Research Article Total Return Swap Valuation with ...

Research ArticleTotal Return Swap Valuation with CounterpartyRisk and Interest Rate Risk

Anjiao Wang1 and Zhongxing Ye1,2

1 School of Business Information, Shanghai University of International Business and Economics, Shanghai 201620, China2Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China

Correspondence should be addressed to Anjiao Wang; [email protected]

Received 17 January 2014; Accepted 28 April 2014; Published 28 May 2014

Academic Editor: Imran Naeem

Copyright © 2014 A. Wang and Z. Ye. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the pricing of total return swap (TRS) under the contagion models with counterparty risk and the interest rate risk. Weassume that interest rate followsHeath-Jarrow-Morton (HJM) forward interest ratemodel and obtain the Libormarket interest rate.The cases where default is related to the interest rate and independent of interest rate are considered. Using the methods of changeof measure and the “total hazard construction,” the joint default probabilities are obtained. Furthermore, we obtain the closed-formformulas of TRS under different contagion models, respectively.

1. Introduction

Total return swap (TRS), as a type of credit derivativesand a financing and leverage tool, is an important off-balance sheet tool, particularly for hedge funds and forbanks seeking additional fee income. The corporate bondsand their credit derivatives are typically financial tools inthe markets which undertake and avoid the credit risk ofthe companies. Therefore, the key of management of creditrisk is the fair pricing of credit derivatives. Especially after2007 financing crisis, the contagion effect of credit riskhas attracted huge attention of financial market regulatorsand financial institutions. To price credit derivatives, defaultcontagion models are developed rapidly in order to makethem more realistic after the subprime mortgage crisis.

The approaches of modeling the pricing of creditderivatives are mainly the value-of-the-firm (or structural)approach and the intensity-based approach (or reduced formapproach). The structural model is based on the work ofMerton [1], Black and Cox [2], and Geske [3]: the defaultoccurs when the firm assets are insufficient to meet paymentson debt or the value of the firm asset falls below a prespecifiedlevel.

Reduced-form models are developed by Artzner andDelbaen [4], Duffie et al. [5], Jarrow and Turnbull [6], and

Madan and Unal [7]. Duffie and Lando [8] show that areduced-formmodel can be obtained from a structuralmodelwith incomplete accounting information.The simplest type ofreduced-form model is where the default time or the creditmigration is the first jump of an exogenously given jumpprocess with an intensity. In Jarrow et al. [9], the intensityfor credit migration is constant; see also Litterman and Iben[10] for a Markov chain model of credit migration. In thepapers by Duffie et al. [5], Duffie and Singleton [11], andLando [12], the intensity of default is a random process. Thecommon feature of the reduced-form models is that defaultcannot be predicted and can occur at any time. Therefore,reduced-form models have been used to price a wide varietyof instruments. In recent years, some papers on estimatingthe parameters of these models are given by Collin-Dufresneand Solnik [13] and Duffee [14]. Jarrow and Yu [15] set upa reduced-form model in which estimation can be based onbond prices as well as credit default swap prices. A systematicdevelopment of mathematical tools for reduced-formmodelshas been given by Elliott et al. [16]. Jamshidian [17] developschange of numeraire methodology for reduced-formmodels.

A TRS is a bilateral financial contract between a totalreturn payer and a total return receiver. One party (the totalreturn payer) pays the total return of a reference security (orreference securities) and receives a form of payment from

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 412890, 12 pageshttp://dx.doi.org/10.1155/2014/412890

2 Abstract and Applied Analysis

Bond issuer

Referenceasset C

Creditprotectionbuyer A

2A

2B 2C2D

2E

TRS payer

Total return

LIBOR + spread s

Payments

TRS recipient

Creditprotectionseller B

Figure 1: The structure of TRS.

the other party (the receiver of the total rate of return).Often payment is a floating rate payment, a spread toLIBOR.The reference assets can be indices, bonds (emergingmarket, sovereign, bank debt, mortgage-backed securities,or corporate), loans (term or revolver), equities, real estatereceivables, lease receivables, or commodities. Ye and Zhuang[18] consider the pricing of TRS only when reference assetdefaults; they obtain TRS pricing formula when default isindependent of the interest rate. When default is related tothe interest rate, using the hybrid model given in Das andSundaram [19], they model default time and the interest rateand give the Monte Carlo simulation result.

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

Suppose that firm 𝐴 (a corporate bond investing firm,credit protection buyer) holds a corporate bond (referenceasset) issued by firm 𝐶 (a corporate bond issuer) (refer to 2Ain Figure 1), and firm𝐶 is subject to default. At bondmaturity,if firm 𝐶 does not default, it will pay the bond principle andinterest to firm 𝐴 (refer to 2B). On the other hand, to hedgethe default risk of firm𝐶, firm𝐴 and firm𝐵 (credit protectionseller, subject to default also) enter into a TRS contract. If firm𝐶 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 aspread 𝑠 to𝐴 (refer to 2D). Firm 𝐵 promises to compensate𝐴for its loss in the event of default of firm 𝐶 (refer to 2E).

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

2. Basic Setup and HJM ForwardInterest Rate Model

We consider a filtered probability space (Ω,F, {F𝑡}𝑇∗

𝑡=0, 𝑃)

which is an uncertain economy with a time horizon of𝑇∗, satisfying the usual conditions of right-continuity and

completeness with respect to 𝑃-null sets, where F = F𝑇∗

and 𝑃 is an equivalent martingale measure under which

discounted bond prices are martingales. We assume theexistence and uniqueness of 𝑃 so that bond markets arecomplete and there is no arbitrage, as shown in discrete timecase by Harrison and Kreps [20] and in continuous time caseby Harrison and Pliska [21]. Subsequent specifications of themodel are all under the equivalent martingale measure (orrisk neutral measure) 𝑃.

On this probability space there is an R𝑑-valued process𝑋𝑡, which presents 𝑑 dimensional economy-wide state vari-

ables. In this paper, we consider the only one state variablewhich is the interest rate denoted by 𝑟

𝑡. There are also two

point processes, 𝑁𝑖 (𝑖 = 𝐵, 𝐶), initialized at 0, representingthe default processes of the firm 𝐵 and firm 𝐶, respectively,such that the default of the firm 𝑖 occurs when𝑁𝑖 jumps from0 to 1.

According to the information contained in the statevariables and the default processes, the enlarged filtration isdefined by

F𝑡= F𝑟

𝑡∨F𝐵

𝑡∨F𝐶

𝑡, (1)

where

F𝑟

𝑡= 𝜎 (𝑟

𝑠, 0 ≤ 𝑠 ≤ 𝑡) ,

F𝑖

𝑡= 𝜎 (𝑁

𝑖

𝑠, 0 ≤ 𝑠 ≤ 𝑡) , 𝑖 = 𝐵, 𝐶,

(2)

are the filtrations generated by 𝑟𝑡and 𝑁𝑖

𝑡(𝑖 = 𝐵, 𝐶), resp-

ectively.Let

G𝐵

𝑡= F𝐵

𝑡∨F𝑟

𝑇∗ ∨F

𝐶

𝑇∗ = F

𝐵

𝑡∨G−𝐵

0,

G𝐶

𝑡= F𝐶

𝑡∨F𝑟

𝑇∗ ∨F

𝐵

𝑇∗ = F

𝐶

𝑡∨G−𝐶

0,

(3)

where

G−𝐵

0= F𝑟

𝑇∗ ∨F

𝐶

𝑇∗ , G

−𝐶

0= F𝑟

𝑇∗ ∨F

𝐵

𝑇∗ . (4)

G−𝑖0(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 𝜏𝑖 denote the default time of firm 𝑖; namely, 𝜏𝑖 is thefirst jump time of𝑁𝑖, which can be defined as

𝜏𝑖= inf {𝑡 : ∫

𝑡

0

𝜆𝑖

𝑠𝑑𝑠 ≥ 𝐸

𝑖} , (5)

where {𝐸𝑖} (𝑖 = 𝐵, 𝐶) is independent of 𝑟𝑡(𝑡 ∈ [0, 𝑇

∗]).

According to the Doob-Meyer decomposition, we havethat

𝑀𝑖

𝑡= 𝑁𝑡− ∫

𝑡∧𝜏𝑖

0

𝜆𝑖

𝑠𝑑𝑠 (6)

is a (𝑃,F𝑡)-martingale.

Due to the impact of counterparty risk, the default inten-sities 𝜆𝐵

𝑡and 𝜆𝐶

𝑡of firms 𝐵 and 𝐶 are no longer independent

under the conditionF𝑟𝑡. The conditional survival probability

Abstract and Applied Analysis 3

and the unconditional survival probability of firm 𝑖 are givenby

𝑃 (𝜏𝑖> 𝑡 | G

−𝑖

0) = exp(−∫

𝑡

0

𝜆𝑗

𝑠𝑑𝑠) , 𝑡 ∈ [0, 𝑇] , (7)

𝑃 (𝜏𝑖> 𝑡) = 𝐸 [exp(−∫

𝑡

0

𝜆𝑖

𝑠𝑑𝑠)] , 𝑡 ∈ [0, 𝑇] , (8)

respectively.As a type of market risk, interest rate risk analysis is

almost always based on simulating movements in one ormore yield curves using the Heath-Jarrow-Morton (HJM)framework to ensure that the yield curvemovements are bothconsistent with current market yield curves and such thatno riskless arbitrage is possible. HJM model was developedin early 1992 by Heath et al. [22]. In this paper, we assumethe default free interest rate is stochastic forward interest rate𝑓(𝑡, 𝑇), which follows the HJM model

𝑑𝑓 (𝑡, 𝑇) = 𝛼 (𝑡, 𝑇) 𝑑𝑡 + 𝜎 (𝑡, 𝑇) 𝑑𝑊𝑡, 0 ≤ 𝑡 ≤ 𝑇, (9)

where𝛼(𝑡, 𝑇) is the drift term and𝜎(𝑡, 𝑇) is the volatility term,and they are both stochastic. For every fixed time 𝑇, they areadapted processes toF

𝑡.

The arbitrage free condition [23] shows that the Brownmovement under each real probability measure can betransformed into the Brownmovement under the risk neutralmeasure. For simplification, in this paper, we assume theBrown movement 𝑊

𝑡is 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

𝑡

𝜎 (𝑡, V) 𝑑V. (10)

The solution of (9) is

𝑓 (𝑡, 𝑇) = 𝑓 (0, 𝑇) + ∫

𝑡

0

𝛼 (𝑢, 𝑇) 𝑑𝑢 + ∫

𝑡

0

𝜎 (𝑢, 𝑇) 𝑑𝑊𝑢, (11)

where 𝑓(0, 𝑇) is the initial forward interest rate curve, whichis known.

At time 𝑡 the instantaneous interest rate 𝑟𝑡= 𝑓(𝑡, 𝑡).

𝐷(𝑡) = exp(− ∫𝑡0𝑟𝑢𝑑𝑢) is the discounted process. Let 𝐵(𝑡, 𝑇)

be the time-𝑡 value of a zero-coupon bond with face value 1 atmaturity 𝑇; we have

𝐵 (𝑡, 𝑇) = exp(−∫𝑇

𝑡

𝑓 (𝑡, V) 𝑑V) , 0 ≤ 𝑡 ≤ 𝑇 ≤ 𝑇. (12)

Under the risk neutral measure 𝑃, 𝐵(𝑡, 𝑇) satisfies the follow-ing differential equation:

𝑑𝐵 (𝑡, 𝑇) = 𝑟𝑡𝐵 (𝑡, 𝑇) 𝑑𝑡 − 𝜎

∗(𝑡, 𝑇) 𝐵 (𝑡, 𝑇) 𝑑𝑊

𝑡. (13)

Denote 𝐿(𝑡, 𝑇) to be the time-𝑡 locked investment yieldcurve from 𝑇 to 𝑇 + 𝛿; then,

𝐿 (𝑡, 𝑇) =𝐵 (𝑡, 𝑇) − 𝐵 (𝑡, 𝑇 + 𝛿)

𝛿𝐵 (𝑡, 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 isthe duration of Libor, normally 0.25 year or half year.

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

3. TRS Valuation When Default IsIndependent of the Interest Rate

Assume that reference asset 𝐶 is a defaultable coupon bondwith face value 1 and the same maturity 𝑇 with TRS contract.𝑇0, 𝑇1, 𝑇2, . . . , 𝑇

𝑛are bond interest payment dates, where 0 =

𝑇0< 𝑇1< ⋅ ⋅ ⋅ < 𝑇

𝑛= 𝑇, for 0 ≤ 𝑖 ≤ 𝑛 − 1, 𝑇

𝑖+1− 𝑇𝑖= Δ𝑇,

𝑛Δ𝑇 = 𝑇.Denote 𝐶

𝑖to be the time-𝑇

𝑖cash flow of TRS payer,

where 𝐶1, . . . , 𝐶

𝑛−1are the interest payments at time 𝑇

𝑖(𝑖 =

1, . . . , 𝑛 − 1) and 𝐶𝑛is the sum of interest payment and value-

added of the bond at time𝑇𝑛= 𝑇, which are all determined at

𝑇0= 0. Let𝑀 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 rateof reference asset 𝐶 is 0.

At time 0, for TRS payment leg, the discounted expecta-tion of the cash flow 𝐹

𝐵is

𝐸 [𝐹𝐵] = 𝐸[

𝑛

∑

𝑖=1

𝐷(𝑇𝑖) 𝐶𝑖1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}] . (15)

At time 0, for TRS recipient leg, the discounted expecta-tion of the cash flow 𝐹

𝐶is

𝐸 [𝐹𝐶] = 𝐸[

𝑛

∑

𝑖=1

𝐷(𝑇𝑖) 𝛿𝑀 (𝐿 (𝑇

𝑖−1, 𝑇𝑖−1) + 𝑠) 1

{𝜏𝐵∧𝜏𝐶>𝑇𝑖}]

+ 𝐸 [𝐷 (𝜏𝐶+ 𝜃) 1

{𝜏𝐶≤𝑇}1{𝜏𝐵>𝜏𝐶+𝜃}] ,

(16)

where 𝐸 is the expectation under risk neutral measure 𝑃, 𝜃 isthe length of settlement period, and 𝜏𝐶 + 𝜃 is settlement date.𝐿(𝑇𝑖−1, 𝑇𝑖−1) is time-𝑇

𝑖−1locked 𝑇

𝑖-Libor interest rate. 𝑠 is the

TRS spread and𝐷(⋅) is the discounted factor.We assume the market is complete; namely, there is no

arbitrage.Therefore, according to the arbitrage-free principle,we have

𝐸 [𝐹𝐵] = 𝐸 [𝐹

𝐶] . (17)


The general pricing formula of TRS is derived as

𝑠 = (𝐸[

𝑛

∑

𝑖=1

𝐷(𝑇𝑖) 𝐶𝑖1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}]

−𝐸[

𝑛

∑

𝑖=1

𝐷(𝑇𝑖) 𝛿𝑀𝐿 (𝑇

𝑖−1, 𝑇𝑖−1) 1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}])

×(𝐸[

𝑛

∑

𝑖=1

𝐷(𝑇𝑖) 𝛿𝑀1

{𝜏𝐵∧𝜏𝐶>𝑇𝑖}])

−1

−

𝐸 [𝐷 (𝜏𝐶+ 𝜃) 1

{𝜏𝐶≤𝑇}1{𝜏𝐵>𝜏𝐶+𝜃}]

𝐸 [∑𝑛

𝑖=1𝐷(𝑇𝑖) 𝛿𝑀1


.

(18)

3.1. TRS Valuation under Two-Firm Looping Default Conta-gion Model. In this subsection we assume that the defaultintensities of credit protection seller 𝐵 and reference asset 𝐶are given, respectively, by

𝜆𝐵

𝑡= 𝑏0+ 𝑏11{𝜏𝐶≤𝑡}, (19)

𝜆𝐶

𝑡= 𝑐0+ 𝑐11{𝜏𝐵≤𝑡}, (20)

where 𝑏0and 𝑏0are nonnegative, satisfying 𝑏

0+ 𝑏1> 0 and

𝑐0+ 𝑐1> 0.

With the result in Leung and Kwok [24], the joint survivalprobability and the joint density of default time (𝜏𝐵, 𝜏𝐶) aregiven by

𝑃 (𝜏𝐵> 𝑡1, 𝜏𝐶> 𝑡2)

=

{{{{{

{{{{{

{

𝑒−𝑏0𝑡1−𝑐0𝑡2 [𝑐0𝑒−𝑏1(𝑡1−𝑡2)− 𝑏1𝑒−𝑐0(𝑡1−𝑡2)

𝑐0− 𝑏1

] , 𝑡2≤ 𝑡1

𝑒−𝑏0𝑡1−𝑐0𝑡2 [𝑐0𝑒−𝑐1(𝑡2−𝑡1)− 𝑐1𝑒−𝑏0(𝑡2−𝑡1)

𝑏0− 𝑐1

] , 𝑡2> 𝑡1,

(21)

𝑓 (𝑡1, 𝑡2)

=

{

{

{

𝑓1(𝑡1, 𝑡2) = 𝑐0(𝑏0+ 𝑏1) 𝑒−(𝑏0+𝑏1)𝑡1−(𝑐0−𝑏1)𝑡2 , 𝑡2≤ 𝑡1

𝑓2(𝑡1, 𝑡2) = 𝑏0(𝑐0+ 𝑐1) 𝑒−(𝑐0+𝑐1)𝑡2−(𝑏0−𝑐1)𝑡1 , 𝑡2> 𝑡1,

(22)

respectively.When the default process and the interest rate process are

independent, by computing formula (18), we can obtain thefollowing result.

Theorem 1. With the default intensity process (19)-(20) andHJM interest rate model (9), the price 𝑠 of TRS is given by

𝑠 =∑𝑛

𝑖=1(𝐶𝑖− 𝛿𝑀𝐿 (𝑇

𝑖−1, 𝑇𝑖−1)) 𝐵 (0, 𝑇

𝑖) 𝑒−(𝑏0+𝑐0)𝑇𝑖

∑𝑛

𝑖=1𝛿𝑀𝑒−(𝑏0+𝑐0)𝑇𝑖

−

𝑐0𝑒−(𝑏0+𝑏1)𝜃∫𝑇

0𝐵 (0, V + 𝜃) 𝑒−(𝑏0+𝑐0)V𝑑V

∑𝑛


.

(23)

Proof. By the arbitrage free principle, the price formula (18)becomes

𝑠 = (𝐸[

𝑛

∑

𝑖=1


−𝐸[

𝑛

∑

𝑖=1


𝑖−1, 𝑇𝑖−1) 1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}])

× (𝐸[

𝑛

∑

𝑖=1



−1

−

𝐸 [𝐷 (𝜏𝐶+ 𝜃) 1


𝐸 [∑𝑛

𝑖=1𝐷(𝑇𝑖)𝑀1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}]

.

(24)

To obtain the analytic solution of 𝑠, we need to compute thefollowing three expectation values:

𝐾1:=

𝑛

∑

𝑖=1

𝐸 [𝐷 (𝑇𝑖) 1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}] ,

𝐾2:=

𝑛

∑

𝑖=1

𝐸 [𝐷 (𝑇𝑖) 𝐿 (𝑇𝑖−1, 𝑇𝑖−1) 1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}] ,

𝐾3:= 𝐸 [𝐷 (𝜏

𝐶+ 𝜃) 1

{𝜏𝐶≤𝑇}1{𝜏𝐵>𝜏𝐶+𝜃}] ,

(25)

where

𝐾1=

𝑛

∑

𝑖=1

𝐸 [𝐷 (𝑇𝑖) 1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}]

=

𝑛

∑

𝑖=1

𝐸[exp(−∫𝑇𝑖

0

𝑟𝑢𝑑𝑢)]𝐸 [1


=

𝑛

∑

𝑖=1

exp(−∫𝑇𝑖

0

𝑓 (0, 𝑢) 𝑑𝑢)𝐸 [1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}]

=

𝑛

∑

𝑖=1

𝐵 (0, 𝑇𝑖) 𝑒−(𝑏0+𝑐0)𝑇𝑖 ,

𝐾2=

𝑛

∑

𝑖=1

𝐸 [𝐷 (𝑇𝑖) 𝐿 (𝑇𝑖−1, 𝑇𝑖−1) 1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}]

=

𝑛

∑

𝑖=1

𝐵 (0, 𝑇𝑖) 𝐿 (𝑇𝑖−1, 𝑇𝑖−1) 𝑒−(𝑏0+𝑐0)𝑇𝑖 ,

𝐾3= 𝐸 [𝐷 (𝜏

𝐶+ 𝜃) 1


= 𝐸 [𝐸 [𝐷 (𝜏𝐶+ 𝜃) 1

{𝜏𝐶≤𝑇}1{𝜏𝐵>𝜏𝐶+𝜃}

| F𝑟

𝑇∗]]

= 𝐸 [𝐵 (0, 𝜏𝐶+ 𝜃) 1



= ∫

𝑇

0

∫

∞

V+𝜃𝐵 (0, V + 𝜃) 𝑓

2(𝑢, V) 𝑑𝑢 𝑑V

= ∫

𝑇

0

𝐵 (0, V + 𝜃)∫∞

V+𝜃𝑐0(𝑏0+ 𝑏1) 𝑒−(𝑏0+𝑏1)𝑡1−(𝑐0−𝑏1)𝑡2𝑑𝑢 𝑑V

= 𝑐0𝑒−(𝑏0+𝑏1)𝜃∫

𝑇

0

𝐵 (0, V + 𝜃) 𝑒−(𝑏0+𝑐0)V𝑑V.

(26)

Substitute𝐾1,𝐾2, and𝐾

3into formula (24), and we obtain

𝑠 =∑𝑛

𝑖=1(𝐶𝑖− 𝛿𝑀𝐿 (𝑇

𝑖−1, 𝑇𝑖−1)) 𝐵 (0, 𝑇


∑𝑛


−

𝑐0𝑒−(𝑏0+𝑏1)𝜃∫𝑇

0𝐵 (0, V + 𝜃) 𝑒−(𝑏0+𝑐0)V𝑑V

∑𝑛


.

(27)

We complete the proof.

Remark 2. From (23), we can conclude that when the length𝜃 of settlement period is zero, the swap rate 𝑠 is only relatedto the systematic factors 𝑏

0and 𝑐0, irrelative to the default

contagion between two firms.

3.2. TRS Valuation under Two-Firm Attenuation ContagionModel. In this subsection, we consider the default contagionto have the hyperbolic attenuation effect. Default intensitiesof firms 𝐵 and 𝐶 have the following forms:

𝜆𝐵

𝑡= 𝑏0+

𝑏2

𝑏1(𝑡 − 𝜏𝐶) + 1

1{𝜏𝐶≤𝑡}, (28)

𝜆𝐶

𝑡= 𝑐0+

𝑐2

𝑐1(𝑡 − 𝜏𝐵) + 1

1{𝜏𝐵≤𝑡}, (29)

where 𝑏0, 𝑐0, 𝑏1, and 𝑐

1are nonnegative, satisfying 𝑏

0+ 𝑏2> 0

and 𝑐0+ 𝑐2> 0. 𝑏

2and 𝑐2reflect the impact strength of the

counterparty and 𝑏1and 𝑐1show 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 𝑏

2= −𝑏1; then, (28) and (29) become, respectively,

𝜆𝐵

𝑡= 𝑏0−

𝑏1

𝑏1(𝑡 − 𝜏𝐶) + 1

1{𝜏𝐶≤𝑡}, (30)

𝜆𝐶

𝑡= 𝑐0−

𝑐1

𝑐1(𝑡 − 𝜏𝐵) + 1

1{𝜏𝐵≤𝑡}. (31)

According to the result in Bai et al. [25], with theintensities (30)-(31), the joint survival probability and thejoint density function of (𝜏𝐵, 𝜏𝐶) are given by

𝑃 (𝜏𝐵> 𝑡1, 𝜏𝐶> 𝑡2)

=

{{{{{{{{{{{{

{{{{{{{{{{{{

{

𝑏1(𝑡1− 𝑡2+1

𝑏1

−1

𝑐0

) 𝑒−(𝑏0𝑡1+𝑐0𝑡2)

+𝑏1

𝑐0

𝑒−(𝑏0+𝑐0)𝑡1 , 𝑡

2≤ 𝑡1

𝑐1(𝑡2− 𝑡1+1

𝑐1

−1

𝑏0

) 𝑒−(𝑏0𝑡1+𝑐0𝑡2)

+𝑐1

𝑏0

𝑒−(𝑏0+𝑐0)𝑡2 , 𝑡

2> 𝑡1,

(32)

𝑓 (𝑡1, 𝑡2)

=

{{{{{{{{

{{{{{{{{

{

𝑓1(𝑡1, 𝑡2) = 𝑐0𝑏0𝑏1[𝑡1− 𝑡2+1

𝑏1

−1

𝑏0

]

×𝑒−(𝑏0𝑡1+𝑐0𝑡2), t

2≤ 𝑡1

𝑓2(𝑡1, 𝑡2) = 𝑏0𝑐0𝑐1[𝑡2− 𝑡1+1

𝑐1

−1

𝑐0

]

×𝑒−(𝑏0𝑡1+𝑐0𝑡2), 𝑡

2> 𝑡1,

(33)

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

Theorem 3. With the default intensities (30)-(31) and HJMinterest rate model (9), 𝑠 is given by

𝑠 =∑𝑛

𝑖=1𝐶𝑖𝐵 (0, 𝑇


∑𝑛

𝑖=1𝛿𝑀𝐵 (0, 𝑇


−∑𝑛

𝑖=1𝐿 (𝑇𝑖−1, 𝑇𝑖−1) 𝛿𝑀𝐵 (0, 𝑇


∑𝑛



−

𝑐0(1 + 𝑏

1𝜃) 𝑒−𝑏0𝜃∫𝑇

0𝐵 (0, V + 𝜃) 𝑒−(𝑏0+𝑐0)V𝑑V

∑𝑛



.

(34)

Proof. To compute (18), we need to only compute the follow-ing three expectation values:

�̃�1:=

𝑛

∑

𝑖=1


�̃�2:=

𝑛

∑

𝑖=1


�̃�3:= 𝐸 [𝐷 (𝜏

𝐶+ 𝛿) 1

{𝜏𝐶≤𝑇}1{𝜏𝐵>𝜏𝐶+𝛿}] ,

(35)


where

�̃�1=

𝑛

∑

𝑖=1


=

𝑛

∑

𝑖=1

𝐵 (0, 𝑇𝑖) 𝑒−(𝑏0+𝑐0)𝑇𝑖 ,

�̃�2=

𝑛

∑

𝑖=1


=

𝑛

∑

𝑖=1

𝐿 (𝑇𝑖−1, 𝑇𝑖−1) 𝐵 (0, 𝑇

𝑖) 𝑒−(𝑏0+𝑐0)𝑇𝑖 ,

�̃�3= 𝐸 [𝐷 (𝜏

𝐶+ 𝛿) 1

{𝜏𝐶≤𝑇}1{𝜏𝐵>𝜏𝐶+𝛿}]

= 𝐸 [𝐸 [𝐷 (𝜏𝐶+ 𝛿) 1

{𝜏𝐶≤𝑇}1{𝜏𝐵>𝜏𝐶+𝛿}

| F𝑟

𝑇∗]]

= 𝐸 [𝐵 (0, 𝜏𝐶+ 𝜃) 1


= ∫

𝑇

0

∫

∞

V+𝜃𝐵 (0, V + 𝜃) 𝑓 (𝑢, V) 𝑑𝑢 𝑑V

= ∫

𝑇

0

𝐵 (0, V + 𝜃)

× ∫

∞

V+𝜃𝑐0𝑏0𝑏1[𝑡1− 𝑡2+1

𝑏1

−1

𝑏0

] 𝑒−(𝑏0𝑡1+𝑐0𝑡2)𝑑𝑢 𝑑V

= 𝑐0(1 + 𝑏

1𝜃) 𝑒−𝑏0𝜃∫

𝑇

0

𝐵 (0, V + 𝜃) 𝑒−(𝑏0+𝑐0)V𝑑V.

(36)

Substituting (36) into (18), we can have

𝑠 =∑𝑛

𝑖=1𝐶𝑖𝐵 (0, 𝑇


∑𝑛



−∑𝑛

𝑖=1𝐿 (𝑇𝑖−1, 𝑇𝑖−1) 𝛿𝑀𝐵 (0, 𝑇


∑𝑛



−

𝑐0(1 + 𝑏

1𝜃) 𝑒−𝑏0𝜃∫𝑇

0𝐵 (0, V + 𝜃) 𝑒−(𝑏0+𝑐0)V𝑑V

∑𝑛

i=1 𝛿𝑀𝐵 (0, 𝑇𝑖) 𝑒−(𝑏0+𝑐0)𝑇𝑖

,

(37)

which is formula (34). The proof is complete.

Remark 4. Combining formula (34) with formula (23), wecan conclude that firm 𝐶’s default contagion on firm 𝐵 haseffect in the pricing of TRS (there is 𝑏

1in formula (34)).

However, firm 𝐵’s default contagion on firm 𝐶 has no effectin the pricing of TRS (there is no 𝑐

1in (34)).Therefore, in the

complex contagion model, we can assume that the referenceasset is the primary firm, and the protection seller is thesecondary firm.

4. TRS Valuation When Default Is Related tothe Interest Rate

Empirical studies show that in the vast majority of casesdefault is related to the interest rate or, in other words, creditrisk is related to interest rate risk. In this section, we assumethat their correlation is described by default intensities. Forsimplification, we assume that the length 𝜃 of the referenceasset’s settlement period is zero.

4.1. TRS Valuation under Two-Firm Looping Default Conta-gion Model. We consider that default intensities of firms 𝐵and 𝐶 have the following form:

𝜆𝐵

𝑡= 𝑏0+ 𝑏𝑟𝑡+ 𝑏11{𝜏𝐶≤𝑡},

𝜆𝐶

𝑡= 𝑐0+ 𝑐𝑟𝑡+ 𝑐11{𝜏𝐵≤𝑡},

(38)

where 𝑏0, 𝑏0, 𝑏, and 𝑐 are nonnegative, satisfying 𝑏

0+ 𝑏1> 0

and 𝑐0+𝑐1> 0. 𝑟𝑡= 𝑓(𝑡, 𝑡)and𝑓(𝑡, 𝑇) satisfies HJMmodel (9).

UsingWang andYe’s result [26], under themodel (38), thejoint conditional distribution of (𝜏𝐵, 𝜏𝐶) is given by

𝑃 (𝜏𝐵> 𝑡1, 𝜏𝐶> 𝑡2| F𝑟

𝑇∗)

= exp (−𝑐0𝑡2− 𝑐𝑅0,𝑡2

)

× [ exp (−𝑐1(𝑡2− 𝑡1)) − exp (− (𝑏

0(𝑡2− 𝑡1) − 𝑏𝑅

𝑡1,𝑡2

))

+ exp (−𝑏0𝑡2− 𝑏𝑅0,𝑡2

)

+ 𝑐1∫

𝑡2

𝑡1

exp (−𝑏0(𝑢 − 𝑡

1) − 𝑐1(𝑡2− 𝑢) − 𝑏𝑅

𝑡1,𝑢) 𝑑𝑢] ,

for 𝑡1< 𝑡2< 𝑇.


𝑇∗)

= exp (−𝑏0𝑡1− 𝑏𝑅0,𝑡1

)

× [ exp (−𝑏1(𝑡1− 𝑡2)) − exp (− (𝑐

0(𝑡1− 𝑡2) − 𝑐𝑅

𝑡2,𝑡1

))

+ exp (−𝑐0𝑡1− 𝑐𝑅0,𝑡1

)

+ 𝑏1∫

𝑡1

𝑡2

exp (−𝑐0(𝑢 − 𝑡

2) − 𝑏1(𝑡1− 𝑢) − 𝑐𝑅

𝑡2,𝑢) 𝑑𝑢] ,

for 𝑡2< 𝑡1< 𝑇.

(39)


The joint conditional density function is

𝑓 (𝑡1, 𝑡2| F𝑟

𝑇∗)

=

{{{{{{

{{{{{{

{

(𝑐0+ 𝑐𝑟𝑡2

) (𝑏0+ 𝑏1+ 𝑏𝑟𝑡1

)

×𝑒−(𝑏0+𝑏1)𝑡1−(𝑐0−𝑏1)𝑡2−𝑏𝑅0,𝑡1

−𝑐𝑅0,𝑡2 , 𝑡2≤ 𝑡1

(𝑏0+ 𝑏𝑟𝑡1

) (𝑐0+ 𝑐1+ 𝑐𝑟𝑡2

)

×𝑒−(𝑏0−𝑐1)𝑡1−(𝑐0+𝑐1)𝑡2−𝑏𝑅0,𝑡1

−𝑐𝑅0,𝑡2 , 𝑡2> 𝑡1,

(40)

where 𝑅0,𝑢

= ∫𝑢

0𝑟V 𝑑V is the cumulative interest rate process

andF𝑟𝑇∗ is the filter generated by 𝑟

𝑡up to 𝑇∗.

When default is related to interest rate, computing expec-tations in formula (18) becomes complicated. We give thefollowing lemma first.

Lemma 5. The interest rate process is given by (9), 𝑅𝑡,𝑠

=

∫𝑠

𝑡𝑟𝑢𝑑𝑢, for 𝑡 < 𝑠. Denote

𝐺1(𝑚1; 𝑡0, 𝑡1) = 𝐸 [exp (−𝑚

1𝑅𝑡0,𝑡1

)] ,

𝐺2(𝑚1, 𝑚2; 𝑡0, 𝑡1, 𝑡2) = 𝐸 [exp (−𝑚

1𝑅𝑡0,𝑡1

− 𝑚2𝑅𝑡1,𝑡2

)] ;

(41)

then,

𝐺1(𝑚1; 𝑡0, 𝑡1)=exp(−𝑚

1∫

𝑡1

𝑡0

𝑓 (𝑡0, 𝑢) 𝑑𝑢

−𝑚1∫

𝑡1

𝑡0

∫

𝑢

𝑡0

𝛼 (𝑠, 𝑢) 𝑑𝑠 𝑑𝑢

+𝑚2

1

2∫

𝑡1

𝑡0

(∫

𝑡1

𝑠

𝜎 (𝑠, 𝑢) 𝑑𝑢)

2

𝑑𝑠) ,

(42)

𝐺2(𝑚1, 𝑚2; 𝑡0, 𝑡1, 𝑡2)

= exp(−𝑚1∫

𝑡1

𝑡0


− 𝑚1∫

𝑡1

𝑡0

∫

𝑢

𝑡0

𝛼 (𝑠, 𝑢) 𝑑𝑠 𝑑𝑢)

⋅ exp(−𝑚2∫

𝑡2

𝑡1


− 𝑚2∫

𝑡2

𝑡1

∫

𝑢

𝑡1

𝛼 (𝑠, 𝑢) 𝑑𝑠 𝑑𝑢)

⋅ exp(𝑚2

1

2∫

𝑡1

𝑡0

(∫

𝑡1

𝑠


2

𝑑𝑠

+𝑚2

2

2∫

𝑡2

𝑡1

(∫

𝑡2

𝑠


2

𝑑𝑠) .

(43)

Proof. Consider

𝑅𝑡0,𝑡1

= ∫

𝑡1

𝑡0

𝑓 (𝑡0, 𝑢) 𝑑𝑢 + ∫

𝑡1

𝑡0

∫

𝑢

𝑡0


+ ∫

𝑡1

𝑡0

∫

𝑢

𝑡0

𝜎 (𝑠, 𝑢) 𝑑𝑊𝑠𝑑𝑢

= ∫

𝑡1

𝑡0

𝑓 (𝑡0, 𝑢) 𝑑𝑢 + ∫

𝑡1

𝑡0

∫

𝑢

𝑡0


+ ∫

𝑡1

𝑡0

∫

𝑡1

𝑠

𝜎 (𝑠, 𝑢) 𝑑𝑢 𝑑𝑊𝑠.

(44)

We can deduce𝐸 [−𝑚

1𝑅𝑡0,𝑡1

] = −𝑚1𝐸 [𝑅𝑡0,𝑡1

]

= −𝑚1∫

𝑡1

𝑡0

𝑓 (𝑡0, 𝑢) 𝑑𝑢 − ∫

𝑡1

𝑡0

∫

𝑢

𝑡0

𝛼 (𝑠, 𝑢) 𝑑𝑠 𝑑𝑢,

Var [−𝑚1𝑅𝑡0,𝑡1

] = 𝑚2

1∫

𝑡1

𝑡0

(∫

𝑡1

𝑠


2

𝑑𝑠.

(45)

By using the formula

𝐸 [exp (−𝑚1𝑅𝑡0,𝑡1

)]

= exp(𝐸 [−𝑚1𝑅𝑡0,𝑡1

] +1

2Var [−𝑚

1𝑅𝑡0,𝑡1

]) .

(46)

And substituting (45),(46) into (47), we can obtain (43).Formula (43) can be rewritten as𝐺2(𝑚1, 𝑚2; 𝑡0, 𝑡1, 𝑡2)

= 𝐸 [exp (−𝑚1𝑅𝑡0,𝑡1

− 𝑚2𝑅𝑡1,𝑡2

)]


) 𝐸𝑡1

[exp (−𝑚2𝑅𝑡1,𝑡2

)]]


) 𝐿1(𝑚2; 𝑡1, 𝑡2)] .

(47)

Similar to the computation of (42), (43) can be obtained, andwe omit it here.

Since the length 𝜃 of settlement period for reference asset𝐶 is 0; namely, if firm 𝐵 has no default, 𝐵 pays compensationfor 𝐶’s loss immediately once 𝐶 defaults, thus (18) becomes

𝑠 = (𝐸[

𝑛

∑

𝑖=1


−𝐸[

𝑛

∑

𝑖=1


𝑖−1, 𝑇𝑖−1) 1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}])

×(𝐸[

𝑛

∑

𝑖=1



−1

−

𝐸 [𝐷 (𝜏𝐶) 1{𝜏𝐶≤𝑇}1{𝜏𝐵>𝜏𝐶}]

𝐸 [∑𝑛


.

(48)

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


Theorem 6. With default intensities (38) and HJMmodel (9),the price 𝑠 of TRS is given by

𝑠 =∑𝑛

𝑖=1𝐶𝑖exp (− (𝑏

0+ 𝑐0) 𝑇𝑖) ⋅ 𝐺1((1 + 𝑏 + 𝑐) ; 0, 𝑇

𝑖)

∑𝑛

𝑖=1𝛿𝑀 exp (− (𝑏

0+ 𝑐0) 𝑇𝑖) ⋅ 𝐺1((1 + 𝑏 + 𝑐) ; 0, 𝑇

𝑖)

− (

𝑛

∑

𝑖=1

M (𝐺2(1 + 𝑏 + 𝑐, 𝑏 + 𝑐; 0, 𝑇

𝑖−1, 𝑇𝑖, 𝑡2)

−𝐺1(1 + 𝑏 + 𝑐; 0, 𝑇

𝑖)))

× (

𝑛

∑

𝑖=1

𝛿𝑀 exp (− (𝑏0+𝑐0) 𝑇𝑖) ⋅ 𝐺1((1+𝑏+ 𝑐) ; 0, 𝑇

𝑖))

−1

−

(𝑐/ (1 + 𝑏 + 𝑐)) (1 − 𝑒−(𝑏+𝑐)𝑇

⋅ 𝐺1(1 + 𝑏 + 𝑐; 0, 𝑇))

∑𝑛


0+ 𝑐0) 𝑇𝑖) ⋅ 𝐺1((1 + 𝑏 + 𝑐) ; 0, 𝑇

𝑖)

−((𝑐0−𝑐 (𝑏0+𝑐0)

1+𝑏+𝑐)×∫

T

0

𝑒−(𝑏0+𝑐0)V⋅ 𝐺1(1+𝑏 +𝑐; 0, V) 𝑑V)

× (

𝑛

∑

𝑖=1

𝛿𝑀 exp (− (𝑏0+𝑐0) 𝑇𝑖) ⋅ 𝐺1((1+𝑏+𝑐) ; 0, 𝑇

𝑖))

−1

.

(49)

Proof. Consider

𝑠 = (𝐸[

𝑛

∑

𝑖=1


− 𝐸[

𝑛

∑

𝑖=1


𝑖−1, 𝑇𝑖−1) 1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}])

× (𝐸[

𝑛

∑

𝑖=1



−1

−


𝐸 [∑𝑛


.

(50)

To compute (48), we need to only deduce the following threeexpectation values:

�̂�1:=

𝑛

∑

𝑖=1


�̂�2:=

𝑛

∑

𝑖=1


�̂�3:= 𝐸 [𝐷 (𝜏

𝐶) 1{𝜏𝐶≤𝑇}1{𝜏𝐵>𝜏𝐶}] ,

(51)

where

�̂�1=

𝑛

∑

𝑖=1


=

𝑛

∑

𝑖=1

𝐸 [𝐸 [𝐷 (𝑇𝑖) 1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}| F𝑟

𝑇∗]]

=

𝑛

∑

𝑖=1

𝐸 [𝑒−𝑅0,𝑇𝑖 ⋅ 𝐸 [1

{𝜏𝐵∧𝜏𝐶>𝑇𝑖}| F𝑟

𝑇∗]]

=

𝑛

∑

𝑖=1

𝐸 [exp (− (𝑏0+ 𝑐0) 𝑇𝑖) ⋅ exp (− (1 + 𝑏 + 𝑐) 𝑅

0,𝑇𝑖

)]

=

𝑛

∑

𝑖=1

exp (− (𝑏0+ 𝑐0) 𝑇𝑖) ⋅ 𝐺1((1 + 𝑏 + 𝑐) ; 0, 𝑇

𝑖) ,

�̂�2=

𝑛

∑

𝑖=1


=

𝑛

∑

𝑖=1

𝐸 [𝐸 [𝐷 (𝑇𝑖) 𝐿 (𝑇𝑖−1, 𝑇𝑖−1) 1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}| F𝑟

𝑇∗]] .

(52)

Substituting the expression (14) of Libor interest rate 𝐿(𝑡, 𝑇)into the formula above, we have

�̂�2=

𝑛

∑

𝑖=1

𝐸 [𝐸 [𝑒−𝑅0,𝑇𝑖

1

𝛿(𝑒𝑅𝑇𝑖−1,𝑇𝑖−1+𝛿 − 1) 1


𝑇∗]]

=

𝑛

∑

𝑖=1

𝐸 [1

𝛿(𝑒−𝑅0,𝑇𝑖−1 − 𝑒

−𝑅0,𝑇𝑖 ) 𝐸 [1


𝑇∗]]

=

𝑛

∑

𝑖=1

1

𝛿(𝐺2(1 + 𝑏 + 𝑐, 𝑏 + 𝑐; 0, 𝑇

𝑖−1, 𝑇𝑖)

−𝐺1(1 + 𝑏 + 𝑐; 0, 𝑇

𝑖)) ,

(53)

�̂�3= 𝐸 [𝐷 (𝜏

𝐶) 1{𝜏𝐶≤𝑇}1{𝜏𝐵>𝜏𝐶}]

= 𝐸 [𝐸 [𝐷 (𝜏𝐶) 1{𝜏𝐶≤𝑇}1{𝜏𝐵>𝜏𝐶}| F𝑟

𝑇∗]]

= 𝐸 [∫

𝑇

0

∫

∞

V𝑒−𝑅0,V𝑓 (𝑢, V | F𝑟

𝑇∗) 𝑑𝑢 𝑑V]

= 𝐸[∫

𝑇

0

(𝑐0+ 𝑐𝑟V) 𝑒

−(𝑐0−𝑏1)V−(1+𝑐)𝑅

0,V

× ∫

∞

V(𝑏0+ 𝑏1+ 𝑏𝑟𝑢) 𝑒−(𝑏0+𝑏1)𝑢−𝑏𝑅

0,𝑢𝑑𝑢 𝑑V]


= 𝐸[(𝑐0−𝑐 (𝑏0+ 𝑐0)

1 + 𝑏 + 𝑐)

×∫

𝑇

0

exp (− (𝑏0+𝑐0) V) exp (− (1+𝑏+𝑐) 𝑅

0,V) 𝑑V

+𝑐

1 + 𝑏 + 𝑐

× ( (1 − exp (− (𝑏0+ 𝑐0) 𝑇 − (1 + 𝑏 + 𝑐) 𝑅

0,𝑇)) ]

=𝑐

1 + 𝑏 + 𝑐(1 − exp (− (𝑏

0+ 𝑐0) 𝑇)

⋅ 𝐺1(1 + 𝑏 + 𝑐; 0, 𝑇))

+ (𝑐0−𝑐 (𝑏0+ 𝑐0)

1 + 𝑏 + 𝑐)

× ∫

𝑇

0

exp (− (𝑏0+ 𝑐0) V) ⋅ 𝐺

1(1 + 𝑏 + 𝑐; 0, V) 𝑑V.

(54)

Thus, substituting (52)–(54) into (48), we obtain

𝑠 =∑𝑛


0+ 𝑐0) 𝑇𝑖) ⋅ 𝐺1((1 + 𝑏 + 𝑐) ; 0, 𝑇

𝑖)

∑𝑛


0+ 𝑐0) 𝑇𝑖) ⋅ 𝐺1((1 + 𝑏 + 𝑐) ; 0, 𝑇

𝑖)

− (

𝑛

∑

𝑖=1

𝑀(𝐺2(1 + 𝑏 + 𝑐, 𝑏 + 𝑐; 0, 𝑇

𝑖−1, 𝑇𝑖)

−𝐺1(1 + 𝑏 + 𝑐; 0, 𝑇

𝑖)))

×(

𝑛

∑

𝑖=1

𝛿𝑀 exp (− (𝑏0+𝑐0) 𝑇𝑖) ⋅ 𝐺1((1+𝑏+𝑐) ; 0, 𝑇

𝑖))

−1

−

(𝑐/ (1 + 𝑏 + 𝑐)) (1 − 𝑒−(𝑏+𝑐)𝑇

⋅ 𝐺1(1 + b + 𝑐; 0, 𝑇))

∑𝑛


0+ 𝑐0) 𝑇𝑖) ⋅ 𝐺1((1 + 𝑏 + 𝑐) ; 0, 𝑇

𝑖)

− ((𝑐0−𝑐 (𝑏0+ 𝑐0)

1 + 𝑏 + 𝑐)

× ∫

𝑇

0

𝑒−(𝑏0+𝑐0)V⋅ 𝐺1(1 + 𝑏 + 𝑐; 0, V) 𝑑V)

×(

𝑛

∑

𝑖=1

𝛿𝑀 exp (−(𝑏0+𝑐0) 𝑇𝑖) ⋅ 𝐺1((1+𝑏+𝑐) ; 0, 𝑇

𝑖))

−1

.

(55)

The proof is complete.

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

4.2. TRS Valuation under Two-Firm Attenuation ContagionModel. In this subsection, besides that default is relatedto interest rate, we assume that default contagion has thehyperbolic attenuation effect. Default intensities of firms 𝐵and 𝐶 are described as follows:

𝜆𝐵

𝑡= 𝑏0+ 𝑏𝑟𝑡+

𝑏2

𝑏1(𝑡 − 𝜏𝐶) + 1

1{𝜏𝐶≤𝑡},

𝜆𝐶

𝑡= 𝑐0+ 𝑐𝑟𝑡+

𝑐2

𝑐1(𝑡 − 𝜏𝐵) + 1

1{𝜏𝐵≤𝑡},

(56)

where 𝑏0, 𝑐0, 𝑏, 𝑐, 𝑏

1, and 𝑐

1are nonnegative, satisfying 𝑏

0+𝑏2>

0 and 𝑐0+𝑐2> 0. When 𝑏

1= 𝑐1= 0, model is simplified to the

looping default model (38).To obtain the analytic solution of 𝑠, we consider the

following simplified model:

𝜆𝐵

𝑡= 𝑏0+ 𝑏𝑟𝑡−

𝑏1

𝑏1(𝑡 − 𝜏𝐶) + 1

1{𝜏𝐶≤𝑡},

𝜆𝐶

𝑡= 𝑐0+ 𝑐𝑟𝑡−

𝑐1

𝑐1(𝑡 − 𝜏𝐵) + 1

1{𝜏𝐵≤𝑡}.

(57)

By the result in [27], the joint conditional survival probabilityof (𝜏𝐵, 𝜏𝐶) under the model (57) is given by


𝑇∗)

= (𝑐1(𝑡2− 𝑡1) + 1) 𝑒

−(𝑐0𝑡2+𝑐𝑅0,𝑡2

)

+ 𝑒−(𝑏0+𝑐0)𝑡2 exp (− (𝑏 + 𝑐) 𝑅

0,𝑡2

)

− 𝑐1𝑒−𝑐0𝑡2 ∫

𝑡2

𝑡1

𝑒−𝑏0(𝑠−𝑡1) exp (−𝑐𝑅

0,𝑡2

− 𝑏𝑅𝑡1,𝑠) 𝑑𝑠

− 𝑒−𝑏0(𝑡2−𝑡1)−𝑐0𝑡2 exp (−𝑐𝑅

0,𝑡2

− 𝑏𝑅𝑡1,𝑡2

)

for 𝑡1≤ 𝑡2≤ 𝑇,


𝑇∗)

= (𝑏1(𝑡1− 𝑡2) + 1) 𝑒

−(𝑏0𝑡1+𝑏𝑅0,𝑡1

)

+ 𝑒−(𝑏0+𝑐0)𝑡1 exp (− (𝑏 + 𝑐) 𝑅

0,𝑡1

)

− 𝑏1𝑒−𝑏0𝑡1 ∫

𝑡1

𝑡2

𝑒−𝑐0(𝑠−𝑡2) exp (−𝑏𝑅

0,𝑡1

− 𝑐𝑅𝑡2,𝑠) 𝑑𝑠

− 𝑒−𝑐0(𝑡1−𝑡2)𝑏0𝑡1 exp (−𝑏𝑅

0,𝑡1

− 𝑐𝑅𝑡2,𝑡1

)

for 𝑡2≤ 𝑡1≤ 𝑇,

(58)


and the joint conditional density function is given by

𝑓 (𝑡1, 𝑡2| F𝑟

𝑇∗)

=

{{{{{{

{{{{{{

{

(𝑏0+ 𝑏𝑟𝑡1

) [(𝑐0+ 𝑐𝑟𝑡2

) (𝑐1(𝑡2− 𝑡1) + 1) − 𝑐

1]

×𝑒−𝑏0𝑡1−𝑐0𝑡2−𝑏𝑅0,𝑡1

−𝑐𝑅0,𝑡2 , 𝑡

1≤ 𝑡2

(c0+ 𝑐𝑟𝑡2

) [(𝑏0+ 𝑏𝑟𝑡1

) (𝑏1(𝑡1− 𝑡2) + 1) − 𝑏

1]

×𝑒−𝑏0𝑡1−𝑐0𝑡2−𝑏𝑅0,𝑡1

−𝑐𝑅0,𝑡2 . 𝑡

2≤ 𝑡1.

(59)

Under the model (57), the price formula of 𝑠 is given by

𝑠 = (𝐸[

𝑛

∑

𝑖=1


− 𝐸[

𝑛

∑

𝑖=1


𝑖−1, 𝑇𝑖−1) 1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}])

× (𝐸[

𝑛

∑

𝑖=1



−1

−


𝐸 [

𝑛

∑

𝑖=1



.

(60)

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

Theorem 8. With default intensities (57) and HJM interestrate model (9), the price 𝑠 of TRS is given by

𝑠 =∑𝑛


0+ 𝑐0) 𝑇𝑖) ⋅ 𝐺1((1 + 𝑏 + 𝑐) ; 0, 𝑇

𝑖)

∑𝑛


0+ 𝑐0) 𝑇𝑖) ⋅ 𝐺1((1 + 𝑏 + 𝑐) ; 0, 𝑇

𝑖)

− (

𝑛

∑

𝑖=1

𝑀(𝐺2(1 + 𝑏 + 𝑐, 𝑏 + 𝑐; 0, 𝑇

𝑖−1, 𝑇𝑖)

− 𝐺1(1 + 𝑏 + 𝑐; 0, 𝑇

𝑖)))

× (

𝑛

∑

𝑖=1

𝛿𝑀 exp (− (𝑏0+𝑐0) 𝑇𝑖) ⋅ 𝐺1((1+𝑏 +𝑐) ; 0, 𝑇

𝑖))

−1

−

(𝑐/ (1 + 𝑏 + 𝑐)) (1 − 𝑒−(𝑏+𝑐)𝑇

⋅ 𝐺1(1 + 𝑏 + 𝑐; 0, 𝑇))

∑𝑛


0+ 𝑐0) 𝑇𝑖) ⋅ 𝐺1((1 + 𝑏 + 𝑐) ; 0, 𝑇

𝑖)

− ((𝑐0−𝑐 (𝑏0+ 𝑐0)

1 + 𝑏 + 𝑐)

× ∫

𝑇

0

𝑒−(𝑏0+𝑐0)V⋅ 𝐺1(1 + 𝑏 + 𝑐; 0, V) 𝑑V)

×(

𝑛

∑

𝑖=1

𝛿𝑀 exp (−(𝑏0+𝑐0) 𝑇𝑖) ⋅ 𝐺1((1+𝑏+𝑐) ; 0, 𝑇

𝑖))

−1

.

(61)

Proof. From (60), to obtain the analytic solution of 𝑠, we needto only compute three expectation values:

𝐾1:=

𝑛

∑

𝑖=1


𝐾2:=

𝑛

∑

𝑖=1


𝐾3:= 𝐸 [𝐷 (𝜏

𝐶) 1{𝜏𝐶≤𝑇}1{𝜏𝐵>𝜏𝐶}] ,

(62)

where

𝐾1=

𝑛

∑

𝑖=1


=

𝑛

∑

𝑖=1

𝐸 [𝐸 [𝐷 (𝑇𝑖) 1{𝜏𝐵∧𝜏𝐶>𝑇𝑖}| F𝑟

𝑇∗]]

=

𝑛

∑

𝑖=1

𝐸 [𝑒−𝑅0,𝑇𝑖 ⋅ 𝐸 [1


𝑇∗]]

=

𝑛

∑

𝑖=1

exp (− (𝑏0+ 𝑐0) 𝑇𝑖) ⋅ 𝐺1((1 + 𝑏 + 𝑐) ; 0, 𝑇

𝑖) ,

𝐾2=

𝑛

∑

𝑖=1


=

𝑛

∑

𝑖=1

𝐸 [𝐸 [1

𝛿𝑒−𝑅0,𝑇𝑖 (𝑒𝑅𝑇𝑖−1,𝑇𝑖−1+𝛿 − 1) 1


𝑇∗]]

=

𝑛

∑

𝑖=1

1

𝛿(𝐺2(1 + 𝑏 + 𝑐, 𝑏 + 𝑐; 0, 𝑇

𝑖−1, 𝑇𝑖)

− 𝐺1(1 + 𝑏 + 𝑐; 0, 𝑇

𝑖)) ,

𝐾3= 𝐸 [𝐷 (𝜏

𝐶) 1{𝜏𝐶≤𝑇}1{𝜏𝐵>𝜏𝐶}]

= 𝐸 [𝐸 [𝐷 (𝜏𝐶) 1{𝜏𝐶≤𝑇}1{𝜏𝐵>𝜏𝐶}| F𝑟

𝑇∗]]

= 𝐸[(𝑐0−𝑐 (𝑏0+ 𝑐0)

1 + 𝑏 + 𝑐)

× ∫

𝑇

0

exp (− (𝑏0+ 𝑐0) V)


× exp (− (1+𝑏 +𝑐) 𝑅0,V) 𝑑V

+𝑐

1 + 𝑏 + 𝑐

× (1 − exp (− (𝑏0+ 𝑐0) 𝑇 − (1 + 𝑏 + 𝑐) 𝑅

0,𝑇) ]

= (𝑐0−𝑐 (𝑏0+ 𝑐0)

1 + 𝑏 + 𝑐)

× ∫

𝑇

0

exp (− (𝑏0+ 𝑐0) V) ⋅ 𝐺

1(1 + 𝑏 + 𝑐; 0, V) 𝑑V

+𝑐

1 + 𝑏 + 𝑐(1 − exp (− (𝑏

0+ 𝑐0) 𝑇)

⋅ 𝐺1(1 + 𝑏 + 𝑐; 0, 𝑇)) .

(63)

Substituting expressions (63) of 𝐾1, 𝐾2, and 𝐾

3into (60), we

can obtain (61). We complete the proof.

Remark 9. From formulas (49) and (61), we conclude thatif the length 𝜃 of the reference asset’s settlement period is0, the price 𝑠 of TRS is only related to interest rate risk andsystematic risk.

5. Conclusion

In this paper, we mainly study the pricing of TRS under theframework of two-firm contagion models and HJM forwardinterest rate model. We obtain the analytic price expressionsof TRS, respectively, based on whether the default is relatedto the interest rate. From these expressions, we claim thatboth default risk and the default-free interest rate risk haveeffects on the valuation of TRS.Moreover, the contagion effectbetween the reference asset and the protection seller is notignorable. Therefore, the models in our paper have certainpractical significance.Wewill further discuss other contagionmodels and interest rate models and compare them with themodels in this paper by using Monte Carlo simulation.

Conflict of Interests

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

Acknowledgments

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

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