Credit Risky Securities Valuation under a Contagion Model with Interacting Intensities

Abstract

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,ℱ,{ℱ_𝑡}^𝑇∗_𝑡=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 ℱ=ℱ_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 ℝ^𝑑-valued process 𝑋_𝑡, which presents 𝑑-dimensional economy-wide state variables. There are also 𝐼 point processes, 𝑁^𝑖(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 𝑁^𝑖 jumps from 0 to 1.

According to the information contained in the state variables and the default processes, the filtration isℱ=ℱ^𝑋𝑡∨ℱ^1𝑡∨ℱ^2𝑡∨⋯∨ℱ^𝐼𝑡,(2.1) whereℱℱ^𝑋𝑡=𝜎𝑋_𝑠,0≤𝑠≤𝑡,(2.2a)_𝑖𝑡=𝜎𝑁^𝑖𝑠,0≤𝑠≤𝑡(2.2b) are the filtrations generated by 𝑋_𝑡 and 𝑁^𝑖𝑡, respectively.

Let ℱ𝒢^𝑡−𝑖=ℱ^1𝑡∨⋯ℱ^𝑡𝑖−1∨ℱ^𝑡𝑖+1⋯∨ℱ^𝐼𝑡,(2.2c)_𝑖𝑡=ℱ^𝑖𝑡∨ℱ^𝑋𝑇_∗∨ℱ^𝑇−𝑖_∗=ℱ^𝑖𝑡∨𝒢^𝑖0,(2.2d)where 𝒢^𝑖0=ℱ^𝑋𝑇_∗∨ℱ^𝑇−𝑖_∗. We know that 𝒢^𝑖0 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 𝒢^𝑖𝑡, it is possible to select a nonnegative, 𝒢^𝑖0-measurable process 𝜆^𝑖𝑡, satisfying ∫^𝑡0𝜆^𝑖𝑠𝑑𝑠<∞,𝑃−𝑎.𝑠. for all 𝑡∈[0,𝑇^∗], so that we can define an inhomogeneous Poisson process 𝑁^𝑖, using the process 𝜆^𝑖𝑡 as its intensity function.

Let 𝜏^𝑖 denote the default time of firm 𝑖, namely, 𝜏^𝑖 be the first jump time of 𝑁^𝑖, in a typical reduced-form model, which can be defined as 𝜏^𝑖=inf𝑡∶^𝑡0𝜆^𝑖𝑠𝑑𝑠≥𝐸^𝑖,(2.3) where {𝐸^𝑖}^𝐼𝑖=1 is independent of 𝑋_𝑡(𝑡∈[0,𝑇^∗]).

According to the Doob-Meyer decomposition, we have that 𝑀^𝑖𝑡=𝑁_𝑡−^{𝑡∧𝜏𝑖}₀𝜆^𝑖𝑠𝑑𝑠,(2.4) is a (𝑃,ℱ_𝑡)-martingale.

Under the above characterization, the conditional survival probability of firm 𝑖 is given by 𝑃𝜏^𝑖>𝑡∣𝒢^𝑖0=exp−^𝑡0𝜆^𝑖𝑠𝑑𝑠,𝑡∈0,𝑇^∗.(2.5)

The unconditional survival probability of firm 𝑖 is given by 𝑃𝜏^𝑖>𝑡=𝐸exp−^𝑡0𝜆^𝑖𝑠𝑑𝑠,𝑡∈0,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 𝜆_𝑖(𝑡∣𝑛) denote the intensity for firm 𝑖 given the observed default times of 𝑛 other firms, 𝑡_𝑘0,𝑡_𝑘1,…,𝑡_𝑘𝑛, where 0=𝑡_𝑘0<𝑡_𝑘1<⋯<𝑡_𝑘𝑛<𝑡<𝜏^𝑖.

The total hazard accumulated by firm 𝑖 by time 𝑡 given 𝑛 observed defaults is defined as 𝜓_𝑖(𝑡∣𝑛)=^𝑛_𝑚₌₁Λ_𝑖𝑡_𝑘𝑚−𝑡_𝑘𝑚−1∣𝑚−1+Λ_𝑖𝑡−𝑡_𝑘𝑛∣𝑛,(2.7) where Λ_𝑖(𝑠∣𝑚)=∫^{𝑡𝑘𝑚𝑡+𝑠}_𝑘𝑚𝜆_𝑖(𝑢∣𝑚)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 𝑡_𝑘𝑚 and 𝑡.

Define the inverse functions Λ^𝑖−1(𝑥∣𝑛)=inf𝑠≥0∶Λ_𝑖(𝑠∣𝑚)≥𝑥(2.8) for 𝑥>0. We can use the following recursive procedure to construct a collection of random variables.

Step 1. Let 𝐸₁,…,𝐸_𝐼 be the i.i.d. unit exponentials and 𝑘₁=argminΛ^𝑖−1𝐸_𝑖∣0∶𝑖=1,…,𝑛,(2.9) and define 𝜏̂¹=Λ^𝑘−1₁𝐸_𝑘1∣0.(2.10)

Step  𝑚+1(𝑚=1,2,…,𝐼−1)
Given that Step 1 up to step 𝑚 have resulted in (𝜏̂¹,…,𝜏̂^𝑚), define the set 𝐼_𝑚={𝑘₁,…,𝑘_𝑚} and 𝐼_𝑚 as the set of firms excluding 𝐼_𝑚. Let 𝑘_𝑚+1=nargminΛ^𝑖−1𝐸_𝑖−𝜓_𝑖(𝜏̂^𝑚∣𝑚)∣𝑚(2.11) and let 𝜏̂^𝑚+1=𝜏̂^𝑚+Λ^𝑘−1_𝑚+1𝐸^𝑘_𝑚+1−𝜓_𝑘𝑚+1(𝜏̂^𝑚∣𝑚)∣𝑚.(2.12)

Norros [22], Shaked and Shanthikumar [23], and Yu [24] prove that 𝜏̂=(𝜏̂¹,…,𝜏̂^𝐼) equals 𝜏=(𝜏¹,…,𝜏^𝐼) in distribution. So we can generate default time 𝜏 by generating 𝜏̂ 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:𝜆𝜆^𝐴𝑡=𝑎₀+𝑎₁𝕝_{{𝜏𝐵≤𝑡}}+𝑎₂𝕝_{{𝜏𝐶≤𝑡}},(2.13a)_𝐵𝑡𝜆=𝑏₀+𝑏₁𝕝_{{𝜏𝐴≤𝑡}}+𝑏₂𝕝_{{𝜏𝐶≤𝑡}},(2.13b)_𝐶𝑡=𝑐₀+𝑐₁𝕝_{{𝜏𝐴≤𝑡}}+𝑐₂𝕝_{{𝜏𝐵≤𝑡}},(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:𝜆𝜆^𝐴𝑡=𝑎₀+𝑎₁𝕝_{{𝜏𝐵≤𝑡,𝜏𝐶>𝑡}}+𝑎₂𝕝_{{𝜏𝐶≤𝑡,𝜏𝐵>𝑡}}+𝑎₃𝕝_{{𝜏𝐵≤𝑡,𝜏𝐶≤𝑡}},(2.14a)_𝐵𝑡𝜆=𝑏₀+𝑏₁𝕝_{{𝜏𝐴≤𝑡,𝜏𝐶>𝑡}}+𝑏₂𝕝_{{𝜏𝐶≤𝑡,𝜏𝐴>𝑡}}+𝑏₃𝕝_{{𝜏𝐴≤𝑡,𝜏𝐶≤𝑡}},(2.14b)_𝐶𝑡=𝑐₀+𝑐₁𝕝_{{𝜏𝐴≤𝑡,𝜏𝐵>𝑡}}+𝑐₂𝕝_{{𝜏𝐵≤𝑡,𝜏𝐴>𝑡}}+𝑐₃𝕝_{{𝜏𝐴≤𝑡,𝜏𝐵≤𝑡}},(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 ℱ_𝑇-measurable random variable, 𝐴=(𝐴_𝑡)_{𝑡∈[0,𝑇]} is an ℱ-adapted, continuous process of finite variation with 𝐴₀=0, 𝑍=(𝑍_𝑡)_{𝑡∈[0,𝑇]} is an ℱ-predictable process, and 𝜏 is a random time.

Definition 3.2. The dividend process 𝐷=(𝐷_𝑡)_𝑡∈ℝ⁺ of the above defaultable claim maturing at 𝑇 equals, for every 𝑡∈ℝ⁺, 𝐷_𝑡=𝑋𝕝_{𝑇<𝜏}𝕝_[𝑇,∞)(𝑡)+_{(0,𝑡∧𝑇]}1−𝑁_𝑢𝑑𝐴_𝑢+_{(0,𝑡∧𝑇]}𝑍_𝑢𝑑𝑁_𝑢,(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], 𝑆_𝑡=𝐸_𝑡_(𝑡,𝑇]𝐵_𝑡𝐵_𝑢𝑑𝐷_𝑢,(3.2) where 𝐵_𝑡∶=𝐵(𝑡)=exp(∫^𝑡0𝑟_𝑠𝑑𝑠) is the money market account, 𝑟_𝑡 is a constant risk-free spot rate, and 𝐸_𝑡 represents the conditional expectation on ℱ_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), 𝑆_𝑡=𝕝_{𝑡<𝜏}𝐵_𝑡𝐺_𝑡𝐸_𝑡𝐵^𝑇−1𝐺_𝑇𝑋+^𝑇𝑡𝐵^𝑢−1𝐺_𝑢𝑍_𝑢𝜆_𝑢𝑑𝑢+𝑑𝐴_𝑢,(3.3) where 𝐺_𝑡=𝑃{𝜏>𝑡∣ℱ_𝑡}.

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 𝐷_𝑡=𝕝_{{𝑡≥𝑇}}. Let 𝑝(𝑡,𝑇) be the time-𝑡 price, then 𝑝(𝑡,𝑇) is given by 𝑝(𝑡,𝑇)=𝐸_𝑡𝐵𝐵^𝑡_𝑇.(3.4)(2)If the dividend process is 𝐷_𝑡=∫^𝑇𝑡𝑟_𝑢𝑑𝑢+𝕝_{{𝑡≥𝑇}}, 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 𝑣^𝑖(𝑡,𝑇) denote the time-𝑡 price, issued by firm 𝑖, 𝛿^𝑖∈[0,1] is the recovery rate of the firm 𝑖, then 𝑣^𝑖(𝑡,𝑇) is given by 𝑣^𝑖(𝑡,𝑇)=𝐸_𝑡exp−^𝑇𝑡=𝐸_𝑡𝑟_𝑢𝑑𝑢𝛿^𝑖𝕝_{{𝜏𝑖≤𝑇}}+𝕝_{{𝜏𝑖>𝑇}}𝐵𝐵^𝑡_𝑇𝛿^𝑖𝕝_{{𝜏𝑖≤𝑇}}+𝕝_{{𝜏𝑖>𝑇}}.(3.5)(4)If the dividend process 𝐷_𝑡=𝑋𝕝_{{𝜏𝑖>𝑇,𝑡≥𝑇}}+∫_{(0,𝑡∧𝑇]}𝑍_𝑢𝑑𝑁_𝑢, using the Doob-Meyer decomposition of 𝑁_𝑡, then the value of the defaultable bond is 𝑆_𝑡=𝐸_𝑡^𝑇𝑡exp−^𝑢𝑡𝑟_𝑣𝑑𝑣𝑍_𝑢𝜆_𝑢𝕝_{𝑢<𝜏}𝑑𝑢+exp−^𝑇𝑡𝑟_𝑣𝑑𝑣𝑋𝕝_{𝜏>𝑇}=𝐵_𝑡𝐸_𝑡^𝑇𝑡𝐵^𝑢−1𝑍_𝑢𝜆_𝑢𝕝_{𝑢<𝜏}𝑑𝑢+𝐵^𝑇−1𝑋𝕝_{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 𝑃^𝑖 which puts zero probability on the paths where default occurs prior to the maturity 𝑇. Specifically, the change of measure is defined by 𝑍_𝑇∶=𝑑𝑃^𝑖𝑑𝑃||||_ℱ𝑇=𝕝_{{𝜏𝑖>𝑇}}exp^𝑇0𝜆^𝑖𝑠𝑑𝑠,(3.7) where 𝑃^𝑖 is a firm-specific (firm 𝑖) probability measure which is absolutely continuous with respect to 𝑃 on the stochastic interval [0,𝜏^𝑖). To proceed the calculations under the measure 𝑃^𝑖, we enlarge the filtration to ℱ^𝑖=(ℱ^𝑖𝑡)_𝑡≥0 as the completion of ℱ=(ℱ_𝑡)_𝑡≥0 by the null sets of the probability measure 𝑃^𝑖.

Applying the result of Jarrow and Yu [15], we know that the defaultable bond price of firm 𝑖 is given by 𝑣^𝑖(𝑡,𝑇)=𝛿^𝑖𝑝(𝑡,𝑇)+𝕝_{{𝜏𝑖>𝑡}}1−𝛿^𝑖𝐸_𝑡exp−^𝑇𝑡𝑟_𝑠+𝜆^𝑖𝑠𝑑𝑠,𝑡≤𝑇(3.8) or 𝑣^𝑖(𝑡,𝑇)𝑝(𝑡,𝑇)=𝛿^𝑖+1−𝛿^𝑖𝕝_{𝜏^𝑖_>𝑡}𝐸_𝑡exp−^𝑇𝑡𝜆^𝑖𝑠𝑑𝑠.(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 𝑣^𝐶(𝑡,𝑇) of the defaultable bond 𝐶 maturity at 𝑇 satisfies 𝑣^𝐶(𝑡,𝑇)=𝑝(𝑡,𝑇)𝛿^𝐶+1−𝛿^𝐶𝕝_{𝜏^𝐶_>𝑡}𝐸_𝑡exp−^𝑇𝑡𝜆^𝐶𝑠𝑑𝑠,(3.10) where 𝐸_𝑡exp−^𝑇𝑡𝜆^𝐶𝑠𝑑𝑠=𝑒^{−𝑐0(𝑇−𝑡)}𝐸_𝑡×exp−^𝑇𝑡𝑐₁𝕝_{{𝜏𝐴≤𝑠,𝜏𝐵>𝑠}}+𝑐₂𝕝_{{𝜏𝐵≤𝑠,𝜏𝐴>𝑠}}+𝑐₃𝕝_{{𝜏𝐴≤𝑠,𝜏𝐵≤𝑠}}𝑑𝑠.(3.11)(I) Conditional on 𝜏^𝐴>𝑡, 𝜏^𝐵>𝑡, namely, neither firm 𝐴 nor firm 𝐵 has defaulted by time 𝑡, the default intensities 𝜆^𝐴𝑡 and 𝜆^𝐵𝑡 under the measure 𝑃^𝐶 and 𝜏^𝐶>𝑡 are given by 𝜆𝜆^𝐴𝑡_𝐵𝑡=𝑎₀+𝑎₁𝕝_{{𝜏𝐵≤𝑡}},=𝑏₀+𝑏₁𝕝_{{𝜏𝐴≤𝑡}}.(3.12) According to the result of Leung and Kwok [18], the conditional joint density function 𝑓_𝑡(𝑡₁,𝑡₂) of (𝜏^𝐴,𝜏^𝐵) is 𝑓_𝑡𝑡₁,𝑡₂=𝑓^1,𝑡𝑡₁𝑓_2,𝑡,𝑡₂,𝑡₃=𝑏₀𝑎₀+𝑎₁𝑒^{−(𝑎0+𝑎1)(𝑡1−𝑡)−(𝑏0−𝑎1)(𝑡2−𝑡)},𝑡₂≤𝑡₁,𝑡₁,𝑡₂,𝑡₃=𝑎₀𝑏₀+𝑏₁𝑒^{−(𝑏0+𝑏1)(𝑡2−𝑡)−(𝑎0−𝑏1)(𝑡1−𝑡)},𝑡₂>𝑡₁.(3.13) The integration region of (3.11) is then appropriately divided into five pieces: 𝐷₁: 𝑡≤𝜏^𝐴≤𝑇, 𝜏^𝐴≤𝜏^𝐵≤𝑇; 𝐷₂: 𝑡≤𝜏^𝐵≤𝑇, 𝜏^𝐵≤𝜏^𝐴≤𝑇; 𝐷₃: 𝑡≤𝜏^𝐴≤𝑇, 𝜏^𝐵≥𝑇; 𝐷₄: 𝑡≤𝜏^𝐵≤𝑇, 𝜏^𝐴≥𝑇; 𝐷₅: 𝜏^𝐴≥𝑇, 𝜏^𝐵≥𝑇: 𝐸_𝑡exp−^𝑇𝑡𝜆^𝐶𝑠𝑑𝑠=𝑒^{−𝑐0(𝑇−𝑡)}𝐸_𝑡=∶×𝑒exp_{−𝑐0(𝑇−𝑡)}−^𝑇𝑡𝑐₁𝕝_{{𝜏𝐴≤𝑠,𝜏𝐵>𝑠}}+𝑐₂𝕝_{{𝜏𝐵≤𝑠,𝜏𝐴>𝑠}}+𝑐₃𝕝_{{𝜏𝐴≤𝑠,𝜏𝐵≤𝑠}}𝑑𝑠=∶𝐽₁𝐼₁+𝐼₂+𝐼₃+𝐼₄+𝐼₅,(3.14) where 𝐼₁=_𝐷1𝑒^{−𝑐1(𝑡2−𝑡1)−𝑐3(𝑇−𝑡2)}𝑓_2,𝑡𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂=𝑎₀𝑏₀+𝑏₁^𝑇𝑡^𝑇𝑡₁𝑒^{−𝑐1(𝑡2−𝑡1)−𝑐3(𝑇−𝑡2)}𝑒^{−(𝑎0−𝑏1)(𝑡1−𝑡)−(𝑏0+𝑏1)(𝑡2−𝑡)}𝑑𝑡₂𝑑𝑡₁=𝑎⁰𝑏₀+𝑏₁𝑏₀+𝑏₁+𝑐₁−𝑐₃𝑒^{−𝑐3(𝑇−𝑡)}−𝑒^{−(𝑎0+𝑏0)(𝑇−𝑡)}𝑎₀+𝑏₀−𝑐₃−𝑒^{−(𝑏0+𝑏1+𝑐1)(𝑇−𝑡)−𝑒00−(𝑎+𝑏)(𝑇−𝑡)}𝐼₂𝑎₀−𝑏₁−𝑐₁,=_𝐷2𝑒^{−𝑐2(𝑡1−𝑡2)−𝑐3(𝑇−𝑡1)}𝑓_1,𝑡𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂=𝑏⁰𝑎₀+𝑎₁𝑎₀+𝑎₁+𝑐₂−𝑐₃𝑒^{−𝑐3(𝑇−𝑡)}−𝑒^{−(𝑎0+𝑏0)(𝑇−𝑡)}𝑎₀+𝑏₀−𝑐₃−𝑒^{−(𝑎0+𝑎1+𝑐2)(𝑇−𝑡)−𝑒00−(𝑎+𝑏)(𝑇−𝑡)}𝐼₃𝑏₀−𝑎₁−𝑐₂,=_𝐷3𝑒^{−(𝑐1(𝑡2−𝑡1))}𝑓_2,𝑡𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂=𝑎⁰𝑏₀+𝑏₁𝑎₀−𝑏₁−𝑐₁𝑏₀+𝑏₁+𝑐₁𝑒^{−(𝑏0+𝑏1+𝑐1)(𝑇−𝑡)}𝐼₄−𝑒^{−(𝑎0+𝑏0)(𝑇−𝑡)},=_𝐷4𝑒^{−(𝑐2(𝑡1−𝑡2))}𝑓_1,𝑡𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂=𝑏⁰𝑎₀+𝑎₁𝑏₀−𝑎₁−𝑐₂𝑎₀+𝑎₁+𝑐₂𝑒^{−(𝑎0+𝑎1+𝑐2)(𝑇−𝑡)}𝐼₅−𝑒^{−(𝑎0+𝑏0)(𝑇−𝑡)},=_𝐷5𝑓_𝑡𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂=𝑒^{−(𝑎0+𝑏0)(𝑇−𝑡)}.(3.15)(II) Conditional on 𝜏^𝐵>𝑡, 𝜏^𝐴≤𝑡, namely, firm 𝐴 has defaulted and firm 𝐵 has not defaulted by time 𝑡, the default intensities 𝜆^𝐵𝑡 are given by 𝜆^𝐵𝑡=𝑏₀+𝑏₁𝕝_{{𝜏𝐴≤𝑡,𝜏𝐶>𝑡}}=𝑏₀+𝑏₁(3.16) and the density function 𝑓_𝑡(𝑡₂) of 𝜏^𝐵 is 𝑓_𝑡𝑡₂=𝑏₀+𝑏₁𝑒^{−(𝑏0+𝑏1)(𝑡2−𝑡)},(3.17) so 𝐸_𝑡exp−^𝑇𝑡𝜆^𝐶𝑠𝑑𝑠=𝑒^{−𝑐0(𝑇−𝑡)}𝐸_𝑡exp−^𝑇𝑡𝑐₁=𝑏𝕝_{{𝜏𝐴≤𝑠,𝜏𝐵>𝑠}}+𝑐₃𝕝_{{𝜏𝐴≤𝑠,𝜏𝐵≤𝑠}}𝑑𝑠₀+𝑏₁𝑒^{−𝑐0(𝑇−𝑡)}×^𝑇𝑡𝑒^{−𝑐1(𝑡2−𝑡)−𝑐3(𝑇−𝑡2)}𝑒^{−(𝑏0+𝑏1)(𝑡2−𝑡)}𝑑𝑡₂+^∞𝑇𝑒^{−𝑐1(𝑇−𝑡)}𝑒^{−(𝑏0+𝑏1)(𝑡2−𝑡)}𝑑𝑡₂=𝑏₀+𝑏₁𝑒^{−𝑐0(𝑇−𝑡)}𝑒^{−𝑐3(𝑇−𝑡)}−𝑒^{−(𝑏0+𝑏1+𝑐1)(𝑇−𝑡)}𝑏₀+𝑏₁+𝑐₁−𝑐₃+𝑒^{−(𝑏0+𝑏1−𝑐1)(𝑇−𝑡)}=∶𝐽₂𝑏₀+𝑏₁.(3.18)(III) Conditional on 𝜏^𝐵≤𝑡, 𝜏^𝐴>𝑡, namely, firm 𝐵 has defaulted and firm 𝐴 has not defaulted by time 𝑡, similar to the computation in (II), we have 𝐸_𝑡exp−^𝑇𝑡𝜆^𝐶𝑠𝑑𝑠=𝑎₀+𝑎₁𝑒^{−𝑐0(𝑇−𝑡)}𝑒^{−𝑐3(𝑇−𝑡)}−𝑒^{−(𝑎0+𝑎1+𝑐2)(𝑇−𝑡)}𝑎₀+𝑎₁+𝑐₂−𝑐₃+𝑒^{−(𝑎0+𝑎1−𝑐2)(𝑇−𝑡)}=∶𝐽₃𝑎₀+𝑎₁,(3.19)(IV) Conditional on 𝜏^𝐴≤𝑡, 𝜏^𝐵≤𝑡, namely, firm 𝐴 and firm 𝐵 have defaulted by time 𝑡, we have 𝐸_𝑡exp−^𝑇𝑡𝜆^𝐶𝑠𝑑𝑠=𝑒^{−(𝑐0+𝑐3)(𝑇−𝑡)}=∶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 𝜆^𝑖𝑡(𝑖=𝐴,𝐵,𝐶) be given by (2.14a)–(2.14c), the time-t defaultable bond price issued by firm 𝐶 is given by 𝑣^𝐶(𝑡,𝑇)=𝑝(𝑡,𝑇)𝛿^𝐶+1−𝛿^𝐶𝕝_{𝜏^𝐶_>𝑡}𝕝_{𝜏^𝐴_>𝑡,𝜏^𝐵_>𝑡}𝐽₁+𝕝_{{𝜏𝐴≤𝑡,𝜏𝐵>𝑡}}𝐽₂+𝕝_{{𝜏𝐴>𝑡,𝜏𝐵≤𝑡}}𝐽₃+𝕝_{{𝜏𝐴≤𝑡,𝜏𝐵≤𝑡}}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.

Figure 1 

The structure of CDS.

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 𝜆^𝑖𝑡(𝑖=𝐴,𝐵,𝐶) are given by model (2.14a)–(2.14c). Then the joint density function of 𝜏=(𝜏^𝐴,𝜏^𝐵,𝜏^𝐶) is given by 𝑓𝑡₁,𝑡₂,𝑡₃=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑓¹𝑓𝑡₁,𝑡₂,𝑡₃,𝑡₁≤𝑡₂≤𝑡₃≤𝑇,₂𝑓𝑡₁,𝑡₂,𝑡₃,𝑡₁≤𝑡₃≤𝑡₂≤𝑇,₃𝑓𝑡₁,𝑡₂,𝑡₃,𝑡₂≤𝑡₁≤𝑡₃≤𝑇,₄𝑓𝑡₁,𝑡₂,𝑡₃,𝑡₂≤𝑡₃≤𝑡₁≤𝑇,₅𝑓𝑡₁,𝑡₂,𝑡₃,𝑡₃≤𝑡₁≤𝑡₂≤𝑇,₆𝑡₁,𝑡₂,𝑡₃,𝑡₃≤𝑡₂≤𝑡₁≤𝑇,(4.1) where 𝑓𝑓₁₂𝑡₁,𝑡₂,𝑡₃=𝑎₀𝑏₀+𝑏₁𝑐₀+𝑐₃𝑒^{−(𝑎0−𝑏1−𝑐1)𝑡1−(𝑏0+𝑏1+𝑐1−𝑐3)𝑡2−(𝑐0+𝑐3)𝑡3},𝑓₃𝑡₁,𝑡₂,𝑡₃=𝑎₀𝑐₀+𝑐₁𝑏₀+𝑏₃𝑒^{−(𝑎0−𝑏1−𝑐1)𝑡1−(𝑏0+𝑏3)𝑡2−(𝑐0+𝑐1+𝑏1−𝑏3)𝑡3},𝑓₄𝑡₁,𝑡₂,𝑡₃=𝑏₀𝑎₀+𝑎₁𝑐₀+𝑐₃𝑒^{−(𝑎0+𝑎1−𝑐1)𝑡1−(𝑏0−𝑎1+𝑐1−𝑐3)𝑡2−(𝑐0+𝑐3)𝑡3},𝑓₅𝑡₁,𝑡₂,𝑡₃=𝑏₀𝑐₀+𝑐₂𝑎₀+𝑎₃𝑒^{−(𝑎0+𝑎3)𝑡1−(𝑏0−𝑐2−𝑎1)𝑡2−(𝑐0+𝑐2+𝑎1−𝑎3)𝑡3},𝑓₆𝑡₁,𝑡₂,𝑡₃=𝑐₀𝑎₀+𝑎₂𝑏₀+𝑏₃𝑒^{−(𝑎0+𝑎2+𝑏2−𝑏3)𝑡1−(𝑏0+𝑏3)𝑡2−(𝑐0−𝑎2−𝑏2)𝑡3},𝑡₁,𝑡₂,𝑡₃=𝑐₀𝑎₀+𝑎₃𝑏₀+𝑏₂𝑒^{−(𝑎0+𝑎3)𝑡1−(𝑏0+𝑏2+𝑎2−𝑎3)𝑡2−(𝑐0−𝑎2−𝑏2)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 𝜏^𝐴<𝜏^𝐵<𝜏^𝐶, then we have 𝐸𝐸₁₂=^𝜏𝐴₀𝑎₀𝑑𝑢=𝑎₀𝜏^𝐴,𝐸₃=^𝜏𝐵₀𝑏₀𝑑𝑢+^𝜏𝐵_𝜏𝐴𝑏₁𝕝_{{𝜏𝐶>𝑢}}𝑑𝑢=𝑏₀𝜏^𝐵+𝑏₁𝜏^𝐵−𝜏^𝐴,==𝑐^𝜏𝐶₀₀𝑐₀𝑑𝑢+^𝜏𝐶_𝜏𝐴𝑐₁𝕝_{{𝜏𝐵>𝑢}}𝑑𝑢+^𝜏𝐶_𝜏𝐵𝑐₂𝕝_{{𝜏𝐴>𝑢}}+𝑐₃𝕝_{{𝜏𝐴≤𝑢}}𝑑𝑢𝜏^𝐶+𝑐₁𝜏^𝐵−𝜏^𝐴+𝑐₃𝜏^𝐶−𝜏^𝐵.(4.3)
The Jacobi determinant of 𝐸 with respect to 𝜏 is given by 𝐶𝜏^𝐴,𝜏^𝐵,𝜏^𝐶=||||𝜕𝐸¹𝜕𝜏^𝐴𝜕𝐸₂𝜕𝜏^𝐵𝜕𝐸₃𝜕𝜏^𝐶||||=𝑎₀𝑏₀+𝑏₁𝑐₀+𝑐₃.(4.4) The density of 𝜏 is therefore 𝑓𝜏^𝐴,𝜏^𝐵,𝜏^𝐶=𝐶𝜏^𝐴,𝜏^𝐵,𝜏^𝐶𝑒^{−(𝐸1+𝐸2+𝐸3)}.(4.5) Substituting 𝐸_𝑖 into 𝑓, we get 𝑓𝑡₁,𝑡₂,𝑡₃=𝑎₀𝑏₀+𝑏₁𝑐₀+𝑐₃𝑒^{−(𝑎0−𝑏1−𝑐1)𝑡1−(𝑏0+𝑏1+𝑐1−𝑐3)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=exp−^{𝜏𝐶0+𝜃}𝑟𝑑𝑡𝕝_{{𝜏𝐴>𝜏𝐶,𝜏𝐵>𝜏𝐶+𝜃,𝜏𝐶≤𝑇}}(4.7) and the value of the fee leg at time 0 is equal to 𝐹=𝑠^𝑇0exp−^𝑡0𝑟𝑑𝑢𝕝_{{𝜏𝐴∧𝜏𝐵∧𝜏𝐶>𝑡}}𝑑𝑡,(4.8) where 𝑇 is the expiration, 𝜃 is the length of the settlement period, and 𝜏^𝐶+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 𝜆^𝑖𝑡(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 𝑠=𝑐⁰𝑎₀+𝑎₂𝑒^{−(𝑏0+𝑏3+𝑟)𝜃}𝑎₀+𝑎₂+𝑏₂−𝑏₃+𝑏₀+𝑏₂−𝑎⁰+𝑎₂𝑏₀+𝑏₃𝑎₀+𝑎₂+𝑏₂−𝑏₃𝑐⁰𝑒^{−(𝑎0+𝑎2+𝑏0+𝑏2+𝑟)𝜃}𝑎₀+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 𝜏^𝐶+𝜃, conditional on default of 𝐶 prior to 𝑇, no default of 𝐴 prior to 𝜏^𝐶, and no default of 𝐵 prior to 𝜏^𝐶+𝜃.
Since it takes no cost to enter a CDS, the value of 𝑠 under this three-firms model is determined by 𝑠𝐸^𝑇0exp−^𝑡0𝑟𝑑𝑢𝕝_{{𝜏𝐴∧𝜏𝐵∧𝜏𝐶>𝑡}}𝑑𝑡=𝐸exp−^{𝜏𝐶0+𝜃}𝑟𝑑𝑡𝕝_{{𝜏𝐴>𝜏𝐶,𝜏𝐵>𝜏𝐶+𝜃,𝜏𝐶≤𝑇}},(4.10) where 𝜏^𝐶+𝜃 represents the settlement date at the end of the settlement period.
Recall that the change of measure is defined by 𝑍_𝑇∶=𝑑𝑃^𝑖𝑑𝑃||||_ℱ𝑇=𝕝_{{𝜏𝑖>𝑇}}exp^𝑇0𝜆^𝑖𝑠𝑑𝑠.(4.11)
Thus, by (4.1), (4.11) and the Fubini Theorem, we can derive the left side of (4.10): 𝑠𝐸=^𝑇0𝑠ex_𝑡0p−^𝑡0𝑟𝑑𝑢𝕝_{{𝜏𝐴∧𝜏𝐵∧𝜏𝐶>𝑡}}𝑑𝑡=𝑠_𝑇0𝑒^−𝑟𝑡𝐸𝕝_{{𝜏𝐴∧𝜏𝐵∧𝜏𝐶>𝑡}}𝑑𝑡𝑒^−𝑟𝑡𝐸^𝐴𝕝^{{𝜏𝐵>𝑡,𝜏𝐶>𝑡}}=𝑠_𝑇0exp−^𝑡0𝜆^𝐴𝑠𝑑𝑠𝑑𝑡=𝑠_𝑇0𝑒^{−(𝑎0+𝑟)𝑡}𝐸^𝐴𝕝_{𝜏^𝐵_>𝑡,𝜏^𝐶_>𝑡}𝑑𝑡𝑒^{−(𝑎0+𝑏0+𝑐0+𝑟)𝑡}𝑑𝑡=𝑠1−𝑒^{−(𝑎0+𝑏0+𝑐0+𝑟)𝑇}𝑎₀+𝑏₀+𝑐₀+𝑟,(4.12) where the fourth equation is according to the two-firms model of Leung and Kwok in [18], 𝐸^𝐴𝕝_{𝜏^𝐵_>𝑡,𝜏^𝐶_>𝑡}=𝑒^{−(𝑏0+𝑐0)𝑡}.(4.13)
The right side of (4.10) equals 𝐸exp−^{𝜏𝐶0+𝜃}𝑟𝑑𝑡𝕝_{{𝜏𝐴>𝜏𝐶,𝜏𝐵>𝜏𝐶+𝜃,𝜏𝐶≤𝑇}}=_𝑇0^∞𝑡_3+𝜃^∞𝑡₃𝑒^{−(𝑡3+𝜃)𝑟}𝑓𝑡₁,𝑡₂,𝑡₃𝑑𝑡₁𝑑𝑡₂𝑑𝑡₃=^𝑇0^∞𝑡_3+𝜃^𝑡2_𝑡3𝑒^{−(𝑡3+𝜃)𝑟}𝑓₅𝑡₁,𝑡₂,𝑡₃𝑑𝑡₁𝑑𝑡₂𝑑𝑡₃+^𝑇0^∞𝑡_3+𝜃^∞𝑡₂𝑒^{−(𝑡3+𝜃)𝑟}𝑓₆𝑡₁,𝑡₂,𝑡₃𝑑𝑡₁𝑑𝑡₂𝑑𝑡₃=∶𝐼₁+𝐼₂,(4.14) where 𝐼₆=^𝑇0^∞𝑡_3+𝜃^𝑡2_𝑡3𝑒^{−(𝑡3+𝜃)𝑟}𝑓₅𝑡₁,𝑡₂,𝑡₃𝑑𝑡₁𝑑𝑡₂𝑑𝑡₃=𝑒^−𝑟𝜃^𝑇0^∞𝑡_3+𝜃^𝑡2_𝑡3𝑒^−𝑟𝑡3𝑐₀𝑎₀+𝑎₂𝑏₀+𝑏₃⋅𝑒^{−(𝑎0+𝑎2+𝑏2−𝑏3)𝑡1−(𝑏0+𝑏3)𝑡2−(𝑐0−𝑎2−𝑏2)𝑡3}𝑑𝑡₁𝑑𝑡₂𝑑𝑡₃=𝑐⁰𝑎₀+𝑎₂𝑏₀+𝑏₃𝑎₀+𝑎₂+𝑏₂−𝑏₃𝑒^−𝑟𝜃×^𝑇0^∞𝑡_3+𝜃𝑒^{−(𝑎0+𝑐0−𝑏3+𝑟)𝑡3}⋅𝑒^{−(𝑏0+𝑏3)𝑡2}−𝑒^{−(𝑐0−𝑎2−𝑏2+𝑟)𝑡3}⋅𝑒^{−(𝑎0+𝑎2+𝑏0+𝑏2)𝑡2}𝑑𝑡₂𝑑𝑡₃=𝑐⁰𝑎₀+𝑎₂𝑒^{−(𝑏0+𝑏3+𝑟)𝜃}𝑎₀+𝑎₂+𝑏₂−𝑏₃−𝑐⁰𝑎₀+𝑎₂𝑏₀+𝑏₃𝑒^{−(𝑎0+𝑎2+𝑏0+𝑏2+𝑟)𝜃}𝑎₀+𝑎₂+𝑏₂−𝑏₃𝑎₀+𝑎₂+𝑏₀+𝑏₂^𝑇0𝑒^{−(𝑎0+𝑏0+𝑐0+𝑟)}𝑑𝑡₃=𝑐⁰𝑎₀+𝑎₂𝑏₀+𝑏₃𝑒^−𝑟𝜃𝑎₀+𝑎₂+𝑏₂−𝑏₃𝑒𝑏₀+𝑏^{−(𝑏0+𝑏3)𝜃}₃−𝑒^{−(𝑎0+𝑎2+𝑏0+𝑏2)𝜃}𝐼𝑎₀+𝑎₂+𝑏₀+𝑏₂⋅1𝑎−⁰+𝑒𝑏₀+𝑐₀+𝑟^{−(𝑎0+𝑏0+𝑐0+𝑟)𝑇},(4.15)₇=^𝑇0^∞𝑡_3+𝜃^∞𝑡₂𝑒^{−(𝑡3+𝜃)𝑟}𝑓₆𝑡₁,𝑡₂,𝑡₃𝑑𝑡₁𝑑𝑡₂𝑑𝑡₃=𝑒^−𝑟𝜃^𝑇0^∞𝑡_3+𝜃𝑐₀𝑏₀+𝑏₂𝑒^{−(𝑐0−𝑎2−𝑏2+𝑟)𝑡3}𝑒^{−(𝑎0+𝑎2+𝑏0+𝑏2)𝑡2}𝑑𝑡₂𝑑𝑡₃=𝑐⁰𝑏₀+𝑏₂𝑒^{−(𝑎0+𝑎2+𝑏0+𝑏2+𝑟)𝜃}𝑎₀+𝑎₂+𝑏₀+𝑏₂^𝑇0𝑒^{−(𝑎0+𝑏0+𝑐0+𝑟)𝑡3}𝑑𝑡₃=𝑐⁰𝑏₀+𝑏₂𝑒^{−(𝑎0+𝑎2+𝑏0+𝑏2+𝑟)𝜃}𝑎₀+𝑎₂+𝑏₀+𝑏₂1−𝑒^{−(𝑎0+𝑏0+𝑐0+𝑟)𝑇}𝑎₀+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,…, 𝑇_𝑛 be the swap payment dates, where 0=𝑇₀<𝑇₁<⋯<𝑇_𝑛=𝑇. We assume that the payment dates are uniformly distributed; that is, 𝑇_𝑖+1−𝑇_𝑖=Δ𝑇 for 1≤𝑖≤𝑛−1 and 𝑛Δ𝑇=𝑇.

The value of the contingent leg at time 0 is given by 𝐶₁=𝑒^{−𝑟(𝜏𝐶+𝜃)}𝕝_{{𝜏𝐶≤𝑇}}𝕝_{{𝜏𝐴>𝜏𝐶}}𝕝_{{𝜏𝐵>𝜏𝐶+𝜃}}.(4.17)

The value of the fee leg at time 0 is given by 𝐹₁=^𝑛_𝑖=1𝑆(Δ𝑇)𝑒^{−𝑟𝑇𝑖}𝕝_{{𝜏𝐴∧𝜏𝐵∧𝜏𝐶>𝑇𝑖}}+𝑒^{−𝑟𝜏𝐶}𝜏^𝐶−𝑇_𝑖−1Δ𝑇𝕝^{{𝑇𝑖−1<𝜏𝐶<𝑇𝑖}}𝕝_{{𝜏𝐴∧𝜏𝐵>𝜏𝐶}}.(4.18) According to the arbitrage-free pricing principle, we have the following theorem.

Theorem 4.5. Let the intensity processes 𝜆^𝑖𝑡(𝑖=𝐴,𝐵,𝐶) be given by (2.14a)–(2.14c), the density function given by (4.1). Then, 𝐸𝐶₁=𝑐⁰𝑒^{−(𝑎0+𝑎2+𝑏0+𝑏2+𝑟)𝜃}𝑎₀+𝑎₂+𝑏₀+𝑏₂𝑏⁰+𝑏₂−𝑎⁰+𝑎₂𝑏₀+𝑏₃𝑎₀+𝑎₂+𝑏₂−𝑏₃+𝑐⁰𝑎₀+𝑎₂𝑒^{−(𝑏0+𝑏3+𝑟)𝜃}𝐸𝐹₁𝑎₀+𝑎₂+𝑏₂−𝑏₃1−𝛼𝑒^−𝛼𝑇,=S(Δ𝑇)𝑐⁰1−𝑒^{−𝛼Δ𝑇}−𝛼Δ𝑇𝑒^{−𝛼Δ𝑇}+𝛼²Δ𝑇𝛼²Δ𝑇11−−𝑒𝑒^{−𝛼𝑇−𝛼Δ𝑇},(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)^𝑛_𝑖=1=𝐸𝑒𝐸𝑒^{−𝑟𝑇𝑖}_{−𝑟(𝜏𝐶+𝜃)}𝕝_{{𝜏𝐴∧𝜏𝐵∧𝜏𝐶>𝑇𝑖}}+𝑆(Δ𝑇)𝐴(Δ𝑇)𝕝_{{𝜏𝐶≤𝑇}}𝕝_{{𝜏𝐴>𝜏𝐶}}𝕝_{{𝜏𝐵>𝜏𝐶+𝜃}},(4.20) where 𝐴(Δ𝑇)=^𝑛_𝑖=1𝐸𝑒^{−𝑟𝜏𝐶}𝜏^𝐶−𝑇_𝑖−1Δ𝑇𝕝^{{𝑇𝑖−1<𝜏𝐶<𝑇𝑖}}𝕝_{{𝜏𝐴∧𝜏𝐵>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 𝜏^𝐶 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: ^𝑛_𝑖=1𝐸𝑒^{−𝑟𝑇𝑖}𝕝_{{𝜏𝐴∧𝜏𝐵∧𝜏𝐶>𝑇𝑖}}=𝑒^{−𝛼Δ𝑇}1−𝑒^{−𝛼𝑛Δ𝑇}1−𝑒^{−𝛼Δ𝑇}=𝑒^{−𝛼Δ𝑇}1−𝑒^−𝛼𝑇1−𝑒^{−𝛼Δ𝑇}𝐸𝑒.(4.22)_{−𝑟𝜏𝐶}𝜏^𝐶−𝑇_𝑖−1Δ𝑇𝕝^{{𝑇𝑖−1<𝜏𝐶<𝑇𝑖}}𝕝_{{𝜏𝐴∧𝜏𝐵>𝜏𝐶}}=1Δ𝑇^𝑇𝑖_𝑇𝑖−1^∞𝑡₃^∞𝑡₃𝑒^−𝑟𝑡3𝑡₃−𝑇_𝑖−1𝑓𝑡₁,𝑡₂,𝑡₃𝑑𝑡₁𝑑𝑡₂𝑑𝑡₃=1Δ𝑇^𝑇𝑖_𝑇𝑖−1^∞𝑡₃^𝑡2_𝑡3𝑒^−𝑟𝑡3𝑡₃−𝑇_𝑖−1𝑓₅𝑡₁,𝑡₂,𝑡₃𝑑𝑡₁𝑑𝑡₂𝑑𝑡₃+^𝑇𝑖_𝑇𝑖−1^∞𝑡₃^∞𝑡₂𝑒^−𝑟𝑡3𝑡₃−𝑇_𝑖−1𝑓₆𝑡₁,𝑡₂,𝑡₃𝑑𝑡₁𝑑𝑡₂𝑑𝑡₃∶=1Δ𝑇𝐼⁸+𝐼₉,(4.23) where 𝐼₈=^𝑇𝑖_𝑇𝑖−1^∞𝑡₃^𝑡2_𝑡3⋅𝑒𝑒^−𝑟𝑡3𝑡₃−𝑇_𝑖−1𝑐₀𝑎₀+𝑎₂𝑏_{0−(𝑎0+𝑎2+𝑏2−𝑏3)𝑡1−(𝑏0+𝑏3)𝑡2−(𝑐0−𝑎2−𝑏2)𝑡3}+𝑏₃𝑑𝑡₁𝑑𝑡₂𝑑𝑡₃=𝑐⁰𝑎₀+𝑎₂𝑎₀+𝑎₂+𝑏₀+𝑏₂𝛼1𝑒^{−𝛼𝑇𝑖}𝑇_𝑖−1−𝑇_𝑖+1𝛼𝑒^{−𝛼𝑇𝑖−1}𝐼−𝑒^{−𝛼𝑇𝑖}₉,=^𝑇𝑖_𝑇𝑖−1^∞𝑡₃^∞𝑡₂⋅𝑒𝑒^−𝑟𝑡3𝑡₃−𝑇_𝑖−1𝑐₀𝑎₀+𝑎₃𝑏_{0−(𝑎0+𝑎3)𝑡1−(𝑏0+𝑏2+𝑎2−𝑎3)𝑡2−(𝑐0−𝑎2−𝑏2)𝑡3}+𝑏₂𝑑𝑡₁𝑑𝑡₂𝑑𝑡₃=𝑐⁰𝑏₀+𝑏₂𝑎₀+𝑎₂+𝑏₀+𝑏₂𝛼1𝑒^{−𝛼𝑇𝑖}𝑇_𝑖−1−𝑇_𝑖+1𝛼𝑒^{−𝛼𝑇𝑖−1}−𝑒^{−𝛼𝑇𝑖}.(4.24)
We can obtain 𝐴(Δ𝑇)=^𝑛_𝑖=1𝐸𝑒^{−𝑟𝜏𝐶}𝜏^𝐶−𝑇_𝑖−1Δ𝑇𝕝^{{𝑇𝑖−1<𝜏𝐶<𝑇𝑖}}𝕝_{{𝜏𝐴∧𝜏𝐵>𝜏𝐶}}=1Δ𝑇^𝑛_𝑖=1𝐼₈=𝑐₀+𝐼₉1−𝑒^{−𝛼Δ𝑇}−𝛼Δ𝑇𝑒^{−𝛼Δ𝑇}𝛼²Δ𝑇11−−𝑒𝑒^{−𝛼𝑇−𝛼Δ𝑇}.(4.25) The right-hand side of (4.20) is 𝐸=𝑒𝑐^{−𝑟(𝜏𝐶+𝜃)}₀𝕝_{{𝜏𝐶≤𝑇}}𝕝_{{𝜏𝐴>𝜏𝐶}}𝕝_{{𝜏𝐵>𝜏𝐶+𝜃}}𝑒^{−(𝑎0+𝑎2+𝑏0+𝑏2+𝑟)𝜃}𝑎₀+𝑎₂+𝑏₀+𝑏₂𝑏⁰+𝑏₂−𝑎⁰+𝑎₂𝑏₀+𝑏₃𝑎₀+𝑎₂+𝑏₂−𝑏₃+𝑐⁰𝑎₀+𝑎₂𝑒^{−(𝑏0+𝑏3+𝑟)𝜃}𝑎₀+𝑎₂+𝑏₂−𝑏₃1−𝛼𝑒^−𝛼𝑇.(4.26) From (4.22)–(4.26), we can obtain the expression of 𝑆(Δ𝑇) from the following equation: 𝑆(Δ𝑇)𝑒^{−𝛼Δ𝑇}+𝑐⁰1−𝑒^{−𝛼Δ𝑇}−𝛼Δ𝑇𝑒^{−𝛼Δ𝑇}𝛼²Δ𝑇𝑒^{−𝛼Δ𝑇}1−𝑒^−𝛼𝑇1−𝑒^{−𝛼Δ𝑇}=𝑐⁰𝑒^{−(𝑎0+𝑎2+𝑏0+𝑏2+𝑟)𝜃}𝑎₀+𝑎₂+𝑏₀+𝑏₂𝑏⁰+𝑏₂−𝑎⁰+𝑎₂𝑏₀+𝑏₃𝑎₀+𝑎₂+𝑏₂−𝑏₃+𝑐⁰𝑎₀+𝑎₂𝑒^{−(𝑏0+𝑏3+𝑟)𝜃}𝑎₀+𝑎₂+𝑏₂−𝑏₃⋅1−𝛼𝑒^−𝛼𝑇.(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.

Acknowledgment

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