The latter is built on the work of Jacod and Skorohod (994 on jumping filtrations of one sequence of marked point processes, while the former is defined through a time change of filtrations. We prove the necessity of differentiating these two types by showing that they are both intuitively and technically distinct. We illustrate via eamples the generality of our definitions. Built on this mathematical framework, we generalize the work by Duffie and Lando (200, Collin-Dufresne et al. (2003, and characterize the eistence of an intensity process for any Markov models, with and without jumps. As in earlier work, we show that delayed information transforms a predictable default time into a totally inaccessible stopping time. Moreover, we derive eplicit analytical connections between the default intensity and the density function of the corresponding first passage time for general continuous Markov models. In addition, we provide a characterization of the intensity process that is useful for empirical estimation (see Chava and Jarrow (2004 and Duffie and Wang (2003 for eisting empirical studies. An outline of this paper is as follows. Section 2 builds the mathematical framework of incomplete information. First we define continuously delayed and discretely delayed filtrations, we then provide general results characterizing the difference of these two types of incomplete information. We then illustrate via eamples how noisy and partial information are unified under this definition. Section 3 characterizes the relation of delayed information with intensity process: this is shown by studying representative models for the asset value process: one-dimensional continuous (strong Markov models, regime switching, and jump diffusion. Eplicit formulas for the default intensities are derived via Meyer s Laplacian approimation, as in Duffie and Lando (200. Section 4 studies the relations of risky debt pricings under different filtrations: complete and incomplete. 2 Delayed filtration: Mathematical definition Intuitively, the continuously delayed filtration allows information to flow continuously, but following a time clock slower than the ordinary one. The discretely delayed filtration, on the other hand, does not allow new information to flow in between two consecutive observation times. Therefore, these two types of filtration delay are distinct. Delayed filtration (continuously delayed. Definition. Given a filtration H = (H t t 0 that satisfies the usual hypotheses, i.e. it is right continuous and contains all the negligible sets. Suppose that (α t t 0 is a time change of H so that for a.s. ω, α t (ω t for all t. The time-changed filtration F = (F t t 0 = (H αt t 0 is called a continuously delayed filtration of H. It s not hard to show that F = (H αt t 0 is well-defined and satisfies the usual hypotheses. See, for eample, He, Wang and Yan (992, Chapter III, 5. Eample. Given a filtration H = (H t t 0 and a constant δ > 0. Let α t = (t δ +, then F t = H αt t 0. That is, (α t t 0 is an increasing, right continuous process, so that α 0 = 0 and α t is an H-stopping time for every 3

4 defines a continuously delayed filtration of H, so that { H0, t δ, F t = H t δ, t > δ. This includes as a special case the eample in Collin-Dufresne, Goldstein, and Helwege (2003, Appendi A, where the filtration H is the natural filtration of Brownian motion. Delayed filtration (discretely delayed. Definition 2. Given a filtration H = (H t t 0 that satisfies the usual hypotheses. Suppose (T k n n ( k K are K strictly increasing 2 sequences of H-stopping times. Suppose also (G i i K i,, i K N is a family of sub -field of H = t 0 H t, so that (i G i i K G j j K, if i j,, i K j K ; (ii G i i K H T i T K i K, where T i T K i K = ma{t i,, T K i K }; (iii For any k, T k n is G j j K -measurable, if n j k. Define and F 0 t = i i K (G i i K {T k + : k K}, F t = i i K ((G i i K (N {T k + : k K}, where N is the collection of all negligible sets. Then F = (F t t 0 is a filtration that satisfies the usual hypotheses and is called a discretely delayed filtration of H. Remark. Discretely delayed filtration arises naturally from discrete observations. For instance, in Duffie and Lando (200, a firm s asset value is observed (with noise at a deterministic and discrete set of times. A natural generalization of their model is that the sequence of observation times can be random as well: the firm s asset value is updated when it issues a press release, or articles in the financial press. Moreover, there could be more than one sequence of observation times. Mathematically speaking, the above definition is the generalization of filtration generated by a marked point process defined in Jacod and Skorohod (994 or filtration of the discrete type as in He, Wang, and Yan (992. It is essentially the filtration generated by finitely many marked point processes. To see Definition 2 is well defined, we have the following lemma. Lemma. Using the notation in Definition 2, we have F 0 = (F 0 t t 0 is a right-continuous sub-filtration of H; 2 Each T k n is a stopping time of F 0 ; 3 F = (F t t 0 is the usual augmentation of F 0. T n+. 2 A sequence (T n n 0 of random variables is strictly increasing, if T 0 = 0, T n and n 0, T n < = T n < 4

5 Proof. First, we check Ft 0 is indeed a -field. (i Ω = i i K (Ω {T k t < T k + : k K} F t 0. (ii A i i K G i i K, ( i i K A i i K {T k + : k K} c = i i K ({T k + : k K} c (A c i i K {T k + : k K} = i i K (A c i i K {T k + : k K} F 0 t (iii If A (l i i K G i i K, then Therefore, Ft 0 is a -field. Moreover, for any A i i K G i i K, l i i K (A (l i i K {T k + : k K} = i i K (( l A (l i i K {T k + : k K} F 0 t. H t, A i i K {T k + : k K} H T i T k {T k + : k K} implying F 0 t H t. To see that (F 0 t t 0 is a filtration, for any s < t, let A F 0 s have the representation for some A i i K G i i K. Then i i K (A i i K {T k s < T k + : k K} ( i i K (A i i K {T k s < T k + : k K} = ( i i K (A i i K K k= {T k s < T k +} {T k j k j k + : k K} = ( i j,,i K j K (A i i K K k= {T k s < T k +}{T k j k j k + : k K} = i j,,i K j K (A i i K ik <j k {T k s < T k +} ik =j k {T k j k s} {T k j k j k + : k K} G j j K {T k j k j k + : k K}. Thus, A F 0 t, implying F 0 s F 0 t. Finally, we show that F 0 is right continuous: let h be n F 0 t+ n -measurable. Then for n, h = h (n i i K {T k ik t+ n <T k + : k K} i i K 5

6 for some h (n i i K G i i K. Define h i i K = lim sup n h (n i i K G i i K. Then for each ω {T k : k K}, there eists n(ω such that m n(ω, T k + ω {T k t + m < T k + : k K} This shows h(ω = h (n i i K G i i K (ω = h i i K G i i K (ω for n large enough. That is, h = i i K h i i K {T k ik t<t k + : k K} F 0 t, t < and Ft 0 = Ft To see that Tn k is an F 0 -stopping time, it suffices to note that {Tn k t} = i i K, n{ti l l t < Ti l l + : l K} F t 0. 3 is obvious, since the operation of completion by negligible sets and the smallest etension to get right-continuity are commutable (see, for eample, Kallenberg (2002, Lemma 7.8. Eample 2. Suppose X is a stochastic process and H = (H t t 0 is the augmented natural filtration of Y. Let (t k k 0 be a strictly increasing sequence of non-negative numbers, and (T n n 0 a strictly increasing sequence of H-stopping times. Then the minimal filtration generated by the two marked point processes (t k, X tk k 0 and (T n, X Tn n 0 is a discretely delayed filtration. For an increasing sequence (t k k 0 of deterministic times, Duffie and Lando (200 considered the discrete observation (Y tk k 0 of the process Y t = z 0 + mt + W t + U t, where W is a Brownian motion and U is a Gaussian process independent of W. This is a special case of Eample 2. Distinction between continuously and discretely delayed filtrations. It is obvious that the two types of delayed filtration defined above are different. To put the difference mathematically precise, we first introduce the notion of a genuinely stochastic process, by which we mean a stochastic process that genuinely leaves its starting position, and in a continuous way. For eample, Brownian motion is genuinely stochastic. Definition 3. Let X be a stochastic process whose state space E is a metric space (E, ρ. X is genuinely stochastic, if E and t > 0, (i the law of X t under P is diffuse; (ii P (ρ(x t, X 0 > ɛ (0, for any ɛ > 0; (iii P (ρ(x t, X 0 > ɛ is continuous in ɛ. Now we can distinguish the discretely delayed filtration from its continuous counterpart. Theorem. Let X be a genuinely stochastic strong Markov process, with H = (H t t 0 being its augmented natural filtration. If T is an unbounded H-stopping time, then the filtration F = (F t t 0, which is defined by F t = (H 0 {t < T } (H T {t T }, is a discretely delayed filtration of H, but not a continuously delayed filtration of H. 6

7 To prove Theorem, it is critical to establish the following lemma and its corollary. Lemma 2. Under the natural filtration H = (H t t 0 of a genuinely stochastic strong Markov process X, if H S H T for stopping times S and T, then S T P -a.s. for all E. Corollary. Under the natural filtration H = (H t t 0 of a genuinely stochastic strong Markov process X, stopping times S and T are equal P -a.s. for all E if and only if H S = H T. Proof of Lemma 2. By assumption, S, X S H T. By the strong Markov property, for any ɛ > 0 and any E, we have {S>T, ρ(xs, X T >ɛ} = {S>T } P (ρ(x S, X T > ɛ H T = {S>T } P X T (ρ(x u, X 0 > ɛ u=s T, P a.s. On the event {S > T }, the right hand side is between 0 and, while the left hand side is either 0 or. So we must have P (S > T = 0. This proved Lemma 2. Proof of Theorem. Obviously, F is a discretely delayed filtration of H. Suppose F is also a continuously delayed filtration of H, with the corresponding time change (α t t 0. We shall show α t = T {T t} and obtain a contradiction. Fi t > 0 and E. For any A H αt = F t, we have A {t < T } F t {t < T } = H 0 {t < T }. By Blumenthal s 0- law, if A {t < T } is not P -negligible, then it must equal to {t < T } P -a.s. Now, we shall show α t = α t {T t}. Assume not, then P (t < T, α t > 0 > 0. Therefore there eists q (0, t such that P (t < T, α t > q > 0. Define F (ɛ = P (t < T, α t > q, ρ(x q, X 0 ɛ, then F is continuous in ɛ, F (0 = 0 and F ( = P (t < T, τ > q > 0. So there eists ɛ 0, such that 0 < F (ɛ 0 < P (t < T. But {α t > q, ρ(x q, X 0 ɛ 0 } H αt = F t, so {α t > q, ρ(x q, X 0 ɛ 0, t < T } = {t < T } P -a.s. However, P (α t > q, ρ(x q, X 0 ɛ 0, t < T = F (ɛ 0 < P (t < T. This is a contradiction. We therefore conclude α t = α t {T t} P -a.s.. By the right continuity of (α t t 0, for P -a.s. ω, α t (ω = α t (ω {T (ω t}, for any t 0. Second, it s easy to see H αt = F t H T. By Lemma 2, α t T P -a.s. ( E. Note {T t} F t = H αt, so (α t {T t} is an H-stopping time. Since and H T{T t} {T > t} = H {T > t} = H (αt {T t} {T > t}, H T{T t} {T t} = H T {T t} = F t {T t} = H αt {T t} = H (αt {T t} {T t}, 7

8 we have H T{T t} = H (αt {T t}. By Corollary and α t = α t {T t}, we conclude α t = T {T t} P -a.s. for ( E. By the right-continuity of (α t t 0, we conclude E, for P -a.s. ω, α t (ω = T (ω {T (ω t}, t 0. Now for any u, t > 0 with u < t, {α t u} H u, since {α t u} = {T t, T u} {T > t} = {T u} {T > t}, and {T u} H u, we must have {T > t} H u. By letting u 0, we have {T > t} u>0 H u = H 0+. By Blumental s 0- law, T is a constant. Contradiction. 3 Delayed filtration and default intensity Given the mathematical framework in the previous section for incomplete information, this section shows how delayed filtration induces default intensity for general Markov models: both for continuous models and for models with jumps. As special eamples, we shall show the results obtained in Duffie and Lando (200 and Collin-Dufresne et. al. (2003. Intensity of a stopping time τ Mathematically, the intensity process (λ t t 0 of a stopping time τ is associated with the compensator A of τ with respect to a given filtration G, relative to which τ is a stopping time. An increasing, right-continuous and adapted process (A t t 0 is called the G-compensator of τ, if A 0 = 0, {τ t} A t is a G-martingale and A is G-predictable. The intensity process (λ t t 0 of τ is then defined as the Radon Nikodym derivative (da t /dt t 0, provided that A is a.s. absolutely continuous with respect to the Lebesgue measure. See Bremaud (98. There are a number of mathematical issues worth noting here. First, for any non-negative random variable τ and a given filtration F = (F t t 0, τ is not necessarily an F-stopping time. Thus, intensity process (λ t t 0 of a stopping time τ is associated with the epanded filtration G = (G t t 0 of F = (F t t 0, where τ is a G-stopping time. There are more than one ways to epand F, and we follow the simplest approach known as the minimal filtration epansion, as in Duffie and Lando (200. Direct computation is fairly hard ecept for some special cases via the route of Radon-Nikodym derivative. We adopt here the methodoy known as the Meyer s Laplacian approimation, also eploited by Duffie and Lando (200. The essence of this approach is to calculate the default intensity λ t by the intuitive definition P (t + h τ > t G t λ t = lim, ( h 0 h Of course, this intuitive definition of a default intensity is not the same as the mathematical definition of the Randon-Nikodym derivative of the compensator of a G-stopping time τ. Under some technical conditions, however, one can show the consistency of these two notions, see Guo, Jarrow and Menn (2006. From a computation perspective, one can first compute Eq. (, and verify it via the Aven s Lemma. (See Appendi A for description of these mathematical tools. 8

9 3. Continuous Markov processes Theorem 2. let X be a one-dimensional, time homogeneous, continuous strong Markov process with X 0 = 0. Let τ := inf{t > 0 : X t y}. Suppose τ has a density f(, y, tdt = P (τ dt, and f(, y, t is jointly continuous in, y and t. Suppose further that the time change (α t t 0 satisfies the following conditions: for a.s. ω, T > 0, δ(t = δ(ω, T > 0, such that t [0, T ], (i α t (ω = 0, if t δ(ω, T ; (ii t α t (ω δ(ω, T, if t > δ(ω, T. Define F = (F t t 0 = (H αt t 0, where H is the augmented natural filtration of X. The minimal epansion of F will be G = (G t t 0 with G t = F t ({τ t} : s t. Then, under G, the intensity of τ is f(x αt, y, t α t P Xα t (τ > u u=t αt {τ>t}. In particular, τ is a totally inaccessible G-stopping time. Before proceeding to the proof, a few remarks are in place: condition (i means a delay from the beginning; condition (ii requires that the delay does not vanish during a finite amount of time. Proof. Without loss of generality, we assume the probability space on which X is defined is the canonical space C[0, of continuous functions. So {ω : τ(ω > t} = {ω : τ(ω > α t (ω, τ θ αt(ω(ω > t α t (ω}. By the strong Markov property, we have P (τ > t F t = P (τ > t H αt = {τ>αt}p Xα t (τ > u u=t αt, and So, P (τ > t + h F t = P (τ > t + h H αt = {τ>αt}p Xα t (τ > u u=t+h αt. h P (t < τ t + h G t = P (t < τ t + h F t h P (τ > t F t {τ>t} = h P Xα t (u < τ u + h {τ>t} P Xα u=t αt t (τ > u λ t := f(x α t, y, t α t {τ>t} P Xα u=t αt as h 0. t (τ > u To show λ t is indeed an intensity process for τ, it suffices to check Aven s conditions. And without loss of generality, we assume that h, so that h P (t < τ t + h G t λ t u+h u f(x αt, y, s f(x αt, y, u ds h {τ>t} P Xα u=t αt t (τ > u sup t α t s t α t+ f(x αt, y, s f(x αt, y, t α t P Xα t (τ > u u=t αt := y t. 9 {τ>t}

10 We now show for a.s. ω, T > 0, T 0 y t(ωdt <. The measurability of (y t t 0 is clear as (y t t 0 is G-adapted and right continuous, hence is optional. Net, let ω be such that it satisfies conditions (i-(ii and α t (ω t for any t 0. For any t [0, T ], if t δ(t, then α t = 0, and If t > δ(t, then it is not hard to verify that where y t 2 sup s T + f( 0, y, s P 0 (τ > T. y t 2 sup s T +, [min s τ δ(t X s,ma s τ δ(t X s] f(, y, s P Y (T, (2 (τ > T Y (T := {X t : t τ δ(t, X t y = Indeed, note τ > t α t τ δ(t, so on the event {τ > t}, and min X s y }. s τ δ(t P Xα t (τ > u u=t αt P Y (T (τ > T, (3 sup f(x αt, y, s f(x αt, y, t α t t α t s t α t+ 2 sup f(, y, s. s T +, [min s τ δ(t X s,ma s τ δ(t X s] Combining the above two cases with the joint continuity of f in (, y, t, it is clear that for a.s. ω, y t (ω is bounded on [0, T ]. So T 0 y tdt < a.s.. In summary, τ has an intensity process λ t = f(x α t, y, t α t P Xα t (τ > u u=t αt {τ>t}. A special case of the above theorem is the following well-known result, which appeared in Collin-Dufresne et. al. (2003. Eample 3. Suppose X is given by the Black-Scholes geometric Brownian motion model with parameters µ,. The delayed filtration F = (W t,, W tk is epanded to G = (W t,, W tk (t τ, then on {τ > t}, the default intensity of G -stopping time τ at time t [t k, t k+ is ψt( µ 2,t t k, X tk ψ( µ 2,t t k, X t k. More generally, if X satisfies a one-dimensional stochastic differential equation dx t = (X t dw t + b(x t dt where b and satisfy some regularity conditions, then for τ y (the first-passage time of X hitting the level y, there is a density P (τ y dt = f(t,, ydt such that f(t,, y is jointly continuous (see Pauwels (987 for details. Hence, the eistence of the corresponding default intensity is implied by our result. 0

11 3.2 Diffusion process with jumps Suppose that the asset value process X follows a jump-diffusion model so that X t = X 0 e (µ 2 2 t+w t ξ ɛ(s, 0<s t, ɛ(s 0 where W is a standard one-dimensional Brownian motion, ɛ is a finite-state continuous-time Markov chain, independent of W and taking values 0,,, S with a known generator (q ij S S, and ɛ(s := ɛ(s ɛ(s. We set q i = j i q ij, and assign to each state i of ɛ (0 i S a positive random variable ξ i with distribution function F i. Finally, we assume that (ξ i i, ɛ and W are all independent. In particular, if ɛ is a standard Poisson process, then X satisfies the SDE dx t ɛ t = µdt + dw t + d( (ξ i. X t Theorem 3. Define the discretely delayed filtration F to be the augmented minimal filtration generated by the (marked point processes {τ t}, n=0 X t k {tk t}, and (ξ ɛ(tn, X Tn, T n n. Here T n is the n-th jump time of ɛ. Take G to be its minimal filtration epansion. Then if τ > t, t k t < t k+ and T n t < T n+, the corresponding default intensity of τ is where θ = µ 2, ψ t (θ, t t k T n, d t = ψ(θ, t t k T n, j ɛ(t X tk X tk + i= 0 F j(dzφ(θ, t t k T n, X tk q ɛ(tj ψ(θ, t t k T n, X tk, zx tk t ψ(θ, t, y = P ( inf W s (θ y (y θs2 > y = e 2s ds for y < 0, 0 s t 0 2πs 3 with W (θ t := W t + θt, and ψ t being the derivative of ψ w.r.t. the variable t. For y y 2, φ(θ, t, y, y 2 = P (inf W s (θ > y, W (θ t y 2 s t = Φ( y 2 θt Φ( y θt e 2θy t t [ Φ( y 2 2y θt t with Φ( being the distribution of a standard normal random variable. 3.3 Regime switching process Φ( y ] θt, t, (4 In a regime-switching model, the asset value process X is assumed to follow a diffusion process given by dx t = X t µ ɛ(t dt + ɛ(t X t dw t,

12 where W is a standard one-dimensional Brownian motion, and ɛ is a finite-state continuous-time Markov chain, independent of W and taking values 0,,, S with a known generator (q ij S S. Finally, the drift and volatility coefficients, µ( and (, are functions of ɛ. We consider the discretely delayed filtration F = (F t t 0 generated by the point process k= X t k {t tk } and the marked point process (ɛ(t n, X Tn, T n n. Let be a constant and τ = inf{t > 0 : X t < }. We epanded F to G so that G t = F t (τ t. Theorem 4. The stopping time τ is totally inaccessible under the filtration G. Moreover, given any t [t k, t k+, if τ > t and T n t < T n+, the default intensity of τ is where ψ t (θ ɛ(t, t t k T n, ɛ(t ψ(θ ɛ(t, t t k T n, ɛ(t X tk X tk, t ψ(θ, t, y = P ( inf W s (θ y (y θs2 > y = e 2s ds for y < 0, 0 s t 0 2πs 3 with W (θ t := W t + θt, and ψ t being the derivative of ψ w.r.t. the variable t. 4 Risky Debt Pricing: relation between complete information and delayed information Consider a zero-coupon bond issued by the firm paying $ at time T if there is no default, and $R at time T if the firm defaults prior to time T. Here R(t, T [0, ] is called the face value of debt recovery rate process (see Jarrow and Turnbull (995. Similar results hold for other recovery rate processes (see Bielecki and Rutkowski (2002 for the relevant alternatives. For simplicity, we assume that the interest rate process is deterministic. Now we investigate the relation between values of defaultable zero recovery rate, zero-coupon bond, under the two different filtration structures, complete and delayed. To this end, let us start with a general and multi-dimensional Markov setting. Let X = (X t t 0 be a Markov process under a risk-neutral measure Q, with a general state space E. Let U be a subset of E and τ := inf{t > 0 : X t U}. 3 We denote by H = (H t t 0 the natural filtration of X. Suppose the maturity date of the bond is T, then it s easy to see from the Markov property that Proposition. Let {t k } k 0 be a strictly increasing sequence of non-negative numbers (t 0 = 0, such that t k. Under the filtration for t k t < t k+, we have G t := (X t,, X tk (t τ V B (t, T = {τ>t} V A (t k, T V A (t k, t. Here V A (t k, T = Q(τ > T H t and V B (t k, T = Q(τ > T G t. 3 Here, X t corresponds to the asset value process, plus any additional processes needed for the interest rate and recovery rate process. 2

13 Proof. Suppose t [t k, t k+, then V B (t, T = {τ>t} Q(τ > T X t,, X tk Q(τ > t X t,, X tk = {τ>t} Q( {τ>tk }Q(X s U, s [t k, T ] H tk X t,, X tk Q( {τ>tk }Q(X s U, s [t k, t] H tk X t,, X tk Q(X s U, s [t k, T ] X tk = {τ>t} Q(X s U, s [t k, t] X tk = V A (t k, T {τ>t} V A (t k, t. Note first that investors A and B have different valuations, ecept at time t i (0 i k. That is, V A (t i, T = V B (t i, T, because at time t i both investors have the same information. Secondly, we do not necessarily have V A (t, T > V B (t, T or V A (t, T < V B (t, T. Indeed, when t k < t, the price for investor B is based on the dated information X tk as compared to that of investor A. This dated information could generate a higher or lower price, depending upon the complete information known only to investor A. Finally, as t t k, V B (t, T V A (t, T, i.e., as investor B s information is updated, his price again agrees with investor A. This relationship is independent of the risk-neutral measure under consideration. References [] M. Ammann. Credit Risk Valuation. Springer, (200. [2] P. Artzner and F. Delbaen. Default risk insurance and incomplete market. Mathematical Finance, 5, 87-95, (995. [3] T. Aven. A theorem for determining the compensator of a counting process. Scandinavian Journal of Statistics, 2, 69-72, (985. [4] A. Bélanger, S. Shreve, and D. Wong. A general framework for pricing credit risk. Mathematical Finance, 4(3, , (2004. [5] T. Bielecki and M. Rutkowski. Credit Risk: Modeling, Valuation, and Hedging. Springer, (2002. [6] F. Black and J. Co. Valuing corporate securities: Some effects of bond indenture provisions. Journal of Finance, 3, , (976. [7] F. Black and M. Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 8, , (973. [8] C. Blanchet-Scalliet and M. Jeanblanc. Hazard rate for credit risk and hedging defaultable contingent claims. Finance and Stochastics, 8, 45-89, (2004. [9] U. Çetin, R. Jarrow, P. Protter, and Y. Yildirim. Modeling credit risk with partial information. Annals of Applied Probability, 4, 67-78, (

14 [0] S. Chava and R. Jarrow. Bankruptcy prediction with industry effects, Review of Finance, 8(4, , (2004. [] P. Collin-Dufresne, R. Goldstein, and J. Helwege. Is credit event risk priced? Modeling contagion via the updating of beliefs. Working paper, Carnegie Mellon University, (2003. [2] Surveys confirm rapid market growth. Creditflu, issue 38, October, (2004. [3] G. R. Duffee. Estimating the price of default risk. The Review of Financial Studies, 2(, , (999. [4] D. Duffie and D. Lando. Term structures and credit spreads with incomplete accounting information. Econometrica, 69, , (200. [5] D. Duffie, S. Schroder, and C. Skiadas. Recursive valuation of defaultable securities and the timing of resolution of uncertainty. Annals of Applied Probability, 6, , (996. [6] D. Duffie and K. Singleton. Modeling term structure of defaultable bonds. Review of Financial Studies, 2, , (999. [7] D. Duffie and K. Singleton, Credit Risk. Princeton University Press, (2003. [8] D. Duffie and K. Wang. Multiperiod corportate failure prediction with stochastic covariates. Working paper, Stanford University, (2003. [9] R. Elliott, M. Jeanblanc, and M. Yor. On models of default risk. Mathematical Finance, 0, 79-95, (2000. [20] K. Giesecke and L. Goldberg. Forecasting default in the face of uncertainty. Journal of Derivatives, 2(, 4-25, (2004. [2] S. W. He, J. G. Wang and J. A. Yan. Semimartingale Theory and Stochastic Calculus. CRC Press, Boca Raton, (992. [22] J. Jacod and A. V. Skorohod. Jumping filtrations and martingales with finite variation. Séminaire de Probabilités XXVIII, 2-35, (994. [23] R. Jarrow, D. Lando, and S. Turnbull. A Markov model for the term structure of credit risk spread. Review of Financial Studies, , (997. [24] R. Jarrow and P. Protter. Structural versus reduced-form models: A new information based perspective. Journal of Investment Management, 2(2, 34-43, (2004. [25] R. Jarrow and S. Turnbull. Credit Risk: Drawing the anay. Risk Magazine, 5 (9, (992. [26] R. Jarrow and S. Turnbull. Pricing options of financial securities subject to default risk. Journal of Finance, 50, 53-86, (995. [27] M. Jeanblanc and M. Rutkowski. Default risk and hazard processes. Mathematical Finance - Bachelier Conference 2000, Springer, (2002. [28] O. Kallenberg. Foundations of Modern Probability. Springer, 2nd edition, (

15 [29] I. Karatzas and S. Shreve. Stochastic Calculus and Brownian Motion. Springer, 2nd edition, (99. [30] S. Kusuoka. A remark on default risk models. Advances in Mathematical Economics,, 69-82, (999. [3] D. Lando. Co processes and credit-risky securities. Review of Derivative Research, 2, 99-20, (998. [32] D. Lando. Credit Risk Modeling: Theory and Applications. Princeton University Press, New Jersey, (2004. [33] F. Longstaff and E. Schwartz. A simple approach to valuing risky fied and floating rate debt. Journal of Finance, 50, , (995. [34] P. A. Meyer. Probability and Potentials, Blasisdell Publishing Company, Massachusetts, (966. [35] D. Madan and H. Unal. Pricing the risk of default. Review of Derivative Research, 2, 2-60, (995. [36] R. Merton. On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29, , (974. [37] E. J. Pauwels. Smooth first-passage densities for one-dimensional diffusions. Journal of Applied Probability, 24, (987. [38] L. C. G. Rogers. Modelling credit risk. Preprint, Univeristy of Bath, (999. [39] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales, Volume 2: Itô Calculus. Cambridge University Press, Cambridge, 2nd edition, (2000. [40] P. Schonbucher. Information driven default contagion. Working paper, ETH Zurich, (2003. [4] P. Protter. Stochastic Integration and Differential Equations. Springer, New York, 2nd edition, (2004. [42] C. Zhou. A jump-diffusion approach to modeling credit risk and valuing defaultable securities. Technical report. Federal Research Board, Washington DC, ( Appendi A Theorem 5. (Meyer s Laplacian Approimation Let Z be a càdlàg positive supermartingale of the class (D (i.e., the set {Z τ : T is a stopping time τ satisfying T < } is uniformly integrable, with lim t E{Z t } = 0. Let Z = M A be its Doob-Meyer decomposition. Define A h t = t 0 Z s E{Z s+h F s } ds. h Then for any stopping time T, A T = lim h 0 A h T, in the sense of the weak topoy (L, L. Proof. See Meyer (966, page 9, T29. 5

16 Lemma 3. (T. Aven (985 Let (Ω, F, (F t t 0, P be a filtered probability space that satisfies usual hypothesis. Let (N t t 0 be a counting process. Assume that E{N t } < for all t. Let {h n } n be a sequence which decreases to zero and let for each n (Yt n t 0 be a measurable version of the process (E{N t+hn N t F t }/h n t 0. Assume that the following statements hold with (λ t t 0 and (y t t 0 being non-negative measurable processes: (i for each t, lim n Yt n = λ t a.s.; (ii for each t, there eists for almost all ω an n 0 = n 0 (t, ω such that Y n s (ω λ s (ω y s (ω, s t, n n 0 (iii t 0 y sds <, a.s. 0 t <. Then N t t 0 λ sds is an F t -martingale, i.e.,( t 0 λ sds t 0 is the compensator of (N t t Appendi B: Proof of Theorem 4 Proof. For ease of eposition, let us denote by F k,n the -field generated by W t,, W tk, T,, T n, ɛ(t,, ɛ(t n, and W T,, W Tn. We further denote P ( F(y k, u n, n, v n for P ( W t = y,, W tk = y k, T = u,, T n = u n, W T =,, W Tn = n, ɛ(t = v,, ɛ(t n = v n. Note on the event {τ > t, T n t < T n+ }, when T,, T n and ɛ(t,, ɛ(t n are known, there is a one-to-one correspondence between (W t,, W tk, W T,, W Tn and (X t,, X tk, X T,, X Tn. We denote by i the value of X T i ( i n and by y j the value of X t j ( j k given F(y k, u n, n, s n. Finally, we let θ i = µ i i i 2 (0 i S, Xi t = ep{(µ i 2 i 2 t + i W t } (0 i S, and (F W t t 0 be the natural filtration of W. Note for t [t k, t k+, one has on the event {τ > t, T n+ > t T n } P (t + h τ > t G t = P (τ > t + h G t = P (τ > t + h, T n+ > t T n F k,n, P (τ > t, T n+ > t T n F k,n by the Bayes formula and the fact that G t {τ > t, T n t < T n+ } = F k,n {τ > t, T n t < T n+ }. Since W is independent of ɛ, by conditioning on Ft W k or Fu W n and by the strong Markov property, we have 6

17 = P (τ > t, T n+ > t T n F(y k, u n, n, v n X s (vi u i s<u i+ X (v i X (vn s X (vn u n > F(y k, u n, n, v n = P ( i inf >, i n, n inf u u n s t i P (T n+ > t T n = u n, ɛ(t n = v n if u n t k, P ( X s (vi i inf u i s u i+ X (v >, i n F(y k, u n, n, v n i u i ψ(θ vn, t u n, if u n < t k, P ( X s (vi i inf u i s u i+ vn e qvn(t un n X (v i u i >, i n, n inf X s (vn u n s t k X (vn u n F(y k, u n, n, v n ψ(θ vn, t t k, vn y k e qvn (t u n. By splitting {T n+ > t T n } into {T n+ > t + h > t T n }, {T n+2 > t + h T n+ > t T n } and {t + h T n+2 > T n+ > t T n }, we get where P (τ > t + h, T n+ > t T n F tk,t n,w Tn P (τ > t, T n+ > t T n F tk,t n,w Tn ψ(θ = e q v n, t + h t k T n, ɛ(tnh vn ψ(θ vn, t t k T n, vn X tk X tk + I + II, I = P (τ > t + h, T n+2 > t + h T n+ > t T n F k,n, P (τ > t, T n+ > t T n F k,n > II = P (τ > t + h, t + h T n+2 > T n+ > t T n F k,n. P (τ > t, T n+ > t T n F k,n The calculation of I is similar to the previous one, ecept that we need to proceed the calculation 7

18 conditioned on T n+ and ɛ(t n+ : = = P (τ > t + h, T n+2 > t + h T n+ > t T n F(y k, u n, n, v n t+h P ( X s (vi X (v n+ s i inf t u i s u i+ X (v >, 0 i n, X un+ inf i u u n+ s t+h i X (v >, n+ u n+ v n+ v n T n+2 > t + h F(u k, u n, n, v n, T n+ = u n+, ɛ(t n+ = v n+ P (T n+ du n+, ɛ(t n+ = v n+ F(y k, u n, n, v n if u n t k, P ( X s (vi i inf u i s u i+ X (v >, i n F(y k, u n, n, v n i u i t+h q vnvn+ e qv n+ (t+h un+ q vn e qvn(u n+ u n q vn v n+ v n t E[ψ(θ vn+, t + h u n+, vn+ v n W (θvn u n n+ u n ; vn+ inf W (θvn s > 0 s u n+ u n vn ]du n+ n if u n < t k, P ( X s (vi i inf u i s u i+ X (v >, i n, i n inf u i F(y k, u n, n, v n v n+ v n t+h t X s (vn u n s t k X (vn u n > q vnv n+ q vn e qv n+ (t+h u n+ q vn e qvn(u n+ u n E[ψ(θ vn+, t + h u n+, vn+ y k v n W (θvn u n+ t k ; vn+ inf W (θvn s > 0 s u n+ t k vn y k ]du n+. Therefore I is j ɛ(t n t+h t q ɛ(tnje q j(t+h u n+ e q ɛ(tn(u n+ T n ψ(θ ɛ(tn, u n+ t k T n, ( ψ(θ ɛ(tn, t t k T n, ɛ(tn X tk. e q ɛ(tn(t T n ψ(θ ɛ(tn, t t k T n, X tk ɛ(tn As h 0, the first term tends to j ɛ(t n q ɛ(t nj = q ɛ(tn. For the second term, notice ( h 0 j v n q vnje[ ψ(θ j, h u, j z vn j W (θvn u+t α ; z < v n W (θvn u+t α ]du, X tk du ɛ(tn n+ where v n = ɛ(t n, z = X tk T n, and α = t k T n. ( epectation is a continuous function of u, so lim h 0 h By the bounded convergence theorem, the = 0. Moreover, it is clear that ( q ɛ(t nh. 8

19 Similarly, to calculate II, recall P (τ > t + h, t + h T n+2 > T n+ > t T n F(y k, u n, n, v n P ( i inf X s (vi u i s u i+ X (v i u i >, i n, n inf u n s t X (vn s X (vn u n > F(t k, u n, n, v n Ch 2, for some constant C. Therefore II Ch 2. In summary, the above calculations show that, on the event {τ > t, T n t < T n+, t k t < t k+ }, ψ t (θ ɛ(tn, t t k T n, lim h 0 h P (t + h τ > t F ɛ(tn t = ψ(θ ɛ(tn, t t k T n, ɛ(tn X tk X tk. Finally, we check that Aven s lemma holds: when τ > t, T n t < T n+ and t k t < t k+, for h, h P (t + h τ > t F t d t ( ψ(θ e q ɛ(t n, t + h t k T n, ɛ(tnh ɛ(tn X tk I II ψ(θ ɛ(tn, t t k T n, ɛ(tn X tk h + d t t t+h ψ s(θ ɛ(tn, s t k T n, ɛ(tn X tk ds hψ(θ ɛ(tn, t t k T n, ɛ(tn X tk + d t ψ(θ ɛ(tn, t + h t k T n, + q ɛ(tn X tk ɛ(t n ψ(θ ɛ(tn, t t k T n, ɛ(tn X tk + I h + Ch ( 2C ψ(θ ɛ(tn, T n+ t k T n, ɛ(tn X tk + 2q ɛ(t n + C 2, where C is the maimum of ψ t (θ ɛ(tn,, X t k ɛ(tn, and C 2 is a constant. This shows the conditions of Aven s lemma are satisfied. 9