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Mathematical Problems in Engineering
Volume 2011, Article ID 412565, 16 pages
http://dx.doi.org/10.1155/2011/412565
Research Article

The Intensity Model for Pricing Credit Securities with Jump Diffusion and Counterparty Risk

Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China

Received 15 January 2011; Accepted 21 February 2011

Academic Editor: Moran Wang

Copyright © 2011 Ruili Hao and Zhongxing Ye. 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.

Abstract

We present an intensity-based model with counterparty risk. We assume the default intensity of firm depends on the stochastic interest rate driven by the jump-diffusion process and the default states of counterparty firms. Furthermore, we make use of the techniques in Park (2008) to compute the conditional distribution of default times and derive the explicit prices of bond and CDS. These are extensions of the models in Jarrow and Yu (2001).

1. Introduction

As credit securities are actively traded and the financial market becomes complex, the valuation of credit securities has called for more effective models according to the real market. Until now, there have been mainly two basic models: the structural model and the reduced-form model. In the first model, the firm's default is governed by the value of its assets and debts, while the default in the reduced-form model is governed by the exogenous factor.

The structural approach was pioneered by Merton [1], then extended by Black and Cox [2] and Longstaff and Schwartz [3], assuming the default before the maturity date and others. In the above models, the asset process was all driven by the Brownian motion. Since the asset value may suffer a sudden drop for the reason of some events in the economy, Zhou [4] provided a jump-diffusion model with credit risk in which jump amplitude followed a log-normal distribution and valuated defaultable securities. In his model, Zhou gave the explicit expressions of defaultable securities' prices when the default occurred at the maturity date 𝑇, but only gave a tractably simulating approach when the firm defaulted before time 𝑇. For the first-passage-time models of credit risk with jump-diffusion process, Steve and Amir [5] and Zhang and Melnik [6] used the approach of Brownian bridge to estimate the jump-diffusion process and priced barrier options. Kou and Wang [7] and Kou et al. [8] made use of the Laplace transform to valuate the credit risk after estimating the jump-diffusion process with an infinitesimal generator. For the problem of the valuation of credit derivatives involving jump-diffusion process, it is still difficult to get explicit results in the event of defaulting before the maturity date, despite using the above approaches. However, it is more convenient to use reduced-form approach for valuating the credit risk in such situation.

Comparing with the structural approach, the reduced-form approach is flexible and tractable in the real market. It is pioneered by Jarrow et al. [9] and Duffie and Singleton [10]. They introduced exogenous mechanism of firm's default. Their models considered the default as a random event which is controlled by a exogenous intensity process.

Davis and Lo [11] firstly proposed the model of credit contagion to account for concentration risk in large portfolios of defaultable securities (DL Model). Later, motivated by a series of events such as the South Korean banking crisis, Long Term Capital Management's potential default and so on, Jarrow and Yu [12] thought the traditionally structural and reduced-form models were full of problems because they all ignored the firm's specific source of credit risk. They made use of the Davis's contagious model and introduced the concept of counterparty risk which is from the default of firm's counterparties. In their models, they paid more attention to the primary-secondary framework in which the intensity of default was influenced by the economy-wide state variables and the default state of the counterparty. Later, there are also other similar applications such as Leung and Kwork [13], Bai et al. [14] and so on. In recent years, some authors applied this into portfolio credit securities such as Yu [15] and Leung and Kwok [16]. Nevertheless, the stochastic interest rate in the above models still was driven by diffusion processes.

At present, aggregate credit risk is still one of the most pervasive threats in the financial markets, which is from the contagious risk caused by business counterparties. In this paper, we mainly discuss the pricing of defaultable securities in primary-secondary framework, extending the models in Jarrow and Yu [12]. We consider that the macroeconomic variables contain the risk-free interest rate which shows the interaction between credit risk and market risk. However, the interest rate may drop suddenly due to some events in the modern economy. Therefore, we allow the stochastic interest rate to follow a jump-diffusion process rather than the continuous diffusion process in Jarrow and Yu [12]. Thus, our model not only reflects the real market much better, but more precisely to identify the impact of counterparty risk on the valuation of credit securities. Moreover, we apply the techniques in Park [17] to the pricing of bond and CDS, so that we avoid solving the PDEs.

2. Model

2.1. The Setting of Model

Let (Ω,,{𝑡}𝑇𝑡=0,𝑃) be the filtered probability space satisfying the usual conditions, where =𝑇, 𝑇 is large enough but finite and 𝑃 is an equivalent martingale measure under which discounted securities' prices are martingales.

On (Ω,,{𝑡}𝑇𝑡=0,𝑃), there is an 𝑑-valued process 𝑋=(𝑋1,,𝑋𝑑), where {𝑋𝑖}𝑑𝑖=1 are Markov processes and represent 𝑑 economy-wide state variables. Beside these, there are 𝑛 companies with 𝑛 point processes {𝑁𝑖}𝑛𝑖=1(𝑁𝑖0=0) which represent the default processes of 𝑛 companies, respectively. When 𝑁𝑖 first jumps from 0 to 1, we call the company 𝑖 defaults and denote 𝜏𝑖 be the default time of company 𝑖. Thus, 𝑁𝑖𝑡=1{𝜏𝑖𝑡}, where 1{} is the indicator function.

The filtration is generated by the state variables and the default processes of 𝑛 companies as follows 𝑡=𝑋𝑡1𝑡𝑛𝑡,(2.1) where 𝑋𝑡𝑋=𝜎𝑠,0𝑠𝑡,𝑖𝑡𝑁=𝜎𝑖𝑠.,0𝑠𝑡(2.2) Denote that 𝑡=1𝑡𝑛𝑡,𝑡𝑖=1𝑡𝑡𝑖1𝑡𝑖+1𝑛𝑡.(2.3) We assume the default time 𝜏𝑖(𝑖=1,,𝑛) possesses a strictly positive 𝑋𝑇𝑇𝑖-adapted intensity process 𝜆𝑖𝑡 satisfying 𝑡0𝜆𝑖𝑠𝑑𝑠<, 𝑃-a.s. for all 𝑡[0,𝑇]. The intensity process 𝜆𝑖𝑡 shows the local default probability in the sense that the default probability of company 𝑖 over a small interval (𝑡,𝑡+Δ𝑡) is equal to 𝜆𝑖𝑡Δ𝑡. These 𝑁𝑖, 1𝑖𝑛 generate the defaults of 𝑛 companies. Their intensity processes 𝜆𝑖, 1𝑖𝑛 depend on state variables and the default states of all other companies. Due to the counterparty risk, {𝜏𝑖}𝑛𝑖=1 may no longer be assumed independent conditionally on 𝑋.

2.2. Primary-Secondary Framework

We divide 𝑛 firms into two mutually exclusive types: 𝑙 primary firms and 𝑛𝑙 secondary firms. Primary firms' default processes only depend on state variables, while secondary firms' default processes depend on the state variables and the default states of the primary firms. This model was proposed by Jarrow and Yu [12]. Now, we provide some assumptions of the model.

Assumption 1 (economy-wide state variables). The state variable 𝑋𝑡 may contain the risk-free spot rate 𝑟𝑡 or other economical variables in the economy environment which may impact on the default probability of the companies.

Assumption 2 (the default times). On (Ω,,{𝑡}𝑇𝑡=0,𝑃), we add several independent unit exponential random variables {𝐸𝑖,1𝑖𝑙} which are independent of 𝑋 under probability measure 𝑃. The default times of 𝑙 primary firms can be defined as 𝜏𝑖=inf𝑡𝑡0𝜆𝑖𝑠𝑑𝑠𝐸𝑖,1𝑖𝑙,(2.4) where 𝜆𝑖𝑡 is adapted to 𝑋𝑡. Then, we add another series independent unit exponential random variables {𝐸𝑗,𝑙+1𝑗𝑛} which are independent of 𝑋 and 𝜏𝑖, 1𝑖𝑙. The default times of 𝑛𝑙 secondary firms can be defined as 𝜏𝑗=inf𝑡𝑡0𝜆𝑗𝑠𝑑𝑠𝐸𝑗,1+𝑙𝑗𝑛,(2.5) where 𝜆𝑗𝑡 is adapted to 𝑋𝑡1𝑡𝑙𝑡.

Assumption 3 (the default probability). Because 𝐸𝑖(1𝑖𝑙) is independent of state variables 𝑋, the conditional and unconditional survival probability distributions of primary firm 𝑖 are given by 𝑃𝜏𝑖>𝑡𝑋𝑇=exp𝑡0𝜆𝑖𝑠,𝑃𝜏𝑑𝑠(2.6)𝑖>𝑡=𝐸exp𝑡0𝜆𝑖𝑠𝑑𝑠,𝑡0,𝑇.(2.7) Similarly, since 𝐸𝑗(1+𝑙𝑗𝑛) is independent of 𝑋 and 𝜏𝑖, 1𝑖𝑙, we have the conditional and unconditional survival probability distributions of secondary firm 𝑗𝑃𝜏𝑗>𝑡𝑋𝑇1𝑇𝑙𝑇=exp𝑡0𝜆𝑗𝑠,𝑃𝜏𝑑𝑠𝑗>𝑡=𝐸exp𝑡0𝜆𝑗𝑠𝑑𝑠,𝑡0,𝑇.(2.8)

Assumption 4 (the default intensity). Because the Primary firms' default processes only depend on macrovariables, we denote their default intensities by 𝜆𝑖𝑡=Λ𝑖0,𝑡,1𝑖𝑙.(2.9) In addition, secondary firms' default processes depend on the macrovariables and the default processes of the primary firms. We denote the intensities by 𝜆𝑗𝑡=Λ𝑗0,𝑡+Σ𝑙𝑘=1Λ𝑗𝑘,𝑡1{𝜏𝑘𝑡},1+𝑙𝑗𝑛,(2.10) where Λ𝑗𝑘,𝑡 is adapted to 𝑋𝑇 for all 𝑘. Λ𝑖0,𝑡 and Λ𝑗0,𝑡 can be constants or stochastic processes which are correlated with the state variables.

Assumption 5 (the risk-free interest rate). The risk-free interest rate 𝑟𝑡 in this framework is stochastic which may follow CIR model, HJM model, Vasicek model or their extensions. It has effect on the defaults of 𝑛 companies.

3. The Pricing of Credit Securities

In this section, we price the defaultable bonds and credit default swap (CDS) in the primary-secondary framework satisfying Assumption 1 to Assumption 5. To obtain some explicit results, we give another specific assumptions. We assume that the state variable 𝑋𝑡 only contains the risk-free spot rate 𝑟𝑡 and the default of a firm is correlated with the default-free term structure. Namely, we will present a one-factor model for credit risk. Furthermore, we mainly consider single counterparty. There are one primary firm and one secondary firm in our pricing model. Counterparty risk may occur when secondary firm holds large amounts of debt issued by the primary firm.

We suppose that the risk-free interest rate follows the jump-diffusion process𝑑𝑟𝑡=𝛼𝐾𝑟𝑡𝑑𝑡+𝜎𝑑𝑊𝑡+𝑞𝑡𝑑𝑌𝑡,(3.1) where 𝑊𝑡 is a standard Brownian motion on the probability space (Ω,,𝑃) and 𝑌𝑡 is a Possion process under 𝑃 with intensity 𝜇. 𝑞𝑡 is a deterministic function and 𝛼, 𝜎, 𝐾 are constants. We assume 𝑊𝑡 and 𝑌𝑡 are mutually independent.

Remark 3.1. In fact, from Park [17], we know (3.1) has the explicit solution as follows: 𝑟𝑡=𝑟0𝑒𝛼𝑡+𝛼𝐾𝑡0𝑒𝛼(𝑡𝑠)𝑑𝑠+𝜎𝑡0𝑒𝛼(𝑡𝑠)𝑑𝑊𝑠+𝑡0𝑞𝑠𝑒𝛼(𝑡𝑠)𝑑𝑌𝑠.(3.2) Moreover, in accordance with the properties of 𝑊𝑠 and 𝑌𝑠, we can check that 𝑟𝑡 is a 𝑟-Markov process, which plays an important role in the following.

3.1. Defaultable Bonds' Pricing

We first give some general pricing formulas for bonds in the primary-secondary framework described in Section 2.2. Suppose that the face value of bond 𝑖(𝑖=1,,𝑛) is 1 dollar. Under the equivalent martingale measure 𝑃, the default-free and defaultable bond's prices are, respectively, given by𝑝(𝑡,𝑇)=𝐸𝑡exp𝑇𝑡𝑟𝑠,𝑉𝑑𝑠(3.3)𝑖(𝑡,𝑇)=𝐸𝑡𝑒𝑇𝑡𝑟𝑠𝑑𝑠𝛽𝑖1{𝜏𝑖𝑇}+1{𝜏𝑖>𝑇},(3.4) where 𝐸𝑡[] represents the conditional expectation with respect to 𝑡, 𝛽𝑖 is the recovery rate of defaultable bond 𝑖, and 𝑇(<𝑇) is the maturity date.

Lemma 3.2 (see [12]). The defaultable bond price can also be expressed as 𝑉𝑖(𝑡,𝑇)=𝛽𝑖𝑝(𝑡,𝑇)+1{𝜏𝑖>𝑡}1𝛽𝑖𝐸𝑡exp𝑇𝑡𝑟𝑠+𝜆𝑖𝑠𝑑𝑠,𝑡𝑇.(3.5)

In the following, we only consider the case with two firms. Firm 𝐴 is the primary firm whose default is independent of the default risk of secondary firm 𝐵 but depends on the interest rate 𝑟, while firm 𝐵's default is correlated with the state of firm 𝐴 and the risk-free interest rate. One assumes their intensity processes, respectively, satisfy some linear relations below:𝜆𝐴𝑡=𝑏𝐴0+𝑏𝐴1𝑟𝑡,𝜆(3.6)𝐵𝑡=𝑏𝐵0+𝑏𝐵1𝑟𝑡+𝑏1{𝜏𝐴𝑡},(3.7) where 𝑏𝐴0, 𝑏𝐴1, 𝑏𝐵0, 𝑏𝐵1, and 𝑏 are positive constants.

Remark 3.3. The interest rate 𝑟𝑡 in our model is an extension of Vasicek model. It may cause negative intensity. We use the similar method in Jarrow and Yu [12] to avoid this case. We can assume 𝜆𝐴𝑡=max{𝑏𝐴0+𝑏𝐴1𝑟𝑡,0}, 𝜆𝐵𝑡=max{𝑏𝐵0+𝑏𝐵1𝑟𝑡+𝑏1{𝜏𝐴𝑡},0}, we will discuss it in other paper.

We price the bonds issued by firm 𝐴 and firm 𝐵. To be convenient, we use time-𝑡 forward interest rate instead of time-0 forward interest rate in (3.2). Let 𝑓(0,𝑢)=𝑟0𝑒𝛼𝑢, then for 𝑢𝑡, (3.2) can be expressed as𝑟𝑢=𝑓(0,𝑢)+𝑡0+𝑢𝑡𝛼𝐾𝑒𝛼(𝑣𝑢)𝑑𝑣+𝑡0+𝑢𝑡𝜎𝑒𝛼(𝑣𝑢)𝑑𝑊𝑣+𝑡0+𝑢𝑡𝑞𝑣𝑒𝛼(𝑣𝑢)𝑑𝑌𝑣=𝑓(𝑡,𝑢)+𝑢𝑡𝛼𝐾𝑒𝛼(𝑣𝑢)𝑑𝑣+𝑢𝑡𝜎𝑒𝛼(𝑣𝑢)𝑑𝑊𝑣+𝑢𝑡𝑞𝑣𝑒𝛼(𝑣𝑢)𝑑𝑌𝑣,(3.8) where𝑓(𝑡,𝑢)=𝑓(0,𝑢)+𝑡0𝛼𝐾𝑒𝛼(𝑣𝑢)𝑑𝑣+𝑡0𝜎𝑒𝛼(𝑣𝑢)𝑑𝑊𝑣+𝑡0𝑞𝑣𝑒𝛼(𝑣𝑢)𝑑𝑌𝑣.(3.9) Now, we present an important theorem in the pricing process of credit securities.

Theorem 3.4. Suppose that 𝑟𝑡 follows (3.1) and 𝑅𝑡,𝑇=𝑇𝑡𝑟𝑠𝑑𝑠 be the cumulative interest from time 𝑡 to 𝑇. Let 𝐸𝑡[𝑒𝑎𝑅𝑡,𝑇]=𝑔(𝑎,𝑡,𝑇) for all 𝑎, then one obtains 𝑔(𝑎,𝑡,𝑇)=exp𝑇𝑡1𝑎𝑓(𝑡,𝑢)+2𝜎2𝑎2𝑐2𝑇𝑒(𝑢)+𝜇𝑎𝑞𝑢𝑐𝑇(𝑢)1𝑑𝑢𝑎𝐾(𝑇𝑡)+𝑎𝐾𝑐𝑇(𝑡),(3.10) where 𝑐𝑣1(𝑢)=𝛼𝑒𝛼(𝑢𝑣)1,0𝑣,𝑢𝑇.(3.11)

Proof. The proving ideas are similar to Jarrow and Yu [12] and Park [17]. Since 𝑟𝑡 follows (3.1), it has an explicit expression as (3.8). Then, we have 𝑎𝑇𝑡𝑟𝑢𝑑𝑢=𝑇𝑡𝑎𝑓(𝑡,𝑢)𝑑𝑢𝑇𝑡𝑑𝑢𝑢𝑡𝑎𝛼𝐾𝑒𝛼(𝑣𝑢)𝑑𝑣𝑇𝑡𝑑𝑢𝑢𝑡𝜎𝑎𝑒𝛼(𝑣𝑢)𝑑𝑊𝑣𝑇𝑡𝑑𝑢𝑢𝑡𝑞𝑣𝑎𝑒𝛼(𝑣𝑢)𝑑𝑌𝑣1(𝑡,𝑇)+2(𝑡,𝑇)+3(𝑡,𝑇)+4(𝑡,𝑇),(3.12) where 𝑎 and 1(𝑡,𝑇)=𝑎𝑇𝑡𝑓(𝑡,𝑢)𝑑𝑢,(3.13)2(𝑡,𝑇)=𝑇𝑡𝑑𝑢𝑢𝑡𝑎𝛼𝐾𝑒𝛼(𝑣𝑢)𝑑𝑣=𝑎𝐾(𝑇𝑡)𝑎𝐾𝛼𝑒𝛼(𝑡𝑇)1,(3.14)3(𝑡,𝑇)=𝑇𝑡𝑑𝑢𝑢𝑡𝜎𝑎𝑒𝛼(𝑣𝑢)𝑑𝑊𝑣,(3.15)4(𝑡,𝑇)=𝑇𝑡𝑑𝑢𝑢𝑡𝑞𝑣𝑎𝑒𝛼(𝑣𝑢)𝑑𝑌𝑣.(3.16) By the Markov property of 𝑟, we have 𝐸𝑡𝑒𝑎𝑅𝑡,𝑇𝑒=𝐸𝑎𝑅𝑡,𝑇𝑟𝑡.(3.17) Hence, we mainly need to obtain 𝐸[𝑒3(𝑡,𝑇)𝑟𝑡] and 𝐸[𝑒4(𝑡,𝑇)𝑟𝑡]. By Fübini's theorem, (3.15) and (3.16) become 3(𝑡,𝑇)=𝑇𝑡𝑑𝑊𝑣𝑇𝑣𝜎𝑎𝑒𝛼(𝑣𝑢)𝑑𝑢=𝑇𝑡𝜎𝑎𝑐𝑇(𝑣)𝑑𝑊𝑣,4(𝑡,𝑇)=𝑇𝑡𝑑𝑌𝑣𝑇𝑣𝑞𝑣𝑎𝑒𝛼(𝑣𝑢)𝑑𝑢=𝑇𝑡𝑎𝑞𝑣𝑐𝑇(𝑣)𝑑𝑌𝑣,(3.18) where 𝑐𝑇(𝑣) is given by (3.11). Moreover, 3(𝑡,𝑇) follows the normal distribution with mean 0 and variance 𝜎2𝑎2𝑇𝑡𝑐2𝑇(𝑣)𝑑𝑣. Therefore, by the independent increments of the diffusion process, 𝐸[𝐴][𝐵])exp(𝐴+𝐵)=exp(Var+Var2.(3.19) So 𝐸𝑒3(𝑡,𝑇)1=exp2𝜎2𝑎2𝑇𝑡𝑐2𝑇.(𝑣)𝑑𝑣(3.20) In addition, using the results in Park [17], based on independent increments for the jump process, 𝐸𝑒4(𝑡,𝑇)𝜇=exp𝑇𝑡𝑒𝑎𝑞𝑣𝑐𝑇(𝑣).1𝑑𝑣(3.21) We substitute (3.13), (3.14), (3.20) and (3.21) into 𝐸𝑡[𝑒𝑎𝑅𝑡,𝑇] and deduce (3.10).

Thus, from Lemma 3.2 and Theorem 3.4, we can derive the pricing formulas of defaultable bonds.

Theorem 3.5. In the primary-secondary framework described as above, the bonds issued by firm 𝐴 and 𝐵 have the same maturity date 𝑇 and recovery rate 𝛽𝐴=𝛽𝐵=0. If the intensity processes 𝜆𝐴𝑡 and 𝜆𝐵𝑡 satisfy (3.6) and (3.7) and no defaults occur up to time 𝑡, then the time-𝑡 price of bond issued by primary firm 𝐴 is 𝑉𝐴(𝑡,𝑇)=𝑔1+𝑏𝐴1,𝑡,𝑇exp𝑏𝐴0(𝑇𝑡)(3.22) and the time-𝑡 price of bond issued by secondary firm 𝐵 is 𝑉𝐵(𝑡,𝑇)=𝑔1+𝑏𝐵1𝑒,𝑡,𝑇(𝑏𝐵0+𝑏)(𝑇𝑡)+𝑏𝑒(𝐾+𝐾𝑏𝐵1+𝑏𝐵0+𝑏)𝑇+(1+𝑏𝐵0+𝑏𝐴0+𝑏𝐵1+𝑏𝐴1)𝑡𝑇𝑡𝑒(1+𝑏𝐵1+𝑏𝐴1)𝑠𝑡𝑓(𝑡,𝑢)𝑑𝑢+(𝑏𝑏𝐴1𝐾𝑏𝐴0)𝑠+(1+𝑏𝐵1+𝑏𝐴1)𝐾𝑐𝑠(𝑡)+(1+𝑏𝐵1)(𝐾𝑟0)𝑑(𝑠,𝑇,0)𝑓1(𝑡,𝑠)+𝑀(𝑠)𝑑𝑠,(3.23) where for for all 𝑘,𝑣,𝑢[0,𝑇], 𝑐𝑣(𝑢) is defined as (3.11) 1𝑑(𝑘,𝑣,𝑢)=𝛼𝑒𝛼𝑢𝑒𝛼𝑣𝑒𝛼𝑘,𝑓(3.24)1(𝑡,𝑠)=𝑡0𝜎1+𝑏𝐵1𝑑(𝑠,𝑇,𝑢)𝑑𝑊𝑢+𝑡01+𝑏𝐵1𝑞𝑢𝑑(𝑠,𝑇,𝑢)𝑑𝑌𝑢,(3.25)𝑀(𝑠)=𝑇𝑠12𝜎21+𝑏𝐵12𝑐2𝑇𝑒(𝑢)+𝜇(1+𝑏𝐵1)𝑞𝑢𝑐𝑇(𝑢)+1𝑑𝑢𝑠𝑡12𝜎21+𝑏𝐵1+𝑏𝐴1𝑐𝑠(𝑢)+1+𝑏𝐵1𝑑(𝑠,𝑇,𝑢)2+𝑑𝑢𝑠𝑡𝜇𝑒𝑞𝑢((1+𝑏𝐵1+𝑏𝐴1)𝑐𝑠(𝑢)+(1+𝑏𝐵1)𝑑(𝑠,𝑇,𝑢))1𝑑𝑢.(3.26)

Proof. Firstly, from Lemma 3.2 and Theorem 3.4, we can easily show that (3.22) holds. Secondly, according to Lemma 3.2, (3.7), and the properties of conditional expectation, we obtain the price of bond issued by firm 𝐵 at time 𝑡𝑉𝐵(𝑡,𝑇)=𝐸𝑡exp𝑇𝑡𝑟𝑠+𝜆𝐵𝑠𝑑𝑠=𝐸𝑡exp𝑏𝐵0(𝑇𝑡)1+𝑏𝐵1𝑅𝑡,𝑇𝑏𝑇𝜏𝐴1{𝜏𝐴𝑇}=𝐸𝑡exp𝑏𝐵0(𝑇𝑡)1+𝑏𝐵1𝑅𝑡,𝑇𝐸exp𝑏𝑇𝜏𝐴1{𝜏𝐴𝑇}𝑡𝑟𝑇.(3.27) By (2.6), the property of conditional expectation and the law of integration by parts, we check that 𝐸exp𝑏𝑇𝜏𝐴1{𝜏𝐴𝑇}𝑡𝑟𝑇=𝑇𝑡+𝑇𝑒𝑏(𝑇𝑠)1{𝑠𝑇}𝑑1𝑒𝑏𝐴0(𝑠𝑡)𝑏𝐴1𝑅𝑡,𝑠=𝑒𝑏(𝑇𝑡)1+𝑏𝑇𝑡𝑒(𝑏𝐴0𝑏)(𝑠𝑡)𝑏𝐴1𝑅𝑡,𝑠.𝑑𝑠(3.28) Hence, 𝑉𝐵(𝑡,𝑇)=𝑒(𝑏𝐵0+𝑏)(𝑇𝑡)𝐸𝑡𝑒(1+𝑏𝐵1)𝑅𝑡,𝑇+𝑏𝑒(𝑏𝐵0+𝑏)𝑇+(𝑏𝐵0+𝑏𝐴0)𝑡𝑇𝑡𝑒(𝑏𝐴0𝑏)𝑠𝐸𝑡𝑒(1+𝑏𝐵1+𝑏𝐴1)𝑅𝑡,𝑠(1+𝑏𝐵1)𝑅𝑠,𝑇𝑑𝑠,(3.29) where (3.29) involves the interchange of the expectation and the integral. Further, using the law of iterated conditional expectations, we have 𝐸𝑡𝑒(1+𝑏𝐵1+𝑏𝐴1)𝑅𝑡,𝑠𝑒(1+𝑏𝐵1)𝑅𝑠,𝑇=𝐸𝑡𝑒(1+𝑏𝐵1+𝑏𝐴1)𝑅𝑡,𝑠𝐸𝑠𝑒(1+𝑏𝐵1)𝑅𝑠,𝑇=𝐸𝑡𝑒(1+𝑏𝐵1+𝑏𝐴1)𝑅𝑡,𝑠𝑔1+𝑏𝐵1,𝑠,𝑇𝐸𝑡𝑒𝐼.(3.30) Denote 1+𝑏𝐵1+𝑏𝐴1=𝑚1 and 1+𝑏𝐵1=𝑚2. Then, from Theorem 3.4, we show that 𝐼𝑚1𝑠𝑡𝑟𝑢𝑑𝑢𝑚2𝑇𝑠𝑓(𝑠,𝑢)𝑑𝑢𝑚2𝐾(𝑇𝑠)+𝐾𝑚2𝑐𝑇+(𝑠)𝑇𝑠12𝜎2𝑚22𝑐2𝑇𝑒(𝑢)+𝜇𝑚2𝑞𝑢𝑐𝑇(𝑢)1𝑑𝑢𝐼1+𝐼2,(3.31) where 𝐼1=𝑚1𝑠𝑡𝑟𝑢𝑑𝑢𝑚2𝑇𝑠𝐼𝑓(𝑠,𝑢)𝑑𝑢,2=𝑇𝑠12𝜎2𝑚22𝑐2𝑇𝑒(𝑢)+𝜇𝑚2𝑞𝑢𝑐𝑇(𝑢)1𝑑𝑢𝑚2𝐾(𝑇𝑠)+𝐾𝑚2𝑐𝑇(𝑠).(3.32) Again, by (3.9), we have 𝑚2𝑇𝑠𝑓(𝑠,𝑢)𝑑𝑢=𝑇𝑠𝑚2𝑓(0,𝑢)𝑑𝑢𝑇𝑠𝑑𝑢𝑠0𝑚2𝛼𝐾𝑒𝛼(𝑣𝑢)𝑑𝑣𝑇𝑠𝑑𝑢𝑠0𝜎𝑚2𝑒𝛼(𝑣𝑢)𝑑𝑊𝑣𝑇𝑠𝑑𝑢𝑠0𝑚2𝑞𝑣𝑒𝛼(𝑣𝑢)𝑑𝑌𝑣1(𝑠,𝑇)+2(𝑠,𝑇)+3(𝑠,𝑇)+4(𝑠,𝑇),(3.33) where 1(𝑠,𝑇)=𝑚2𝑇𝑠𝑓(0,𝑢)𝑑𝑢,2(𝑠,𝑇)=𝑇𝑠𝑑𝑢𝑠0𝑚2𝛼𝐾𝑒𝛼(𝑣𝑢)𝑑𝑣,3(𝑠,𝑇)=𝑇𝑠𝑑𝑢𝑠0𝜎𝑚2𝑒𝛼(𝑣𝑢)𝑑𝑊𝑣,4(𝑠,𝑇)=𝑇𝑠𝑑𝑢𝑠0𝑚2𝑞𝑣𝑒𝛼(𝑣𝑢)𝑑𝑌𝑣.(3.34) We easily check that 1(𝑠,𝑇)=𝑚2𝑟0𝑑(𝑠,𝑇,0),2(𝑠,𝑇)=𝑚2𝐾𝑐𝑇(𝑠)+𝑚2𝐾𝑑(𝑠,𝑇,0),(3.35) where 𝑑(𝑠,𝑇,𝑢) is given by (3.24). Moreover, using Fübini's theorem, we obtain 3(𝑠,𝑇)=𝑠0𝜎𝑚2𝑑(𝑠,𝑇,𝑢)𝑑𝑊𝑢,4(𝑠,𝑇)=𝑠0𝑚2𝑞𝑢𝑑(𝑠,𝑇,𝑢)𝑑𝑌𝑢.(3.36) Therefore, we give a different expression below: 𝑚2𝑇𝑠𝑓(𝑠,𝑢)𝑑𝑢=𝑚2𝑟0𝑑(𝑠,𝑇,0)𝑚2𝐾𝑐𝑇(𝑠)+𝑚2𝐾𝑑(𝑠,𝑇,0)𝑠0𝜎𝑚2𝑑(𝑠,𝑇,𝑢)𝑑𝑊𝑢𝑠0𝑚2𝑞𝑢𝑑(𝑠,𝑇,𝑢)𝑑𝑌𝑢.(3.37) Then, from (3.12) and (3.37), we find that 𝐼1=𝑚1𝑠𝑡𝑟𝑢𝑑𝑢𝑚2𝑇𝑠𝑓(𝑠,𝑢)𝑑𝑢=𝑚1𝑠𝑡𝑓(𝑡,𝑢)𝑑𝑢𝑚1𝐾(𝑠𝑡)+𝑚1𝐾𝑐𝑠(𝑡)+𝑚2𝐾𝑟0𝑑(𝑠,𝑇,0)𝑚2𝐾𝑐𝑇(𝑠)𝑡0𝜎𝑚2𝑑(𝑠,𝑇,𝑢)𝑑𝑊𝑢𝑡0𝑚2𝑞𝑢𝑑(𝑠,𝑇,𝑢)𝑑𝑌𝑢𝑠𝑡𝜎𝑚1𝑐𝑠(𝑢)+𝑚2𝑑(𝑠,𝑇,𝑢)𝑑𝑊𝑢𝑠𝑡𝑞𝑢𝑚1𝑐𝑠(𝑢)+𝑚2𝑑(𝑠,𝑇,𝑢)𝑑𝑌𝑢.(3.38) In addition, applying (3.19) and the results in Park [17], we have 𝐸𝑒𝑠𝑡𝜎(𝑚1𝑐𝑠(𝑢)+𝑚2𝑑(𝑠,𝑇,𝑢))𝑑𝑊𝑢1=exp2𝜎2𝑠𝑡𝑚1𝑐𝑠(𝑢)+𝑚2𝑑(𝑠,𝑇,𝑢)2,𝐸𝑒𝑑𝑢𝑠𝑡𝑞𝑢(𝑚1𝑐𝑠(𝑢)+𝑚2𝑑(𝑠,𝑇,𝑢))𝑑𝑌𝑢𝜇=exp𝑠𝑡𝑒𝑞𝑢(𝑚1𝑐𝑠(𝑢)+𝑚2𝑑(𝑠,𝑇,𝑢)).1𝑑𝑢(3.39) Therefore, combining (3.39) and (3.31), we obtain 𝐸𝑡𝑒𝐼=𝑒𝐼2𝐸𝑡𝑒𝐼1(i)=exp1+𝑏𝐵1+𝑏𝐴1𝑠𝑡𝑓(𝑡,𝑢)𝑑𝑢𝑏𝐴1𝐾𝑠+1+𝑏𝐵1+𝑏𝐴1𝐾𝑡1+𝑏𝐵1𝐾𝑇exp1+𝑏𝐵1+𝑏𝐴1𝐾𝑐𝑠(𝑡)+1+𝑏𝐵1𝐾𝑟0𝑑(𝑠,𝑇,0)𝑓1(𝑡,𝑠)𝐸𝑡𝑒𝑠𝑡𝜎[(1+𝑏𝐵1+𝑏𝐴1)𝑐𝑠(𝑢)+(1+𝑏𝐵1)𝑑(𝑠,𝑇,𝑢)]𝑑𝑊𝑢𝐸𝑡𝑒𝑠𝑡𝑞𝑢[(1+𝑏𝐵1+𝑏𝐴1)𝑐𝑠(𝑢)+(1+𝑏𝐵1)𝑑(𝑠,𝑇,𝑢)]𝑑𝑌𝑢(ii)=exp1+𝑏𝐵1+𝑏𝐴1𝑠𝑡𝑓(𝑡,𝑢)𝑑𝑢𝑏𝐴1𝐾𝑠+1+𝑏𝐵1+𝑏𝐴1𝐾𝑡1+𝑏𝐵1𝐾𝑇exp1+𝑏𝐵1+𝑏𝐴1𝐾𝑐𝑠(𝑡)+1+𝑏𝐵1𝐾𝑟0𝑑(𝑠,𝑇,0)𝑓11(𝑡,𝑠)exp2𝜎2𝑠𝑡1+𝑏𝐵1+𝑏𝐴1𝑐𝑠(𝑢)+1+𝑏𝐵1𝑑(𝑠,𝑇,𝑢)2𝜇𝑑𝑢exp𝑠𝑡𝑒𝑞𝑢((1+𝑏𝐵1+𝑏𝐴1)𝑐𝑠(𝑢)+(1+𝑏𝐵1)𝑑(𝑠,𝑇,𝑢))1𝑑𝑢exp𝑇𝑠12𝜎21+𝑏𝐵12𝑐2𝑇𝑒(𝑢)+𝜇(1+𝑏𝐵1)𝑞𝑢𝑐𝑇(𝑢),1𝑑𝑢(3.40) where (i) follows from the independence of 𝑊𝑢 and 𝑌𝑢 and the property of conditional expectation and (ii) holds from the Markov property and the independent increment property of 𝑊𝑢 and 𝑌𝑢. Finally, we substitute 𝐸𝑡[𝑒𝐼] into (3.29) and obtain (3.23). The proof is completed.

3.2. CDS's Pricing

Firm 𝐶 holds a bond issued by the reference firm 𝐴 with the maturity date 𝑇1. To decrease the possible loss, firm 𝐶 buys protection with the maturity date 𝑇2(𝑇2𝑇1) from firm 𝐵 on condition that firm 𝐶 gives the payments to firm 𝐵 at a fixed swap rate in time while firm 𝐵 promises to make up firm 𝐶 for the loss caused by the default of firm 𝐴 at a certain rate. Each party has the obligation to make payments until its own default. The source of credit risk may be from three parties: the issuer of bond, the buyer of CDS and the seller of CDS.

In the following, we consider a simple situation which only contains the risk from reference firm 𝐴 and firm 𝐵. At the same time, to make the calculation convenient, we suppose the recovery rate of the bond issued by firm 𝐴 is zero and the face value is 1 dollar. In the event of firm 𝐴's default, firm 𝐵 compensates firm 𝐶 for 1 dollar if he does not default, otherwise 0 dollar. There are four cases for the defaults of firm 𝐴 and firm 𝐵.

Case 1. The defaults of firm 𝐴 and firm 𝐵 are mutually independent conditional on the risk-free interest rate.

Case 2. Firm 𝐴 is the primary party whose default only depends on the risk-free interest rate (the only economy state variable) and the firm 𝐵 is the secondary party whose default depends on the risk-free interest rate and the default state of firm 𝐴.

Case 3. Firm 𝐵 is the primary party and the firm 𝐴 is the secondary party.

Case 4. The defaults of firm 𝐴, and firm 𝐵 are mutually contagious (looping default).

Now, we make use of the results in previous sections to price the CDS in Case 2. We assume firm 𝐴 is the primary party and the firm 𝐵 is the secondary party. Denoted the swap rate by a constant 𝑐 and interest rate by 𝑟𝑡, let the default times of firm 𝐴 and 𝐵 be 𝜏𝐴 with the intensity 𝜆𝐴 and 𝜏𝐵 with the intensity 𝜆𝐵, respectively.

Theorem 3.6. Suppose the risk-free interest rate 𝑟𝑡 satisfies (3.1) and the intensities 𝜆𝐴 and 𝜆𝐵 satisfy (3.6) and (3.7), respectively. Then, the swap rate 𝑐 has the following expression: 𝑉𝑐=𝐵0,𝑇2𝑒(𝑏𝐵0+𝑏𝐴0)𝑇2𝑔1+𝑏𝐵1+𝑏𝐴1,0,𝑇2𝑇20𝑔(1,0,𝑠)𝑑𝑠,(3.41) where 𝑔(,,) and 𝑉𝐵(0,𝑇2) are given by Theorems 3.4 and 3.5, respectively.

Proof. Firstly, the time-0 market value of buyer 𝐶's payments to seller 𝐵 is 𝐸𝑇20𝑐𝑒𝑠0𝑟𝑢𝑑𝑢𝑑𝑠=𝑐𝑇20𝐸𝑒𝑅0,𝑠𝑑𝑠,(3.42) where 𝑅0,𝑠 is defined as Theorem 3.4.
Secondly, the time-0 market value of firm 𝐵's promised payoff in case of firm 𝐴's default is𝐸1{𝜏𝐴𝑇2}𝑒𝑇20𝑟𝑢𝑑𝑢1{𝜏𝐵>𝑇2}.(3.43) Thus, in accordance with the arbitrage-free principle, we obtain 𝐸1𝑐={𝜏𝐴𝑇2}𝑒𝑇20𝑟𝑢𝑑𝑢1{𝜏𝐵>𝑇2}𝑇20𝐸𝑒𝑅0,𝑠.𝑑𝑠(3.44) Further, we can use the properties of conditional expectation to simplify (3.44) as follows: 𝐸𝐸1𝑐={𝜏𝐴𝑇2}𝑟𝑇𝐵𝑇𝑒𝑇20𝑟𝑢𝑑𝑢1{𝜏𝐵>𝑇2}𝑇20𝐸𝑒𝑅0,𝑠=𝐸1𝑑𝑠{𝜏𝐵>𝑇2}𝑒𝑇20𝑟𝑢𝑑𝑢1𝐸{𝜏𝐴>𝑇2}𝑒𝑇20𝑟𝑢𝑑𝑢1{𝜏𝐵>𝑇2}𝑇20𝐸𝑒𝑅0,𝑠=𝑉𝑑𝑠𝐵0,𝑇21𝐸{𝜏𝐴>𝑇2}𝑒𝑇20(𝑟𝑢+𝜆𝐵𝑢)𝑑𝑢𝑇20𝐸𝑒𝑅0,𝑠,𝑑𝑠(3.45) where the last one is obtained by (3.4). Note that 𝑉𝐵(0,𝑇2) can be obtained by (3.23) and 𝐸[𝑒𝑅0,𝑠]=𝑔(1,0,𝑠) by Theorem 3.4. We substitute (3.7) into the above expectation term 𝐸1{𝜏𝐴>𝑇2}exp𝑇20𝑟𝑢+𝜆𝐵𝑢1𝑑𝑢=𝐸{𝜏𝐴>𝑇2}exp𝑇20𝑏𝐵0+1+𝑏𝐵1𝑟𝑢+𝑏1{𝜏𝐴𝑢}1𝑑𝑢=𝐸{𝜏𝐴>𝑇2}exp𝑏𝐵0𝑇21+𝑏𝐵1𝑇20𝑟𝑢𝑇𝑑𝑢𝑏2𝜏𝐴1{𝜏𝐴𝑇2}1=𝐸{𝜏𝐴>𝑇2}𝑒𝑏𝐵0𝑇2(1+𝑏𝐵1)𝑅20,𝑇(i)𝐸1=𝐸{𝜏𝐴>𝑇2}𝑟𝑇𝑒𝑏𝐵0𝑇2(1+𝑏𝐵1)𝑅20,𝑇(ii)𝑒=𝐸𝑏𝐵0𝑇2(1+𝑏𝐵1)𝑅20,𝑇𝑇20𝜆𝐴𝑢𝑑𝑢(iii)=𝑒(𝑏𝐵0+𝑏𝐴0)𝑇2𝐸𝑒(1+𝑏𝐵1+𝑏𝐴1)𝑇20𝑟𝑢𝑑𝑢(iv)=𝑒(𝑏𝐵0+𝑏𝐴0)𝑇2𝑔1+𝑏𝐵1+𝑏𝐴1,0,𝑇2,(3.46) where (i) involves the property of conditional expectation, (ii) follows from (2.6), and (iv) follows from Theorem 3.4. Now, substituting these results into (3.45), we show (3.41) holds. The proof is complete.

Remark 3.7. The model in Case 1 can be considered a special case of primary-secondary model and the price of CDS can be derived by the similar method. The pricing of CDS in Case 4 will be discussed in another paper. In Case 3, if 𝜆𝐴𝑡 and 𝜆𝐵𝑡 satisfy the below relations: 𝜆𝐴𝑡=𝑏𝐴0+𝑏𝐴1𝑟𝑡+𝑏1{𝜏𝐵𝑡},𝜆𝐵𝑡=𝑏𝐵0+𝑏𝐵1𝑟𝑡,(3.47) where 𝑏𝐴0, 𝑏𝐴1, 𝑏𝐵0, 𝑏𝐵1, and 𝑏 are positive constants, then the swap rate 𝑔𝑐=1+𝑏𝐵1,0,𝑇2𝑒𝑏𝐵0𝑇2𝑒(𝑏𝐵0+𝑏𝐴0)𝑇2𝑔1+𝑏𝐵1+𝑏𝐴1,0,𝑇2𝑇20,𝑔(1,0,𝑠)𝑑𝑠(3.48) where 𝑔(,,) are given by Theorem 3.4. The deriving process is similar to Theorem 3.6, so we omit it.

Remark 3.8. In our models, to make the expressions comparatively simple, we all assume that the recovery rates are zero. When the relevant recovery rates are nonzero constant, the pricing formulas are still easily obtained from Lemma 3.2 because we can get 𝑝(𝑡,𝑇)=𝑔(1,𝑡,𝑇) from Theorem 3.4. We omit the process.

4. Conclusion

This paper gives the pricing formulas of defaultable bonds and CDSs. In our model, we consider the case that the default intensity is correlated with the risk-free interest rate following jump-diffusion process and the counterparty's default, which is more realistic. We involve the jump risk of risk-free interest rate in the pricing, generalizing the contagious model in Jarrow and Yu [12].

In fact, we only consider the comparatively simple situation. We can further study the more general model. For example, we consider the case that the relevant recovery rates are stochastic and the interest rate satisfies more general jump-diffusion process. Moreover, the model in this paper is actually one-factor model with one state variable, while we can discuss multifactor models in which there are several state variables. In a word, the contagious model of credit security with counterparty risk is very necessary to be further discussed in the future.

Acknowledgments

The authors gratefully acknowledge the support from the National Basic Research Program of China (973 Program no. 2007CB814903) and thank the reviewers for their valued comments.

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