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

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,ℱ,{ℱ_𝑡}^𝑇∗_𝑡=0,𝑃) be the filtered probability space satisfying the usual conditions, where ℱ=ℱ_𝑇^∗, 𝑇^∗ is large enough but finite and 𝑃 is an equivalent martingale measure under which discounted securities' prices are martingales.

On (Ω,ℱ,{ℱ_𝑡}^𝑇∗_𝑡=0,𝑃), there is an ℛ^𝑑-valued process 𝑋=(𝑋¹,…,𝑋^𝑑), where {𝑋^𝑖}^𝑑𝑖=1 are Markov processes and represent 𝑑 economy-wide state variables. Beside these, there are 𝑛 companies with 𝑛 point processes {𝑁^𝑖}^𝑛𝑖=1(𝑁^𝑖0=0) which represent the default processes of 𝑛 companies, respectively. When 𝑁^𝑖 first jumps from 0 to 1, we call the company 𝑖 defaults and denote 𝜏^𝑖 be the default time of company 𝑖. Thus, 𝑁^𝑖𝑡=1_{{𝜏𝑖≤𝑡}}, where 1_{⋅} is the indicator function.

The filtration ℱ is generated by the state variables and the default processes of 𝑛 companies as follows ℱ_𝑡=ℱ^𝑋𝑡∨ℋ^1𝑡∨⋯∨ℋ^𝑛𝑡,(2.1) where ℱ^𝑋𝑡=𝜎𝑋_𝑠,0≤𝑠≤𝑡,ℋ^𝑖𝑡=𝜎𝑁^𝑖𝑠,0≤𝑠≤𝑡.(2.2) Denote that ℋ_𝑡=ℋ^1𝑡∨⋯∨ℋ^𝑛𝑡,ℋ^𝑡−𝑖=ℋ^1𝑡∨⋯∨ℋ^𝑡𝑖−1∨ℋ^𝑡𝑖+1∨⋯∨ℋ^𝑛𝑡.(2.3) We assume the default time 𝜏^𝑖(𝑖=1,…,𝑛) possesses a strictly positive ℱ^𝑋𝑇_∗∨ℋ^𝑇−𝑖_∗-adapted intensity process 𝜆^𝑖𝑡 satisfying ∫^𝑡0𝜆^𝑖𝑠𝑑𝑠<∞, 𝑃-a.s. for all 𝑡∈[0,𝑇^∗]. The intensity process 𝜆^𝑖𝑡 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 𝜆^𝑖𝑡Δ𝑡. These 𝑁^𝑖, 1≤𝑖≤𝑛 generate the defaults of 𝑛 companies. Their intensity processes 𝜆^𝑖, 1≤𝑖≤𝑛 depend on state variables and the default states of all other companies. Due to the counterparty risk, {𝜏^𝑖}^𝑛𝑖=1 may no longer be assumed independent conditionally on ℱ^𝑋.

2.2. Primary-Secondary Framework

We divide 𝑛 firms into two mutually exclusive types: 𝑙 primary firms and 𝑛−𝑙 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 𝑋_𝑡 may contain the risk-free spot rate 𝑟_𝑡 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,ℱ,{ℱ_𝑡}^𝑇∗_𝑡=0,𝑃), we add several independent unit exponential random variables {𝐸^𝑖,1≤𝑖≤𝑙} which are independent of 𝑋 under probability measure 𝑃. The default times of 𝑙 primary firms can be defined as 𝜏^𝑖=inf𝑡∶^𝑡0𝜆^𝑖𝑠𝑑𝑠≥𝐸^𝑖,1≤𝑖≤𝑙,(2.4) where 𝜆^𝑖𝑡 is adapted to ℱ^𝑋𝑡. Then, we add another series independent unit exponential random variables {𝐸^𝑗,𝑙+1≤𝑗≤𝑛} which are independent of 𝑋 and 𝜏^𝑖, 1≤𝑖≤𝑙. The default times of 𝑛−𝑙 secondary firms can be defined as 𝜏^𝑗=inf𝑡∶^𝑡0𝜆^𝑗𝑠𝑑𝑠≥𝐸^𝑗,1+𝑙≤𝑗≤𝑛,(2.5) where 𝜆^𝑗𝑡 is adapted to ℱ^𝑋𝑡∨ℋ^1𝑡∨⋯∨ℋ^𝑙𝑡.

Assumption 3 (the default probability). Because 𝐸^𝑖(1≤𝑖≤𝑙) is independent of state variables 𝑋, the conditional and unconditional survival probability distributions of primary firm 𝑖 are given by 𝑃𝜏𝑃𝜏^𝑖>𝑡∣ℱ^𝑋𝑇_∗=exp−^𝑡0𝜆^𝑖𝑠𝑑𝑠,(2.6)_𝑖>𝑡=𝐸exp−^𝑡0𝜆^𝑖𝑠𝑑𝑠,𝑡∈0,𝑇^∗.(2.7) Similarly, since 𝐸^𝑗(1+𝑙≤𝑗≤𝑛) is independent of 𝑋 and 𝜏^𝑖, 1≤𝑖≤𝑙, we have the conditional and unconditional survival probability distributions of secondary firm 𝑗𝑃𝜏𝑃^𝑗>𝜏𝑡∣ℱ^𝑋𝑇_∗∨ℋ^1𝑇_∗∨⋯∨ℋ^𝑙𝑇_∗=exp−^𝑡0𝜆^𝑗𝑠𝑑𝑠_𝑗,>𝑡=𝐸exp−^𝑡0𝜆^𝑗𝑠𝑑𝑠,𝑡∈0,𝑇^∗.(2.8)

Assumption 4 (the default intensity). Because the Primary firms' default processes only depend on macrovariables, we denote their default intensities by 𝜆^𝑖𝑡=Λ^𝑖0,𝑡,1≤𝑖≤𝑙.(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 𝜆^𝑗𝑡=Λ^𝑗0,𝑡+Σ^𝑙𝑘=1Λ^{𝑗𝑘,𝑡}1_{{𝜏𝑘≤𝑡}},1+𝑙≤𝑗≤𝑛,(2.10) where Λ^{𝑗𝑘,𝑡} is adapted to ℱ^𝑋𝑇_∗ for all 𝑘. Λ^𝑖0,𝑡 and Λ^𝑗0,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 𝑟_𝑡 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 𝑋_𝑡 only contains the risk-free spot rate 𝑟_𝑡 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 process𝑑𝑟_𝑡=𝛼𝐾−𝑟_𝑡𝑑𝑡+𝜎𝑑𝑊_𝑡+𝑞_𝑡𝑑𝑌_𝑡,(3.1) where 𝑊_𝑡 is a standard Brownian motion on the probability space (Ω,ℱ,𝑃) and 𝑌_𝑡 is a Possion process under 𝑃 with intensity 𝜇. 𝑞_𝑡 is a deterministic function and 𝛼, 𝜎, 𝐾 are constants. We assume 𝑊_𝑡 and 𝑌_𝑡 are mutually independent.

Remark 3.1. In fact, from Park [17], we know (3.1) has the explicit solution as follows: 𝑟_𝑡=𝑟₀𝑒^−𝛼𝑡+𝛼𝐾^𝑡0𝑒^{−𝛼(𝑡−𝑠)}𝑑𝑠+𝜎^𝑡0𝑒^{−𝛼(𝑡−𝑠)}𝑑𝑊_𝑠+^𝑡0𝑞_𝑠𝑒^{−𝛼(𝑡−𝑠)}𝑑𝑌_𝑠.(3.2) Moreover, in accordance with the properties of 𝑊_𝑠 and 𝑌_𝑠, we can check that 𝑟_𝑡 is a ℱ^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)=𝐸_𝑡exp−^𝑇𝑡𝑉𝑟_𝑠𝑑𝑠,(3.3)_𝑖(𝑡,𝑇)=𝐸_𝑡𝑒^{−∫𝑇𝑡𝑟𝑠𝑑𝑠}𝛽^𝑖1_{{𝜏𝑖≤𝑇}}+1_{{𝜏𝑖>𝑇}},(3.4) where 𝐸_𝑡[⋅] represents the conditional expectation with respect to ℱ_𝑡, 𝛽^𝑖 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 𝑉^𝑖(𝑡,𝑇)=𝛽^𝑖𝑝(𝑡,𝑇)+1_{{𝜏𝑖>𝑡}}1−𝛽^𝑖𝐸_𝑡exp−^𝑇𝑡𝑟_𝑠+𝜆^𝑖𝑠𝑑𝑠,𝑡≤𝑇.(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:𝜆𝜆^𝐴𝑡=𝑏^𝐴0+𝑏^𝐴1𝑟_𝑡,(3.6)_𝐵𝑡=𝑏^𝐵0+𝑏^𝐵1𝑟_𝑡+𝑏1_{{𝜏𝐴≤𝑡}},(3.7) where 𝑏^𝐴0, 𝑏^𝐴1, 𝑏^𝐵0, 𝑏^𝐵1, and 𝑏 are positive constants.

Remark 3.3. The interest rate 𝑟_𝑡 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 𝜆^𝐴𝑡=max{𝑏^𝐴0+𝑏^𝐴1𝑟_𝑡,0}, 𝜆^𝐵𝑡=max{𝑏^𝐵0+𝑏^𝐵1𝑟_𝑡+𝑏1_{{𝜏𝐴≤𝑡}},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)=𝑟₀𝑒^−𝛼𝑢, then for 𝑢≥𝑡, (3.2) can be expressed as𝑟_𝑢=𝑓(0,𝑢)+^𝑡0+^𝑢𝑡𝛼𝐾𝑒^{𝛼(𝑣−𝑢)}𝑑𝑣+^𝑡0+^𝑢𝑡𝜎𝑒^{𝛼(𝑣−𝑢)}𝑑𝑊_𝑣+^𝑡0+^𝑢𝑡𝑞^𝑣𝑒^{𝛼(𝑣−𝑢)}𝑑𝑌_𝑣=𝑓(𝑡,𝑢)+^𝑢𝑡𝛼𝐾𝑒^{𝛼(𝑣−𝑢)}𝑑𝑣+^𝑢𝑡𝜎𝑒^{𝛼(𝑣−𝑢)}𝑑𝑊_𝑣+^𝑢𝑡𝑞_𝑣𝑒^{𝛼(𝑣−𝑢)}𝑑𝑌_𝑣,(3.8) where𝑓(𝑡,𝑢)=𝑓(0,𝑢)+^𝑡0𝛼𝐾𝑒^{𝛼(𝑣−𝑢)}𝑑𝑣+^𝑡0𝜎𝑒^{𝛼(𝑣−𝑢)}𝑑𝑊_𝑣+^𝑡0𝑞_𝑣𝑒^{𝛼(𝑣−𝑢)}𝑑𝑌_𝑣.(3.9) Now, we present an important theorem in the pricing process of credit securities.

Theorem 3.4. Suppose that 𝑟_𝑡 follows (3.1) and 𝑅_𝑡,𝑇=∫^𝑇𝑡𝑟_𝑠𝑑𝑠 be the cumulative interest from time 𝑡 to 𝑇. Let 𝐸_𝑡[𝑒^{−𝑎𝑅𝑡,𝑇}]=𝑔(𝑎,𝑡,𝑇) for all 𝑎∈ℛ, then one obtains 𝑔(𝑎,𝑡,𝑇)=exp^𝑇𝑡−𝑎𝑓(𝑡,𝑢)+12𝜎²𝑎²𝑐^2𝑇(𝑢)+𝜇𝑒^{−𝑎𝑞𝑢𝑐𝑇(𝑢)}−1𝑑𝑢−𝑎𝐾(𝑇−𝑡)+𝑎𝐾𝑐_𝑇(𝑡),(3.10) where 𝑐_𝑣(𝑢)=−1𝛼𝑒^{𝛼(𝑢−𝑣)}−1,0≤𝑣,𝑢≤𝑇.(3.11)

Proof. The proving ideas are similar to Jarrow and Yu [12] and Park [17]. Since 𝑟_𝑡 follows (3.1), it has an explicit expression as (3.8). Then, we have −𝑎^𝑇𝑡𝑟_𝑢𝑑𝑢=−−^𝑇𝑡𝑎𝑓(𝑡,𝑢)𝑑𝑢−^𝑇𝑡𝑑𝑢^𝑢𝑡𝑎𝛼𝐾𝑒^{𝛼(𝑣−𝑢)}𝑑𝑣_𝑇𝑡𝑑𝑢^𝑢𝑡𝜎𝑎𝑒^{𝛼(𝑣−𝑢)}𝑑𝑊_𝑣−^𝑇𝑡𝑑𝑢^𝑢𝑡𝑞_𝑣𝑎𝑒^{𝛼(𝑣−𝑢)}𝑑𝑌_𝑣≜ℎ₁(𝑡,𝑇)+ℎ₂(𝑡,𝑇)+ℎ₃(𝑡,𝑇)+ℎ₄(𝑡,𝑇),(3.12) where 𝑎∈ℛ and ℎℎ₁(𝑡,𝑇)=−𝑎^𝑇𝑡𝑓(𝑡,𝑢)𝑑𝑢,(3.13)₂(𝑡,𝑇)=−^𝑇𝑡𝑑𝑢^𝑢𝑡𝑎𝛼𝐾𝑒^{𝛼(𝑣−𝑢)}𝑑𝑣=−𝑎𝐾(𝑇−𝑡)−𝑎𝐾𝛼𝑒^{𝛼(𝑡−𝑇)}ℎ−1,(3.14)₃ℎ(𝑡,𝑇)=−^𝑇𝑡𝑑𝑢^𝑢𝑡𝜎𝑎𝑒^{𝛼(𝑣−𝑢)}𝑑𝑊_𝑣,(3.15)₄(𝑡,𝑇)=−^𝑇𝑡𝑑𝑢^𝑢𝑡𝑞_𝑣𝑎𝑒^{𝛼(𝑣−𝑢)}𝑑𝑌_𝑣.(3.16) By the Markov property of 𝑟, we have 𝐸_𝑡𝑒^{−𝑎𝑅𝑡,𝑇}=𝐸𝑒^{−𝑎𝑅𝑡,𝑇}∣𝑟_𝑡.(3.17) Hence, we mainly need to obtain 𝐸[𝑒^{ℎ3(𝑡,𝑇)}∣𝑟_𝑡] and 𝐸[𝑒^{ℎ4(𝑡,𝑇)}∣𝑟_𝑡]. By Fübini's theorem, (3.15) and (3.16) become ℎ₃(𝑡,𝑇)=−^𝑇𝑡𝑑𝑊_𝑣^𝑇𝑣ℎ₄𝜎𝑎𝑒^{𝛼(𝑣−𝑢)}𝑑𝑢=−^𝑇𝑡𝜎𝑎𝑐_𝑇(𝑣)𝑑𝑊_𝑣,(𝑡,𝑇)=−^𝑇𝑡𝑑𝑌_𝑣^𝑇𝑣𝑞_𝑣𝑎𝑒^{𝛼(𝑣−𝑢)}𝑑𝑢=−^𝑇𝑡𝑎𝑞_𝑣𝑐_𝑇(𝑣)𝑑𝑌_𝑣,(3.18) where 𝑐_𝑇(𝑣) is given by (3.11). Moreover, ℎ₃(𝑡,𝑇) follows the normal distribution with mean 0 and variance 𝜎²𝑎²∫^𝑇𝑡𝑐^2𝑇(𝑣)𝑑𝑣. Therefore, by the independent increments of the diffusion process, 𝐸exp(𝐴+𝐵)=exp(Var[𝐴]+Var[𝐵])2.(3.19) So 𝐸𝑒^{ℎ3(𝑡,𝑇)}=exp12𝜎²𝑎²^𝑇𝑡𝑐^2𝑇(𝑣)𝑑𝑣.(3.20) In addition, using the results in Park [17], based on independent increments for the jump process, 𝐸𝑒^{ℎ4(𝑡,𝑇)}=exp𝜇^𝑇𝑡𝑒^{−𝑎𝑞𝑣𝑐𝑇(𝑣)}−1𝑑𝑣.(3.21) We substitute (3.13), (3.14), (3.20) and (3.21) into 𝐸_𝑡[𝑒^{−𝑎𝑅𝑡,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 𝛽^𝐴=𝛽^𝐵=0. If the intensity processes 𝜆^𝐴𝑡 and 𝜆^𝐵𝑡 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 𝑉^𝐴(𝑡,𝑇)=𝑔1+𝑏^𝐴1,𝑡,𝑇exp−𝑏^𝐴0(𝑇−𝑡)(3.22) and the time-𝑡 price of bond issued by secondary firm 𝐵 is 𝑉^𝐵(𝑡,𝑇)=𝑔1+𝑏^𝐵1,𝑡,𝑇𝑒^{−(𝑏𝐵0+𝑏)(𝑇−𝑡)}+𝑏𝑒^{−(𝐾+𝐾𝑏𝐵1+𝑏𝐵0+𝑏)𝑇+(1+𝑏𝐵0+𝑏𝐴0+𝑏𝐵1+𝑏𝐴1)𝑡}⋅^𝑇𝑡𝑒^{−(1+𝑏𝐵1+𝑏𝐴1)∫𝑠𝑡𝑓(𝑡,𝑢)𝑑𝑢+(𝑏−𝑏𝐴1𝐾−𝑏𝐴0)𝑠+(1+𝑏𝐵1+𝑏𝐴1)𝐾𝑐𝑠(𝑡)+(1+𝑏𝐵1)(𝐾−𝑟0)𝑑(𝑠,𝑇,0)−𝑓1(𝑡,𝑠)+𝑀(𝑠)}𝑑𝑠,(3.23) where for for all 𝑘,𝑣,𝑢∈[0,𝑇], 𝑐_𝑣(𝑢) is defined as (3.11) 𝑑(𝑘,𝑣,𝑢)=−1𝛼𝑒^𝛼𝑢𝑒^−𝛼𝑣𝑓−𝑒^−𝛼𝑘,(3.24)₁𝑀(𝑡,𝑠(𝑠))==^𝑡0𝜎_𝑇𝑠1+𝑏^𝐵1𝑑(𝑠,𝑇,𝑢)𝑑𝑊_𝑢+^𝑡01+𝑏^𝐵1𝑞_𝑢𝑑(𝑠,𝑇,𝑢)𝑑𝑌_𝑢,(3.25)+12𝜎²1+𝑏^𝐵1²𝑐^2𝑇(𝑢)+𝜇𝑒^{−(1+𝑏𝐵1)𝑞𝑢𝑐𝑇(𝑢)}−1𝑑𝑢_𝑠𝑡12𝜎²1+𝑏^𝐵1++𝑏^𝐴1𝑐_𝑠(𝑢)+1+𝑏^𝐵1𝑑(𝑠,𝑇,𝑢)²𝑑𝑢_𝑠𝑡𝜇𝑒^{−𝑞𝑢((1+𝑏𝐵1+𝑏𝐴1)𝑐𝑠(𝑢)+(1+𝑏𝐵1)𝑑(𝑠,𝑇,𝑢))}−1𝑑𝑢.(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 𝑡𝑉^𝐵(𝑡,𝑇)=𝐸_𝑡exp−^𝑇𝑡𝑟_𝑠=𝐸_𝑡+𝜆^𝐵𝑠𝑑𝑠=𝐸_𝑡exp−𝑏^𝐵0(𝑇−𝑡)−1+𝑏^𝐵1𝑅_𝑡,𝑇−𝑏𝑇−𝜏^𝐴1_{𝜏^𝐴_≤𝑇}exp−𝑏^𝐵0(𝑇−𝑡)−1+𝑏^𝐵1𝑅_𝑡,𝑇⋅𝐸exp−𝑏𝑇−𝜏^𝐴1_{𝜏^𝐴_≤𝑇}∣ℱ_𝑡∨ℱ^𝑟𝑇_∗.(3.27) By (2.6), the property of conditional expectation and the law of integration by parts, we check that 𝐸ex=p−_𝑇𝑡𝑏𝑇−𝜏^𝐴1_{𝜏^𝐴_≤𝑇}∣ℱ_𝑡∨ℱ^𝑟𝑇_∗+^∞𝑇𝑒^{−𝑏(𝑇−𝑠)1{𝑠≤𝑇}}=𝑒_{−𝑏(𝑇−𝑡)}𝑑1−𝑒^{−𝑏𝐴0(𝑠−𝑡)−𝑏𝐴1𝑅𝑡,𝑠}1+𝑏^𝑇𝑡𝑒^{−(𝑏𝐴0−𝑏)(𝑠−𝑡)−𝑏𝐴1𝑅𝑡,𝑠}𝑑𝑠.(3.28) Hence, 𝑉^𝐵(𝑡,𝑇)=𝑒^{−(𝑏𝐵0+𝑏)(𝑇−𝑡)}𝐸_𝑡𝑒^{−(1+𝑏𝐵1)𝑅𝑡,𝑇}+𝑏𝑒^{−(𝑏𝐵0+𝑏)𝑇+(𝑏𝐵0+𝑏𝐴0)𝑡}⋅^𝑇𝑡𝑒^{−(𝑏𝐴0−𝑏)𝑠}𝐸_𝑡𝑒^{−(1+𝑏𝐵1+𝑏𝐴1)𝑅𝑡,𝑠−(1+𝑏𝐵1)𝑅𝑠,𝑇}𝑑𝑠,(3.29) where (3.29) involves the interchange of the expectation and the integral. Further, using the law of iterated conditional expectations, we have 𝐸_𝑡𝑒=𝐸^{−(1+𝑏𝐵1+𝑏𝐴1)𝑅𝑡,𝑠}_𝑡⋅𝑒^{−(1+𝑏𝐵1)𝑅𝑠,𝑇}=𝐸_𝑡𝑒^{−(1+𝑏𝐵1+𝑏𝐴1)𝑅𝑡,𝑠}⋅𝐸_𝑠𝑒^{−(1+𝑏𝐵1)𝑅𝑠,𝑇}≜𝐸_𝑡𝑒^{−(1+𝑏𝐵1+𝑏𝐴1)𝑅𝑡,𝑠}⋅𝑔1+𝑏^𝐵1,𝑠,𝑇𝑒^𝐼.(3.30) Denote 1+𝑏^𝐵1+𝑏^𝐴1=𝑚₁ and 1+𝑏^𝐵1=𝑚₂. Then, from Theorem 3.4, we show that 𝐼≜−𝑚₁^𝑠𝑡𝑟_𝑢𝑑𝑢−𝑚₂^𝑇𝑠+𝑓(𝑠,𝑢)𝑑𝑢−𝑚₂𝐾(𝑇−𝑠)+𝐾𝑚₂𝑐_𝑇(𝑠)_𝑇𝑠≜𝐼₁21𝜎²𝑚²²𝑐^2𝑇(𝑢)+𝜇𝑒^{−𝑚2𝑞𝑢𝑐𝑇(𝑢)}−1𝑑𝑢+𝐼₂,(3.31) where 𝐼₁=−𝑚₁^𝑠𝑡𝑟_𝑢𝑑𝑢−𝑚₂^𝑇𝑠𝐼𝑓(𝑠,𝑢)𝑑𝑢,₂=^𝑇𝑠12𝜎²−𝑚₂𝑚²²𝑐^2𝑇(𝑢)+𝜇𝑒^{−𝑚2𝑞𝑢𝑐𝑇(𝑢)}−1𝑑𝑢𝐾(𝑇−𝑠)+𝐾𝑚₂𝑐_𝑇(𝑠).(3.32) Again, by (3.9), we have −𝑚₂^𝑇𝑠𝑓(𝑠,𝑢)𝑑𝑢=−−^𝑇𝑠𝑚₂𝑓(0,𝑢)𝑑𝑢−^𝑇𝑠𝑑𝑢^𝑠0𝑚₂𝛼𝐾𝑒^{𝛼(𝑣−𝑢)}𝑑𝑣_𝑇𝑠𝑑𝑢^𝑠0𝜎𝑚₂𝑒^{𝛼(𝑣−𝑢)}𝑑𝑊_𝑣−^𝑇𝑠𝑑𝑢^𝑠0𝑚₂𝑞_𝑣𝑒^{𝛼(𝑣−𝑢)}𝑑𝑌_𝑣≜ℎ^1(𝑠,𝑇)+ℎ^2(𝑠,𝑇)+ℎ^3(𝑠,𝑇)+ℎ^4(𝑠,𝑇),(3.33) where ℎ^1(𝑠,𝑇)=−𝑚₂^𝑇𝑠ℎ𝑓(0,𝑢)𝑑𝑢,_2ℎ(𝑠,𝑇)=−^𝑇𝑠𝑑𝑢^𝑠0𝑚₂𝛼𝐾𝑒^{𝛼(𝑣−𝑢)}𝑑𝑣,_3ℎ_4(𝑠,𝑇)=−^𝑇𝑠𝑑𝑢^𝑠0𝜎𝑚₂𝑒^{𝛼(𝑣−𝑢)}𝑑𝑊_𝑣,(𝑠,𝑇)=−^𝑇𝑠𝑑𝑢^𝑠0𝑚₂𝑞_𝑣𝑒^{𝛼(𝑣−𝑢)}𝑑𝑌_𝑣.(3.34) We easily check that ℎℎ^1(𝑠,𝑇)=−𝑚₂𝑟₀𝑑(𝑠,𝑇,0),_2(𝑠,𝑇)=−𝑚₂𝐾𝑐_𝑇(𝑠)+𝑚₂𝐾𝑑(𝑠,𝑇,0),(3.35) where 𝑑(𝑠,𝑇,𝑢) is given by (3.24). Moreover, using Fübini's theorem, we obtain ℎℎ^3_4(𝑠,𝑇)=−^𝑠0𝜎𝑚₂𝑑(𝑠,𝑇,𝑢)𝑑𝑊_𝑢,(𝑠,𝑇)=−^𝑠0𝑚₂𝑞_𝑢𝑑(𝑠,𝑇,𝑢)𝑑𝑌_𝑢.(3.36) Therefore, we give a different expression below: −𝑚₂^𝑇𝑠𝑓(𝑠,𝑢)𝑑𝑢=−−𝑚₂𝑟₀𝑑(𝑠,𝑇,0)−𝑚₂𝐾𝑐_𝑇(𝑠)+𝑚₂𝐾𝑑(𝑠,𝑇,0)_𝑠0𝜎𝑚₂𝑑(𝑠,𝑇,𝑢)𝑑𝑊_𝑢−^𝑠0𝑚₂𝑞_𝑢𝑑(𝑠,𝑇,𝑢)𝑑𝑌_𝑢.(3.37) Then, from (3.12) and (3.37), we find that 𝐼₁=−𝑚₁^𝑠𝑡𝑟_𝑢𝑑𝑢−𝑚₂^𝑇𝑠=−𝑚₁𝑓(𝑠,𝑢)𝑑𝑢^𝑠𝑡−𝑚₂𝑓(𝑡,𝑢)𝑑𝑢−𝑚₁𝐾(𝑠−𝑡)+𝑚₁𝐾𝑐_𝑠(𝑡)+𝑚₂𝐾−𝑟₀𝑑(𝑠,𝑇,0)𝐾𝑐_𝑇(𝑠)−^𝑡0𝜎𝑚₂𝑑(𝑠,𝑇,𝑢)𝑑𝑊_𝑢−^𝑡0𝑚₂𝑞_𝑢𝑑(𝑠,𝑇,𝑢)𝑑𝑌_𝑢−^𝑠𝑡𝜎𝑚₁𝑐_𝑠(𝑢)+𝑚₂𝑑(𝑠,𝑇,𝑢)𝑑𝑊_𝑢−^𝑠𝑡𝑞_𝑢𝑚₁𝑐_𝑠(𝑢)+𝑚₂𝑑(𝑠,𝑇,𝑢)𝑑𝑌_𝑢.(3.38) In addition, applying (3.19) and the results in Park [17], we have 𝐸𝑒^{−∫𝑠𝑡𝜎(𝑚1𝑐𝑠(𝑢)+𝑚2𝑑(𝑠,𝑇,𝑢))𝑑𝑊𝑢}=exp12𝜎²^𝑠𝑡𝑚₁𝐸𝑒𝑐_𝑠(𝑢)+𝑚₂𝑑(𝑠,𝑇,𝑢)²𝑑𝑢_{−∫𝑠𝑡𝑞𝑢(𝑚1𝑐𝑠(𝑢)+𝑚2𝑑(𝑠,𝑇,𝑢))𝑑𝑌𝑢},=exp𝜇^𝑠𝑡𝑒^{−𝑞𝑢(𝑚1𝑐𝑠(𝑢)+𝑚2𝑑(𝑠,𝑇,𝑢))}−1𝑑𝑢.(3.39) Therefore, combining (3.39) and (3.31), we obtain 𝐸_𝑡𝑒^𝐼=𝑒^𝐼2𝐸_𝑡𝑒^𝐼1_(i)=⋅eexxpp−1+1𝑏+_𝐵1𝑏^𝐵1+𝑏^𝐴1^𝑠𝑡𝑓(𝑡,𝑢)𝑑𝑢−𝑏^𝐴1𝐾𝑠+1+𝑏^𝐵1+𝑏^𝐴1𝐾𝑡−1+𝑏^𝐵1𝐾𝑇⋅𝐸_𝑡+𝑏^𝐴1𝐾𝑐_𝑠(𝑡)+1+𝑏^𝐵1𝐾−𝑟₀𝑑(𝑠,𝑇,0)−𝑓₁(𝑡,𝑠)𝑒^{−∫𝑠𝑡𝜎[(1+𝑏𝐵1+𝑏𝐴1)𝑐𝑠(𝑢)+(1+𝑏𝐵1)𝑑(𝑠,𝑇,𝑢)]𝑑𝑊𝑢}⋅𝐸^𝑡𝑒^{−∫𝑠𝑡𝑞𝑢[(1+𝑏𝐵1+𝑏𝐴1)𝑐𝑠(𝑢)+(1+𝑏𝐵1)𝑑(𝑠,𝑇,𝑢)]𝑑𝑌𝑢}_(ii)=⋅eexxpp−1+1𝑏+_𝐵1𝑏^𝐵1+𝑏^𝐴1^𝑠𝑡𝑓(𝑡,𝑢)𝑑𝑢−𝑏^𝐴1𝐾𝑠+1+𝑏^𝐵1+𝑏^𝐴1𝐾𝑡−1+𝑏^𝐵1𝐾𝑇⋅exp1+𝑏^𝐴1𝐾𝑐_𝑠(𝑡)+1+𝑏^𝐵1𝐾−𝑟₀𝑑(𝑠,𝑇,0)−𝑓₁(𝑡,𝑠)2𝜎²^𝑠𝑡1+𝑏^𝐵1⋅exp𝜇_𝑠𝑡+𝑏^𝐴1𝑐_𝑠(𝑢)+1+𝑏^𝐵1𝑑(𝑠,𝑇,𝑢)²𝑑𝑢⋅exp_𝑇𝑠𝑒^{−𝑞𝑢((1+𝑏𝐵1+𝑏𝐴1)𝑐𝑠(𝑢)+(1+𝑏𝐵1)𝑑(𝑠,𝑇,𝑢))}−1𝑑𝑢12𝜎²1+𝑏^𝐵1²𝑐^2𝑇(𝑢)+𝜇𝑒^{−(1+𝑏𝐵1)𝑞𝑢𝑐𝑇(𝑢)}−1𝑑𝑢,(3.40) where (i) follows from the independence of 𝑊_𝑢 and 𝑌_𝑢 and the property of conditional expectation and (ii) holds from the Markov property and the independent increment property of 𝑊_𝑢 and 𝑌_𝑢. Finally, we substitute 𝐸_𝑡[𝑒^𝐼] 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(𝑇₂≤𝑇₁) 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 𝑟_𝑡, let the default times of firm 𝐴 and 𝐵 be 𝜏^𝐴 with the intensity 𝜆^𝐴 and 𝜏^𝐵 with the intensity 𝜆^𝐵, respectively.

Theorem 3.6. Suppose the risk-free interest rate 𝑟_𝑡 satisfies (3.1) and the intensities 𝜆^𝐴 and 𝜆^𝐵 satisfy (3.6) and (3.7), respectively. Then, the swap rate 𝑐 has the following expression: 𝑐=𝑉^𝐵0,𝑇₂−𝑒^{−(𝑏𝐵0+𝑏𝐴0)𝑇2}𝑔1+𝑏^𝐵1+𝑏^𝐴1,0,𝑇₂∫^𝑇2₀𝑔(1,0,𝑠)𝑑𝑠,(3.41) where 𝑔(⋅,⋅,⋅) and 𝑉^𝐵(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 𝐸^𝑇2₀𝑐𝑒^{−∫𝑠0𝑟𝑢𝑑𝑢}𝑑𝑠=𝑐^𝑇2₀𝐸𝑒^{−𝑅0,𝑠}𝑑𝑠,(3.42) where 𝑅_0,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 is𝐸1_{{𝜏𝐴≤𝑇2}}𝑒^{−∫𝑇20𝑟𝑢𝑑𝑢}1_{{𝜏𝐵>𝑇2}}.(3.43) Thus, in accordance with the arbitrage-free principle, we obtain 𝑐=𝐸1^{{𝜏𝐴≤𝑇2}}𝑒^{−∫𝑇20𝑟𝑢𝑑𝑢}1_{{𝜏𝐵>𝑇2}}∫^𝑇2₀𝐸𝑒^{−𝑅0,𝑠}𝑑𝑠.(3.44) Further, we can use the properties of conditional expectation to simplify (3.44) as follows: 𝑐=𝐸𝐸1^{{𝜏𝐴≤𝑇2}}∣ℱ^𝑟𝑇_∗∨ℋ^𝐵𝑇_∗𝑒^{−∫𝑇20𝑟𝑢𝑑𝑢}1_{{𝜏𝐵>𝑇2}}=𝐸1∫^𝑇2₀𝐸𝑒^{−𝑅0,𝑠}𝑑𝑠_{{𝜏𝐵>𝑇2}}𝑒^{−∫𝑇20𝑟𝑢𝑑𝑢}−𝐸1^{{𝜏𝐴>𝑇}₂^}𝑒^{−∫𝑇20𝑟𝑢𝑑𝑢}1_{{𝜏𝐵>𝑇2}}=𝑉∫^𝑇2₀𝐸𝑒^{−𝑅0,𝑠}𝑑𝑠_𝐵0,𝑇₂−𝐸1_{𝜏^𝐴_>𝑇2}𝑒^{−∫𝑇20(𝑟𝑢+𝜆𝐵𝑢)𝑑𝑢}∫^𝑇2₀𝐸𝑒^{−𝑅0,𝑠}𝑑𝑠,(3.45) where the last one is obtained by (3.4). Note that 𝑉^𝐵(0,𝑇₂) can be obtained by (3.23) and 𝐸[𝑒^{−𝑅0,𝑠}]=𝑔(1,0,𝑠) by Theorem 3.4. We substitute (3.7) into the above expectation term 𝐸1=𝐸_{𝜏^𝐴_>𝑇2}1exp_{{𝜏𝐴>𝑇2}}−^𝑇2₀𝑟_𝑢+𝜆^𝐵𝑢𝑑𝑢=𝐸1_{{𝜏𝐴>𝑇2}}exp−^𝑇2₀𝑏^𝐵0+1+𝑏^𝐵1𝑟_𝑢+𝑏1_{{𝜏𝐴≤𝑢}}𝑑𝑢exp−𝑏^𝐵0𝑇₂−1+𝑏^𝐵1^𝑇2₀𝑟_𝑢𝑑𝑢−𝑏𝑇₂−𝜏^𝐴1_{𝜏^𝐴_≤𝑇2}=𝐸1_{{𝜏𝐴>𝑇2}}𝑒^{−𝑏𝐵0𝑇2−(1+𝑏𝐵1)𝑅20,𝑇}_(i)=𝐸𝐸1_{{𝜏𝐴>𝑇2}}∣ℱ^𝑟𝑇_∗𝑒^{−𝑏𝐵0𝑇2−(1+𝑏𝐵1)𝑅20,𝑇}_(ii)=𝐸𝑒^{−𝑏𝐵0𝑇2−(1+𝑏𝐵1)𝑅20,𝑇−∫𝑇20𝜆𝐴𝑢𝑑𝑢}_(iii)=𝑒^{−(𝑏𝐵0+𝑏𝐴0)𝑇2}𝐸𝑒^{−(1+𝑏𝐵1+𝑏𝐴1)∫𝑇20𝑟𝑢𝑑𝑢}_(iv)=𝑒^{−(𝑏𝐵0+𝑏𝐴0)𝑇2}𝑔1+𝑏^𝐵1+𝑏^𝐴1,0,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 𝜆^𝐴𝑡 and 𝜆^𝐵𝑡 satisfy the below relations: 𝜆^𝐴𝑡=𝜆𝑏^𝐴0_𝐵𝑡+𝑏^𝐴1𝑟_𝑡+𝑏1_{{𝜏𝐵≤𝑡}},=𝑏^𝐵0+𝑏^𝐵1𝑟_𝑡,(3.47) where 𝑏^𝐴0, 𝑏^𝐴1, 𝑏^𝐵0, 𝑏^𝐵1, and 𝑏 are positive constants, then the swap rate 𝑐=𝑔1+𝑏^𝐵1,0,𝑇₂𝑒^{−𝑏𝐵0𝑇2}−𝑒^{−(𝑏𝐵0+𝑏𝐴0)𝑇2}𝑔1+𝑏^𝐵1+𝑏^𝐴1,0,𝑇₂∫^𝑇2₀𝑔(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.