Total Return Swap Valuation with Counterparty Risk and Interest Rate Risk

Abstract

We study the pricing of total return swap (TRS) under the contagion models with counterparty risk and the interest rate risk. We assume that interest rate follows Heath-Jarrow-Morton (HJM) forward interest rate model and obtain the Libor market interest rate. The cases where default is related to the interest rate and independent of interest rate are considered. Using the methods of change of measure and the “total hazard construction,” the joint default probabilities are obtained. Furthermore, we obtain the closed-form formulas of TRS under different contagion models, respectively.

1. Introduction

Total return swap (TRS), as a type of credit derivatives and a financing and leverage tool, is an important off-balance sheet tool, particularly for hedge funds and for banks seeking additional fee income. The corporate bonds and their credit derivatives are typically financial tools in the markets which undertake and avoid the credit risk of the companies. Therefore, the key of management of credit risk is the fair pricing of credit derivatives. Especially after 2007 financing crisis, the contagion effect of credit risk has attracted huge attention of financial market regulators and financial institutions. To price credit derivatives, default contagion models are developed rapidly in order to make them more realistic after the subprime mortgage crisis.

The approaches of modeling the pricing of credit derivatives are mainly the value-of-the-firm (or structural) approach and the intensity-based approach (or reduced form 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.

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 where 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 given by 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]. Jamshidian [17] develops change of numeraire methodology for reduced-form models.

A TRS is a bilateral financial contract between a total return payer and a total return receiver. One party (the total return payer) pays the total return of a reference security (or reference securities) and receives a form of payment from the other party (the receiver of the total rate of return). Often payment is a floating rate payment, a spread to LIBOR. The reference assets can be indices, bonds (emerging market, sovereign, bank debt, mortgage-backed securities, or corporate), loans (term or revolver), equities, real estate receivables, lease receivables, or commodities. Ye and Zhuang [18] consider the pricing of TRS only when reference asset defaults; they obtain TRS pricing formula when default is independent of the interest rate. When default is related to the interest rate, using the hybrid model given in Das and Sundaram [19], they model default time and the interest rate and give the Monte Carlo simulation result.

In this paper, we consider the counterparty default contagion risk between total return receiver (firm 𝐵) and the reference asset (firm 𝐶) and study two-firm contagion models. The cash flow of a TRS in this model is provided by Figure 1.

Figure 1 

The structure of TRS.

Suppose that firm 𝐴 (a corporate bond investing firm, credit protection buyer) holds a corporate bond (reference asset) issued by firm 𝐶 (a corporate bond issuer) (refer to 2A in Figure 1), and firm 𝐶 is subject to default. At bond maturity, if firm 𝐶 does not default, it will pay the bond principle and interest to firm 𝐴 (refer to 2B). On the other hand, to hedge the default risk of firm 𝐶, firm 𝐴 and firm 𝐵 (credit protection seller, subject to default also) enter into a TRS contract. If firm 𝐶 has no default, firm 𝐴 will make its total return to firm 𝐵 (refer to 2C), and, in exchange, firm 𝐵 gives the Libor plus a spread 𝑠 to 𝐴 (refer to 2D). Firm 𝐵 promises to compensate 𝐴 for its loss in the event of default of firm 𝐶 (refer to 2E).

The structure of this paper is organized as follows. In Section 2, we give the basic setup and HJM forward interest rate model. In Section 3, we study the two-firm contagion models when default is independent of stochastic interest rate and obtain the closed-form formulas of TRS. In Section 4, we consider the case that default is related to interest rate. Using the “total hazard” approach, joint survival probabilities are derived and analytic formulas are obtained under two-firm contagion models. Section 5 is the conclusion.

2. Basic Setup and HJM Forward Interest Rate Model

We consider a filtered probability space (Ω,𝓕,{ℱ_𝑡}^𝑇∗_𝑡=0,𝑃) which is an uncertain economy with a time horizon of 𝑇^∗, 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 martingales. We assume the existence and uniqueness of 𝑃 so that bond markets are complete and there is no arbitrage, as shown in discrete time case by Harrison and Kreps [20] and in continuous time case by Harrison and Pliska [21]. Subsequent specifications of the model are all under the equivalent martingale measure (or risk neutral measure) 𝑃.

On this probability space there is an ℝ^𝑑-valued process 𝑋_𝑡, which presents 𝑑 dimensional economy-wide state variables. In this paper, we consider the only one state variable which is the interest rate denoted by 𝑟_𝑡. There are also two point processes, 𝑁^𝑖(𝑖=𝐵,𝐶), initialized at 0, representing the default processes of the firm 𝐵 and firm 𝐶, respectively, such that the default of the firm 𝑖 occurs when 𝑁^𝑖 jumps from 0 to 1.

According to the information contained in the state variables and the default processes, the enlarged filtration is defined by 𝓕_𝑡=ℱ^𝑟_𝑡∨ℱ^𝐵_𝑡∨ℱ^𝐶_𝑡, (1) where ℱ^𝑟_𝑡=𝜎𝑟_𝑠,ℱ,0≤𝑠≤𝑡_𝑖𝑡=𝜎𝑁^𝑖_𝑠,0≤𝑠≤𝑡,𝑖=𝐵,𝐶, (2) are the filtrations generated by 𝑟_𝑡 and 𝑁^𝑖_𝑡(𝑖=𝐵,𝐶), respectively.

Let 𝓖^𝐵_𝑡=ℱ^𝐵_𝑡∨ℱ^𝑟_𝑇^∗∨ℱ^𝐶_𝑇^∗=ℱ^𝐵_𝑡∨𝓖₀^−𝐵,𝓖^𝐶_𝑡=ℱ^𝐶_𝑡∨ℱ^𝑟_𝑇^∗∨ℱ^𝐵_𝑇^∗=ℱ^𝐶_𝑡∨𝓖₀^−𝐶, (3) where 𝓖₀^−𝐵=ℱ^𝑟_𝑇^∗∨ℱ^𝐶_𝑇^∗,𝓖₀^−𝐶=ℱ^𝑟_𝑇^∗∨ℱ^𝐵_𝑇^∗. (4)𝓖₀^−𝑖(𝑖=𝐵,𝐶) contains complete information on the state variables and the default processes of all firms up to time 𝑇^∗ other than that of the 𝑖th firm.

Let 𝜏^𝑖 denote the default time of firm 𝑖; namely, 𝜏^𝑖 is the first jump time of 𝑁^𝑖, which can be defined as 𝜏^𝑖=inf𝑡∶^𝑡₀𝜆^𝑖_𝑠𝑑𝑠≥𝐸^𝑖, (5) where {𝐸^𝑖}(𝑖=𝐵,𝐶) is independent of 𝑟_𝑡(𝑡∈[0,𝑇^∗]).

According to the Doob-Meyer decomposition, we have that 𝑀^𝑖_𝑡=𝑁_𝑡−^{𝑡∧𝜏𝑖}₀𝜆^𝑖_𝑠𝑑𝑠 (6) is a (𝑃,𝓕_𝑡)-martingale.

Due to the impact of counterparty risk, the default intensities 𝜆^𝐵_𝑡 and 𝜆^𝐶_𝑡 of firms 𝐵 and 𝐶 are no longer independent under the condition ℱ^𝑟_𝑡. The conditional survival probability and the unconditional survival probability of firm 𝑖 are given by 𝑃𝜏^𝑖>𝑡∣𝓖₀^−𝑖=exp−^𝑡₀𝜆^𝑗_𝑠𝑑𝑠,𝑡∈[0,𝑇],𝑃𝜏^𝑖>𝑡=𝐸exp−^𝑡₀𝜆^𝑖_𝑠𝑑𝑠,𝑡∈[0,𝑇], (7)(8) respectively.

As a type of market risk, interest rate risk analysis is almost always based on simulating movements in one or more yield curves using the Heath-Jarrow-Morton (HJM) framework to ensure that the yield curve movements are both consistent with current market yield curves and such that no riskless arbitrage is possible. HJM model was developed in early 1992 by Heath et al. [22]. In this paper, we assume the default free interest rate is stochastic forward interest rate 𝑓(𝑡,𝑇), which follows the HJM model 𝑑𝑓(𝑡,𝑇)=𝛼(𝑡,𝑇)𝑑𝑡+𝜎(𝑡,𝑇)𝑑𝑊_𝑡,0≤𝑡≤𝑇, (9) where 𝛼(𝑡,𝑇) is the drift term and 𝜎(𝑡,𝑇) is the volatility term, and they are both stochastic. For every fixed time 𝑇, they are adapted processes to 𝓕_𝑡.

The arbitrage free condition [23] shows that the Brown movement under each real probability measure can be transformed into the Brown movement under the risk neutral measure. For simplification, in this paper, we assume the Brown movement 𝑊_𝑡 is standard under the risk neutral measure 𝑃, where 𝛼(𝑡,𝑇) satisfies 𝛼(𝑡,𝑇)=𝜎(𝑡,𝑇)𝜎^∗(𝑡,𝑇)=𝜎(𝑡,𝑇)^𝑇_𝑡𝜎(𝑡,𝑣)𝑑𝑣. (10)

The solution of (9) is 𝑓(𝑡,𝑇)=𝑓(0,𝑇)+^𝑡₀𝛼(𝑢,𝑇)𝑑𝑢+^𝑡₀𝜎(𝑢,𝑇)𝑑𝑊_𝑢, (11) where 𝑓(0,𝑇) is the initial forward interest rate curve, which is known.

At time 𝑡 the instantaneous interest rate 𝑟_𝑡=𝑓(𝑡,𝑡). 𝐷(𝑡)=exp(−∫^𝑡₀𝑟_𝑢𝑑𝑢) is the discounted process. Let 𝐵(𝑡,𝑇) be the time-𝑡 value of a zero-coupon bond with face value 1 at maturity 𝑇; we have 𝐵(𝑡,𝑇)=exp−^𝑇_𝑡𝑓(𝑡,𝑣)𝑑𝑣,0≤𝑡≤𝑇≤𝑇. (12) Under the risk neutral measure 𝑃, 𝐵(𝑡,𝑇) satisfies the following differential equation: 𝑑𝐵(𝑡,𝑇)=𝑟_𝑡𝐵(𝑡,𝑇)𝑑𝑡−𝜎^∗(𝑡,𝑇)𝐵(𝑡,𝑇)𝑑𝑊_𝑡. (13)

Denote 𝐿(𝑡,𝑇) to be the time-𝑡 locked investment yield curve from 𝑇 to 𝑇+𝛿; then, 𝐿()=𝐵()()𝑡,𝑇𝑡,𝑇−𝐵𝑡,𝑇+𝛿().𝛿𝐵𝑡,𝑇+𝛿 (14) For 0≤𝑡<𝑇, 𝐿(𝑡,𝑇) is called forward Libor interest rate, and, for 𝑡=𝑇, 𝐿(𝑡,𝑇) is called spot Libor interest rate. 𝛿 is the duration of Libor, normally 0.25 year or half year.

In the next two sections of this paper, we discuss the pricing of TRS with default risk and interest rate risk under different contagion models.

3. TRS Valuation When Default Is Independent of the Interest Rate

Assume that reference asset 𝐶 is a defaultable coupon bond with face value 1 and the same maturity 𝑇 with TRS contract. 𝑇₀,𝑇₁,𝑇₂,…,𝑇_𝑛 are bond interest payment dates, where 0=𝑇₀<𝑇₁<⋯<𝑇_𝑛=𝑇, for 0≤𝑖≤𝑛−1, 𝑇_𝑖+1−𝑇_𝑖=Δ𝑇, 𝑛Δ𝑇=𝑇.

Denote 𝐶_𝑖 to be the time-𝑇_𝑖 cash flow of TRS payer, where 𝐶₁,…,𝐶_𝑛−1 are the interest payments at time 𝑇_𝑖(𝑖=1,…,𝑛−1) and 𝐶_𝑛 is the sum of interest payment and value-added of the bond at time 𝑇_𝑛=𝑇, which are all determined at 𝑇₀=0. Let 𝑀 be notional principal, and 𝛿 is the maturity of Libor interest rate, the same with bond’s payment cycle Δ𝑇, namely, 𝛿=Δ𝑇. For simplification, the default recovery rate of reference asset 𝐶 is 0.

At time 0, for TRS payment leg, the discounted expectation of the cash flow 𝐹_𝐵 is 𝐸𝐹_𝐵=𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝐶_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}. (15)

At time 0, for TRS recipient leg, the discounted expectation of the cash flow 𝐹_𝐶 is 𝐸𝐹_𝐶=𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝛿𝑀𝐿𝑇_𝑖−1,𝑇_𝑖−1+𝑠1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}+𝐸𝐷𝜏^𝐶+𝜃1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶+𝜃}}, (16) where 𝐸 is the expectation under risk neutral measure 𝑃, 𝜃 is the length of settlement period, and 𝜏^𝐶+𝜃 is settlement date. 𝐿(𝑇_𝑖−1,𝑇_𝑖−1) is time-𝑇_𝑖−1 locked 𝑇_𝑖-Libor interest rate. 𝑠 is the TRS spread and 𝐷(⋅) is the discounted factor.

We assume the market is complete; namely, there is no arbitrage. Therefore, according to the arbitrage-free principle, we have 𝐸𝐹_𝐵=𝐸𝐹_𝐶. (17) The general pricing formula of TRS is derived as 𝑠=𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝐶_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}−𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝛿𝑀𝐿𝑇_𝑖−1,𝑇_𝑖−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}×𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝛿𝑀1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}⁻¹−𝐸𝐷𝜏^𝐶+𝜃1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶+𝜃}}𝐸∑^𝑛_𝑖=1𝐷𝑇_𝑖𝛿𝑀1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}. (18)

3.1. TRS Valuation under Two-Firm Looping Default Contagion Model

In this subsection we assume that the default intensities of credit protection seller 𝐵 and reference asset 𝐶 are given, respectively, by 𝜆^𝐵_𝑡=𝑏₀+𝑏₁1_{{𝜏𝐶≤𝑡}},𝜆^𝐶_𝑡=𝑐₀+𝑐₁1_{{𝜏𝐵≤𝑡}}, (19)(20) where 𝑏₀ and 𝑏₀ are nonnegative, satisfying 𝑏₀+𝑏₁>0 and 𝑐₀+𝑐₁>0.

With the result in Leung and Kwok [24], the joint survival probability and the joint density of default time (𝜏^𝐵,𝜏^𝐶) are given by 𝑃𝜏^𝐵>𝑡₁,𝜏^𝐶>𝑡₂=𝑒^{−𝑏0𝑡1−𝑐0𝑡2}𝑐₀𝑒^{−𝑏1(𝑡1−𝑡2)}−𝑏₁𝑒^{−𝑐0(𝑡1−𝑡2)}𝑐₀−𝑏₁,𝑡₂≤𝑡₁𝑒^{−𝑏0𝑡1−𝑐0𝑡2}𝑐₀𝑒^{−𝑐1(𝑡2−𝑡1)}−𝑐₁𝑒^{−𝑏0(𝑡2−𝑡1)}𝑏₀−𝑐₁,𝑡₂>𝑡₁,𝑓𝑡₁,𝑡₂=𝑓₁𝑡₁,𝑡₂=𝑐₀𝑏₀+𝑏₁𝑒^{−(𝑏0+𝑏1)𝑡1−(𝑐0−𝑏1)𝑡2},𝑡₂≤𝑡₁𝑓₂𝑡₁,𝑡₂=𝑏₀𝑐₀+𝑐₁𝑒^{−(𝑐0+𝑐1)𝑡2−(𝑏0−𝑐1)𝑡1},𝑡₂>𝑡₁, (21)(22) respectively.

When the default process and the interest rate process are independent, by computing formula (18), we can obtain the following result.

Theorem 1. With the default intensity process (19)-(20) and HJM interest rate model (9), the price 𝑠 of TRS is given by ∑𝑠=^𝑛𝑖=1𝐶_𝑖−𝛿𝑀𝐿𝑇_𝑖−1,𝑇_𝑖−1𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖}∑^𝑛_𝑖=1𝛿𝑀𝑒^{−(𝑏0+𝑐0)𝑇𝑖}−𝑐₀𝑒^{−(𝑏0+𝑏1)𝜃}∫^𝑇₀𝐵(0,𝑣+𝜃)𝑒^{−(𝑏0+𝑐0)𝑣}𝑑𝑣∑^𝑛_𝑖=1𝛿𝑀𝑒^{−(𝑏0+𝑐0)𝑇𝑖}. (23)

Proof. By the arbitrage free principle, the price formula (18) becomes 𝑠=𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝐶_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}−𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝛿𝑀𝐿𝑇_𝑖−1,𝑇_𝑖−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}×𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝛿𝑀1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}⁻¹−𝐸𝐷𝜏^𝐶+𝜃1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶+𝜃}}𝐸∑^𝑛_𝑖=1𝐷𝑇_𝑖𝑀1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}. (24) To obtain the analytic solution of 𝑠, we need to compute the following three expectation values: 𝐾₁∶=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}},𝐾₂∶=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖𝐿𝑇_𝑖−1,𝑇_𝑖−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}},𝐾₃∶=𝐸𝐷𝜏^𝐶+𝜃1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶+𝜃}}, (25) where 𝐾₁=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}=^𝑛_𝑖=1𝐸exp−^𝑇𝑖₀𝑟_𝑢𝑑𝑢𝐸1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}=^𝑛_𝑖=1exp−^𝑇𝑖₀𝑓(0,𝑢)𝑑𝑢𝐸1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}=^𝑛_𝑖=1𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖},𝐾₂=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖𝐿𝑇_𝑖−1,𝑇_𝑖−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}=^𝑛_𝑖=1𝐵0,𝑇_𝑖𝐿𝑇_𝑖−1,𝑇_𝑖−1𝑒^{−(𝑏0+𝑐0)𝑇𝑖},𝐾₃=𝐸𝐷𝜏^𝐶+𝜃1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶+𝜃}}=𝐸𝐸𝐷𝜏^𝐶+𝜃1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶+𝜃}}∣ℱ^𝑟_𝑇^∗=𝐸𝐵0,𝜏^𝐶+𝜃1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶+𝜃}}=^𝑇₀^∞_𝑣+𝜃𝐵(0,𝑣+𝜃)𝑓₂()=𝑢,𝑣𝑑𝑢𝑑𝑣_𝑇0𝐵(0,𝑣+𝜃)^∞_𝑣+𝜃𝑐₀𝑏₀+𝑏₁𝑒^{−(𝑏0+𝑏1)𝑡1−(𝑐0−𝑏1)𝑡2}𝑑𝑢𝑑𝑣=𝑐₀𝑒^{−(𝑏0+𝑏1)𝜃𝑇}₀𝐵(0,𝑣+𝜃)𝑒^{−(𝑏0+𝑐0)𝑣}𝑑𝑣. (26) Substitute 𝐾₁, 𝐾₂, and 𝐾₃ into formula (24), and we obtain ∑𝑠=^𝑛𝑖=1𝐶_𝑖−𝛿𝑀𝐿𝑇_𝑖−1,𝑇_𝑖−1𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖}∑^𝑛_𝑖=1𝛿𝑀𝑒^{−(𝑏0+𝑐0)𝑇𝑖}−𝑐₀𝑒^{−(𝑏0+𝑏1)𝜃}∫^𝑇₀𝐵(0,𝑣+𝜃)𝑒^{−(𝑏0+𝑐0)𝑣}𝑑𝑣∑^𝑛_𝑖=1𝛿𝑀𝑒^{−(𝑏0+𝑐0)𝑇𝑖}. (27) We complete the proof.

Remark 2. From (23), we can conclude that when the length 𝜃 of settlement period is zero, the swap rate 𝑠 is only related to the systematic factors 𝑏₀ and 𝑐₀, irrelative to the default contagion between two firms.

3.2. TRS Valuation under Two-Firm Attenuation Contagion Model

In this subsection, we consider the default contagion to have the hyperbolic attenuation effect. Default intensities of firms 𝐵 and 𝐶 have the following forms: 𝜆^𝐵_𝑡=𝑏₀+𝑏₂𝑏₁𝑡−𝜏^𝐶1+1^{{𝜏𝐶≤𝑡}},𝜆^𝐶_𝑡=𝑐₀+𝑐₂𝑐₁𝑡−𝜏^𝐵1+1^{{𝜏𝐵≤𝑡}}, (28)(29) where 𝑏₀,𝑐₀, 𝑏₁, and 𝑐₁ are nonnegative, satisfying 𝑏₀+𝑏₂>0 and 𝑐₀+𝑐₂>0. 𝑏₂ and 𝑐₂ reflect the impact strength of the counterparty and 𝑏₁ and 𝑐₁ show the attenuation speed of one party default on the other party. For 𝑏₁=𝑐₁=0, the model becomes the looping default model (19)-(20).

Under the above model, the analytic solutions of the joint survival probability cannot be obtained. For simplification, we assume 𝑏₂=−𝑏₁; then, (28) and (29) become, respectively, 𝜆^𝐵_𝑡=𝑏₀−𝑏₁𝑏₁𝑡−𝜏^𝐶1+1^{{𝜏𝐶≤𝑡}},𝜆^𝐶_𝑡=𝑐₀−𝑐₁𝑐₁𝑡−𝜏^𝐵1+1^{{𝜏𝐵≤𝑡}}. (30)(31)

According to the result in Bai et al. [25], with the intensities (30)-(31), the joint survival probability and the joint density function of (𝜏^𝐵,𝜏^𝐶) are given by 𝑃𝜏^𝐵>𝑡₁,𝜏^𝐶>𝑡₂=𝑏₁𝑡₁−𝑡₂+1𝑏₁−1𝑐₀𝑒^{−(𝑏0𝑡1+𝑐0𝑡2)}+𝑏₁𝑐₀𝑒^{−(𝑏0+𝑐0)𝑡1},𝑡₂≤𝑡₁𝑐₁𝑡₂−𝑡₁+1𝑐₁−1𝑏₀𝑒^{−(𝑏0𝑡1+𝑐0𝑡2)}+𝑐₁𝑏₀𝑒^{−(𝑏0+𝑐0)𝑡2},𝑡₂>𝑡₁,𝑓𝑡₁,𝑡₂=𝑓₁𝑡₁,𝑡₂=𝑐₀𝑏₀𝑏₁𝑡₁−𝑡₂+1𝑏₁−1𝑏₀×𝑒^{−(𝑏0𝑡1+𝑐0𝑡2)},t₂≤𝑡₁𝑓₂𝑡₁,𝑡₂=𝑏₀𝑐₀𝑐₁𝑡₂−𝑡₁+1𝑐₁−1𝑐₀×𝑒^{−(𝑏0𝑡1+𝑐0𝑡2)},𝑡₂>𝑡₁, (32)(33) respectively.

Then the expression of 𝑠 is given by the following theorem.

Theorem 3. With the default intensities (30)-(31) and HJM interest rate model (9), 𝑠 is given by ∑𝑠=^𝑛𝑖=1𝐶_𝑖𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖}∑^𝑛_𝑖=1𝛿𝑀𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖}−∑^𝑛_𝑖=1𝐿𝑇_𝑖−1,𝑇_𝑖−1𝛿𝑀𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖}∑^𝑛_𝑖=1𝛿𝑀𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖}−𝑐₀1+𝑏₁𝜃𝑒^−𝑏0𝜃∫^𝑇₀𝐵(0,𝑣+𝜃)𝑒^{−(𝑏0+𝑐0)𝑣}𝑑𝑣∑^𝑛_𝑖=1𝛿𝑀𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖}. (34)

Proof. To compute (18), we need to only compute the following three expectation values: 𝐾₁∶=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}},𝐾₂∶=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖𝐿𝑇_𝑖−1,𝑇_𝑖−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}},𝐾₃∶=𝐸𝐷𝜏^𝐶+𝛿1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶+𝛿}}, (35) where 𝐾₁=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}=^𝑛_𝑖=1𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖},𝐾₂=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖𝐿𝑇_𝑖−1,𝑇_𝑖−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}=^𝑛_𝑖=1𝐿𝑇_𝑖−1,𝑇_𝑖−1𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖},𝐾₃=𝐸𝐷𝜏^𝐶+𝛿1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶+𝛿}}=𝐸𝐸𝐷𝜏^𝐶+𝛿1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶+𝛿}}∣ℱ^𝑟_𝑇^∗=𝐸𝐵0,𝜏^𝐶+𝜃1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶+𝜃}}=^𝑇₀^∞_𝑣+𝜃𝐵()𝑓()=0,𝑣+𝜃𝑢,𝑣𝑑𝑢𝑑𝑣_𝑇0𝐵()×0,𝑣+𝜃_∞𝑣+𝜃𝑐₀𝑏₀𝑏₁𝑡₁−𝑡₂+1𝑏₁−1𝑏₀𝑒^{−(𝑏0𝑡1+𝑐0𝑡2)}𝑑𝑢𝑑𝑣=𝑐₀1+𝑏₁𝜃𝑒^{−𝑏0𝜃𝑇}₀𝐵(0,𝑣+𝜃)𝑒^{−(𝑏0+𝑐0)𝑣}𝑑𝑣. (36) Substituting (36) into (18), we can have ∑𝑠=^𝑛𝑖=1𝐶_𝑖𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖}∑^𝑛_𝑖=1𝛿𝑀𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖}−∑^𝑛_𝑖=1𝐿𝑇_𝑖−1,𝑇_𝑖−1𝛿𝑀𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖}∑^𝑛_𝑖=1𝛿𝑀𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖}−𝑐₀1+𝑏₁𝜃𝑒^−𝑏0𝜃∫^𝑇₀𝐵(0,𝑣+𝜃)𝑒^{−(𝑏0+𝑐0)𝑣}𝑑𝑣∑^𝑛_i=1𝛿𝑀𝐵0,𝑇_𝑖𝑒^{−(𝑏0+𝑐0)𝑇𝑖}, (37) which is formula (34). The proof is complete.

Remark 4. Combining formula (34) with formula (23), we can conclude that firm 𝐶’s default contagion on firm 𝐵 has effect in the pricing of TRS (there is 𝑏₁ in formula (34)). However, firm 𝐵’s default contagion on firm 𝐶 has no effect in the pricing of TRS (there is no 𝑐₁ in (34)). Therefore, in the complex contagion model, we can assume that the reference asset is the primary firm, and the protection seller is the secondary firm.

4. TRS Valuation When Default Is Related to the Interest Rate

Empirical studies show that in the vast majority of cases default is related to the interest rate or, in other words, credit risk is related to interest rate risk. In this section, we assume that their correlation is described by default intensities. For simplification, we assume that the length 𝜃 of the reference asset’s settlement period is zero.

4.1. TRS Valuation under Two-Firm Looping Default Contagion Model

We consider that default intensities of firms 𝐵 and 𝐶 have the following form: 𝜆^𝐵_𝑡=𝑏₀+𝑏𝑟_𝑡+𝑏₁1_{{𝜏𝐶≤𝑡}},𝜆^𝐶_𝑡=𝑐₀+𝑐𝑟_𝑡+𝑐₁1_{{𝜏𝐵≤𝑡}}, (38) where 𝑏₀, 𝑏₀,𝑏, and 𝑐 are nonnegative, satisfying 𝑏₀+𝑏₁>0 and 𝑐₀+𝑐₁>0. 𝑟_𝑡=𝑓(𝑡,𝑡)and𝑓(𝑡,𝑇) satisfies HJM model (9).

Using Wang and Ye’s result [26], under the model (38), the joint conditional distribution of (𝜏^𝐵,𝜏^𝐶) is given by 𝑃𝜏^𝐵>𝑡₁,𝜏^𝐶>𝑡₂∣ℱ^𝑟_𝑇^∗=exp−𝑐₀𝑡₂−𝑐𝑅_0,𝑡2×exp−𝑐₁𝑡₂−𝑡₁−exp−𝑏₀𝑡₂−𝑡₁−𝑏𝑅_𝑡1,𝑡2+exp−𝑏₀𝑡₂−𝑏𝑅_0,𝑡2+𝑐₁^𝑡2_𝑡1exp−𝑏₀𝑢−𝑡₁−𝑐₁𝑡₂−𝑢−𝑏𝑅_𝑡1,𝑢𝑑𝑢,for𝑡₁<𝑡₂𝑃𝜏<𝑇._𝐵>𝑡₁,𝜏^𝐶>𝑡₂∣ℱ^𝑟_𝑇^∗=exp−𝑏₀𝑡₁−𝑏𝑅_0,𝑡1×exp−𝑏₁𝑡₁−𝑡₂−exp−𝑐₀𝑡₁−𝑡₂−𝑐𝑅_𝑡2,𝑡1+exp−𝑐₀𝑡₁−𝑐𝑅_0,𝑡1+𝑏₁^𝑡1_𝑡2exp−𝑐₀𝑢−𝑡₂−𝑏₁𝑡₁−𝑢−𝑐𝑅_𝑡2,𝑢𝑑𝑢,for𝑡₂<𝑡₁<𝑇. (39)

The joint conditional density function is 𝑓𝑡₁,𝑡₂∣ℱ^𝑟_𝑇^∗=𝑐₀+𝑐𝑟_𝑡2𝑏₀+𝑏₁+𝑏𝑟_𝑡1×𝑒^{−(𝑏0+𝑏1)𝑡1−(𝑐0−𝑏1)𝑡2−𝑏𝑅10,𝑡−𝑐𝑅20,𝑡},𝑡₂≤𝑡₁𝑏₀+𝑏𝑟_𝑡1𝑐₀+𝑐₁+𝑐𝑟_𝑡2×𝑒^{−(𝑏0−𝑐1)𝑡1−(𝑐0+𝑐1)𝑡2−𝑏𝑅10,𝑡−𝑐𝑅20,𝑡},𝑡₂>𝑡₁, (40) where 𝑅_0,𝑢=∫^𝑢₀𝑟_𝑣𝑑𝑣 is the cumulative interest rate process and ℱ^𝑟_𝑇^∗ is the filter generated by 𝑟_𝑡 up to 𝑇^∗.

When default is related to interest rate, computing expectations in formula (18) becomes complicated. We give the following lemma first.

Lemma 5. The interest rate process is given by (9), 𝑅_𝑡,𝑠=∫^𝑠_𝑡𝑟_𝑢𝑑𝑢, for 𝑡<𝑠. Denote 𝐺₁𝑚₁;𝑡₀,𝑡₁=𝐸exp−𝑚₁𝑅_𝑡0,𝑡1,𝐺₂𝑚₁,𝑚₂;𝑡₀,𝑡₁,𝑡₂=𝐸exp−𝑚₁𝑅_𝑡0,𝑡1−𝑚₂𝑅_𝑡1,𝑡2; (41) then, 𝐺₁𝑚₁;𝑡₀,𝑡₁=exp−𝑚₁^𝑡1_𝑡0𝑓𝑡₀,𝑢𝑑𝑢−𝑚₁^𝑡1_𝑡0^𝑢_𝑡0𝛼()+𝑚𝑠,𝑢𝑑𝑠𝑑𝑢₂₁2^𝑡1_𝑡0^𝑡1_𝑠𝜎(𝑠,𝑢)𝑑𝑢²𝑑𝑠,𝐺₂𝑚₁,𝑚₂;𝑡₀,𝑡₁,𝑡₂=exp−𝑚₁^𝑡1_𝑡0𝑓𝑡₀,𝑢𝑑𝑢−𝑚₁^𝑡1_𝑡0^𝑢_𝑡0𝛼(𝑠,𝑢)𝑑𝑠𝑑𝑢⋅exp−𝑚₂^𝑡2_𝑡1𝑓𝑡₁,𝑢𝑑𝑢−𝑚₂^𝑡2_𝑡1^𝑢_𝑡1𝛼()𝑚𝑠,𝑢𝑑𝑠𝑑𝑢⋅exp²¹2^𝑡1_𝑡0^𝑡1_𝑠𝜎(𝑠,𝑢)𝑑𝑢²+𝑚𝑑𝑠₂₂2^𝑡2_𝑡1^𝑡2_𝑠𝜎(𝑠,𝑢)𝑑𝑢²𝑑𝑠. (42)(43)

Proof. Consider 𝑅_𝑡0,𝑡1=^𝑡1_𝑡0𝑓𝑡₀,𝑢𝑑𝑢+^𝑡1_𝑡0^𝑢_𝑡0𝛼()+𝑠,𝑢𝑑𝑠𝑑𝑢_{𝑡1𝑡0𝑢𝑡0}𝜎(𝑠,𝑢)𝑑𝑊_𝑠=𝑑𝑢_𝑡1𝑡0𝑓𝑡₀,𝑢𝑑𝑢+^𝑡1_𝑡0^𝑢_𝑡0𝛼()+𝑠,𝑢𝑑𝑠𝑑𝑢_{𝑡1𝑡0𝑡1𝑠}𝜎(𝑠,𝑢)𝑑𝑢𝑑𝑊_𝑠. (44) We can deduce 𝐸−𝑚₁𝑅_𝑡0,𝑡1=−𝑚₁𝐸𝑅_𝑡0,𝑡1=−𝑚₁^𝑡1_𝑡0𝑓𝑡₀,𝑢𝑑𝑢−^𝑡1_𝑡0^𝑢_𝑡0𝛼(𝑠,𝑢)𝑑𝑠𝑑𝑢,Var−𝑚₁𝑅_𝑡0,𝑡1=𝑚²₁^𝑡1_𝑡0^𝑡1_𝑠𝜎(𝑠,𝑢)𝑑𝑢²𝑑𝑠. (45)By using the formula𝐸exp−𝑚₁𝑅_𝑡0,𝑡1=exp𝐸−𝑚₁𝑅_𝑡0,𝑡1+12Var−𝑚₁𝑅_𝑡0,𝑡1. (46) And substituting (45), (46) into (47), we can obtain (43).
Formula (43) can be rewritten as 𝐺₂𝑚₁,𝑚₂;𝑡₀,𝑡₁,𝑡₂=𝐸exp−𝑚₁𝑅_𝑡0,𝑡1−𝑚₂𝑅_𝑡1,𝑡2=𝐸exp−𝑚₁𝑅_𝑡0,𝑡1𝐸_𝑡1exp−𝑚₂𝑅_𝑡1,𝑡2=𝐸exp−𝑚₁𝑅_𝑡0,𝑡1𝐿₁𝑚₂;𝑡₁,𝑡₂. (47) Similar to the computation of (42), (43) can be obtained, and we omit it here.

Since the length 𝜃 of settlement period for reference asset 𝐶 is 0; namely, if firm 𝐵 has no default, 𝐵 pays compensation for 𝐶’s loss immediately once 𝐶 defaults, thus (18) becomes 𝑠=𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝐶_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}−𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝛿𝑀𝐿𝑇_𝑖−1,𝑇_𝑖−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}×𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝛿𝑀1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}⁻¹−𝐸𝐷𝜏^𝐶1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶}}𝐸∑^𝑛_𝑖=1𝐷𝑇_𝑖𝑀1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}. (48)

The price 𝑠 of TRS is given by the theorem below.

Theorem 6. With default intensities (38) and HJM model (9), the price 𝑠 of TRS is given by ∑𝑠=^𝑛𝑖=1𝐶_𝑖exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖∑^𝑛_𝑖=1𝛿𝑀exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖−^𝑛_𝑖=1M𝐺₂1+𝑏+𝑐,𝑏+𝑐;0,𝑇_𝑖−1,𝑇_𝑖,𝑡₂−𝐺₁1+𝑏+𝑐;0,𝑇_𝑖×^𝑛_𝑖=1𝛿𝑀exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖⁻¹−(𝑐/(1+𝑏+𝑐))1−𝑒^{−(𝑏+𝑐)𝑇}⋅𝐺₁(1+𝑏+𝑐;0,𝑇)∑^𝑛_𝑖=1𝛿𝑀exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖−𝑐₀−𝑐𝑏₀+𝑐₀1+𝑏+𝑐×^T₀𝑒^{−(𝑏0+𝑐0)𝑣}⋅𝐺₁()×1+𝑏+𝑐;0,𝑣𝑑𝑣_𝑛𝑖=1𝛿𝑀exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖⁻¹. (49)

Proof. Consider 𝑠=𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝐶_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}−𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝛿𝑀𝐿𝑇_𝑖−1,𝑇_𝑖−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}×𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝛿𝑀1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}⁻¹−𝐸𝐷𝜏^𝐶1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶}}𝐸∑^𝑛_𝑖=1𝐷𝑇_𝑖𝑀1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}. (50) To compute (48), we need to only deduce the following three expectation values: 𝐾₁∶=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}},𝐾₂∶=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖𝐿𝑇_𝑖−1,𝑇_𝑖−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}},𝐾₃∶=𝐸𝐷𝜏^𝐶1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶}}, (51) where 𝐾₁=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}=^𝑛_𝑖=1𝐸𝐸𝐷𝑇_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}∣ℱ^𝑟_𝑇^∗=^𝑛_𝑖=1𝐸𝑒^{−𝑅𝑖0,𝑇}⋅𝐸1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}∣ℱ^𝑟_𝑇^∗=^𝑛_𝑖=1𝐸exp−𝑏₀+𝑐₀𝑇_𝑖⋅exp−(1+𝑏+𝑐)𝑅_0,𝑇𝑖=^𝑛_𝑖=1exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖,𝐾₂=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖𝐿𝑇_𝑖−1,𝑇_𝑖−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}=^𝑛_𝑖=1𝐸𝐸𝐷𝑇_𝑖𝐿𝑇_𝑖−1,𝑇_𝑖−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}∣ℱ^𝑟_𝑇^∗. (52) Substituting the expression (14) of Libor interest rate 𝐿(𝑡,𝑇) into the formula above, we have 𝐾₂=^𝑛_𝑖=1𝐸𝐸𝑒^{−𝑅𝑖0,𝑇}1𝛿𝑒^{𝑅𝑇𝑖−1,𝑇𝑖−1+𝛿}−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}∣ℱ^𝑟_𝑇^∗=^𝑛_𝑖=1𝐸1𝛿𝑒^{−𝑅0,𝑇𝑖−1}−𝑒^{−𝑅𝑖0,𝑇}𝐸1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}∣ℱ^𝑟_𝑇^∗=^𝑛_𝑖=11𝛿𝐺₂1+𝑏+𝑐,𝑏+𝑐;0,𝑇_𝑖−1,𝑇_𝑖−𝐺₁1+𝑏+𝑐;0,𝑇_𝑖,𝐾₃=𝐸𝐷𝜏^𝐶1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶}}=𝐸𝐸𝐷𝜏^𝐶1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶}}∣ℱ^𝑟_𝑇^∗=𝐸^𝑇₀^∞_𝑣𝑒^{−𝑅0,𝑣}𝑓𝑢,𝑣∣ℱ^𝑟_𝑇^∗𝑑𝑢𝑑𝑣=𝐸^𝑇₀𝑐₀+𝑐𝑟_𝑣𝑒^{−(𝑐0−𝑏1)𝑣−(1+𝑐)𝑅0,𝑣}×^∞_𝑣𝑏₀+𝑏₁+𝑏𝑟_𝑢𝑒^{−(𝑏0+𝑏1)𝑢−𝑏𝑅0,𝑢}𝑐𝑑𝑢𝑑𝑣=𝐸₀−𝑐𝑏₀+𝑐₀1+𝑏+𝑐×^𝑇₀exp−𝑏₀+𝑐₀𝑣exp−(1+𝑏+𝑐)𝑅_0,𝑣+𝑐𝑑𝑣×(−𝑏1+𝑏+𝑐1−exp₀+𝑐₀𝑇−(1+𝑏+𝑐)𝑅_0,𝑇=𝑐−𝑏1+𝑏+𝑐1−exp₀+𝑐₀𝑇⋅𝐺₁(+𝑐1+𝑏+𝑐;0,𝑇)₀−𝑐𝑏₀+𝑐₀1+𝑏+𝑐×^𝑇₀exp−𝑏₀+𝑐₀𝑣⋅𝐺₁(1+𝑏+𝑐;0,𝑣)𝑑𝑣. (53)(54) Thus, substituting (52)–(54) into (48), we obtain ∑𝑠=^𝑛𝑖=1𝐶_𝑖exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖∑^𝑛_𝑖=1𝛿𝑀exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖−^𝑛_𝑖=1𝑀𝐺₂1+𝑏+𝑐,𝑏+𝑐;0,𝑇_𝑖−1,𝑇_𝑖−𝐺₁1+𝑏+𝑐;0,𝑇_𝑖×^𝑛_𝑖=1𝛿𝑀exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖⁻¹−(𝑐/(1+𝑏+𝑐))1−𝑒^{−(𝑏+𝑐)𝑇}⋅𝐺₁(1+b+𝑐;0,𝑇)∑^𝑛_𝑖=1𝛿𝑀exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖−𝑐₀−𝑐𝑏₀+𝑐₀×1+𝑏+𝑐_𝑇0𝑒^{−(𝑏0+𝑐0)𝑣}⋅𝐺₁()×1+𝑏+𝑐;0,𝑣𝑑𝑣_𝑛𝑖=1𝛿𝑀exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖⁻¹. (55) The proof is complete.

Remark 7. From the expression (49), the price 𝑠 of TRS is related to the default free interest rate risk (because there are 𝑏 and 𝑐). So the interest rate risk is not negligible in the pricing of credit derivatives.

4.2. TRS Valuation under Two-Firm Attenuation Contagion Model

In this subsection, besides that default is related to interest rate, we assume that default contagion has the hyperbolic attenuation effect. Default intensities of firms 𝐵 and 𝐶 are described as follows:𝜆^𝐵_𝑡=𝑏₀+𝑏𝑟_𝑡+𝑏₂𝑏₁𝑡−𝜏^𝐶1+1^{{𝜏𝐶≤𝑡}},𝜆^𝐶_𝑡=𝑐₀+𝑐𝑟_𝑡+𝑐₂𝑐₁𝑡−𝜏^𝐵1+1^{{𝜏𝐵≤𝑡}}, (56) where 𝑏₀, 𝑐₀, 𝑏, 𝑐, 𝑏₁, and 𝑐₁ are nonnegative, satisfying 𝑏₀+𝑏₂>0 and 𝑐₀+𝑐₂>0. When 𝑏₁=𝑐₁=0, model is simplified to the looping default model (38).

To obtain the analytic solution of 𝑠, we consider the following simplified model: 𝜆^𝐵_𝑡=𝑏₀+𝑏𝑟_𝑡−𝑏₁𝑏₁𝑡−𝜏^𝐶1+1^{{𝜏𝐶≤𝑡}},𝜆^𝐶_𝑡=𝑐₀+𝑐𝑟_𝑡−𝑐₁𝑐₁𝑡−𝜏^𝐵1+1^{{𝜏𝐵≤𝑡}}. (57) By the result in [27], the joint conditional survival probability of (𝜏^𝐵,𝜏^𝐶) under the model (57) is given by 𝑃𝜏^𝐵>𝑡₁,𝜏^𝐶>𝑡₂∣ℱ^𝑟_𝑇^∗=𝑐₁𝑡₂−𝑡₁+1𝑒^{−(𝑐0𝑡2+𝑐𝑅20,𝑡)}+𝑒^{−(𝑏0+𝑐0)𝑡2}exp−(𝑏+𝑐)𝑅_0,𝑡2−𝑐₁𝑒^{−𝑐0𝑡2𝑡2}_𝑡1𝑒^{−𝑏0(𝑠−𝑡1)}exp−𝑐𝑅_0,𝑡2−𝑏𝑅_𝑡1,𝑠𝑑𝑠−𝑒^{−𝑏0(𝑡2−𝑡1)−𝑐0𝑡2}exp−𝑐𝑅_0,𝑡2−𝑏𝑅_𝑡1,𝑡2for𝑡₁≤𝑡₂𝑃𝜏≤𝑇,_𝐵>𝑡₁,𝜏^𝐶>𝑡₂∣ℱ^𝑟_𝑇^∗=𝑏₁𝑡₁−𝑡₂+1𝑒^{−(𝑏0𝑡1+𝑏𝑅10,𝑡)}+𝑒^{−(𝑏0+𝑐0)𝑡1}exp−(𝑏+𝑐)𝑅_0,𝑡1−𝑏₁𝑒^{−𝑏0𝑡1𝑡1}_𝑡2𝑒^{−𝑐0(𝑠−𝑡2)}exp−𝑏𝑅_0,𝑡1−𝑐𝑅_𝑡2,𝑠𝑑𝑠−𝑒^{−𝑐0(𝑡1−𝑡2)𝑏0𝑡1}exp−𝑏𝑅_0,𝑡1−𝑐𝑅_𝑡2,𝑡1for𝑡₂≤𝑡₁≤𝑇, (58) and the joint conditional density function is given by 𝑓𝑡₁,𝑡₂∣ℱ^𝑟_𝑇^∗=𝑏₀+𝑏𝑟_𝑡1𝑐₀+𝑐𝑟_𝑡2𝑐₁𝑡₂−𝑡₁+1−𝑐₁×𝑒^{−𝑏0𝑡1−𝑐0𝑡2−𝑏𝑅10,𝑡−𝑐𝑅20,𝑡},𝑡₁≤𝑡₂c₀+𝑐𝑟_𝑡2𝑏₀+𝑏𝑟_𝑡1𝑏₁𝑡₁−𝑡₂+1−𝑏₁×𝑒^{−𝑏0𝑡1−𝑐0𝑡2−𝑏𝑅10,𝑡−𝑐𝑅20,𝑡}.𝑡₂≤𝑡₁. (59)

Under the model (57), the price formula of 𝑠 is given by 𝑠=𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝐶_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}−𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝛿𝑀𝐿𝑇_𝑖−1,𝑇_𝑖−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}×𝐸^𝑛_𝑖=1𝐷𝑇_𝑖𝛿𝑀1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}⁻¹−𝐸𝐷𝜏^𝐶1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶}}𝐸∑^𝑛_𝑖=1𝐷𝑇_𝑖𝛿𝑀1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}. (60)

Combining with Lemma 5, we can obtain the price 𝑠 of TRS, and the result is the following theorem.

Theorem 8. With default intensities (57) and HJM interest rate model (9), the price 𝑠 of TRS is given by ∑𝑠=^𝑛𝑖=1𝐶_𝑖exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖∑^𝑛_𝑖=1𝛿𝑀exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖−^𝑛_𝑖=1𝑀𝐺₂1+𝑏+𝑐,𝑏+𝑐;0,𝑇_𝑖−1,𝑇_𝑖−𝐺₁1+𝑏+𝑐;0,𝑇_𝑖×^𝑛_𝑖=1𝛿𝑀exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖⁻¹−(𝑐/(1+𝑏+𝑐))1−𝑒^{−(𝑏+𝑐)𝑇}⋅𝐺₁(1+𝑏+𝑐;0,𝑇)∑^𝑛_𝑖=1𝛿𝑀exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖−𝑐₀−𝑐𝑏₀+𝑐₀×1+𝑏+𝑐_𝑇0𝑒^{−(𝑏0+𝑐0)𝑣}⋅𝐺₁()×1+𝑏+𝑐;0,𝑣𝑑𝑣_𝑛𝑖=1𝛿𝑀exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖⁻¹. (61)

Proof. From (60), to obtain the analytic solution of 𝑠, we need to only compute three expectation values: 𝐾₁∶=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}},𝐾₂∶=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖𝐿𝑇_𝑖−1,𝑇_𝑖−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}},𝐾₃∶=𝐸𝐷𝜏^𝐶1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶}}, (62) where 𝐾₁=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}=^𝑛_𝑖=1𝐸𝐸𝐷𝑇_𝑖1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}∣ℱ^𝑟_𝑇^∗=^𝑛_𝑖=1𝐸𝑒^{−𝑅𝑖0,𝑇}⋅𝐸1_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}∣ℱ^𝑟_𝑇^∗=^𝑛_𝑖=1exp−𝑏₀+𝑐₀𝑇_𝑖⋅𝐺₁(1+𝑏+𝑐);0,𝑇_𝑖,𝐾₂=^𝑛_𝑖=1𝐸𝐷𝑇_𝑖𝐿𝑇_𝑖−1,𝑇_𝑖−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}=^𝑛_𝑖=1𝐸𝐸1𝛿𝑒^{−𝑅𝑖0,𝑇}𝑒^{𝑅𝑇𝑖−1,𝑇𝑖−1+𝛿}−11_{{𝜏𝐵∧𝜏𝐶>𝑇𝑖}}∣ℱ^𝑟_𝑇^∗=^𝑛_𝑖=11𝛿𝐺₂1+𝑏+𝑐,𝑏+𝑐;0,𝑇_𝑖−1,𝑇_𝑖−𝐺₁1+𝑏+𝑐;0,𝑇_𝑖,𝐾₃=𝐸𝐷𝜏^𝐶1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶}}=𝐸𝐸𝐷𝜏^𝐶1_{{𝜏𝐶≤𝑇}}1_{{𝜏𝐵>𝜏𝐶}}∣ℱ^𝑟_𝑇^∗=𝐸𝑐₀−𝑐𝑏₀+𝑐₀×1+𝑏+𝑐_𝑇0exp−𝑏₀+𝑐₀𝑣×exp−(1+𝑏+𝑐)𝑅_0,𝑣+𝑐𝑑𝑣×−𝑏1+𝑏+𝑐1−exp₀+𝑐₀𝑇−(1+𝑏+𝑐)𝑅_0,𝑇=𝑐₀−𝑐𝑏₀+𝑐₀×1+𝑏+𝑐_𝑇0exp−𝑏₀+𝑐₀𝑣⋅𝐺₁()+𝑐1+𝑏+𝑐;0,𝑣𝑑𝑣−𝑏1+𝑏+𝑐1−exp₀+𝑐₀𝑇⋅𝐺₁(1+𝑏+𝑐;0,𝑇). (63) Substituting expressions (63) of 𝐾₁, 𝐾₂, and 𝐾₃ into (60), we can obtain (61). We complete the proof.

Remark 9. From formulas (49) and (61), we conclude that if the length 𝜃 of the reference asset’s settlement period is 0, the price 𝑠 of TRS is only related to interest rate risk and systematic risk.

5. Conclusion

In this paper, we mainly study the pricing of TRS under the framework of two-firm contagion models and HJM forward interest rate model. We obtain the analytic price expressions of TRS, respectively, based on whether the default is related to the interest rate. From these expressions, we claim that both default risk and the default-free interest rate risk have effects on the valuation of TRS. Moreover, the contagion effect between the reference asset and the protection seller is not ignorable. Therefore, the models in our paper have certain practical significance. We will further discuss other contagion models and interest rate models and compare them with the models in this paper by using Monte Carlo simulation.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research was supported by the National Natural Science Foundation of China under Grants no. 11326168 and no. 11171215 and Yangcai Project (no. YC-XK-13106).