We study a three-firm contagion model with counterparty risk and apply this model to price defaultable bonds and credit default swap (CDS). This model assumes that default intensities are driven by external common factors as well as other defaults in the system. Using the “total hazard” approach, default times can be generated and the joint density function is obtained. We represent the pricing method of defaultable bonds and obtain the closed-form pricing formulas. By the approach of “change of measure,” analytical solutions of CDS swap rate (swap premuim) are derived in the continuous time framework and the discrete time framework, respectively.

1. Introduction

The corporate bonds and their credit derivatives are typical financial tools in the markets which undertake and avoid the credit risk of the companies. There are two basic approaches to modeling the pricing of defaultable securities: the value-of-the-firm (or structural) approach and the intensity-based approach. The structural model is based on the work of Merton [1], Black and Cox [2], and Geske [3]: the default occurs when the firm assets are insufficient to meet payments on debt or the value of the firm asset falls below a prespecified level.

Nevertheless, the value of the firm assets is not observable, which brings difficulties to the pricing of credit derivatives. Reduced-form approach for credit risks avoids the disadvantage of structural approach which models the firm's value directly. They use risk-neutral pricing principle of contingent claims and take the time of default or other credit events as an exogenous random variable.

Reduced-form models are developed by Artzner and Delbaen [4], Duffie et al. [5], Jarrow and Turnbull [6], and Madan and Unal [7]. Duffie and Lando [8] show that a reduced-form model can be obtained from a structural model with incomplete accounting information. The simplest type of reduced-form model is that the default time or the credit migration is the first jump of an exogenously given jump process with an intensity. In Jarrow et al. [9], the intensity for credit migration is constant; see also Litterman and Iben [10] for a Markov chain model of credit migration. In the papers by Duffie et al. [5], Duffie and Singleton [11], and Lando [12], the intensity of default is a random process. The common feature of the reduced-form models is that default cannot be predicted and can occur at any time. Therefore, reduced-form models have been used to price a wide variety of instruments. In recent years, some papers on estimating the parameters of these models are Collin-Dufresne and Solnik [13] and Duffee [14]. Jarrow and Yu [15] set up a reduced-form model in which estimation can be based on bond prices as well as credit default swap prices. A systematic development of mathematical tools for reduced-form models has been given by Elliott et al. [16], and Jamshidian [17] develops change of numeraire methodology for reduced-form models.

In this paper, we mainly discuss the pricing of the defaultable zero-coupon bonds and CDS based on the intensity model with correlated default. The structure of this paper is organized as follows: in Section 2, we give the basic setup and the three-firm contagion model with an interaction term, comparing it with the model in Leung and Kwok [18]. In Section 3, we give the general pricing formulas in various cases. In this general framework, a pricing formula of defaultable bonds is provided for three-firm model. In Section 4, using the approach of “total hazard” and “change of measure,” we present the construction of default time, derive the joint density function, and obtain the closed forms of CDS swap rate (swap premium) in the continuous time and the discrete time framework, respectively. We conclude this paper with Section 5.

2. Basic Setup and Three-Firm Contagion Model

2.1. Basic Setup and Construction of Default Time

We consider an uncertain economy with a time horizon of 𝑇 described by a filtered probability space (Ω,,{𝑡}𝑇𝑡=0,𝑃) (in this paper we follow the symbols and notations of Jarrow and Yu [15]) satisfying the usual conditions of right continuity and completeness with respect to 𝑃-null sets, where =𝑇 and 𝑃 is an equivalent martingale measure under which discounted bond prices are martingale. We assume the existence and uniqueness of 𝑃, so that bond markets are complete and no arbitrage, as shown in discrete time by Harrison and Kreps [19] and in continuous time by Harrison and Pliska [20]. Subsequent specifications of the model are all under the equivalent martingale measure 𝑃.

On this probability space, there is an 𝑑-valued process 𝑋𝑡, which presents 𝑑-dimensional economy-wide state variables. There are also 𝐼 point processes, 𝑁𝑖(𝑖=1,2,,𝐼), initialized at 0. These represent the default processes of the firms in the economy such that the default of the 𝑖th firm occurs when 𝑁𝑖 jumps from 0 to 1.

According to the information contained in the state variables and the default processes, the filtration is=𝑋𝑡1𝑡2𝑡𝐼𝑡,(2.1) where𝑋𝑡𝑋=𝜎𝑠,,0𝑠𝑡(2.2a)𝑖𝑡𝑁=𝜎𝑖𝑠,0𝑠𝑡(2.2b) are the filtrations generated by 𝑋𝑡 and 𝑁𝑖𝑡, respectively.

Let 𝑡𝑖=1𝑡𝑡𝑖1𝑡𝑖+1𝐼𝑡,𝒢(2.2c)𝑖𝑡=𝑖𝑡𝑋𝑇𝑇𝑖=𝑖𝑡𝒢𝑖0,(2.2d)where 𝒢𝑖0=𝑋𝑇𝑇𝑖. We know that 𝒢𝑖0 contains complete information on the state variables and the default processes of all firms other than that of the 𝑖th, all the way up to time 𝑇.

According to the filtration 𝒢𝑖𝑡, it is possible to select a nonnegative, 𝒢𝑖0-measurable process 𝜆𝑖𝑡, satisfying 𝑡0𝜆𝑖𝑠𝑑𝑠<,𝑃𝑎.𝑠. for all 𝑡[0,𝑇], so that we can define an inhomogeneous Poisson process 𝑁𝑖, using the process 𝜆𝑖𝑡 as its intensity function.

Let 𝜏𝑖 denote the default time of firm 𝑖, namely, 𝜏𝑖 be the first jump time of 𝑁𝑖, in a typical reduced-form model, which can be defined as 𝜏𝑖=inf𝑡𝑡0𝜆𝑖𝑠𝑑𝑠𝐸𝑖,(2.3) where {𝐸𝑖}𝐼𝑖=1 is independent of 𝑋𝑡(𝑡[0,𝑇]).

According to the Doob-Meyer decomposition, we have that 𝑀𝑖𝑡=𝑁𝑡𝑡𝜏𝑖0𝜆𝑖𝑠𝑑𝑠,(2.4) is a (𝑃,𝑡)-martingale.

Under the above characterization, the conditional survival probability of firm 𝑖 is given by 𝑃𝜏𝑖>𝑡𝒢𝑖0=exp𝑡0𝜆𝑖𝑠𝑑𝑠,𝑡0,𝑇.(2.5)

The unconditional survival probability of firm 𝑖 is given by 𝑃𝜏𝑖>𝑡=𝐸exp𝑡0𝜆𝑖𝑠𝑑𝑠,𝑡0,𝑇.(2.6)

Now, we give the recursive construction of default time as Yu in his paper [21]. Specifically, we start with the case of no state variable. Let the notation 𝜆𝑖(𝑡𝑛) denote the intensity for firm 𝑖 given the observed default times of 𝑛 other firms, 𝑡𝑘0,𝑡𝑘1,,𝑡𝑘𝑛, where 0=𝑡𝑘0<𝑡𝑘1<<𝑡𝑘𝑛<𝑡<𝜏𝑖.

The total hazard accumulated by firm 𝑖 by time 𝑡 given 𝑛 observed defaults is defined as 𝜓𝑖(𝑡𝑛)=𝑛𝑚=1Λ𝑖𝑡𝑘𝑚𝑡𝑘𝑚1𝑚1+Λ𝑖𝑡𝑡𝑘𝑛𝑛,(2.7) where Λ𝑖(𝑠𝑚)=𝑡𝑘𝑚𝑡+𝑠𝑘𝑚𝜆𝑖(𝑢𝑚)𝑑𝑢 is the total hazard accumulated by firm 𝑖 for a period of length 𝑠 following the 𝑚th default. At the same time, we assume that there is no default between 𝑡𝑘𝑚 and 𝑡.

Define the inverse functions Λ𝑖1(𝑥𝑛)=inf𝑠0Λ𝑖(𝑠𝑚)𝑥(2.8) for 𝑥>0. We can use the following recursive procedure to construct a collection of random variables.

Step 1. Let 𝐸1,,𝐸𝐼 be the i.i.d. unit exponentials and 𝑘1Λ=argmin𝑖1𝐸𝑖0𝑖=1,,𝑛,(2.9) and define ̂𝜏1=Λ𝑘11𝐸𝑘10.(2.10)

Step  𝑚+1(𝑚=1,2,,𝐼1)
Given that Step 1 up to step 𝑚 have resulted in (̂𝜏1,,̂𝜏𝑚), define the set 𝐼𝑚={𝑘1,,𝑘𝑚} and 𝐼𝑚 as the set of firms excluding 𝐼𝑚. Let 𝑘𝑚+1Λ=nargmin𝑖1𝐸𝑖𝜓𝑖(̂𝜏𝑚𝑚)𝑚(2.11) and let ̂𝜏𝑚+1=̂𝜏𝑚+Λ𝑘1𝑚+1𝐸𝑘𝑚+1𝜓𝑘𝑚+1(̂𝜏𝑚𝑚)𝑚.(2.12)

Norros [22], Shaked and Shanthikumar [23], and Yu [24] prove that ̂𝜏=(̂𝜏1,,̂𝜏𝐼) equals 𝜏=(𝜏1,,𝜏𝐼) in distribution. So we can generate default time 𝜏 by generating ̂𝜏 and we will not distinguish them from now on.

2.2. Three-Firms Contagion Model

In this subsection, we explore the three-firms contagion model with an interaction term. Consider the case where the default intensity of one firm is affected by the default of other two firms, so that when one firm defaults the default probabilities of other two firms will jump. In the three-firms contagion model, the interdependent structure between firm 𝐴, firm 𝐵, and firm 𝐶 is characterized by the correlated default intensities.

Recall Leung and Kwok's three-firms model:𝜆𝐴𝑡=𝑎0+𝑎1𝕝{𝜏𝐵𝑡}+𝑎2𝕝{𝜏𝐶𝑡},𝜆(2.13a)𝐵𝑡=𝑏0+𝑏1𝕝{𝜏𝐴𝑡}+𝑏2𝕝{𝜏𝐶𝑡}𝜆,(2.13b)𝐶𝑡=𝑐0+𝑐1𝕝{𝜏𝐴𝑡}+𝑐2𝕝{𝜏𝐵𝑡},(2.13c)where 𝑎0>0, 𝑏0>0, 𝑐0>0 and satisfying 𝑎0+𝑎1+𝑎2>0, 𝑏0+𝑏1+𝑏2>0, 𝑐0+𝑐1+𝑐2>0. Similarly, 𝑎0>0, 𝑏0>0, and 𝑐0>0 reflect the effect of macroeconomic factor and itself on firms 𝐴, 𝐵, 𝐶, respectively.

Nevertheless, Leung and Kwok have not allow the effect of two parties' simultaneous default on the third party, namely, there is not an interaction term in their model. Thus, if three firms are copartners, then the default risk of each firm may be overestimated and the asset value may be underestimated because there exists the case in which the default events might overlap. If they are competitors, then the case is contrary.

For the above reason, we allow the following three-firms contagion model:𝜆𝐴𝑡=𝑎0+𝑎1𝕝{𝜏𝐵𝑡,𝜏𝐶>𝑡}+𝑎2𝕝{𝜏𝐶𝑡,𝜏𝐵>𝑡}+𝑎3𝕝{𝜏𝐵𝑡,𝜏𝐶𝑡},𝜆(2.14a)𝐵𝑡=𝑏0+𝑏1𝕝{𝜏𝐴𝑡,𝜏𝐶>𝑡}+𝑏2𝕝{𝜏𝐶𝑡,𝜏𝐴>𝑡}+𝑏3𝕝{𝜏𝐴𝑡,𝜏𝐶𝑡}𝜆,(2.14b)𝐶𝑡=𝑐0+𝑐1𝕝{𝜏𝐴𝑡,𝜏𝐵>𝑡}+𝑐2𝕝{𝜏𝐵𝑡,𝜏𝐴>𝑡}+𝑐3𝕝{𝜏𝐴𝑡,𝜏𝐵𝑡},(2.14c)where 𝑎0>0, 𝑏0>0, 𝑐0>0 and satisfying 𝑎0+𝑎1+𝑎2+𝑎3>0, 𝑏0+𝑏1+𝑏2+𝑏3>0, 𝑐0+𝑐1+𝑐2+𝑐3>0. 𝑎0>0, 𝑏0>0, and 𝑐0>0 reflect the effect of macroeconomic factor and itself on firms 𝐴, 𝐵, 𝐶, respectively.

Nextly, we employ the three-firms model specified by (2.14a)–(2.14c) to price defaultable bonds and CDS swap rate.

3. Bond Pricing under Three-Firm Model

3.1. The General Pricing Formulas

Definition 3.1. A defaultable claim maturing at 𝑇 is the quadruple (𝑋,𝐴,𝑍,𝜏), where 𝑋 is an 𝑇-measurable random variable, 𝐴=(𝐴𝑡)𝑡[0,𝑇] is an -adapted, continuous process of finite variation with 𝐴0=0, 𝑍=(𝑍𝑡)𝑡[0,𝑇] is an -predictable process, and 𝜏 is a random time.

Definition 3.2. The dividend process 𝐷=(𝐷𝑡)𝑡+ of the above defaultable claim maturing at 𝑇 equals, for every 𝑡+, 𝐷𝑡=𝑋𝕝{𝑇<𝜏}𝕝[𝑇,)(𝑡)+(0,𝑡𝑇]1𝑁𝑢𝑑𝐴𝑢+(0,𝑡𝑇]𝑍𝑢𝑑𝑁𝑢,(3.1) where 𝑋 is the promised payoff, 𝐴 represents the process of promised dividends, and the process 𝑍 is the recovery process.

Definition 3.3. The exdividend price process 𝑆 of a defaultable claim (𝑋,𝐴,𝑍,𝜏) equals, for every 𝑡[0,𝑇], 𝑆𝑡=𝐸𝑡(𝑡,𝑇]𝐵𝑡𝐵𝑢𝑑𝐷𝑢,(3.2) where 𝐵𝑡=𝐵(𝑡)=exp(𝑡0𝑟𝑠𝑑𝑠) is the money market account, 𝑟𝑡 is a constant risk-free spot rate, and 𝐸𝑡 represents the conditional expectation on 𝑡 under the equivalent martingale measure 𝑃.

By Definitions 3.2 and 3.3, the exdividend price of a defaultable claim (𝑋,𝐴,𝑍,𝜏) is given by the following.

Lemma 3.4. The exdividend price of the defaultable claim (𝑋,𝐴,𝑍,𝜏) equals, for 𝑡[0,𝑇), 𝑆𝑡=𝕝{𝑡<𝜏}𝐵𝑡𝐺𝑡𝐸𝑡𝐵𝑇1𝐺𝑇𝑋+𝑇𝑡𝐵𝑢1𝐺𝑢𝑍𝑢𝜆𝑢𝑑𝑢+𝑑𝐴𝑢,(3.3) where 𝐺𝑡=𝑃{𝜏>𝑡𝑡}.

From Lemma 3.4, We can explore the following special cases.(1) For the default-free zero-coupon bond which pays one dollar, the dividend process is 𝐷𝑡=𝕝{𝑡𝑇}. Let 𝑝(𝑡,𝑇) be the time-𝑡 price, then 𝑝(𝑡,𝑇) is given by 𝑝(𝑡,𝑇)=𝐸𝑡𝐵𝑡𝐵𝑇.(3.4)(2)If the dividend process is 𝐷𝑡=𝑇𝑡𝑟𝑢𝑑𝑢+𝕝{𝑡𝑇}, then the value of the bond is always 1.(3)For the defaultable zero-coupon bond which pays one dollar if not default and pays 𝛿 times the price of a default-free bond at maturity, where 𝛿 is introduced by Jarrow and Turnbull [6] and Jarrow et al. [9] as “recovery of Treasury,” let 𝑣𝑖(𝑡,𝑇) denote the time-𝑡 price, issued by firm 𝑖, 𝛿𝑖[0,1] is the recovery rate of the firm 𝑖, then 𝑣𝑖(𝑡,𝑇) is given by 𝑣𝑖(𝑡,𝑇)=𝐸𝑡exp𝑇𝑡𝑟𝑢𝛿𝑑𝑢𝑖𝕝{𝜏𝑖𝑇}+𝕝{𝜏𝑖>𝑇}=𝐸𝑡𝐵𝑡𝐵𝑇𝛿𝑖𝕝{𝜏𝑖𝑇}+𝕝{𝜏𝑖>𝑇}.(3.5)(4)If the dividend process 𝐷𝑡=𝑋𝕝{𝜏𝑖>𝑇,𝑡𝑇}+(0,𝑡𝑇]𝑍𝑢𝑑𝑁𝑢, using the Doob-Meyer decomposition of 𝑁𝑡, then the value of the defaultable bond is 𝑆𝑡=𝐸𝑡𝑇𝑡exp𝑢𝑡𝑟𝑣𝑍𝑑𝑣𝑢𝜆𝑢𝕝{𝑢<𝜏}𝑑𝑢+exp𝑇𝑡𝑟𝑣𝑑𝑣𝑋𝕝{𝜏>𝑇}=𝐵𝑡𝐸𝑡𝑇𝑡𝐵𝑢1𝑍𝑢𝜆𝑢𝕝{𝑢<𝜏}𝑑𝑢+𝐵𝑇1𝑋𝕝{𝜏>𝑇}.(3.6)

3.2. Bond Pricing under Three-Firms Model

We assume that there are three firms 𝐴, 𝐵, and 𝐶. Now, we consider the case that each firm holds the other two firms' defaultable bonds, so that when one party defaults, the other two parties' default probability will jump. The default intensities are described as (2.14a)–(2.14c).

We adopt the change of measure introduced by Collin-Dufresne et al. [25] to define a firm-specific probability measure 𝑃𝑖 which puts zero probability on the paths where default occurs prior to the maturity 𝑇. Specifically, the change of measure is defined by 𝑍𝑇=𝑑𝑃𝑖||||𝑑𝑃𝑇=𝕝{𝜏𝑖>𝑇}exp𝑇0𝜆𝑖𝑠𝑑𝑠,(3.7) where 𝑃𝑖 is a firm-specific (firm 𝑖) probability measure which is absolutely continuous with respect to 𝑃 on the stochastic interval [0,𝜏𝑖). To proceed the calculations under the measure 𝑃𝑖, we enlarge the filtration to 𝑖=(𝑖𝑡)𝑡0 as the completion of =(𝑡)𝑡0 by the null sets of the probability measure 𝑃𝑖.

Applying the result of Jarrow and Yu [15], we know that the defaultable bond price of firm 𝑖 is given by 𝑣𝑖(𝑡,𝑇)=𝛿𝑖𝑝(𝑡,𝑇)+𝕝{𝜏𝑖>𝑡}1𝛿𝑖𝐸𝑡exp𝑇𝑡𝑟𝑠+𝜆𝑖𝑠𝑑𝑠,𝑡𝑇(3.8) or 𝑣𝑖(𝑡,𝑇)𝑝(𝑡,𝑇)=𝛿𝑖+1𝛿𝑖𝕝{𝜏𝑖>𝑡}𝐸𝑡exp𝑇𝑡𝜆𝑖𝑠𝑑𝑠.(3.9)

Because of the symmetry of default intensities, we need only to compute one firm's value of the three firms. In the remainder of this subsection, we will derive the closed-form pricing formula of firm 𝐶.

For firm 𝐶, the time-𝑡 value 𝑣𝐶(𝑡,𝑇) of the defaultable bond 𝐶 maturity at 𝑇 satisfies 𝑣𝐶𝛿(𝑡,𝑇)=𝑝(𝑡,𝑇)𝐶+1𝛿𝐶𝕝{𝜏𝐶>𝑡}𝐸𝑡exp𝑇𝑡𝜆𝐶𝑠𝑑𝑠,(3.10) where 𝐸𝑡exp𝑇𝑡𝜆𝐶𝑠𝑑𝑠=𝑒𝑐0(𝑇𝑡)𝐸𝑡×exp𝑇𝑡𝑐1𝕝{𝜏𝐴𝑠,𝜏𝐵>𝑠}+𝑐2𝕝{𝜏𝐵𝑠,𝜏𝐴>𝑠}+𝑐3𝕝{𝜏𝐴𝑠,𝜏𝐵𝑠}.𝑑𝑠(3.11)(I) Conditional on 𝜏𝐴>𝑡, 𝜏𝐵>𝑡, namely, neither firm 𝐴 nor firm 𝐵 has defaulted by time 𝑡, the default intensities 𝜆𝐴𝑡 and 𝜆𝐵𝑡 under the measure 𝑃𝐶 and 𝜏𝐶>𝑡 are given by 𝜆𝐴𝑡=𝑎0+𝑎1𝕝{𝜏𝐵𝑡},𝜆𝐵𝑡=𝑏0+𝑏1𝕝{𝜏𝐴𝑡}.(3.12) According to the result of Leung and Kwok [18], the conditional joint density function 𝑓𝑡(𝑡1,𝑡2) of (𝜏𝐴,𝜏𝐵) is 𝑓𝑡𝑡1,𝑡2=𝑓1,𝑡𝑡1,𝑡2,𝑡3=𝑏0𝑎0+𝑎1𝑒(𝑎0+𝑎1)(𝑡1𝑡)(𝑏0𝑎1)(𝑡2𝑡),𝑡2𝑡1,𝑓2,𝑡𝑡1,𝑡2,𝑡3=𝑎0𝑏0+𝑏1𝑒(𝑏0+𝑏1)(𝑡2𝑡)(𝑎0𝑏1)(𝑡1𝑡),𝑡2>𝑡1.(3.13) The integration region of (3.11) is then appropriately divided into five pieces: 𝐷1: 𝑡𝜏𝐴𝑇, 𝜏𝐴𝜏𝐵𝑇; 𝐷2: 𝑡𝜏𝐵𝑇, 𝜏𝐵𝜏𝐴𝑇; 𝐷3: 𝑡𝜏𝐴𝑇, 𝜏𝐵𝑇; 𝐷4: 𝑡𝜏𝐵𝑇, 𝜏𝐴𝑇; 𝐷5: 𝜏𝐴𝑇, 𝜏𝐵𝑇: 𝐸𝑡exp𝑇𝑡𝜆𝐶𝑠𝑑𝑠=𝑒𝑐0(𝑇𝑡)𝐸𝑡×exp𝑇𝑡𝑐1𝕝{𝜏𝐴𝑠,𝜏𝐵>𝑠}+𝑐2𝕝{𝜏𝐵𝑠,𝜏𝐴>𝑠}+𝑐3𝕝{𝜏𝐴𝑠,𝜏𝐵𝑠}𝑑𝑠=𝑒𝑐0(𝑇𝑡)𝐼1+𝐼2+𝐼3+𝐼4+𝐼5=𝐽1,(3.14) where 𝐼1=𝐷1𝑒𝑐1(𝑡2𝑡1)𝑐3(𝑇𝑡2)𝑓2,𝑡𝑡1,𝑡2𝑑𝑡1𝑑𝑡2=𝑎0𝑏0+𝑏1𝑇𝑡𝑇𝑡1𝑒𝑐1(𝑡2𝑡1)𝑐3(𝑇𝑡2)𝑒(𝑎0𝑏1)(𝑡1𝑡)(𝑏0+𝑏1)(𝑡2𝑡)𝑑𝑡2𝑑𝑡1=𝑎0𝑏0+𝑏1𝑏0+𝑏1+𝑐1𝑐3𝑒𝑐3(𝑇𝑡)𝑒(𝑎0+𝑏0)(𝑇𝑡)𝑎0+𝑏0𝑐3𝑒(𝑏0+𝑏1+𝑐1)(𝑇𝑡)𝑒00(𝑎+𝑏)(𝑇𝑡)𝑎0𝑏1𝑐1,𝐼2=𝐷2𝑒𝑐2(𝑡1𝑡2)𝑐3(𝑇𝑡1)𝑓1,𝑡𝑡1,𝑡2𝑑𝑡1𝑑𝑡2=𝑏0𝑎0+𝑎1𝑎0+𝑎1+𝑐2𝑐3𝑒𝑐3(𝑇𝑡)𝑒(𝑎0+𝑏0)(𝑇𝑡)𝑎0+𝑏0𝑐3𝑒(𝑎0+𝑎1+𝑐2)(𝑇𝑡)𝑒00(𝑎+𝑏)(𝑇𝑡)𝑏0𝑎1𝑐2,𝐼3=𝐷3𝑒(𝑐1(𝑡2𝑡1))𝑓2,𝑡𝑡1,𝑡2𝑑𝑡1𝑑𝑡2=𝑎0𝑏0+𝑏1𝑎0𝑏1𝑐1𝑏0+𝑏1+𝑐1𝑒(𝑏0+𝑏1+𝑐1)(𝑇𝑡)𝑒(𝑎0+𝑏0)(𝑇𝑡),𝐼4=𝐷4𝑒(𝑐2(𝑡1𝑡2))𝑓1,𝑡𝑡1,𝑡2𝑑𝑡1𝑑𝑡2=𝑏0𝑎0+𝑎1𝑏0𝑎1𝑐2𝑎0+𝑎1+𝑐2𝑒(𝑎0+𝑎1+𝑐2)(𝑇𝑡)𝑒(𝑎0+𝑏0)(𝑇𝑡),𝐼5=𝐷5𝑓𝑡𝑡1,𝑡2𝑑𝑡1𝑑𝑡2=𝑒(𝑎0+𝑏0)(𝑇𝑡).(3.15)(II) Conditional on 𝜏𝐵>𝑡, 𝜏𝐴𝑡, namely, firm 𝐴 has defaulted and firm 𝐵 has not defaulted by time 𝑡, the default intensities 𝜆𝐵𝑡 are given by 𝜆𝐵𝑡=𝑏0+𝑏1𝕝{𝜏𝐴𝑡,𝜏𝐶>𝑡}=𝑏0+𝑏1(3.16) and the density function 𝑓𝑡(𝑡2) of 𝜏𝐵 is 𝑓𝑡𝑡2=𝑏0+𝑏1𝑒(𝑏0+𝑏1)(𝑡2𝑡),(3.17) so 𝐸𝑡exp𝑇𝑡𝜆𝐶𝑠𝑑𝑠=𝑒𝑐0(𝑇𝑡)𝐸𝑡exp𝑇𝑡𝑐1𝕝{𝜏𝐴𝑠,𝜏𝐵>𝑠}+𝑐3𝕝{𝜏𝐴𝑠,𝜏𝐵𝑠}=𝑏𝑑𝑠0+𝑏1𝑒𝑐0(𝑇𝑡)×𝑇𝑡𝑒𝑐1(𝑡2𝑡)𝑐3(𝑇𝑡2)𝑒(𝑏0+𝑏1)(𝑡2𝑡)𝑑𝑡2+𝑇𝑒𝑐1(𝑇𝑡)𝑒(𝑏0+𝑏1)(𝑡2𝑡)𝑑𝑡2=𝑏0+𝑏1𝑒𝑐0(𝑇𝑡)𝑒𝑐3(𝑇𝑡)𝑒(𝑏0+𝑏1+𝑐1)(𝑇𝑡)𝑏0+𝑏1+𝑐1𝑐3+𝑒(𝑏0+𝑏1𝑐1)(𝑇𝑡)𝑏0+𝑏1=𝐽2.(3.18)(III) Conditional on 𝜏𝐵𝑡, 𝜏𝐴>𝑡, namely, firm 𝐵 has defaulted and firm 𝐴 has not defaulted by time 𝑡, similar to the computation in (II), we have 𝐸𝑡exp𝑇𝑡𝜆𝐶𝑠=𝑎𝑑𝑠0+𝑎1𝑒𝑐0(𝑇𝑡)𝑒𝑐3(𝑇𝑡)𝑒(𝑎0+𝑎1+𝑐2)(𝑇𝑡)𝑎0+𝑎1+𝑐2𝑐3+𝑒(𝑎0+𝑎1𝑐2)(𝑇𝑡)𝑎0+𝑎1=𝐽3,(3.19)(IV) Conditional on 𝜏𝐴𝑡, 𝜏𝐵𝑡, namely, firm 𝐴 and firm 𝐵 have defaulted by time 𝑡, we have 𝐸𝑡exp𝑇𝑡𝜆𝐶𝑠𝑑𝑠=𝑒(𝑐0+𝑐3)(𝑇𝑡)=𝐽4.(3.20) From (3.10), (3.11), and the discussions in (I)–(IV), we have the following theorem.

Theorem 3.5. Let intensity processes 𝜆𝑖𝑡(𝑖=𝐴,𝐵,𝐶) be given by (2.14a)–(2.14c), the time-t defaultable bond price issued by firm 𝐶 is given by 𝑣𝐶𝛿(𝑡,𝑇)=𝑝(𝑡,𝑇)𝐶+1𝛿𝐶𝕝{𝜏𝐶>𝑡}𝕝{𝜏𝐴>𝑡,𝜏𝐵>𝑡}𝐽1+𝕝{𝜏𝐴𝑡,𝜏𝐵>𝑡}𝐽2+𝕝{𝜏𝐴>𝑡,𝜏𝐵𝑡}𝐽3+𝕝{𝜏𝐴𝑡,𝜏𝐵𝑡}𝐽4,(3.21) where 𝐽1, 𝐽2, 𝐽3, 𝐽4 are given by (3.14)–(3.20).

4. CDS Valuation under Three-Firms Model

4.1. The Basics

As one of the important credit derivatives, CDS is a contract agreement which allows the transfer of credit risk of a risky asset (basket of risky assets) from one party to the other. A financial institution may use a CDS to transfer credit risk of a risky asset while continues to retain the legal ownership of the asset. To determine a fair swap rate of a CDS in the presence of counterparty risks, the interdependent default risk structures between these parties must be considered simultaneously.

On CDS valuation, there have been numerous works in recent years. Based on the reduced-form approach with correlated market and credit risks, the closed-form valuation formula for the swap rate of a CDS is obtained by Jarrow and Yildirim [26]. They assume that the default intensity is “almost” linear in the short interest rate. Recently, considering the impact of counterparty risk on the pricing of a CDS, Jarrow and Yu in [15] assume an interdependent default structure that avoids “looping default” by involving primary-secondary framework and simplifies the payoff structure where the protection seller's compensation is made only at the maturity of the swap contract. They discover that the default risk of the protection seller and reference entity is ignored. Hull and White [27] apply the credit index model for valuing CDS with counterparty risk. M. A. Kim and T. S. Kim[28] conclude that if the default correlation between the counterparty and reference bond is ignored, then the pricing error in a CDS can be quite substantial. Chen and Filpovic in their paper [29] develop a generalized affine model to price credit default swaps under default correlations and counterparty risk. Yu [21] uses the “total hazard” approach to construct the default process from independent and identically distributed exponential random variables and obtains an analytic expression of the joint distribution of default times in his two-firms and three-firms contagion models. Leung and Kwok [18] use the “change of measure” approach introduced by Collin-Dufresne et al. [25] to price the CDS in two-firms model and three-firms contagion model, respectively, and obtain the closed-form formulas.

We assume that party 𝐴 (CDS protection buyer) holds a corporate bond of party C (reference asset) and party 𝐶 is subject to default. Party 𝐴 faces the credit risk arising from default of party 𝐶. To seek protection against such default risk, party 𝐴 enters a CDS contract in which he agrees to make premium payments, known as the swap premium to party 𝐵(CDS protection seller). In exchange, party 𝐵 promises to compensate 𝐴 for its loss in the event of default of the bond (reference asset).

Similar to the description in [30], a diagrammatic overview of CDS under the three-firms contagion model (2.14a)–(2.14c) is provided by Figure 1.

We employ the three-firms model specified by (2.14a)–(2.14c) to price the CDS and study the effect of the default of each party on the swap rate. Suppose that party 𝐴 (a corporate bond investing firm) holds a corporate bond (reference asset) issued by party 𝐶 (a corporate bond issuer) (refer to 1𝐴 in Figure 1) and firm 𝐶 is subject to default. At bond maturity, if firm 𝐶 does not default, then it will pay the bond principle and interest to firm 𝐴 (see 1𝐵). Otherwise, it has no payments (refer to 1𝐶). On the other hand, to hedge the default risk of firm 𝐶, firm 𝐴, and firm 𝐵 (the protection seller, such as a monoline insurer), enter into a CDS contract. Firm 𝐴 and 𝐵 are also subject to default. If firm 𝐶 and 𝐴 have no default, then firm 𝐴 makes fixed premium payments, known as the swap premium to firm 𝐵 (see 1𝐷). Either firm 𝐴 or firm 𝐶 defaults, there is no premium payments to firm 𝐵 (refer to 1𝐸). In exchange, firm 𝐵 promises to compensate 𝐴 (if 𝐴 does not default) for its loss in the event of default of the bond 𝐶 (reference asset) as long as 𝐵 does not default (refer to 1𝐹). If the protection seller 𝐵 defaults prior to the default of either the reference asset 𝐶 or the protection buyer 𝐴, then the protection seller 𝐵 can simply walk away from the contract and has no obligation to pay the compensation to the protection buyer (see 1𝐺).

In this section, we will analyze the effect of correlated risks between three parties in a CDS using a similar contagion model as in Leung and Kwok's model [18]. Differently from their model on CDS valuation with counterparty risk, we allow an interaction term in the default intensity model for three firms, namely, we discuss how the simultaneous default of two parties impacts on the third one.

4.2. The Joint Density Function for Three-Firms Model

To price CDS swap rate 𝑠, we firstly need to provide the joint density function 𝑓(𝑡1,𝑡2,𝑡3) of three firms 𝐴, 𝐵, and 𝐶. We adopt the “total hazard” approach by Yu [21] and Zheng and Jiang [31] as description in Section 2.1; the result is the following Lemma.

Lemma 4.1. Assume that 𝜆𝑖𝑡(𝑖=𝐴,𝐵,𝐶) are given by model (2.14a)–(2.14c). Then the joint density function of 𝜏=(𝜏𝐴,𝜏𝐵,𝜏𝐶) is given by 𝑓𝑡1,𝑡2,𝑡3=𝑓1𝑡1,𝑡2,𝑡3,𝑡1𝑡2𝑡3𝑓𝑇,2𝑡1,𝑡2,𝑡3,𝑡1𝑡3𝑡2𝑓𝑇,3𝑡1,𝑡2,𝑡3,𝑡2𝑡1𝑡3𝑓𝑇,4𝑡1,𝑡2,𝑡3,𝑡2𝑡3𝑡1𝑓𝑇,5𝑡1,𝑡2,𝑡3,𝑡3𝑡1𝑡2𝑓𝑇,6𝑡1,𝑡2,𝑡3,𝑡3𝑡2𝑡1𝑇,(4.1) where 𝑓1𝑡1,𝑡2,𝑡3=𝑎0𝑏0+𝑏1𝑐0+𝑐3𝑒(𝑎0𝑏1𝑐1)𝑡1(𝑏0+𝑏1+𝑐1𝑐3)𝑡2(𝑐0+𝑐3)𝑡3,𝑓2𝑡1,𝑡2,𝑡3=𝑎0𝑐0+𝑐1𝑏0+𝑏3𝑒(𝑎0𝑏1𝑐1)𝑡1(𝑏0+𝑏3)𝑡2(𝑐0+𝑐1+𝑏1𝑏3)𝑡3,𝑓3𝑡1,𝑡2,𝑡3=𝑏0𝑎0+𝑎1𝑐0+𝑐3𝑒(𝑎0+𝑎1𝑐1)𝑡1(𝑏0𝑎1+𝑐1𝑐3)𝑡2(𝑐0+𝑐3)𝑡3,𝑓4𝑡1,𝑡2,𝑡3=𝑏0𝑐0+𝑐2𝑎0+𝑎3𝑒(𝑎0+𝑎3)𝑡1(𝑏0𝑐2𝑎1)𝑡2(𝑐0+𝑐2+𝑎1𝑎3)𝑡3,𝑓5𝑡1,𝑡2,𝑡3=𝑐0𝑎0+𝑎2𝑏0+𝑏3𝑒(𝑎0+𝑎2+𝑏2𝑏3)𝑡1(𝑏0+𝑏3)𝑡2(𝑐0𝑎2𝑏2)𝑡3,𝑓6𝑡1,𝑡2,𝑡3=𝑐0𝑎0+𝑎3𝑏0+𝑏2𝑒(𝑎0+𝑎3)𝑡1(𝑏0+𝑏2+𝑎2𝑎3)𝑡2(𝑐0𝑎2𝑏2)𝑡3.(4.2)

Proof. With the total hazard method (2.10) and (2.12) introduced in Section 2, we can express default time 𝜏 in terms of standard exponential variables 𝐸, and vice versa. If 𝜏𝐴<𝜏𝐵<𝜏𝐶, then we have 𝐸1=𝜏𝐴0𝑎0𝑑𝑢=𝑎0𝜏𝐴,𝐸2=𝜏𝐵0𝑏0𝑑𝑢+𝜏𝐵𝜏𝐴𝑏1𝕝{𝜏𝐶>𝑢}𝑑𝑢=𝑏0𝜏𝐵+𝑏1𝜏𝐵𝜏𝐴,𝐸3=𝜏𝐶0𝑐0𝑑𝑢+𝜏𝐶𝜏𝐴𝑐1𝕝{𝜏𝐵>𝑢}𝑑𝑢+𝜏𝐶𝜏𝐵𝑐2𝕝{𝜏𝐴>𝑢}+𝑐3𝕝{𝜏𝐴𝑢}𝑑𝑢=𝑐0𝜏𝐶+𝑐1𝜏𝐵𝜏𝐴+𝑐3𝜏𝐶𝜏𝐵.(4.3)
The Jacobi determinant of 𝐸 with respect to 𝜏 is given by 𝐶𝜏𝐴,𝜏𝐵,𝜏𝐶=||||𝜕𝐸1𝜕𝜏𝐴𝜕𝐸2𝜕𝜏𝐵𝜕𝐸3𝜕𝜏𝐶||||=𝑎0𝑏0+𝑏1𝑐0+𝑐3.(4.4) The density of 𝜏 is therefore 𝑓𝜏𝐴,𝜏𝐵,𝜏𝐶𝜏=𝐶𝐴,𝜏𝐵,𝜏𝐶𝑒(𝐸1+𝐸2+𝐸3).(4.5) Substituting 𝐸𝑖 into 𝑓, we get 𝑓𝑡1,𝑡2,𝑡3=𝑎0𝑏0+𝑏1𝑐0+𝑐3𝑒(𝑎0𝑏1𝑐1)𝑡1(𝑏0+𝑏1+𝑐1𝑐3)𝑡2(𝑐0+𝑐3)𝑡3,for0<𝑡1<𝑡2<𝑡3.(4.6)
The density function 𝑓 in other regions can be expressed similarly with permutation. Thus, we complete the proof of the lemma.

4.3. CDS Valuation

In this subsection, we employ the three-firms model specified by (2.14a)–(2.14c) to price the CDS swap rate 𝑠 (or swap premium) in continuous time framework and discrete time framework, respectively. We assume that the recovery rate is zero and the risk-free spot rate 𝑟 is a constant.

4.3.1. In Continuous Time Framework

In this framework, the value of the contingent leg at time 0 is equal to 𝐶=exp𝜏𝐶0+𝜃𝕝𝑟𝑑𝑡{𝜏𝐴>𝜏𝐶,𝜏𝐵>𝜏𝐶+𝜃,𝜏𝐶𝑇}(4.7) and the value of the fee leg at time 0 is equal to 𝐹=𝑠𝑇0exp𝑡0𝕝𝑟𝑑𝑢{𝜏𝐴𝜏𝐵𝜏𝐶>𝑡}𝑑𝑡,(4.8) where 𝑇 is the expiration, 𝜃 is the length of the settlement period, and 𝜏𝐶+𝜃 represents the settlement date at the end of the settlement period.

We can derive 𝑠 by computing the expectation of 𝐶 and 𝐹; the result is the following theorem.

Theorem 4.2. Let the intensity processes 𝜆𝑖𝑡(𝑖=𝐴,𝐵,𝐶) be given by (2.14a)–(2.14c), the density function given by (4.1). Then, the swap rate 𝑠 is given by 𝑐𝑠=0𝑎0+𝑎2𝑒(𝑏0+𝑏3+𝑟)𝜃𝑎0+𝑎2+𝑏2𝑏3+𝑏0+𝑏2𝑎0+𝑎2𝑏0+𝑏3𝑎0+𝑎2+𝑏2𝑏3𝑐0𝑒(𝑎0+𝑎2+𝑏0+𝑏2+𝑟)𝜃𝑎0+𝑎2+𝑏0+𝑏2.(4.9)

Proof. According to the arbitrage-free principle, we set the present value of protection buyer's payment equal to the present value of the compensation payment made at 𝜏𝐶+𝜃, conditional on default of 𝐶 prior to 𝑇, no default of 𝐴 prior to 𝜏𝐶, and no default of 𝐵 prior to 𝜏𝐶+𝜃.
Since it takes no cost to enter a CDS, the value of 𝑠 under this three-firms model is determined by 𝑠𝐸𝑇0exp𝑡0𝕝𝑟𝑑𝑢{𝜏𝐴𝜏𝐵𝜏𝐶>𝑡}𝑑𝑡=𝐸exp𝜏𝐶0+𝜃𝕝𝑟𝑑𝑡{𝜏𝐴>𝜏𝐶,𝜏𝐵>𝜏𝐶+𝜃,𝜏𝐶𝑇},(4.10) where 𝜏𝐶+𝜃 represents the settlement date at the end of the settlement period.
Recall that the change of measure is defined by 𝑍𝑇=𝑑𝑃𝑖||||𝑑𝑃𝑇=𝕝{𝜏𝑖>𝑇}exp𝑇0𝜆𝑖𝑠𝑑𝑠.(4.11)
Thus, by (4.1), (4.11) and the Fubini Theorem, we can derive the left side of (4.10): 𝑠𝐸𝑇0exp𝑡0𝕝𝑟𝑑𝑢{𝜏𝐴𝜏𝐵𝜏𝐶>𝑡}𝑑𝑡=𝑠𝑡0𝑒𝑟𝑡𝐸𝕝{𝜏𝐴𝜏𝐵𝜏𝐶>𝑡}𝑑𝑡=𝑠𝑇0𝑒𝑟𝑡𝐸𝐴𝕝{𝜏𝐵>𝑡,𝜏𝐶>𝑡}exp𝑡0𝜆𝐴𝑠𝑑𝑠𝑑𝑡=𝑠𝑇0𝑒(𝑎0+𝑟)𝑡𝐸𝐴𝕝{𝜏𝐵>𝑡,𝜏𝐶>𝑡}𝑑𝑡=𝑠𝑇0𝑒(𝑎0+𝑏0+𝑐0+𝑟)𝑡𝑑𝑡=𝑠1𝑒(𝑎0+𝑏0+𝑐0+𝑟)𝑇𝑎0+𝑏0+𝑐0,+𝑟(4.12) where the fourth equation is according to the two-firms model of Leung and Kwok in [18], 𝐸𝐴𝕝{𝜏𝐵>𝑡,𝜏𝐶>𝑡}=𝑒(𝑏0+𝑐0)𝑡.(4.13)
The right side of (4.10) equals 𝐸exp𝜏𝐶0+𝜃𝕝𝑟𝑑𝑡{𝜏𝐴>𝜏𝐶,𝜏𝐵>𝜏𝐶+𝜃,𝜏𝐶𝑇}=𝑇0𝑡3+𝜃𝑡3𝑒(𝑡3+𝜃)𝑟𝑓𝑡1,𝑡2,𝑡3𝑑𝑡1𝑑𝑡2𝑑𝑡3=𝑇0𝑡3+𝜃𝑡2𝑡3𝑒(𝑡3+𝜃)𝑟𝑓5𝑡1,𝑡2,𝑡3𝑑𝑡1𝑑𝑡2𝑑𝑡3+𝑇0𝑡3+𝜃𝑡2𝑒(𝑡3+𝜃)𝑟𝑓6𝑡1,𝑡2,𝑡3𝑑𝑡1𝑑𝑡2𝑑𝑡3=𝐼1+𝐼2,(4.14) where 𝐼6=𝑇0𝑡3+𝜃𝑡2𝑡3𝑒(𝑡3+𝜃)𝑟𝑓5𝑡1,𝑡2,𝑡3𝑑𝑡1𝑑𝑡2𝑑𝑡3=𝑒𝑟𝜃𝑇0𝑡3+𝜃𝑡2𝑡3𝑒𝑟𝑡3𝑐0𝑎0+𝑎2𝑏0+𝑏3𝑒(𝑎0+𝑎2+𝑏2𝑏3)𝑡1(𝑏0+𝑏3)𝑡2(𝑐0𝑎2𝑏2)𝑡3𝑑𝑡1𝑑𝑡2𝑑𝑡3=𝑐0𝑎0+𝑎2𝑏0+𝑏3𝑎0+𝑎2+𝑏2𝑏3𝑒𝑟𝜃×𝑇0𝑡3+𝜃𝑒(𝑎0+𝑐0𝑏3+𝑟)𝑡3𝑒(𝑏0+𝑏3)𝑡2𝑒(𝑐0𝑎2𝑏2+𝑟)𝑡3𝑒(𝑎0+𝑎2+𝑏0+𝑏2)𝑡2𝑑𝑡2𝑑𝑡3=𝑐0𝑎0+𝑎2𝑒(𝑏0+𝑏3+𝑟)𝜃𝑎0+𝑎2+𝑏2𝑏3𝑐0𝑎0+𝑎2𝑏0+𝑏3𝑒(𝑎0+𝑎2+𝑏0+𝑏2+𝑟)𝜃𝑎0+𝑎2+𝑏2𝑏3𝑎0+𝑎2+𝑏0+𝑏2𝑇0𝑒(𝑎0+𝑏0+𝑐0+𝑟)𝑑𝑡3=𝑐0𝑎0+𝑎2𝑏0+𝑏3𝑒𝑟𝜃𝑎0+𝑎2+𝑏2𝑏3𝑒(𝑏0+𝑏3)𝜃𝑏0+𝑏3𝑒(𝑎0+𝑎2+𝑏0+𝑏2)𝜃𝑎0+𝑎2+𝑏0+𝑏21𝑒(𝑎0+𝑏0+𝑐0+𝑟)𝑇𝑎0+𝑏0+𝑐0,𝐼+𝑟(4.15)7=𝑇0𝑡3+𝜃𝑡2𝑒(𝑡3+𝜃)𝑟𝑓6𝑡1,𝑡2,𝑡3𝑑𝑡1𝑑𝑡2𝑑𝑡3=𝑒𝑟𝜃𝑇0𝑡3+𝜃𝑐0𝑏0+𝑏2𝑒(𝑐0𝑎2𝑏2+𝑟)𝑡3𝑒(𝑎0+𝑎2+𝑏0+𝑏2)𝑡2𝑑𝑡2𝑑𝑡3=𝑐0𝑏0+𝑏2𝑒(𝑎0+𝑎2+𝑏0+𝑏2+𝑟)𝜃𝑎0+𝑎2+𝑏0+𝑏2𝑇0𝑒(𝑎0+𝑏0+𝑐0+𝑟)𝑡3𝑑𝑡3=𝑐0𝑏0+𝑏2𝑒(𝑎0+𝑎2+𝑏0+𝑏2+𝑟)𝜃𝑎0+𝑎2+𝑏0+𝑏21𝑒(𝑎0+𝑏0+𝑐0+𝑟)𝑇𝑎0+𝑏0+𝑐0.+𝑟(4.16)
By (4.10) and (4.12)–(4.16), we obtain expression (4.9) of 𝑠.

Remark 4.3. From (4.9), we can see that the swap rate s is not dependent on the expiration date T. The default of the buyer impacts on the swap rate 𝑠, so it is not strict if assuming that the buyer has no default throughout the process though the default risk of the protection buyer has little impact on the swap rate. The reference asset's default risk proxied by 𝑐0 gives the most significant impact on the swap rate, and an increasing higher value of 𝑐0 gives rise to a higher swap rate. The contagion effect of the protection buyer and the protection seller on the reference asset has no effect on the swap rate 𝑠(there are no 𝑐1, 𝑐2, and 𝑐3 terms). This shows that when valuating CDS in “loop-default” models, without loss of generality, we can assume that the reference asset is the primary firm and the protection buyer and the seller are secondary firms.

Remark 4.4. From (4.9), if the settlement period 𝜃 is zero, the swap rate 𝑠 is 𝑐0, which is the default intensity of reference asset determined by macroeconomic factor and itself, and has nothing to do with the credit risk of the protection buyer and seller.

4.4. Valuation of CDS in the Discrete Time Framework

In the discrete time framework, let 𝑇0, 𝑇1, 𝑇2,, 𝑇𝑛 be the swap payment dates, where 0=𝑇0<𝑇1<<𝑇𝑛=𝑇. We assume that the payment dates are uniformly distributed; that is, 𝑇𝑖+1𝑇𝑖=Δ𝑇 for 1𝑖𝑛1 and 𝑛Δ𝑇=𝑇.

The value of the contingent leg at time 0 is given by 𝐶1=𝑒𝑟(𝜏𝐶+𝜃)𝕝{𝜏𝐶𝑇}𝕝{𝜏𝐴>𝜏𝐶}𝕝{𝜏𝐵>𝜏𝐶+𝜃}.(4.17)

The value of the fee leg at time 0 is given by 𝐹1=𝑛𝑖=1𝑒𝑆(Δ𝑇)𝑟𝑇𝑖𝕝{𝜏𝐴𝜏𝐵𝜏𝐶>𝑇𝑖}+𝑒𝑟𝜏𝐶𝜏𝐶𝑇𝑖1𝕝Δ𝑇{𝑇𝑖1<𝜏𝐶<𝑇𝑖}𝕝{𝜏𝐴𝜏𝐵>𝜏𝐶}.(4.18) According to the arbitrage-free pricing principle, we have the following theorem.

Theorem 4.5. Let the intensity processes 𝜆𝑖𝑡(𝑖=𝐴,𝐵,𝐶) be given by (2.14a)–(2.14c), the density function given by (4.1). Then, 𝐸𝐶1=𝑐0𝑒(𝑎0+𝑎2+𝑏0+𝑏2+𝑟)𝜃𝑎0+𝑎2+𝑏0+𝑏2𝑏0+𝑏2𝑎0+𝑎2𝑏0+𝑏3𝑎0+𝑎2+𝑏2𝑏3+𝑐0𝑎0+𝑎2𝑒(𝑏0+𝑏3+𝑟)𝜃𝑎0+𝑎2+𝑏2𝑏31𝑒𝛼𝑇𝛼,𝐸𝐹1𝑐=S(Δ𝑇)01𝑒𝛼Δ𝑇𝛼Δ𝑇𝑒𝛼Δ𝑇+𝛼2Δ𝑇𝛼2Δ𝑇1𝑒𝛼𝑇1𝑒𝛼Δ𝑇,(4.19) where 𝛼=𝑎0+𝑏0+𝑐0+𝑟. The swap rate 𝑆(Δ𝑇) is given by equating 𝐸[𝐶1] and 𝐸[𝐹1].

Proof. Similar to the discussion in the continuous time, since it takes no cost to enter a CDS, the value of the swap rate 𝑆(Δ𝑇) under this three-firms model is determined by 𝑆(Δ𝑇)𝑛𝑖=1𝐸𝑒𝑟𝑇𝑖𝕝{𝜏𝐴𝜏𝐵𝜏𝐶>𝑇𝑖}𝑒+𝑆(Δ𝑇)𝐴(Δ𝑇)=𝐸𝑟(𝜏𝐶+𝜃)𝕝{𝜏𝐶𝑇}𝕝{𝜏𝐴>𝜏𝐶}𝕝{𝜏𝐵>𝜏𝐶+𝜃},(4.20) where 𝐴(Δ𝑇)=𝑛𝑖=1𝐸𝑒𝑟𝜏𝐶𝜏𝐶𝑇𝑖1𝕝Δ𝑇{𝑇𝑖1<𝜏𝐶<𝑇𝑖}𝕝{𝜏𝐴𝜏𝐵>𝜏𝐶},(4.21) where 𝜃 is still the length of the settlement period. The first term in (4.20) gives the present value of the sum of periodic swap payments (terminated when either 𝐴, 𝐵, or 𝐶 defaults or at maturity), and 𝑆(Δ𝑇)𝐴(Δ𝑇) is the present value of the accrued swap premium for the fraction of period between 𝜏𝐶 and the last payment date. The right term represents the present value which the protection seller (𝐵) pays if the reference asset defaults prior to the maturity: 𝑛𝑖=1𝐸𝑒𝑟𝑇𝑖𝕝{𝜏𝐴𝜏𝐵𝜏𝐶>𝑇𝑖}=𝑒𝛼Δ𝑇1𝑒𝛼𝑛Δ𝑇1𝑒𝛼Δ𝑇=𝑒𝛼Δ𝑇1𝑒𝛼𝑇1𝑒𝛼Δ𝑇.𝐸𝑒(4.22)𝑟𝜏𝐶𝜏𝐶𝑇𝑖1𝕝Δ𝑇{𝑇𝑖1<𝜏𝐶<𝑇𝑖}𝕝{𝜏𝐴𝜏𝐵>𝜏𝐶}=1Δ𝑇𝑇𝑖𝑇𝑖1𝑡3𝑡3𝑒𝑟𝑡3𝑡3𝑇𝑖1𝑓𝑡1,𝑡2,𝑡3𝑑𝑡1𝑑𝑡2𝑑𝑡3=1Δ𝑇𝑇𝑖𝑇𝑖1𝑡3𝑡2𝑡3𝑒𝑟𝑡3𝑡3𝑇𝑖1𝑓5𝑡1,𝑡2,𝑡3𝑑𝑡1𝑑𝑡2𝑑𝑡3+𝑇𝑖𝑇𝑖1𝑡3𝑡2𝑒𝑟𝑡3𝑡3𝑇𝑖1𝑓6𝑡1,𝑡2,𝑡3𝑑𝑡1𝑑𝑡2𝑑𝑡31=𝐼Δ𝑇8+𝐼9,(4.23) where 𝐼8=𝑇𝑖𝑇𝑖1𝑡3𝑡2𝑡3𝑒𝑟𝑡3𝑡3𝑇𝑖1𝑐0𝑎0+𝑎2𝑏0+𝑏3𝑒(𝑎0+𝑎2+𝑏2𝑏3)𝑡1(𝑏0+𝑏3)𝑡2(𝑐0𝑎2𝑏2)𝑡3𝑑𝑡1𝑑𝑡2𝑑𝑡3=𝑐0𝑎0+𝑎2𝑎0+𝑎2+𝑏0+𝑏21𝛼𝑒𝛼𝑇𝑖𝑇𝑖1𝑇𝑖+1𝛼𝑒𝛼𝑇𝑖1𝑒𝛼𝑇𝑖,𝐼9=𝑇𝑖𝑇𝑖1𝑡3𝑡2𝑒𝑟𝑡3𝑡3𝑇𝑖1𝑐0𝑎0+𝑎3𝑏0+𝑏2𝑒(𝑎0+𝑎3)𝑡1(𝑏0+𝑏2+𝑎2𝑎3)𝑡2(𝑐0𝑎2𝑏2)𝑡3𝑑𝑡1𝑑𝑡2𝑑𝑡3=𝑐0𝑏0+𝑏2𝑎0+𝑎2+𝑏0+𝑏21𝛼𝑒𝛼𝑇𝑖𝑇𝑖1𝑇𝑖+1𝛼𝑒𝛼𝑇𝑖1𝑒𝛼𝑇𝑖.(4.24)
We can obtain 𝐴(Δ𝑇)=𝑛𝑖=1𝐸𝑒𝑟𝜏𝐶𝜏𝐶𝑇𝑖1𝕝Δ𝑇{𝑇𝑖1<𝜏𝐶<𝑇𝑖}𝕝{𝜏𝐴𝜏𝐵>𝜏𝐶}=1Δ𝑇𝑛𝑖=1𝐼8+𝐼9=𝑐01𝑒𝛼Δ𝑇𝛼Δ𝑇𝑒𝛼Δ𝑇𝛼2Δ𝑇1𝑒𝛼𝑇1𝑒𝛼Δ𝑇.(4.25) The right-hand side of (4.20) is 𝐸𝑒𝑟(𝜏𝐶+𝜃)𝕝{𝜏𝐶𝑇}𝕝{𝜏𝐴>𝜏𝐶}𝕝{𝜏𝐵>𝜏𝐶+𝜃}=𝑐0𝑒(𝑎0+𝑎2+𝑏0+𝑏2+𝑟)𝜃𝑎0+𝑎2+𝑏0+𝑏2𝑏0+𝑏2𝑎0+𝑎2𝑏0+𝑏3𝑎0+𝑎2+𝑏2𝑏3+𝑐0𝑎0+𝑎2𝑒(𝑏0+𝑏3+𝑟)𝜃𝑎0+𝑎2+𝑏2𝑏31𝑒𝛼𝑇𝛼.(4.26) From (4.22)–(4.26), we can obtain the expression of 𝑆(Δ𝑇) from the following equation: 𝑒𝑆(Δ𝑇)𝛼Δ𝑇+𝑐01𝑒𝛼Δ𝑇𝛼Δ𝑇𝑒𝛼Δ𝑇𝛼2𝑒Δ𝑇𝛼Δ𝑇1𝑒𝛼𝑇1𝑒𝛼Δ𝑇=𝑐0𝑒(𝑎0+𝑎2+𝑏0+𝑏2+𝑟)𝜃𝑎0+𝑎2+𝑏0+𝑏2𝑏0+𝑏2𝑎0+𝑎2𝑏0+𝑏3𝑎0+𝑎2+𝑏2𝑏3+𝑐0𝑎0+𝑎2𝑒(𝑏0+𝑏3+𝑟)𝜃𝑎0+𝑎2+𝑏2𝑏31𝑒𝛼𝑇𝛼.(4.27)

Remark 4.6. As analyzed in the continuous time, the expression for the swap premium 𝑆(Δ𝑇) in (4.27) shows no dependence on 𝑎1, 𝑎3, 𝑏1, 𝑐1, 𝑐2, 𝑐3. In the financial sense, prior to the default of the underlying asset, the default event of the protection buyer or the protection seller will terminate the contract. This is why 𝑎1, 𝑎3, 𝑏1, 𝑐1, 𝑐2, 𝑐3 have no influence on the swap premium. Moreover, we discover that the swap premium is also insensitive to maturity.

5. Conclusion

In this paper, we present a three-firms contagion model with an interaction term which is an improvement in the model of Leung and Kwok [18]. Under this model, we analyze the pricing of defaultable bonds and obtain the closed forms. We also discuss the CDS valuation in continuous time and discrete time framework, respectively. The analytical solutions of CDS swap rate (swap premium) are obtained by the approaches of “total hazard construction” and “change of measure.” Besides, we analyze the effect of the default of the protection buyer, the protection seller, and the reference asset on the swap rate.

Our model has its actual background. For example, before and during the global financial crisis, as default risk of the reference asset issuer increased, the protection seller collected higher CDS swap premiums. Thus, default risk of the protection buyer increased since more CDS swap premiums were payed. On the other hand, the protection seller compensated more and more for the loss of reference asset (if it defaulted). When the protection seller (such as a monoline insurer) had no ability to compensate for the loss of reference asset, it went bankrupt. All of these could be important reasons for the financial crisis. So our model is of some significance.


This paper was supported by the National Basic Research Program of China (973 program 2007CB814903).