Abstract

We assume that the filtration is generated by a -dimensional Brownian motion as well as an integer-valued random measure . The random variable is the default time and is the default loss. Let be the progressive enlargement of by ; that is, is the smallest filtration including such that is a -stopping time and is -measurable. We mainly consider the forward CDS with loss in the framework of stochastic interest rates whose term structures are modeled by the Heath-Jarrow-Morton approach with jumps under the general conditional density hypothesis. We describe the dynamics of the defaultable bond in and the forward CDS with random loss explicitly by the BSDEs method.

1. Introduction

The credit default swap (CDS) is one of the most crucial credit derivatives in the financial market and has received many attentions in recent years; see [1–5] and so forth. In the existing literature, the recovery rate of the CDS is either a constant (see [1–3], etc.) or a stochastic process. There is nothing to do with the default loss (see [4], etc.). However, the fact that once the default happens, the default loss is immediately generated and the default loss should depend on the default time is noted. From the view of the buyer of the forward CDS, he may be more interested in the credit derivative whose recovery rate depends on the default loss; that is, he hopes to obtain the higher recovery rate to avoid higher loss. More precisely, let be the default time and let be the default loss; both and are random variables. The recovery rate of the CDS is a function of the default loss, saying ; that is to say, if default occurs in causing the buyer with the default loss , then he can obtain for the reimburse at the default time . In this paper, we will consider this new kind of CDS in the framework of stochastic interest rates.

First, we assume that the filtration without default is generated by a -dimensional Brownian motion as well as an integer-valued random measure . In general, the default time is not an -stopping time; thus, the filtration of the investor is given by , the smallest filtration including such that is a -stopping time and is -measurable.

We introduce the concept of forward CDS with random loss . For any fixed with , let be a fixed tenor structure of forward CDS; we assume that the recovery rate of the forward CDS is , then the discounted payoff of the protection leg for the buyer is given by where is the short rate of the default-free bonds. If no default occurs before , he needs to pay the fee at , for . It is notable that the fee must be determined at time ; thus, the discounted payoff of the fee leg (also known as the premium leg) equals We assume that itself is an equivalent martingale measure; then the fair spread of the forward CDS for the protection buyer at , , is defined as a -adapted process such that

Since in the interval of premiums, the short rate usually changes, we assume that the short rate of the default-free bonds is a -adapted process whose term structure is determined by the arbitrage-free HJM model with jumps (see Björk et al. [6]). In this paper, we will describe the dynamic of in the frame of random interest rates by the BSDE approach.

Similar works can be found in Xiong and Kohlmann [5], in which they considered the forward CDS without default loss and their default time is modeled by the Cox model. Different from Xiong and Kohlmann [5], we assume that satisfies the following more general conditional density hypothesis (see in Jacod [7], Amendinger [8], and Callegaro et al. [9]):

Assumption 1 (the conditional density hypothesis). (i) Let be the law of and the -regular conditional law of is equivalent to the law of , that is,
(ii)   has no atoms.

Under this assumption, the immersion property is no longer valid, which leads to the problem of the enlargement of the filtration; that is, a -martingale may be not a -martingale, which is a fundamental problem in stochastic analysis and has been widely studied in [9–14] and so forth. By this assumption, we introduce both of which depend on the default parameters , , and . One can see from Tian, Xiong, and Ye (2012), see Theorem 4.6, Corollary 4.7 and Corollary 4.8 that is a -Brownian motion and is the compensator of with respect to . Further, any -martingale can be represented into the following form for some -predictable process , -measurable function , and a -measurable function .

Because of the more general conditional density hypothesis, the method of Xiong and Kohlmann [5] is no longer applicable to our case, so we extend their results and introduce a different BSDEs system depending on the given default parameters , , and and the term structure parameters and ; see (45) for more details. We show that the BSDEs system (45) always has a solution , which helps us describe the predefault value of the defaultable bond explicitly. Then we introduce another BSDE depending on the recovery rate function and : which has a solution . By the solutions of and , we describe the dynamic of the spread of the CDS more accurately.

In short, the main results are:(1)an analysis of the conditional survival process which satisfies a special BSDE under some mild conditions; see Theorem 7;(2)under the generally conditional density assumption, Theorems 15 and 19 describe the dynamics of the defaultable and default-free bond in , respectively;(3)by Theorem 25, the dynamics of the spreads of the forward CDS is obtained, which helps us gain a deeper understanding of CDS.

The paper is organized as follows. In Section 2, we set up the stronger Jacod’s hypothesis with further discussion and review martingale representation in . In Section 3, we introduce a BSDE system to describe the predefault value of the defaultable bonds, which makes us describe the dynamic of the price of the defaultable bonds more explicitly, and as a byproduct, we can also describe the dynamic of the default-free bonds in . In Section 4, we discuss the dynamics of CDS as another application of where we assume that the CDS depends on the loss random variable . Conclusions are given at the end.

2. The Setup and Notations

We assume that is the inference filtration on carrying a -dimensional Brownian motion as well as an integer-valued random measure on , where is a Blackwell space. We use to describe the change of macroeconomic policy, environment, and so forth.

Assumption 2. The filtration is the natural filtration generated by and , that is, where is the collection of -null sets from .

In the following, we let be the family of all -predictable processes and . We assume that the compensator of is given by , where is a transition kernel from into . For any local -martingale , one knows that has the representation property where is an -valued -predictable process with and is a -measurable function with (see Lemma 4.24 in page 185 of Jacod and Shiryaev [15] for more details).

Remark 3. It is easy to see that there exists an -optional process and a sequence of stopping times such that for all positive -measurable function , Furthermore, as the compensator of is , so the filtration is quasi-left continuous.

Let be the default time and let be the default loss, which are random variables. We are interested in the new enlarged filtration which is the smallest filtration including such that is a -stopping time and is -measurable. We assume that satisfies the usual conditions and is called the progressive enlargement of by . We can easily see that is given by The information means that once the default happens, the default loss is immediately generated. The similar progressive enlargement of filtration can also be found in Dellacherie and Meyer [10] and Kchia et al. [16, 17]. It is notable that the results of this progressive enlargement filtration may be seen as the further extension and different from the traditional progressive enlargement of filtration in the literature; see, for example, in Jeulin [18], Jacod [7], Jeanblanc et al. [19], El Karoui et al. [11], Jeanblanc and Le Cam [12], Jeanblanc and Song [13], Callegaro et al. [9], and Jeanblanc and Song [14], among many others.

Hereinafter, we suppose that Assumptions 1 and 2 always hold. The following lemmas come from Pham [20].

Lemma 4. is a -measurable random variable (r.v.) if and only if where is a -measurable r.v. and is an -measurable function.

Lemma 5. (1) Any -predictable process is represented as where is a -predictable process and where is a -measurable function.
(2) Any -optional process is represented as where is an -optional process and where is a -measurable function.

We can see that the default time is not -stopping times, that is, for every -stopping time .

From Assumption 1, one can see from Jacod [7] or Amendinger [8] that there exists a strictly positive -measurable function , called the -conditional density of with respect to , such that for every , is a càdlàg -martingale and for any ,

Since is a strictly positive martingale, from Assumption 2 one can see that can be represented in the following form: where and where is an -valued -predictable process with and is a -measurable function with . The following theorem, from Tian et al. [21], reveals the relation of , , and .

Theorem 6. For any given bounded , , and , is the density process of a pair with respect to if and only if

Let be the conditional survival process; one can easily see that process is a -supermartingale.

Theorem 7. For any given bounded , , and satisfying (19), let and then is the solution of the following BSDE:

Remark 8. It is notable that the initial value is and the terminal value is , which is different from the traditional BSDE. In general, the BSDE (22) may not have a solution, but if , , and satisfy the condition (19), the BSDE (22) always has a solution.

Proof. Let and then is a strictly positive semimartingale with , a.s. for each . Further more, and one can see that is the solution of the following SDE: and then (19) is equivalent to the following condition: that is to say, is a solution of the following BSDE: From the definition of the conditional survival process , one can see that and thus the theorem holds.

Remark 9. Let be any solution of the BSDE (27); we introduce then , and satisfy (19).

Example 10. If let and ; then one can see that , , and satisfy (19). Hence and , which implies

2.1. The Martingale Representation in

From Tian et al. [21], we have the following martingale representation theorem.

Theorem 11. Let be a u.i. -martingale; then there exists a -predictable process and a -measurable function such that where is a -Brownian motion, and where is the compensator of with respect to .

Corollary 12. Let be a local -martingale; then there exist a -predictable process , a -measurable function , and a -measurable function such that

3. The Dynamics of the Default-Free Bond in

We now discuss the market of the defaultable-free bonds. Since is the filtration without default, it is natural to assume that the price processes of the default-free bonds are semimartingale. We assume that itself is an equivalent martingale measure and the short rate is modeled by the arbitrage-free HJM model with jumps (see Björk et al. [6]). More precisely, let be the price at time of the default-free bond maturing at time (a -bond), then is an semimartingale with for each . To discuss the term structure of the default-free bond, we assume that, for every fixed , is -a.s. continuously differentiable in the -variable, and introduce the instantaneous forward rate and the short rate , as in [6]. If the object is continuously differentiable in the -variable, then we denote the partial -derivative by .

For the simplicity, we assume that is an equivalent martingale measure for for any maturity ; that is, the discount process is a -martingale, and the dynamics of is given by where is a bounded -predictable process on , is a bounded -valued -predictable process on and where is a bounded -measurable function on for each . Since it is arbitrage free, one can see from [6] or [22] that where The following lemma can be found in Proposition 2.4 of [6] or Lemma 2.3 of [22].

Lemma 13. If the dynamics of the forward rate is given by (36), then the short rate satisfies where , , and the dynamics is given by

Remark 14. From (40), one can see that If we take , one gets

3.1. The Price Process of the Defaultable Bond

Let be the time- price of the defaultable zero coupon bond with zero recovery and with maturity , that is, where is the predefault value of the default bond .

We first review the following case which is usually called the Cox model. Given a nonnegative bounded -predictable process with , a.s. Let be a random variable (r.v.) independent to with unit exponential law; then the default time is defined as see Lando [23]. For Cox model, is the so-called intensity process and ; also see Lando [23]. However, under the conditional density assumption, is still not clear and we want to describe the dynamic of , so we introduce the following BSDEs system: The solution of the system of BSDEs (45) is a triplet satisfying (45) such that for any , is an -valued -predictable process and is a -measurable function such that the stochastic exponential is a uniformly integrable -martingale on .

Theorem 15. (i) The system of BSDEs (45) has a solution.
(ii) Let be any solution of the system of BSDEs (45); then for any maturity and for any , satisfies the following SDE

The proof is given in Appendix.

Remark 16. From (46), one can see that can be rewritten as the follows: where is the spread of the defaultable bond with respect to the default-free bond , which may be negative.

Remark 17. If as in Example 10, then the BSDEs system (45) can be simplified into the following form: which can be transformed into the BSDEs system (1.3) of Xiong and Kohlmann [5], so the BSDEs system (45) may be viewed as an extension of [5].

Corollary 18. Let be any solution of the system of BSDEs (45), then for any maturity and for any , satisfies the following SDE:

Proof. Since , one can see that and ; then SDE (51) follows from .

3.2. The Price Process of the Default-Free Bond in

It is easy to see that and from (40), one can see that is completely determined by , , and . Although is the filtration without default, the filtration can be observed by all investor; thus, it is also interesting to consider the dynamics of , where We need to introduce the following BSDEs system: The solution of the BSDEs system (54) is such that it satisfies (54) such that for any , is an -valued -predictable process, is a -measurable function, and is a -measurable function such that that the stochastic exponential       is a uniformly integrable -martingale on .

Theorem 19. (i) The BSDEs system (54) has a solution.
(ii) Let be any solution of the system of BSDEs (54); then satisfies the following SDE:

The proof is given in Appendix.

Remark 20. Since the BSDEs system (54) not only depends on , but also depends on , , , and , the dynamics of the price process of the default-free bond in is affected by both the default time and the loss.

Remark 21. Although is a semimartingale, which is not affected by the default, there still exists a jump in at the default time.

4. The Dynamics of the Forward CDS

In this section, we will describe the dynamics of the spreads of the forward CDS. For any fixed with , let be a fixed tenor structure. We assume that the recovery rate is determined by a process with , for ; that is, if default occurs between the dates and , the buyer of the CDS will receive the protection payment at time . The random variable depends on the loss when the default occurs, which is different from [3, 5]. For example, ; that is to say, if , the recovery rate is , and if , the recovery rate is , ( ).

We assume that and the forward credit default swap (forward CDS) issued at time , with the unit notional. The discounted payoff of the protection leg equals and the discounted payoff of the fee leg (also known as the premium leg) equals The following definition is adapted from [3].

Definition 22. The fair spread of the forward CDS with enlarged filtration for the protection buyer equals, for every , is defined as a -adapted process such that ; that is,

We have the following lemma.

Lemma 23. Let be the predefault value of the defaultable bond, and let and then we have, for ,

Proof. Since, for , (60) follows from the definition of .

To describe the dynamic of the spread, we need to introduce the BSDE (7) depending on .

The solution of the BSDE (7) is where is an -valued -predictable process and is a -measurable function, such that the stochastic exponential is a uniformly integrable -martingale on .

One can see that the BSDE (7) always has a solution, such that

Remark 24. Let be the solution of the BSDE (7), then which implies that the restricted process is a -martingale; that is to say, for , satisfies the following SDE:

Theorem 25. Assume that is the solution of the BSDE (7) and is the solution of the system of BSDEs (45); let and assume that is the predefault vaule of the CDS, that is, , and then for , satisfies the following SDE where .

From (66) one can see the dynamic of the spread of the forward CDS on and the proof is given in Appendix.

5. Conclusion

In this paper, we extend the results of Xiong and Kohlmann [5] and consider the problem of how to model the dynamics of the fair spread of the forward CDS whose recovery rate depends on the loss in the jump-Heath-Jarrow-Morton framework. Since is known at the default time , we consider the problem in the filtration including such that is a -stopping time and is -measurable. Because of the more general conditional density hypothesis, we introduce a new BSDEs system (45) depending on the default parameters , , and and the term structure parameters and . By the solution of the BSDEs system (45), we can explicitly describe the term structure of the defaultable bond. As a byproduct, we can also describe the term structure of the default-free bond in the larger filtration . By introducing another BSDE (7) more simpler than Xiong and Kohlmann [5], we explicitly describe the dynamics of the fair spread of the forward CDS with the recovery rate .

Appendix

Proof of Theorem 15. (i) One can see that is a strictly positive -martingale, which can be rewriten as the follows: where is an -predictable process and is a -measurable function. Since and since one can see that thus, If we let then is a solution of the BSDEs system (45).
(ii) Let be any solution of the system of BSDEs (45), and let be the solution of the following SDE: One can see from Itô's formula that satisfies the SDE: Thus, the process is a uniformly integrable martingale. Furthermore, once again from Itô's formula, one can see that and thus . Therefore, which implies

Proof of Theorem 19. (i) Let ; one can see that is a strictly positive -martingale; according to Corollary 12, one can see that can be represented into the following: where is an -valued -predictable process, is a -measurable function with , and is a -measurable function . It is notable that , , and may depend on . Thus, which implies Since , then and it follows from (42) that if we let then is a solution of the BSDEs system (54).
(ii) Let be any solution of the system of BSDEs (54), and let be the solution of the following SDE: One can see that the process is a uniformly integrable martingale, and from Itô's formula, we see that and thus and from which one can see that , a.s., which completes the proof.

Proof of Theorem 25. Since , one can see that is a strictly positive -martingale, which can be rewriten as the following: Let ; then, and thus, for , From Itô's formula, one can see that which implies which is (66).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 11171215), National Natural Science foundation of Shanghai (no. 13ZR1422000), and Yang Cai Project (no. YC-XK-13106).