The Dynamic Spread of the Forward CDS with General Random Loss

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 580713, 17 pages http://dx.doi.org/10.1155/2014/580713 Research Article The Dynamic Spread of the Forward CDS with General Random Loss Kun Tian,1 Dewen Xiong,1 and Zhongxing Ye1,2 1 2 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China School of Business Information, Shanghai University of International Business and Economics, Shanghai 201620, China Correspondence should be addressed to Kun Tian; Received 24 February 2014; Accepted 3 May 2014; Published 27 May 2014 Academic Editor: Igor Leite Freire Copyright © 2014 Kun Tian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We assume that the filtration F is generated by a 𝑑-dimensional Brownian motion 𝑊 = (𝑊1 , . . . , 𝑊𝑑 )󸀠 as well as an integer-valued random measure 𝜇(𝑑𝑢, 𝑑𝑦). The random variable 𝜏̃ is the default time and 𝐿 is the default loss. Let G = {G𝑡 ; 𝑡 ≥ 0} be the progressive enlargement of F by (̃ 𝜏, 𝐿); that is, G is the smallest filtration including F such that 𝜏̃ is a G-stopping time and 𝐿 is G𝜏̃ -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 G 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 𝜏̃ be the default time and let 𝐿 be the default loss; both 𝜏̃ and 𝐿 are random variables. The recovery rate of the CDS is a function of the default loss, saying ℎ(𝐿); that is to say, if default occurs in [𝑎, 𝑏] causing the buyer with the default loss 𝐿, then he can obtain ℎ(𝐿) for the reimburse at the default time 𝜏̃. In this paper, we will consider this new kind of CDS in the framework of stochastic interest rates. First, we assume that the filtration F without default is generated by a 𝑑-dimensional Brownian motion 𝑊 = (𝑊1 , . . . , 𝑊𝑑 )󸀠 as well as an integer-valued random measure 𝜇(𝑑𝑢, 𝑑𝑦). In general, the default time 𝜏̃ is not an F-stopping time; thus, the filtration of the investor is given by G = {G𝑡 ; 𝑡 ≥ 0}, the smallest filtration including F such that 𝜏̃ is a G-stopping time and 𝐿 is G𝜏̃-measurable. We introduce the concept of forward CDS with random loss 𝐿. For any fixed 𝑎, 𝑏 with 0 < 𝑎 < 𝑏 < 𝑇∗ , let T = {𝑎 = 𝑇0 < 𝑇1 < ⋅ ⋅ ⋅ < 𝑇𝑛 = 𝑏} be a fixed tenor structure of forward CDS; we assume that the recovery rate of the forward CDS is ℎ(𝐿), then the discounted payoff of the protection leg for the buyer is given by 𝑛 𝜏̃ $𝑃 (𝑡) = ∑ ℎ (𝐿) 𝑒− ∫𝑡 𝑟(𝑢)𝑑𝑢 I𝑇𝑘−1 <̃𝜏≤𝑇𝑘 𝑘=1 (1) 𝜏̃ − ∫𝑡 𝑟(𝑢)𝑑𝑢 = ℎ (𝐿) 𝑒 I𝑎<̃𝜏≤𝑏 , where 𝑟(𝑢) is the short rate of the default-free bonds. If no default occurs before 𝑇𝑘 , he needs to pay the fee R𝑡 at 𝑇𝑘 , for 𝑘 = 0, 1, . . . , 𝑛 − 1. It is notable that the fee R𝑡 must be 2 Abstract and Applied Analysis determined at time 𝑡 ∈ [0, 𝑇0 ); thus, the discounted payoff of the fee leg (also known as the premium leg) equals 𝑛 𝑇𝑘 $𝐿 (𝑡) = ∑ R𝑡 𝑒− ∫𝑡 𝑟𝑢 𝑑𝑢 I𝜏̃>𝑇𝑘 . 𝑡 (2) 𝑘=1 We assume that 𝑃 itself is an equivalent martingale measure; then the fair spread of the forward CDS for the protection buyer at 𝑡, 𝑡 ∈ [0, 𝑇0 ], is defined as a G-adapted process R𝑡 such that 𝐸 ($𝑃 (𝑡) − $𝐿 (𝑡) | G𝑡 ) = 0. (3) Since in the interval of premiums, the short rate usually changes, we assume that the short rate of the default-free bonds 𝑟(𝑡) is a F-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 R𝑡 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 (̃ 𝜏, 𝐿) 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 𝜂(𝑑𝑙)𝑑𝑠 be the law of (̃ 𝜏, 𝐿) and the F-regular conditional law of (̃ 𝜏, 𝐿) is equivalent to the law of (̃ 𝜏, 𝐿), that is, 𝑃 (̃ 𝜏 ∈ 𝑑𝑠, 𝐿 ∈ 𝑑𝑙 | F𝑡 ) ∼ 𝜂 (𝑑𝑙) 𝑑𝑠, (𝑃, G). Further, any (𝑃, G)-martingale 𝑀𝑡 can be represented into the following form 𝑀𝑡 = 𝑀0 + ∫ 𝜉(𝑢)󸀠 𝑑𝑊G (𝑢) 0 𝑡 + ∫ ∫ 𝜁 (𝑢, 𝑦) {𝜇 (𝑑𝑢, 𝑑𝑦) − ]G (𝑑𝑢, 𝑑𝑦)} 0 𝐸 + ℎ (̃ 𝜏; 𝜏̃, 𝐿) I𝑡≥̃𝜏 −∫ 𝑡∧̃ 𝜏 0 ℎ (𝑠; 𝑠, 𝑙) 𝑝𝑠 (𝑠, 𝑙) 𝜂 (𝑑𝑙) 𝑑𝑠, 𝐺𝑠− 𝑅 ∫ ̃ for some G-predictable process 𝜉, P(G)-measurable function 𝜁(𝑢, 𝑦), and a O(F) ⊗ B(R+ ) ⊗ B(R)-measurable function ℎ(𝑢; 𝑠, 𝑙). 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 𝑝𝑠 (𝑠, 𝑙), 𝜃1 (𝑢; 𝑠, 𝑙)I𝑢>𝑠 , and 𝜃2 (𝑢, 𝑦; 𝑠, 𝑙)I𝑢>𝑠 and the term structure parameters 𝐶(𝑢, 𝑇) and 𝐽(𝑢, 𝑦, 𝑇); see (45) for more details. We show that the BSDEs system (45) always has a solution (𝑋𝑑,𝑇 , 𝜉1𝑑,𝑇 , 𝜉2𝑑,𝑇 ), which helps us describe the predefault value of the defaultable bond explicitly. Then we introduce another BSDE depending on the recovery rate function ℎ(𝑙) and 𝑝𝑠 (𝑠, 𝑙): 𝑡 𝑠 𝑌𝑡 = 𝑌0 − ∫ ∫ ℎ (𝑙) 𝑒− ∫0 𝑟(𝑢)𝑑𝑢 0 R × 𝑝𝑠 (𝑠, 𝑙) I𝑠>𝑎 𝜂 (𝑑𝑙) 𝑑𝑠 for every 𝑡 ≥ 0; (4) 𝑡 (ii) 𝜂(𝑑𝑠, 𝑑𝑙) has no atoms. + ∫ 𝑌𝑢− 𝜁1 (𝑢)󸀠 𝑑𝑊 (𝑢) 0 Under this assumption, the immersion property is no longer valid, which leads to the problem of the enlargement of the filtration; that is, a (𝑃, F)-martingale may be not a (𝑃, G)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 𝑡 𝑊G (𝑡) := 𝑊 (𝑡) + ∫ {{ 0 (6) 1 − 1} 𝜃̃1 (𝑢) I𝑢≤̃𝜏 𝐺𝑢− −𝜃1 (𝑢; 𝜏̃, 𝐿) I𝑢>̃𝜏} 𝑑𝑢, 1 ] (𝑑𝑢, 𝑑𝑦) := {1 − { − 1} 𝜃̃2 (𝑢, 𝑦) I𝑢≤̃𝜏 𝐺𝑢− (5) G +𝜃2 (𝑢, 𝑦; 𝜏̃, 𝐿) I𝑢>̃𝜏} 𝐹𝑢 (𝑑𝑦) 𝑑𝑢, both of which depend on the default parameters 𝑝𝑠 (𝑠, 𝑙), 𝜃1 (𝑢; 𝑠, 𝑙)I𝑢>𝑠 , and 𝜃2 (𝑢, 𝑦; 𝑠, 𝑙)I𝑢>𝑠 . One can se (...truncated)