RATING BASED LÉVY LIBOR MODEL

Transcription

1 RATING BASED LÉVY LIBOR MODEL ERNST EBERLEIN AND ZORANA GRBAC Abstract. In this paper we consider modeling of credit risk within the Libor market models. We extend the classical definition of the defaultfree forward Libor rate to defaultable bonds with credit ratings and develop the rating based Libor market model. As driving processes for the dynamics of the default-free and the pre-default term structure of Libor rates time-inhomogeneous Lévy processes are used. Credit migration is modeled by a conditional Markov chain, whose properties are preserved under different forward Libor measures. Conditions for absence of arbitrage in the model are derived and valuation formulae for some common credit derivatives in this setup are presented. 1. Introduction Due to the recent credit crisis, interest rate markets have experienced some dramatic changes and a number of anomalies have appeared. In particular, the Libor rates that have always been assumed to be essentially default-free rates, in these days reflect also the credit risk of the interbanking sector see recent papers by Mercurio 29, Morini 29, Henrard 29, and many others. However, in the present literature there exist many defaultable extensions of the Heath Jarrow Morton HJM framework for modeling of the term structure of instantaneous continuously compounded forward rates, whereas credit risk within the Libor market models seems to be far less studied. To mention just some of the papers proposing the defaultable HJM models, we begin with Bielecki and Rutkowski 2, 24, who introduced an extension of the Gaussian HJM model to defaultable bonds with credit migration. Eberlein and Özkan 23 developed the defaultable HJM model with credit migration based on Lévy processes. More recently, Özkan and Schmidt 25 and Jakubowski and Nieweglowski 29a consider infinite dimensional Lévy processes for credit risk modeling within the HJM framework. On the other side, the first extension of the log-normal Libor model to defaultable contracts is due to Lotz and Schlögl 2, who use a deterministic hazard rate to model the time of default. Schönbucher 2 extended the log-normal Libor model by adding defaultable forward Libor rates to the model for default-free Libor rates. Following his ideas, Eberlein, Kluge, and Schönbucher 26 constructed the Lévy Libor model with default risk, driven by time-inhomogeneous Lévy processes. None of these models takes into account that in markets subject to credit risk, there usually exists a multitude of credit rating classes. A detailed account on different approaches to Key words and phrases. credit risk, ratings, time-inhomogeneous Lévy process, Libor, conditional Markov chain. 1

2 2 ERNST EBERLEIN AND ZORANA GRBAC credit risk modeling can be found in Bielecki and Rutkowski 22, Lando 24, Duffie and Singleton 23 and McNeil, Frey, and Embrechts 25, Chapters 8 and 9. In this paper we develop an arbitrage-free model for defaultable forward Libor rates related to defaultable bonds with credit ratings. As driving processes time-inhomogeneous Lévy processes are used. We call this model the rating based Lévy Libor model. The modeling objects in any Libor market model are the discretely compounded forward Libor rates, whose dynamics are modeled under forward martingale measures with maturities corresponding to the tenor structure. To develop the rating based Libor model, we begin by constructing a family of default-free forward Libor rates and forward measures, according to the Lévy Libor model Eberlein and Özkan 25. In addition, we model the pre-default term structure, i.e. we specify the dynamics of the forward Libor rates for every rating class. These rates are not modeled directly, instead the modeling objects are the inter-rating spreads, which are assumed to evolve as exponential semimartingales driven by time-inhomogeneous Lévy processes. By specifying the inter-rating spreads as positive processes, we ensure automatically that higher interest rates correspond to worse credit ratings, thus reflecting the increased investment risk. Credit migration of a defaultable bond is modeled by a conditional Markov chain with a finite number of states representing different rating classes. This process is constructed in a canonical way by enlarging the reference probability space which carries the default-free information. Due to this canonical construction and the fact that any two forward measures are related via the Radon Nikodym density process that is adapted to the reference filtration, we are able to show that the conditional Markov property is preserved under all forward measures. Moreover, we prove that the progressive enlargement of the default-free reference filtration with the natural filtration of the conditional Markov chain has the immersion property under all forward measures, i.e. local martingales with respect to the reference filtration remain local martingales with respect to the enlarged filtration. The paper is structured as follows. In Section 2 we introduce the setting and the main ingredients for rating based Libor modeling, in particular we introduce the defaultable and the rating-dependent forward Libor rates and associated spreads. Section 3 presents a detailed construction of the pre-default term structure of the rating-dependent Libor rates under corresponding forward measures. The credit migration between rating classes is introduced in Section 4, using the classical conditional Markov chain approach, which is in this paper adapted to the modeling directly under forward measures. In Section 5 we put all these building blocks together and derive necessary and sufficient conditions for the absence of arbitrage in the model. Finally, Section 6 is devoted to the valuation of credit derivatives in the rating based Libor model. We derive expressions for the price of a defaultable bond and a credit default swap. Furthermore, we introduce the defaultable forward measures which are useful tools for valuation of interest rate derivatives such as forward rate agreements, swaps and caps/floors

3 RATING BASED LÉVY LIBOR MODEL 3 on the defaultable Libor rate. As an example we provide a formula for the defaultable Libor rate caps. 2. Definitions and notation Let us consider a fixed time horizon T and a discrete tenor structure = T <... < T n = T, where δ k = T k1 T k, for k =,..., n 1. Assume that default-free and defaultable zero coupon bonds with maturities T 1,..., T n are traded in the market. We denote by Bt, T k and B C t, T k the time-t prices of a default-free and a defaultable zero coupon bond with maturity T k, respectively. Note that BT k, T k = 1 and B C T k, T k 1, as the defaultable bond may default before maturity. Moreover, we assume that the defaultable bond is rated, i.e. at each time point it has a certain credit rating that reflects the credit quality of its issuer. The migration between various classes of a credit rating system will be described by a stochastic process C; the subscript C in the defaultable bond price emphasizes its dependence on the credit migration process. The credit ratings are identified with elements of a finite set denoted by K = {1, 2,..., K}, where 1 stands for the best possible rating and K corresponds to the default event. The process C is assumed to be a continuoustime conditional Markov chain with state space K. The default state K is an absorbing state for C and the default time τ is modeled as the first time when C reaches this state, i.e. τ = inf {t > : C t = K}. We assume C K a.s. A defaultable bond pays to its holder 1 unit of cash at maturity only if default does not occur before that date. In case of default, the holder of the bond receives a reduced payment called the recovery payment. There exist several different recovery schemes describing the amount and timing of the recovery payment for a detailed overview see Bielecki and Rutkowski 22, Sections and or McNeil, Frey, and Embrechts 25, Section In this work we adopt the fractional recovery of treasury value scheme: in case of default prior to maturity, a fixed fraction of the face value of the bond is paid to the bond holder at the maturity date. This fraction depends on the rating class from which the bond has defaulted and is represented by a vector q = q 1, q 2,..., q of recovery rates, where q i [, 1] for every i. Therefore, the payoff of the defaultable bond at maturity is given by B C T k, T k = 1 {τ>tk } 1 {τ Tk }q Cτ = 1 {CTk =i} 1 {CTk =K}q Cτ, where C τ denotes the pre-default rating. The defaultable bond price process B C t, T k t Tk can be written as B C t, T k = B i t, T k 1 {Ct=i} q Cτ Bt, T k 1 {Ct=K}, 1

4 4 ERNST EBERLEIN AND ZORANA GRBAC where B i t, T k denotes the pre-default price of the defaultable bond at time t given that the bond is in the rating class i during the time interval [, t], where i {1,..., K 1}. We have B i T k, T k = 1, for each i. Our goal is to build up in this discrete tenor setting a model for the evolution of discretely compounded forward interest rates related to defaultable bonds. Let us first recall that the default-free forward Libor rate at time t T k for the accrual period [T k, T k1 ] is defined as Lt, T k := 1 δ k Bt, Tk Bt, T k1 1 In addition, we introduce the concept of a discretely compounded forward interest rate related to defaultable bond prices. The idea is to generalize the above definition by using the defaultable bond prices instead of the defaultfree bond prices. For a detailed discussion on this concept we refer to Bielecki and Rutkowski 22, Section , page 431, where a defaultable forward rate agreement FRA which yields this rate is described. We call it the defaultable forward Libor rate. The default risk in this context means the risk of default of the underlying instrument. It does not mean the counterparty credit risk. The defaultable forward Libor rate is a rate that one can contract for at time t T k, on a defaultable forward investment of one unit of cash from T k to T k1. The settlement scheme prescribes that default prior to the reset date T k of the FRA results only in the reduction of the principal value and the contract then becomes default-free. The defaultable forward Libor rate is defined at time t T k for the accrual period [T k, T k1 ] as. L C t, T k := 1 δ k BC t, T k B C t, T k Note that it depends on the present state C t of the migration process. Furthermore, making use of the bond price process B i, T k, we define the forward Libor rate for rating class i at time t T k for the accrual period [T k, T k1 ] by L i t, T k := 1 Bi t, T k δ k B i t, T k1 1, 3 for each i = 1,..., K 1. The discrete-tenor forward inter-rating spreads between two rating classes are given by H i t, T k := L it, T k L i 1 t, T k, i = 1, 2,..., K 1, 4 1 δ k L i 1 t, T k where we set L, T k := L, T k. Combining 3 and 4 we establish the following connection between the inter-rating spreads and the bond prices H i t, T k = 1 Bi t, T k B i 1 t, T k1 1 δ k B i 1 t, T k B i t, T k1. 5 Remark 2.1. Observe that the quantities H i t, T k represent the discretetenor analogs of the inter-rating spreads g i t, T g i 1 t, T in the defaultable HJM framework, i.e. the differences between instantaneous continuously compounded forward rates for rating classes i and i 1 see Bielecki and

5 RATING BASED LÉVY LIBOR MODEL 5 Rutkowski 22, page 46. In the HJM framework, the bond price B i t, T k is given by the following formula B i t, T k = exp T k t g i t, sds, where g i t, s is the instantaneous forward rate for the rating class i = 1,..., K 1. Inserting this into 5 yields H i t, T k = 1 exp Tk1 T k g i t, sds δ 1 k Tk1 exp T k g i 1 t, sds T = 1 k1 exp g i t, s g i 1 t, sds 1 δ k 1 δ k T k1 T k g i t, s g i 1 t, sds. T k Therefore, H i t, T k can be thought of as the average inter-rating spread over the interval [T k, T k1 ], which explains why we refer to it as the discrete-tenor inter-rating spread. 3. Pre-default term structure of Libor rates Our goal is to develop an arbitrage-free model for the evolution of defaultable forward Libor rates. In order to do so, we are going to specify the pre-default term structure of rating-dependent Libor rates L i, T k for each credit rating i, where i {1,..., K 1}. We require that Lt, T k L 1 t, T k L t, T k to reflect the empirical fact that higher interest rates correspond to worse credit ratings, as a compensation for the increased investment risk. Making use of relation i 1 δ k L i t, T k = 1 δ k Lt, T k 1 δ k H j t, T k, 6 which follows from 4, we choose not to model the Libor rates L i, T k directly. Instead, we model the forward inter-rating spreads H j, T k as positive processes and therefore, by 6, ensure automatically the monotonicity of Libor rates with respect to credit ratings. To model the default-free Libor rates L, T k we shall adopt the Lévy Libor model of Eberlein and Özkan 25. Our construction is presented below in detail The driving process. Let Ω, F = F T, F = F t t T, P T be a complete stochastic basis and let X t t T be an -valued time-inhomogeneous Lévy process, also known as PIIAC process with independent increments and absolutely continuous characteristics. For a precise definition and main properties of these processes we refer the reader to Eberlein and Kluge 26 and Eberlein, Jacod, and Raible 25. We assume that the j=1

6 6 ERNST EBERLEIN AND ZORANA GRBAC filtration F is the completed, natural filtration of X. The probability measure P T plays the role of the forward measure associated with the terminal tenor date T. The triplet of local semimartingale characteristics of X is denoted by b t, c t, F T t t T and we make the following Assumption SUP. The triplets b t, c t, F T t satisfy sup b t c t x 2 1F T t dx < t T and there exist constants M, ε > such that sup exp u, x F T t dx < t T x >1 for every u [ 1 εm, 1 εm] d. The definition of the local characteristics of a semimartingale, as well as other results from general semimartingale theory that we use throughout the paper are taken from Jacod and Shiryaev 23, whose notation we adopt. Other books such as Protter 24 or Métivier 1982 can also be used as references for semimartingale theory. Note that Assumption SUP implies the existence of exponential moments of X cf. Lemma 6 in Eberlein and Kluge 26. It also makes X a special semimartingale with the following canonical representation X t = t cs dw T s T xµ ν ds, dx, where W T denotes a standard Brownian motion with respect to P T, µ is the random measure of jumps of X and ν T ds, dx = F T s dxds is the compensator of µ. Note that we assumed that X is driftless, i.e. b =, as the drift term will be included separately in the model The default-free Lévy Libor model. Here we outline briefly the construction of the default-free Lévy Libor model. For details we refer to Eberlein and Özkan 25. The model is driven by a time-inhomogeneous Lévy process and is built up using backward induction a standard procedure for Libor market models; see the seminal papers by Miltersen, Sandmann, and Sondermann 1997, Brace, G atarek, and Musiela 1997 and Musiela and Rutkowski The following assumptions are made: L.1 For every T k there is a deterministic, Borel measurable function σ, T k : [, T ], which represents the volatility of the forward Libor rate L, T k. We assume that n 1 σ j s, T k M, k=1 for all s [, T ] and every coordinate j {1,..., d}, where M > is the constant from Assumption SUP. If s > T k, then σs, T k =.

7 RATING BASED LÉVY LIBOR MODEL 7 L.2 The initial term structure B, T k is strictly positive and strictly decreasing in k. The backward induction is started by specifying the dynamics of the most distant Libor rate L, T n 1 under P T. In each step of the construction a new forward measure P Tk1 is constructed and the next Libor rate is then specified under this measure as Lt, T k = L, T k exp b L s, T k ds with initial condition L, T k = 1 B, Tk δ k B, T k1 1. σs, T k dx T k1 s 7 The drift term b L s, T k is chosen to make L, T k a P Tk1 - martingale, i.e. b L s, T k = 1 2 σs, T k, c s σs, T k 8 e σs,tk,x 1 σs, T k, x F T k1 s dx. The process X T k1 is obtained from the driving process X in such a way that it is driftless under the forward measure P Tk1 associated with the tenor date T k1. More precisely, the measure P Tk1 is given by dp Tk1 = B, T n 1 1 δ j Lt, T j = B, T Bt, T k1 dp T B, T k1 B, T k1 Bt, T, 9 Ft and X T k1 where j=k1 is a special semimartingale with canonical decomposition X T k1 t = W T k1 t := W T t t cs dw T k1 s cs n 1 j=k1 T xµ ν k1 ds, dx, 1 ls, T j σs, T j ds 11 is a standard d-dimensional Brownian motion with respect to P Tk1 ν T k1 ds, dx := n 1 j=k1 =: F T k1 s dxds and βs, x, T j ν T ds, dx 12 is the P Tk1 -compensator of µ. Here we used for short βs, x, T j := 1 ls, T j e σs,tj,x 1 13 with ls, T j := δ jls, T j 1 δ j Ls, T j. 14

8 8 ERNST EBERLEIN AND ZORANA GRBAC This construction guarantees that the discounted processes B,T j B,T k are martingales with respect to the forward measure P Tk for all j, k. The default-free Libor model is thus free of arbitrage and the time-t price πt Y of a contingent claim with payoff Y at maturity T k is given by π Y t = Bt, T k E PTk [Y F t ] The pre-default term structure of rating-dependent Libor rates. In this subsection we proceed by modeling the pre-default term structure of the rating-dependent Libor rates, or equivalently of the forward inter-rating spreads. We make the following additional assumptions: RL.1 For every rating class i {1,..., K 1} and every maturity T k there is a deterministic, Borel measurable function γ i, T k : [, T ], which represents the volatility of the inter-rating spread H i, T k. We assume that γ i s, T k = for s > T k and that n 1 σ j s, T k γ j 1 s, T k γ j s, T k M, k=1 for all s [, T ] and every coordinate j {1,..., d}. RL.2 The initial term structure L i, T k, i = 1,..., K 1, of forward Libor rates satisfies < L, T k L 1, T k L, T k, for all k =, 1,..., n 1, i.e. < B, T k B, T k1 B 1, T k B 1, T k1 B, T k B, T k1. We postulate that the forward inter-rating spread H i, T k for the rating class i, i = 1,..., K 1, and the tenor date T k, k = 1,..., n 1, is an exponential semimartingale whose dynamics under the forward measure P Tk1 is given by H i t, T k = H i, T k exp b H i s, T k ds γ i s, T k dx T k1 s with initial condition H i, T k = 1 Bi, T k B i 1, T k1 δ k B i 1, T k B i, T k The drift term b H i, T k will be specified in the forthcoming section. We assume that b H i s, T k = for s T k, i.e. H i t, T k = H i T k, T k for t T k. In the following theorem we deduce the dynamics of the rating-dependent forward Libor rates L i, T k under the corresponding forward measures, which is implied by specification 15. Theorem 3.1. Assume that L.1, L.2, RL.1 and RL.2 are in force. For each k = 1,..., n 1, let L, T k and H i, T k, i {1,..., K 1}, be given by 7 and 15, respectively. Then:

9 RATING BASED LÉVY LIBOR MODEL 9 a The rating-dependent forward Libor rates L i, T k satisfy for every T k and any t T k Lt, T k L 1 t, T k L t, T k, i.e. Libor rates are monotone with respect to credit ratings. b The dynamics of the Libor rate L i, T k under P Tk1 is given by L i t, T k = L i, T k exp where b L i s, T k ds cs σ i s, T k dw T k1 s T S i s, x, T k µ ν k1 ds, dx, 16 σ i s, T k := l i s, T k 1 l i 1 s, T k σ i 1 s, T k h i s, T k γ i s, T k = l i s, T k 1[ ls, T k σs, T k i j=1 represents the volatility of the Brownian part and ] h j s, T k γ j s, T k S i s, x, T k := ln 1 l i s, T k 1 β i s, x, T k 1 17 controls the jump size. Here we have used h i s, T k := δ kh i s, T k 1 δ k H i s, T k, 18 and l i s, T k := δ kl i s, T k 1 δ k L i s, T k, 19 β i s, x, T k := β i 1 s, x, T k 1 h i s, T k e γ is,t k,x 1 = 1 ls, T k e σs,tk,x 1 i j=1 1 h j s, T k e γ js,t k,x 1. 2

10 1 ERNST EBERLEIN AND ZORANA GRBAC The drift term in 16 is given by b L i s, T k = l 1 i s, T k i h j s, T k b H j s, T k j=1 1 2 l 1 i s, T k ls, T k c s σs, T k l 1 i s, T k i h j s, T k h j s, T k 2 c s γ j s, T k 2 j=1 1 2 l is, T k 1 c s σ j s, T k 2 S i s, x, T k l 1 i s, T k βs, x, T k 1 i j=1 h j s, T k γ j s, T k, x F T k1 s dx. 21 Proof: The proof is deferred to the appendix. Remark 3.2. Let us compare the expressions for the dynamics of the ratingdependent Libor rate L i, T k and the dynamics of the default-free Libor rate L, T k under P Tk1. Note that 7 can be written as where Lt, T k = L, T k exp Ss, x, T k := ln b L s, T k ds cs σs, T k dw T k1 s 22 T Ss, x, T k µ ν k1 dt, dx, 1 ls, T k 1 βs, x, T k 1 = σs, T k, x, with βs, x, T k defined in 13 and ls, T k in 14. Therefore, we observe that the equation 16 describing the dynamics of the Libor rate for the rating i is of the same form as the default-free Libor rate dynamics 22, naturally with the appropriate specifications of σ i, T k and S i,, T k. In Figure 1 we represent graphically the connections between different rating-dependent Libor rates. Having established the pre-default term structure, the next step is to study migration between different rating classes in order to obtain an arbitrage-free model for the evolution of defaultable Libor rates. The credit migration process is assumed to be a canonically constructed conditional Markov process C, which we study in the next section.

11 RATING BASED LÉVY LIBOR MODEL 11 Lt, T n 1 L i 1 t, T n 1 H i t,t n 1 L i t, T n 1 Lt, T k L i 1 t, T k H i t,t k L i t, T k Lt, T k 1 L i 1 t, T k 1 H i t,t k 1 L i t, T k 1 Lt, T 1 L i 1 t, T 1 H i t,t 1 L i t, T 1 Default-free Rating i 1 Rating i Figure 1. Connection between subsequent Libor rates 4. Credit migration under forward measures 4.1. Conditional Markov chains and their main properties. Let us describe the appropriate probabilistic setting required for a model that allows credit migration. As pointed out in the introduction, rating classes are typically identified with elements of a finite set, denoted by K. We assume that K = {1, 2,..., K}, where 1 denotes the best possible rating and K corresponds to the default event. In credit risk theory credit migration is usually modeled by a conditional Markov chain C with continuous time parameter and the state space K. We adopt the same idea here. Recall that in this setting the default state K is an absorbing state of C and the default time τ is modeled as the first hitting time of this state, i.e. τ = inf {t > : C t = K}. To construct such a process, we are going to use the canonical construction based on a given reference filtration F and a stochastic infinitesimal generator Λ. This construction can be found in Bielecki and Rutkowski 22 or Eberlein and Özkan 23. Our underlying probability space is Ω, F, P T with a given filtration F = F t t T, which is generated by the time-inhomogeneous Lévy process X driving the default-free and the predefault term structure of Libor rates. Furthermore, let Λ = Λt t T be a matrix-valued stochastic process

12 12 ERNST EBERLEIN AND ZORANA GRBAC λ 11 t λ 12 t... λ 1K t λ 21 t λ 22 t... λ 2K t Λt = where λ ij : Ω [, T ] R are bounded, F-progressively measurable stochastic processes. For every i, j K, i j, the processes λ ij are non-negative and λ ii t = j K\{i} λ ijt, for t [, T ]. The last row of Λ contains only zeros since the state K is an absorbing state of C. Let µ = µ 1,..., µ K be a probability distribution on K, which is the initial distribution of the process C, i.e. the distribution of C. In credit risk applications µ is a one-point mass on the rating class observed at time t =. The process C is constructed from the initial distribution and the F- adapted infinitesimal generator Λ by enlarging the underlying probability space Ω, F, P T to a probability space denoted in the sequel by Ω, G, Q T. The new probability space is obtained as a product space of the underlying one with a probability space supporting the initial distribution µ of C and a probability space supporting a sequence of uniformly distributed random variables, which control, together with the entries of the infinitesimal generator Λ, the laws of jump times τ n n N of C and jump heights. Note that by using a product space we obtain a certain independence which will be crucial for many properties of C. We denote by F its trivial extension from the original probability space Ω, F, P T to Ω, G, Q T. Moreover, all stochastic processes are extended to the new probability space by retaining their names and setting for example X ω := Xω, and similarly for other processes. Remark 4.1. We recall that this canonical construction is a generalization of the classical Cox construction, which is used in credit risk theory to model the default time with a given F-intensity λ see Jeanblanc and Rutkowski 2 or Bielecki and Rutkowski 22. Indeed, when K = 2, the conditional Markov chain has only two states which have the interpretation of a non-default state 1 and the default state 2 and the above construction becomes the Cox construction of a default time with intensity λt = λ 11 t. The process C obtained by the canonical construction is an F-conditional Markov chain, i.e. if we denote by F C the natural filtration of the process C, the conditional Markov property E QT [hc s F t F C t ] = E QT [hc s F t σc t ] 24 is satisfied for every t s T and any function h : K R. It is important to mention that the process C possesses also the following property: E QT [hc s F u F C t ] = E QT [hc s F u σc t ], 25 for every t s u T and any function h : K R. We will refer to 25 too as the conditional Markov property, even though it is a stronger property which implies property 24. Note that in general, not all conditional Markov chains possess property 25, but the canonically

13 RATING BASED LÉVY LIBOR MODEL 13 constructed process C satisfies both 24 and 25 compare Bielecki and Rutkowski 22, formula and comments thereafter. For every fixed t, the σ-algebra σc t is finitely generated, as C t takes its values in a finite set K. This enables us to establish the following decomposition of conditional expectations it is a counterpart in the conditional Markov setting of Corollary in Bielecki and Rutkowski 22 or Lemma 1 in Elliot, Jeanblanc, and Yor 2: Lemma 4.2. If Y is a G-measurable random variable, then K E QT [Y 1 {Ct=i} F s ] E QT [Y F s σc t ] = 1 {Ct=i} E QT [1 {Ct=i} F s ], 26 for every t s and i K. Proof: The proof is straightforward combining the definition of conditional expectation and the aforementioned fact that σc t is generated by atoms {C t = j}, j = 1,..., K. Note that the right-hand side in 26 is well-defined since { E QT [1 {Ct=i} F s ] = } {C t = i} c Q T -a.s. cf. Last and Brandt 1995, Lemma A3.17. The conditional expectation on the left-hand side in 26 is equal to the right-hand side if and only if [ ] K E QT [1 {Ct=i}Y F s ] E QT [1 F 1 {Ct=j}Y ] = E QT 1 F 1 {Ct=j} = E QT [ 1 F 1 {Ct=j} 1 {Ct=i} E QT [1 {Ct=j}Y F s ] E QT [1 {Ct=j} F s ] E QT [1 {Ct=i} F s ] ], for every F F s and every {C t = j}, j K, since the σ-algebra F s σc t is generated by finite intersections F {C t = j}. We have [ ] E QT [1 {Ct=j}Y F s ] E QT 1 F 1 {Ct=j} E QT [1 {Ct=j} F s ] [ [ E QT [1 {Ct=j}Y F s ] ]] = E QT E QT 1 F 1 {Ct=j} F s E QT [1 {Ct=j} F s ] [ ] EQT [1 F 1 {Ct=j}Y F s ] = E QT E QT [1 E QT [1 {Ct=j} F s ] {Ct=j} F s ] = E QT [E QT [1 F 1 {Ct=j}Y F s ]] = E QT [1 F 1 {Ct=j}Y ], which is what we had to show. In view of this result, the conditional Markov property takes the following form: K E QT [hc s F t Ft C E QT [hc s 1 {Ct=i} F t ] ] = 1 {Ct=i}, 27 E QT [1 {Ct=i} F t ] for every t s and any function h : K R. Let us now examine the most important properties of the process C. Due to the canonical construction, each random state C s is actually influenced

14 14 ERNST EBERLEIN AND ZORANA GRBAC by information from the filtration F only up to time s. A precise formulation of this property is contained in the following proposition. Proposition 4.3. Let C be a conditional Markov chain obtained by the canonical construction. Then: a For every t s u T and j K E QT [1 {Cs=j} F u F C t ] = E QT [1 {Cs=j} F s F C t ]. b More generally, for any t s 1 s 2 u T and j 1, j 2 K E QT [1 {Cs1 =j 1 }1 {Cs2 =j 2 } F u F C t ] = E QT [1 {Cs1 =j 1 }1 {Cs2 =j 2 } F s2 F C t ]. Proof: The proof relies on the canonical construction of C and its jump times and the independence we mentioned earlier. We refer to Grbac 21, Proposition 2.18 for details. The σ-algebra Ft C corollary shows. can be omitted in the above results, as the following Corollary 4.4. Let C be a canonically constructed conditional Markov chain. Then: a For every s u T and j K E QT [1 {Cs=j} F u ] = E QT [1 {Cs=j} F s ]. b For any s 1 s 2 u T and j 1, j 2 K E QT [1 {Cs1 =j 1 }1 {Cs2 =j 2 } F u ] = E QT [1 {Cs1 =j 1 }1 {Cs2 =j 2 } F s2 ]. Proof: This follows by inserting t = into the previous proposition and applying Lemma 4.2. The independence between the initial distribution µ of C and F implies E QT [1 {C =i} F u ] = E QT [1 {C =i} F s ] = E QT [1 {C =i}] = µ i, which yields both claims. In the remainder of the subsection we study the transition probabilities of a canonically constructed conditional Markov chain C. It turns out that the usual properties of transition probabilities of an ordinary Markov chain, such as the Chapman Kolmogorov equation, will remain valid, but will be expressed in terms of F-conditional expectations. To fix the notation, let us denote E QT [Y F s ; C t = i] := E Q T [Y 1 {Ct=i} F s ] E QT [1 {Ct=i} F s ], where Y is a G-measurable random variable and t s T. Hence, by Lemma 4.2, E QT [Y F s ; C t = i] is an F s -measurable random variable that agrees with E QT [Y F s σc t ] on the set {C t = i}. Bielecki and Rutkowski 22 use a slightly different notation E QT [Y F s {C t = i}] instead, but we prefer the above notation to make a clear distinction from conditioning with respect to the smallest σ-algebra generated by F s and the set {C t = i}.

15 For Y = 1 {Cs=j}, the expression RATING BASED LÉVY LIBOR MODEL 15 E QT [1 {Cs=j} F s ; C t = i] = Q T C s = j F s ; C t = i denotes the conditional probability with respect to F of the process C being in state j at time s if it was in state i at time t. Definition 4.5. The F-conditional transition probability matrix of C is defined as P t, s = [p ij t, s] i,j=1,...,k, t s T, where p ij t, s = Q T C s = j F s ; C t = i = Q T C s = j F T ; C t = i. 28 Note that the second equality in 28 follows from Corollary 4.4. Remark 4.6. The process C together with the family of stochastic matrices P t, s t s T is an F-doubly stochastic Markov chain in the sense of Definition 2.1 introduced in Jakubowski and Nieweglowski 29b; see Remark 2.16 and Theorem 2.14 in their paper. F-doubly stochastic Markov chains form a subclass of the class of F-conditional Markov chains and are particularly suitable for applications in credit risk. Processes that are typically used for this purpose such as an ordinary Markov chain, a compound Poisson process, a Cox process and a canonically constructed conditional Markov chain belong to this class. The main properties of F-doubly stochastic Markov chains are studied in Jakubowski and Nieweglowski 29b. We conclude this section by formulating the appropriate F-conditional versions of the Chapman Kolmogorov equation and the forward Kolmogorov equation for C. Proposition 4.7. Let C be a canonically constructed F-conditional Markov chain and P t, s t s T its family of conditional transition probability matrices. Then: a P, satisfies the F-conditional Chapman Kolmogorov equation P t, s = P t, up u, s, t u s. 29 b P, satisfies the F-conditional forward Kolmogorov equation dp t, s ds = P t, sλs, P t, t = Id. Proof: Part a follows from Theorem 2.7 and part b from Definition 2.8 in Jakubowski and Nieweglowski 29b. Clearly, both claims can also be established directly, without the notion of an F-doubly stochastic Markov chain; see Grbac 21, Proposition The immersion property in the conditional Markov chain setting. Starting from the reference filtration F and constructing a conditional Markov chain C, we obtain an enlargement of F that we denote by G, where G = G t t T with G t = F t Ft C and completed. A natural question, that we examine in this subsection, is if the immersion property or a so-called H-hypothesis is satisfied for this enlargement, i.e. if F-local martingales

16 16 ERNST EBERLEIN AND ZORANA GRBAC remain G-local martingales. In case when a conditional Markov chain is constructed canonically, the answer to this question is affirmative, as we show in the sequel. Proposition 4.8. Let C be a canonically constructed conditional Markov chain. Then for every B Fs C and s u T E QT [1 B F u ] = E QT [1 B F s ]. Proof: We must verify the claim for any B Fs C = σ t s σc t. Since Fs C is generated by finite intersections of sets of the form {C t = j}, j K, t [, s], it is enough to prove the claim for an arbitrary set of the form {C s1 = i} {C s2 = j}, for some s 1 s 2 s and i, j K. But this is exactly Corollary 4.4 and now by standard arguments monotone class theorem the claim follows for any B Fs C. Theorem 4.9 Hypotheses H1, H2 and H3. Let C be a canonically constructed conditional Markov chain. Furthermore, let X be a bounded F T - measurable random variable and Y a bounded F C s -measurable random variable, s [, T ]. Then the following three equivalent statements hold: H1 E QT [XY F s ] = E QT [X F s ] E QT [Y F s ], i.e. the σ-fields F T and F C s are conditionally independent given F s. H2 E QT [Y F T ] = E QT [Y F s ]. H3 E QT [X F s F C s ] = E QT [X F s ]. Proof: Making use of Proposition 4.8 we see that H2 holds for Y = 1 B, where B Fs C. This implies that it is also true for any bounded Fs C - measurable Y since every Fs C -measurable random variable can be written as a limit of a sequence of elementary Fs C -measurable random variables. Applying the dominated convergence theorem for conditional expectation we establish H2. The equivalences between H1, H2 and H3 follow from Theorem II.45 in Dellacherie and Meyer Consequently, we obtain the following result. Theorem 4.1 H- hypothesis. Let Ω, G, Q T be a probability space with a given filtration F and let C be a canonically constructed F-conditional Markov chain. Then the immersion property holds for the filtrations F and G = F F C. Proof: Follows directly from the statements in Theorem 4.9 and the wellknown fact that they are equivalent to the immersion property of the enlargement see for example Brémaud and Yor 1978 or Jeanblanc and Rutkowski 2. Remark Since σc s F C s, it immediately follows that all statements from Theorem 4.9 remain valid when we replace F C s with σc s.

17 RATING BASED LÉVY LIBOR MODEL 17 The immersion property of the enlargement obviously implies that every F-semimartingale X remains a G-semimartingale. Moreover, if X is a timeinhomogeneous Lévy process with respect to F, then it remains a timeinhomogeneous Lévy process with respect to G. Proposition Suppose that the assumptions of Theorem 4.1 are in force and let X be a time-inhomogeneous Lévy process with respect to the filtration F with the triplet of predictable characteristics B, C, ν. Then X remains a time-inhomogeneous Lévy process with respect to G with the same predictable characteristics. Proof: Clearly, X is a G-semimartingale such that X =. According to Theorem II.2.42 and Corollary II.2.48 in Jacod and Shiryaev 23, B, C, ν is its triplet of predictable characteristics if and only if the process e i u,x Giu M locg, with G defined in Theorem II.2.27 in Jacod and Shiryaev 23. By assumption, B, C, ν is the triplet of predictable characteristics of X with respect to F; hence, this process is in M loc F. Due to the immersion property it is also in M loc G, and therefore B, C, ν is the triplet of predictable characteristics with respect to G. Moreover, X has independent increments with respect to G if B, C, ν is deterministic which is the case because its increments are independent with respect to F cf. Jacod and Shiryaev 23, Theorem II Finally, equation 4.16 from the same theorem shows that the characteristic function of X t takes the form given in the definition of a time-inhomogeneous Lévy process in Eberlein and Kluge 26, Section 2.1 and the proof is completed. We conclude this section by introducing a certain generalization of the H-hypothesis in the conditional Markov setting. Definition Let C be an F-conditional Markov chain with the natural filtration F C and G = F F C. Furthermore, for a fixed r, denote for every s r F s := F s F C r and Fr := F s s r. We say that the enlargement G of F r satisfies the H r -hypothesis if H r Every F s s r -local martingale is a G s s r -local martingale. Remark Note that G is indeed an enlargement of F r since for every s r we have F s = F s F C r F s F C s = G s. Proposition Let C be a canonically constructed F-conditional Markov chain. Then the H r -hypothesis holds between the filtrations F r and G. Moreover, hypothesis H r implies hypothesis H for this enlargement. If, in addition, we assume that the reference filtration F is a natural filtration of a time-inhomogeneous Lévy process, then the converse statement is also true, i.e. H implies H r, for every r.

18 18 ERNST EBERLEIN AND ZORANA GRBAC Proof: Let us fix an arbitrary r and show that H r holds for F r and G. Note that for every s r G s = F s F C s = F s F C s, and therefore, H r is equivalent to showing that E[Y F T ] = E[Y F s ], for every bounded F C s -measurable random variable. By definition of F r, this is actually E[Y F T F C r ] = E[Y F s F C r ]. Now we are done since this follows from Proposition 4.3 exactly in the same way as H2 follows from Corollary 4.4. Moreover, the implication H r H is obvious, since it can be easily proved that every F-local martingale is an F F C r -local martingale, which is then, by hypothesis H r, a G-local martingale. We simply have to note that, due to the canonical construction of C, we have F C r F r E r, 3 where E r is a σ-algebra independent from F. This means that F s F C r = F s E r, for s r. Thus, the enlargement of F to F r, given as F F C r = F E r, clearly possesses the immersion property this is simply the initial enlargement of F with an independent σ-algebra E r. Conversely, let us assume that the filtration F is the natural filtration of a time-inhomogeneous Lévy process X and let us prove that H implies H r, for every r. The proof relies on the representation theorem for local martingales and the fact that X remains a time-inhomogeneous Lévy process with the same characteristics also with respect to the enlarged filtration G. Let us fix some r and establish H r. As we pointed out above, the filtration F r may be thought of as the initial enlargement of F with an independent σ-algebra E r. This allows us to make use of the representation theorem for local martingales in filtrations generated by processes with independent increments Jacod and Shiryaev 23, Theorem III More precisely, if M r is a local martingale with respect to F r, it can be written as M r = M r r H r X c W r µ X ν X, for some processes H r L 2 loc Xc and W r G loc µ, where X c is the continuous martingale part of X and µ X the random measure of jumps of X with compensator ν X for the definitions of L 2 loc Xc and G loc µ see pages 48 and 72 of Jacod and Shiryaev 23. Hence, thanks to the Proposition 4.12, which ensures that both the continuous martingale part and the purely discontinuous martingale part of X remain the same with respect to G, M r is a G-local martingale and H r is verified Conditional Markov property under forward Libor measures. This subsection is devoted to the study of the conditional Markov property under different forward Libor measures. To develop a rating based Lévy Libor model with a migration process which is a canonically constructed conditional Markov process, we have to know how changes of the forward

19 RATING BASED LÉVY LIBOR MODEL 19 measure affect this process and the immersion property of the associated enlargement. Recall that the conditional Markov chain C is constructed starting with Ω, F, P T and using the canonical construction. The enlarged probability space on which C is defined is denoted by Ω, G, Q T. Moreover, we denote by Q Tk the forward measure defined on Ω, G Tk which is obtained from Q T in the same way as P Tk is constructed from P T. The Radon Nikodym derivative of Q Tk with respect to Q T is therefore dq Tk dq T =: ψ k, 31 where ψ k is a positive, F Tk -measurable random variable with expectation 1 more precisely it is given in equation 9. By construction, C satisfies the F-conditional Markov property under the terminal forward measure Q T. In the theorem below we show that the conditional Markov property of C is preserved under all forward measures Q Tk, k = 1,..., n 1. Theorem Let C be a canonically constructed conditional Markov chain with respect to Q T and let Q Tk, k = 1,..., n 1, be the forward measures given by 31. Then C is a conditional Markov chain with respect to every Q Tk, k = 1,..., n 1, i.e. E QTk [hc s F u F C t ] = E QTk [hc s F u σc t ], 32 for all t s u T k and any function h : K R. Furthermore, the matrices of conditional transition probabilities under Q T and Q Tk are the same, i.e. p Q T k ij t, s = p Q T ij t, s, 33 for all i, j K and t s T k, where p Q T k ij t, s and p Q T ij t, s are defined by 28. Proof: Let us fix a k {1, 2,..., n 1} and establish the conditional Markov property 32. By assumption C is a conditional Markov chain under Q T.

20 2 ERNST EBERLEIN AND ZORANA GRBAC Therefore, we obtain the following sequence of equalities: E QTk [hc s F u F C t ] = E Q T [ψ k hc s F u F C t ] E QT [ψ k F u F C t ] = E Q T [E QT [ψ k hc s F u F C s ] F u F C t ] E QT [ψ k F u ] = E Q T [hc s E QT [ψ k F u F C s ] F u F C t ] E QT [ψ k F u ] = E Q T [hc s E QT [ψ k F u ] F u F C t ] E QT [ψ k F u ] = E Q T [ψ k F u ]E QT [hc s F u F C t ] E QT [ψ k F u ] = E QT [hc s F u F C t ] = E QT [hc s F u σc t ] =... same reasoning backwards... = E QTk [hc s F u σc t ], where we have applied the abstract Bayes rule for the first equality and the second one follows from H3 in Theorem 4.9 plus the dominated convergence theorem. The third equality is obvious since hc s is F C s -measurable, the fourth one is again a consequence of H3, and finally, the fifth equality is the conditional Markov property 25. For the remaining equalities we use Remark 4.11 and the same reasoning backwards. Thus, we have shown 32. It remains to prove the second claim in the proposition. From the above calculation, it is obvious that Q Tk C s = j F s σc t = Q T C s = j F s σc t and then in particular also Q Tk C s = j F s ; C t = i = Q T C s = j F s ; C t = i, on the set {C t = i}, for all i, j K and t s T k. These are by definition the transition probabilities under the measures Q Tk and Q T and hence, 33 is proved. Generally speaking, the immersion property of some filtration enlargement is not always preserved under an equivalent change of probability measure. This usually depends on the component of the Radon Nikodym density corresponding to the filtration F C, which has to satisfy some conditions. However, in the case of the forward Libor measures, this component is trivial since the Radon Nikodym densities are adapted to F. Thus, it turns out that the immersion property of the enlargement is indeed preserved under all forward measures. Theorem Let C be a canonically constructed conditional Markov chain with respect to Q T and let Q Tk, k = 1,..., n 1, be the forward measures given by 31. Then the immersion property holds under all Q Tk, i.e. every F, Q Tk -local martingale is a G, Q Tk -local martingale.

21 RATING BASED LÉVY LIBOR MODEL 21 Proof: The proof relies on the following result which can be found for example in Coculescu and Nikeghbali 27: Let P and Q be two equivalent probability measures and assume that the H-hypothesis holds under P. Denote ψ := dq dp, ψ t := dq, ψt G := dq. dp Ft dp Gt Then hypothesis H holds under Q if and only if for every X F T, X E P [Xψ G t ] ψ G t = E P[Xψ F t ] ψ t. In our case, dq T k dq T is F T k -measurable, which implies ψ t = ψ G t and hence, the condition is trivially fulfilled since E QT [Xψ G t ] = E QT [Xψ F t ], by H3. Therefore, we conclude that the immersion property is satisfied under all Q Tk. 5. Absence of arbitrage in the rating based Lévy Libor model From arbitrage pricing theory we know that in order to have an arbitragefree model, the forward prices of defaultable bonds B C,T k B,T, where the default-free bond with maturity T is used as a numeraire, must be local martingales with respect to the forward measure Q T. When the default-free bonds with other maturities are used as numeraires, the forward defaultable bond price processes have to be local martingales with respect to the corresponding forward measures as well. It can be shown that it is enough to require that B C,T k B,T k is a local martingale with respect to the forward measure Q Tk, for every k = 1,..., n 1. To see this, let us fix some k, l {1,..., n} and assume that l k the other case is treated similarly. We have where Bt,Tk Bt,T l t T k B C t, T k Bt, T l = B Ct, T k Bt, T k Bt, T k Bt, T l, is the density process of the change of measure from Q Tk to Q Tl up to a norming constant; compare equation 9. Hence, B C,T k B,T l is a Q Tl -local martingale if and only if B C,T k B,T k is a Q Tk -local martingale cf. Proposition III.3.8a in Jacod and Shiryaev 23. So far we have not specified directly the bond prices in the model, but as we have already specified the inter-rating spreads H j, T k, j = 1,..., K 1, any bond price specification we make must be consistent with relation 5 connecting the bond prices and the inter-rating spreads. Let us explore the consequences of this for the bond prices. For a fixed t [, T k ], we obtain recursively from 5 B j t, T k B j 1 t, T k = k 1 1 δ l H j t, T l l=l 1 B jt, T l B j 1 t, T l,

22 22 ERNST EBERLEIN AND ZORANA GRBAC where T l is a tenor date such that t T l 1, T l ]. Consequently, for each rating i it follows B i t, T k Bt, T k = B 1t, T k i B j t, T k i k 1 Bt, T k B j 1 t, T k = B it, T l 1 δ l H j t, T l Bt, T l. j=2 1 j=1 l=l Since every bond price specification must be consistent with the above relation, we postulate Assumption B. For every i {1, 2,..., K 1} B i t, T k i k 1 Bt, T k = 1 t e λisds, 34 1 δ l H j t, T l j=1 l= for some integrable, F-adapted stochastic process λ i that satisfies Tk e λ i sds = i k 1 1 δ l H j T l, T l. 35 j=1 l= It is easily checked that the above specification is indeed consistent and moreover, B it k,t k BT k,t k = 1 due to 35. Recall that by assumption H jt, T l = H j T l, T l, for t T l. Under Assumption B, the forward bond price process B C,T k B,T k is given by B C t, T k Bt, T k where = = i k 1 j=1 l= 1 e λisds 1 1 δ l H j t, T l {Ct=i} q 1 Cτ {C t=k} Ht, T k, ie λ isds 1 {Ct=i} q Cτ 1 {C t=k}, 36 Ht, T k, i := i k 1 j=1 l= 1 1 δ l H j t, T l. In the sequel we are going to provide necessary and sufficient conditions for the local martingality of the forward defaultable bond price process B C,T k B,T k. Let us begin by stating the main result. Theorem 5.1. Let T k be a tenor date. Assume that the processes H j, T k, j = 1,..., K 1, are given by 15 and that Assumption B holds. Then the process B C,T k B,T k defined in 36 is a local martingale with respect to the forward measure Q Tk and the filtration G if and only if the following condition is satisfied:

23 RATING BASED LÉVY LIBOR MODEL 23 for almost all t T k on the set {C t K} b H e λ C t sds t, T k, C t λ Ct t = 1 q Ct λ CtKt 37 Ht, T k, C t 1 Ht, T k, je t λ jsds j=1,j C t Ht, T k, C t e λ C t sds λ Ctjt. Before proving this theorem, we need some auxiliary results. In the following lemma we deduce the dynamics of the process H, T k, i for each i under the measure Q Tk. Lemma 5.2. Let T k be a tenor date and assume that H j, T k are given by 15. The process H, T k, i defined in 36 has the following dynamics under Q Tk Ht, T k, i = H, T k, i E t b H s, T k, ids i k 1 j=1 l=1 i k 1 h j s, T l c s γ j s, T l dw T k s 38 j=1 l=1 1 1 h j s, T l e γ js,t l,x 1 1 µ ν T k ds, dx where h j s, T l is defined in 18 and b H s, T k, i := i k 1 h j s, T l b H j s, T l j=1 l=1 i k 1 h j s, T l γ j s, T l, j=1 l=1 i k 1 j=1 l=1 k 1 m=l1 ls, T m c s σs, T m 1 2 h js, T l h j s, T l 2 c s γ j s, T l 2 1 i k 1 h j s, T l c s γ j s, T l j=1 l=1 [ i k h j s, T l e γ js,t l,x 1 1 j=1 l=1 i k 1 h j s, T l γ j s, T l, x j=1 l=1 k 1 m=l1 Proof: The proof is deferred to the appendix. 1 ls, T m e σs,tm,x 1 ] F T k s dx.,

24 24 ERNST EBERLEIN AND ZORANA GRBAC Furthermore, we make the following observation: the processes H, T k, i and C do not jump simultaneously, i.e. Ht, T k, i C t = Q Tk -a.s. for t [, T ]. This property is a consequence of the canonical construction of C. A similar result concerning a canonically constructed default time is stated in Jakubowski and Nieweglowski 29a, Proposition 2. The proposition below is a slight generalization since we deal with a series of jump times and, in addition, require the property to hold under all forward measures Q Tk. Proposition 5.3. Let Y t t T be an F-adapted semimartingale and τ n a sequence of random times representing the jump times of a conditional Markov chain constructed by the canonical construction. Then Q Tk Y τn =, n N, for every forward measure Q Tk 1 k n. Proof: See Appendix B. Using these results, we are now able to prove the main theorem. Proof of Theorem 5.1: Recall from Theorem 4.16 that C is a conditional Markov chain under every forward measure Q Tk, k = 1,..., n. According to Bielecki and Rutkowski 22, Proposition , for each i the process Mt i = 1 {Ct=i} λ Csisds 4 is a Q Tk -martingale. Moreover, we will make use of an auxiliary process H ij, for i, j K, i j, defined in Bielecki and Rutkowski 22, page 333: = H ij t <u t 1 {Cu =i}1 {Cu=j}. This process counts the number of jumps of C from the state i to the state j in the time interval [, t] and it is known that M ij t = H ij t λ ij u1 {Cu=i}du 41 is a Q Tk -martingale; see Bielecki and Rutkowski 22, page 47. In particular, the process H ik is useful for us since it takes the following values: it equals 1 if and only if C jumped from i to the default state K remember that this state is absorbing during the time interval [, t] and otherwise it equals zero. Therefore, we can use it to rewrite the defaultable bond price process in the following way: B C t, T k = Bi t, T k 1 {Ct=i} q i Bt, T k Ht ik.