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 Β© 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 βtotal hazard construction,β 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 π΅) andthe reference asset (firm πΆ) and study two-firm contagionmodels. The cash flow of a TRS in this model is provided byFigure 1.
Suppose that firm π΄ (a corporate bond investing firm,credit protection buyer) holds a corporate bond (referenceasset) issued by firm πΆ (a corporate bond issuer) (refer to 2Ain Figure 1), and firmπΆ is subject to default. At bondmaturity,if firm πΆ does not default, it will pay the bond principle andinterest to firm π΄ (refer to 2B). On the other hand, to hedgethe default risk of firmπΆ, firmπ΄ and firmπ΅ (credit protectionseller, subject to default also) enter into a TRS contract. If firmπΆ has no default, firm π΄ will make its total return to firm π΅
(refer to 2C), and, in exchange, firm π΅ gives the Libor plus aspread π toπ΄ (refer to 2D). Firm π΅ promises to compensateπ΄for its loss in the event of default of firm πΆ (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 βtotal hazardβ 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 (Ξ©,F, {Fπ‘}πβ
π‘=0, π)
which is an uncertain economy with a time horizon ofπβ, satisfying the usual conditions of right-continuity and
completeness with respect to π-null sets, where F = Fπβ
and π is an equivalent martingale measure under which
discounted bond prices are martingales. We assume theexistence and uniqueness of π 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) π.
On this probability space there is an Rπ-valued processππ‘, which presents π dimensional economy-wide state vari-
ables. In this paper, we consider the only one state variablewhich is the interest rate denoted by π
π‘. There are also two
point processes, ππ (π = π΅, πΆ), initialized at 0, representingthe default processes of the firm π΅ and firm πΆ, respectively,such that the default of the firm π occurs whenππ jumps from0 to 1.
According to the information contained in the statevariables and the default processes, the enlarged filtration isdefined by
Fπ‘= Fπ
π‘β¨Fπ΅
π‘β¨FπΆ
π‘, (1)
where
Fπ
π‘= π (π
π , 0 β€ π β€ π‘) ,
Fπ
π‘= π (π
π
π , 0 β€ π β€ π‘) , π = π΅, πΆ,
(2)
are the filtrations generated by ππ‘and ππ
π‘(π = π΅, πΆ), resp-
ectively.Let
Gπ΅
π‘= Fπ΅
π‘β¨Fπ
πβ β¨F
πΆ
πβ = F
π΅
π‘β¨Gβπ΅
0,
GπΆ
π‘= FπΆ
π‘β¨Fπ
πβ β¨F
π΅
πβ = F
πΆ
π‘β¨GβπΆ
0,
(3)
where
Gβπ΅
0= Fπ
πβ β¨F
πΆ
πβ , G
βπΆ
0= Fπ
πβ β¨F
π΅
πβ . (4)
Gβπ0(π = π΅, πΆ) contains complete information on the state
variables and the default processes of all firms up to time πβother than that of the πth firm.
Let ππ denote the default time of firm π; namely, ππ is thefirst jump time ofππ, which can be defined as
ππ= inf {π‘ : β«
π‘
0
ππ
π ππ β₯ πΈ
π} , (5)
where {πΈπ} (π = π΅, πΆ) is independent of ππ‘(π‘ β [0, π
β]).
According to the Doob-Meyer decomposition, we havethat
ππ
π‘= ππ‘β β«
π‘β§ππ
0
ππ
π ππ (6)
is a (π,Fπ‘)-martingale.
Due to the impact of counterparty risk, the default inten-sities ππ΅
π‘and ππΆ
π‘of firms π΅ and πΆ are no longer independent
under the conditionFππ‘. The conditional survival probability
Abstract and Applied Analysis 3
and the unconditional survival probability of firm π are givenby
π (ππ> π‘ | G
βπ
0) = exp(ββ«
π‘
0
ππ
π ππ ) , π‘ β [0, π] , (7)
π (ππ> π‘) = πΈ [exp(ββ«
π‘
0
ππ
π ππ )] , π‘ β [0, π] , (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 rateπ(π‘, π), which follows the HJM model
ππ (π‘, π) = πΌ (π‘, π) ππ‘ + π (π‘, π) πππ‘, 0 β€ π‘ β€ π, (9)
whereπΌ(π‘, π) is the drift term andπ(π‘, π) is the volatility term,and they are both stochastic. For every fixed time π, they areadapted processes toF
π‘.
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 π
π‘is standard under the risk neutral
measure π, where πΌ(π‘, π) satisfies
πΌ (π‘, π) = π (π‘, π) πβ(π‘, π) = π (π‘, π) β«
π
π‘
π (π‘, V) πV. (10)
The solution of (9) is
π (π‘, π) = π (0, π) + β«
π‘
0
πΌ (π’, π) ππ’ + β«
π‘
0
π (π’, π) πππ’, (11)
where π(0, π) is the initial forward interest rate curve, whichis known.
At time π‘ the instantaneous interest rate ππ‘= π(π‘, π‘).
π·(π‘) = exp(β β«π‘0ππ’ππ’) is the discounted process. Let π΅(π‘, π)
be the time-π‘ value of a zero-coupon bond with face value 1 atmaturity π; we have
π΅ (π‘, π) = exp(ββ«π
π‘
π (π‘, V) πV) , 0 β€ π‘ β€ π β€ π. (12)
Under the risk neutral measure π, π΅(π‘, π) satisfies the follow-ing differential equation:
ππ΅ (π‘, π) = ππ‘π΅ (π‘, π) ππ‘ β π
β(π‘, π) π΅ (π‘, π) ππ
π‘. (13)
Denote πΏ(π‘, π) to be the time-π‘ locked investment yieldcurve from π to π + πΏ; then,
πΏ (π‘, π) =π΅ (π‘, π) β π΅ (π‘, π + πΏ)
πΏπ΅ (π‘, π + πΏ). (14)
For 0 β€ π‘ < π, πΏ(π‘, π) is called forward Libor interest rate,and, for π‘ = π, πΏ(π‘, π) is called spot Libor interest rate. πΏ 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 πΆ is a defaultable coupon bondwith face value 1 and the same maturity π with TRS contract.π0, π1, π2, . . . , π
πare bond interest payment dates, where 0 =
π0< π1< β β β < π
π= π, for 0 β€ π β€ π β 1, π
π+1β ππ= Ξπ,
πΞπ = π.Denote πΆ
πto be the time-π
πcash flow of TRS payer,
where πΆ1, . . . , πΆ
πβ1are the interest payments at time π
π(π =
1, . . . , π β 1) and πΆπis the sum of interest payment and value-
added of the bond at timeππ= π, which are all determined at
π0= 0. Letπ be notional principal, and πΏ is the maturity of
Libor interest rate, the same with bondβs payment cycle Ξπ,namely, πΏ = Ξπ. For simplification, the default recovery rateof reference asset πΆ is 0.
At time 0, for TRS payment leg, the discounted expecta-tion of the cash flow πΉ
π΅is
πΈ [πΉπ΅] = πΈ[
π
β
π=1
π·(ππ) πΆπ1{ππ΅β§ππΆ>ππ}] . (15)
At time 0, for TRS recipient leg, the discounted expecta-tion of the cash flow πΉ
πΆis
πΈ [πΉπΆ] = πΈ[
π
β
π=1
π·(ππ) πΏπ (πΏ (π
πβ1, ππβ1) + π ) 1
{ππ΅β§ππΆ>ππ}]
+ πΈ [π· (ππΆ+ π) 1
{ππΆβ€π}1{ππ΅>ππΆ+π}] ,
(16)
where πΈ is the expectation under risk neutral measure π, π isthe length of settlement period, and ππΆ + π is settlement date.πΏ(ππβ1, ππβ1) is time-π
πβ1locked π
π-Libor interest rate. π is the
TRS spread andπ·(β ) is the discounted factor.We assume the market is complete; namely, there is no
arbitrage.Therefore, according to the arbitrage-free principle,we have
πΈ [πΉπ΅] = πΈ [πΉ
πΆ] . (17)
4 Abstract and Applied Analysis
The general pricing formula of TRS is derived as
π = (πΈ[
π
β
π=1
π·(ππ) πΆπ1{ππ΅β§ππΆ>ππ}]
βπΈ[
π
β
π=1
π·(ππ) πΏππΏ (π
πβ1, ππβ1) 1{ππ΅β§ππΆ>ππ}])
Γ(πΈ[
π
β
π=1
π·(ππ) πΏπ1
{ππ΅β§ππΆ>ππ}])
β1
β
πΈ [π· (ππΆ+ π) 1
{ππΆβ€π}1{ππ΅>ππΆ+π}]
πΈ [βπ
π=1π·(ππ) πΏπ1
{ππ΅β§ππΆ>ππ}]
.
(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 π΅ and reference asset πΆare given, respectively, by
ππ΅
π‘= π0+ π11{ππΆβ€π‘}, (19)
ππΆ
π‘= π0+ π11{ππ΅β€π‘}, (20)
where π0and π0are nonnegative, satisfying π
0+ π1> 0 and
π0+ π1> 0.
With the result in Leung and Kwok [24], the joint survivalprobability and the joint density of default time (ππ΅, ππΆ) aregiven by
π (ππ΅> π‘1, ππΆ> π‘2)
=
{{{{{
{{{{{
{
πβπ0π‘1βπ0π‘2 [π0πβπ1(π‘1βπ‘2)β π1πβπ0(π‘1βπ‘2)
π0β π1
] , π‘2β€ π‘1
πβπ0π‘1βπ0π‘2 [π0πβπ1(π‘2βπ‘1)β π1πβπ0(π‘2βπ‘1)
π0β π1
] , π‘2> π‘1,
(21)
π (π‘1, π‘2)
=
{
{
{
π1(π‘1, π‘2) = π0(π0+ π1) πβ(π0+π1)π‘1β(π0βπ1)π‘2 , π‘2β€ π‘1
π2(π‘1, π‘2) = π0(π0+ π1) πβ(π0+π1)π‘2β(π0βπ1)π‘1 , π‘2> π‘1,
(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 π of TRS is given by
π =βπ
π=1(πΆπβ πΏππΏ (π
πβ1, ππβ1)) π΅ (0, π
π) πβ(π0+π0)ππ
βπ
π=1πΏππβ(π0+π0)ππ
β
π0πβ(π0+π1)πβ«π
0π΅ (0, V + π) πβ(π0+π0)VπV
βπ
π=1πΏππβ(π0+π0)ππ
.
(23)
Proof. By the arbitrage free principle, the price formula (18)becomes
π = (πΈ[
π
β
π=1
π·(ππ) πΆπ1{ππ΅β§ππΆ>ππ}]
βπΈ[
π
β
π=1
π·(ππ) πΏππΏ (π
πβ1, ππβ1) 1{ππ΅β§ππΆ>ππ}])
Γ (πΈ[
π
β
π=1
π·(ππ) πΏπ1
{ππ΅β§ππΆ>ππ}])
β1
β
πΈ [π· (ππΆ+ π) 1
{ππΆβ€π}1{ππ΅>ππΆ+π}]
πΈ [βπ
π=1π·(ππ)π1{ππ΅β§ππΆ>ππ}]
.
(24)
To obtain the analytic solution of π , we need to compute thefollowing three expectation values:
πΎ1:=
π
β
π=1
πΈ [π· (ππ) 1{ππ΅β§ππΆ>ππ}] ,
πΎ2:=
π
β
π=1
πΈ [π· (ππ) πΏ (ππβ1, ππβ1) 1{ππ΅β§ππΆ>ππ}] ,
πΎ3:= πΈ [π· (π
πΆ+ π) 1
{ππΆβ€π}1{ππ΅>ππΆ+π}] ,
(25)
where
πΎ1=
π
β
π=1
πΈ [π· (ππ) 1{ππ΅β§ππΆ>ππ}]
=
π
β
π=1
πΈ[exp(ββ«ππ
0
ππ’ππ’)]πΈ [1
{ππ΅β§ππΆ>ππ}]
=
π
β
π=1
exp(ββ«ππ
0
π (0, π’) ππ’)πΈ [1{ππ΅β§ππΆ>ππ}]
=
π
β
π=1
π΅ (0, ππ) πβ(π0+π0)ππ ,
πΎ2=
π
β
π=1
πΈ [π· (ππ) πΏ (ππβ1, ππβ1) 1{ππ΅β§ππΆ>ππ}]
=
π
β
π=1
π΅ (0, ππ) πΏ (ππβ1, ππβ1) πβ(π0+π0)ππ ,
πΎ3= πΈ [π· (π
πΆ+ π) 1
{ππΆβ€π}1{ππ΅>ππΆ+π}]
= πΈ [πΈ [π· (ππΆ+ π) 1
{ππΆβ€π}1{ππ΅>ππΆ+π}
| Fπ
πβ]]
= πΈ [π΅ (0, ππΆ+ π) 1
{ππΆβ€π}1{ππ΅>ππΆ+π}]
Abstract and Applied Analysis 5
= β«
π
0
β«
β
V+ππ΅ (0, V + π) π
2(π’, V) ππ’ πV
= β«
π
0
π΅ (0, V + π)β«β
V+ππ0(π0+ π1) πβ(π0+π1)π‘1β(π0βπ1)π‘2ππ’ πV
= π0πβ(π0+π1)πβ«
π
0
π΅ (0, V + π) πβ(π0+π0)VπV.
(26)
SubstituteπΎ1,πΎ2, andπΎ
3into formula (24), and we obtain
π =βπ
π=1(πΆπβ πΏππΏ (π
πβ1, ππβ1)) π΅ (0, π
π) πβ(π0+π0)ππ
βπ
π=1πΏππβ(π0+π0)ππ
β
π0πβ(π0+π1)πβ«π
0π΅ (0, V + π) πβ(π0+π0)VπV
βπ
π=1πΏππβ(π0+π0)ππ
.
(27)
We complete the proof.
Remark 2. From (23), we can conclude that when the lengthπ of settlement period is zero, the swap rate π is only relatedto the systematic factors π
0and π0, 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 π΅ and πΆ have the following forms:
ππ΅
π‘= π0+
π2
π1(π‘ β ππΆ) + 1
1{ππΆβ€π‘}, (28)
ππΆ
π‘= π0+
π2
π1(π‘ β ππ΅) + 1
1{ππ΅β€π‘}, (29)
where π0, π0, π1, and π
1are nonnegative, satisfying π
0+ π2> 0
and π0+ π2> 0. π
2and π2reflect the impact strength of the
counterparty and π1and π1show the attenuation speed of one
party default on the other party. For π1= π1= 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 π
2= βπ1; then, (28) and (29) become, respectively,
ππ΅
π‘= π0β
π1
π1(π‘ β ππΆ) + 1
1{ππΆβ€π‘}, (30)
ππΆ
π‘= π0β
π1
π1(π‘ β ππ΅) + 1
1{ππ΅β€π‘}. (31)
According to the result in Bai et al. [25], with theintensities (30)-(31), the joint survival probability and thejoint density function of (ππ΅, ππΆ) are given by
π (ππ΅> π‘1, ππΆ> π‘2)
=
{{{{{{{{{{{{
{{{{{{{{{{{{
{
π1(π‘1β π‘2+1
π1
β1
π0
) πβ(π0π‘1+π0π‘2)
+π1
π0
πβ(π0+π0)π‘1 , π‘
2β€ π‘1
π1(π‘2β π‘1+1
π1
β1
π0
) πβ(π0π‘1+π0π‘2)
+π1
π0
πβ(π0+π0)π‘2 , π‘
2> π‘1,
(32)
π (π‘1, π‘2)
=
{{{{{{{{
{{{{{{{{
{
π1(π‘1, π‘2) = π0π0π1[π‘1β π‘2+1
π1
β1
π0
]
Γπβ(π0π‘1+π0π‘2), t
2β€ π‘1
π2(π‘1, π‘2) = π0π0π1[π‘2β π‘1+1
π1
β1
π0
]
Γπβ(π0π‘1+π0π‘2), π‘
2> π‘1,
(33)
respectively.Then the expression of π is given by the following theorem.
Theorem 3. With the default intensities (30)-(31) and HJMinterest rate model (9), π is given by
π =βπ
π=1πΆππ΅ (0, π
π) πβ(π0+π0)ππ
βπ
π=1πΏππ΅ (0, π
π) πβ(π0+π0)ππ
ββπ
π=1πΏ (ππβ1, ππβ1) πΏππ΅ (0, π
π) πβ(π0+π0)ππ
βπ
π=1πΏππ΅ (0, π
π) πβ(π0+π0)ππ
β
π0(1 + π
1π) πβπ0πβ«π
0π΅ (0, V + π) πβ(π0+π0)VπV
βπ
π=1πΏππ΅ (0, π
π) πβ(π0+π0)ππ
.
(34)
Proof. To compute (18), we need to only compute the follow-ing three expectation values:
οΏ½ΜοΏ½1:=
π
β
π=1
πΈ [π· (ππ) 1{ππ΅β§ππΆ>ππ}] ,
οΏ½ΜοΏ½2:=
π
β
π=1
πΈ [π· (ππ) πΏ (ππβ1, ππβ1) 1{ππ΅β§ππΆ>ππ}] ,
οΏ½ΜοΏ½3:= πΈ [π· (π
πΆ+ πΏ) 1
{ππΆβ€π}1{ππ΅>ππΆ+πΏ}] ,
(35)
6 Abstract and Applied Analysis
where
οΏ½ΜοΏ½1=
π
β
π=1
πΈ [π· (ππ) 1{ππ΅β§ππΆ>ππ}]
=
π
β
π=1
π΅ (0, ππ) πβ(π0+π0)ππ ,
οΏ½ΜοΏ½2=
π
β
π=1
πΈ [π· (ππ) πΏ (ππβ1, ππβ1) 1{ππ΅β§ππΆ>ππ}]
=
π
β
π=1
πΏ (ππβ1, ππβ1) π΅ (0, π
π) πβ(π0+π0)ππ ,
οΏ½ΜοΏ½3= πΈ [π· (π
πΆ+ πΏ) 1
{ππΆβ€π}1{ππ΅>ππΆ+πΏ}]
= πΈ [πΈ [π· (ππΆ+ πΏ) 1
{ππΆβ€π}1{ππ΅>ππΆ+πΏ}
| Fπ
πβ]]
= πΈ [π΅ (0, ππΆ+ π) 1
{ππΆβ€π}1{ππ΅>ππΆ+π}]
= β«
π
0
β«
β
V+ππ΅ (0, V + π) π (π’, V) ππ’ πV
= β«
π
0
π΅ (0, V + π)
Γ β«
β
V+ππ0π0π1[π‘1β π‘2+1
π1
β1
π0
] πβ(π0π‘1+π0π‘2)ππ’ πV
= π0(1 + π
1π) πβπ0πβ«
π
0
π΅ (0, V + π) πβ(π0+π0)VπV.
(36)
Substituting (36) into (18), we can have
π =βπ
π=1πΆππ΅ (0, π
π) πβ(π0+π0)ππ
βπ
π=1πΏππ΅ (0, π
π) πβ(π0+π0)ππ
ββπ
π=1πΏ (ππβ1, ππβ1) πΏππ΅ (0, π
π) πβ(π0+π0)ππ
βπ
π=1πΏππ΅ (0, π
π) πβ(π0+π0)ππ
β
π0(1 + π
1π) πβπ0πβ«π
0π΅ (0, V + π) πβ(π0+π0)VπV
βπ
i=1 πΏππ΅ (0, ππ) πβ(π0+π0)ππ
,
(37)
which is formula (34). The proof is complete.
Remark 4. Combining formula (34) with formula (23), wecan conclude that firm πΆβs default contagion on firm π΅ haseffect in the pricing of TRS (there is π
1in formula (34)).
However, firm π΅βs default contagion on firm πΆ has no effectin the pricing of TRS (there is no π
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 π of the referenceassetβs settlement period is zero.
4.1. TRS Valuation under Two-Firm Looping Default Conta-gion Model. We consider that default intensities of firms π΅and πΆ have the following form:
ππ΅
π‘= π0+ πππ‘+ π11{ππΆβ€π‘},
ππΆ
π‘= π0+ πππ‘+ π11{ππ΅β€π‘},
(38)
where π0, π0, π, and π are nonnegative, satisfying π
0+ π1> 0
and π0+π1> 0. ππ‘= π(π‘, π‘)andπ(π‘, π) satisfies HJMmodel (9).
UsingWang andYeβs result [26], under themodel (38), thejoint conditional distribution of (ππ΅, ππΆ) is given by
π (ππ΅> π‘1, ππΆ> π‘2| Fπ
πβ)
= exp (βπ0π‘2β ππ 0,π‘2
)
Γ [ exp (βπ1(π‘2β π‘1)) β exp (β (π
0(π‘2β π‘1) β ππ
π‘1,π‘2
))
+ exp (βπ0π‘2β ππ 0,π‘2
)
+ π1β«
π‘2
π‘1
exp (βπ0(π’ β π‘
1) β π1(π‘2β π’) β ππ
π‘1,π’) ππ’] ,
for π‘1< π‘2< π.
π (ππ΅> π‘1, ππΆ> π‘2| Fπ
πβ)
= exp (βπ0π‘1β ππ 0,π‘1
)
Γ [ exp (βπ1(π‘1β π‘2)) β exp (β (π
0(π‘1β π‘2) β ππ
π‘2,π‘1
))
+ exp (βπ0π‘1β ππ 0,π‘1
)
+ π1β«
π‘1
π‘2
exp (βπ0(π’ β π‘
2) β π1(π‘1β π’) β ππ
π‘2,π’) ππ’] ,
for π‘2< π‘1< π.
(39)
Abstract and Applied Analysis 7
The joint conditional density function is
π (π‘1, π‘2| Fπ
πβ)
=
{{{{{{
{{{{{{
{
(π0+ πππ‘2
) (π0+ π1+ πππ‘1
)
Γπβ(π0+π1)π‘1β(π0βπ1)π‘2βππ 0,π‘1
βππ 0,π‘2 , π‘2β€ π‘1
(π0+ πππ‘1
) (π0+ π1+ πππ‘2
)
Γπβ(π0βπ1)π‘1β(π0+π1)π‘2βππ 0,π‘1
βππ 0,π‘2 , π‘2> π‘1,
(40)
where π 0,π’
= β«π’
0πV πV is the cumulative interest rate process
andFππβ is the filter generated by π
π‘up to πβ.
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), π π‘,π
=
β«π
π‘ππ’ππ’, for π‘ < π . Denote
πΊ1(π1; π‘0, π‘1) = πΈ [exp (βπ
1π π‘0,π‘1
)] ,
πΊ2(π1, π2; π‘0, π‘1, π‘2) = πΈ [exp (βπ
1π π‘0,π‘1
β π2π π‘1,π‘2
)] ;
(41)
then,
πΊ1(π1; π‘0, π‘1)=exp(βπ
1β«
π‘1
π‘0
π (π‘0, π’) ππ’
βπ1β«
π‘1
π‘0
β«
π’
π‘0
πΌ (π , π’) ππ ππ’
+π2
1
2β«
π‘1
π‘0
(β«
π‘1
π
π (π , π’) ππ’)
2
ππ ) ,
(42)
πΊ2(π1, π2; π‘0, π‘1, π‘2)
= exp(βπ1β«
π‘1
π‘0
π (π‘0, π’) ππ’
β π1β«
π‘1
π‘0
β«
π’
π‘0
πΌ (π , π’) ππ ππ’)
β exp(βπ2β«
π‘2
π‘1
π (π‘1, π’) ππ’
β π2β«
π‘2
π‘1
β«
π’
π‘1
πΌ (π , π’) ππ ππ’)
β exp(π2
1
2β«
π‘1
π‘0
(β«
π‘1
π
π (π , π’) ππ’)
2
ππ
+π2
2
2β«
π‘2
π‘1
(β«
π‘2
π
π (π , π’) ππ’)
2
ππ ) .
(43)
Proof. Consider
π π‘0,π‘1
= β«
π‘1
π‘0
π (π‘0, π’) ππ’ + β«
π‘1
π‘0
β«
π’
π‘0
πΌ (π , π’) ππ ππ’
+ β«
π‘1
π‘0
β«
π’
π‘0
π (π , π’) πππ ππ’
= β«
π‘1
π‘0
π (π‘0, π’) ππ’ + β«
π‘1
π‘0
β«
π’
π‘0
πΌ (π , π’) ππ ππ’
+ β«
π‘1
π‘0
β«
π‘1
π
π (π , π’) ππ’ πππ .
(44)
We can deduceπΈ [βπ
1π π‘0,π‘1
] = βπ1πΈ [π π‘0,π‘1
]
= βπ1β«
π‘1
π‘0
π (π‘0, π’) ππ’ β β«
π‘1
π‘0
β«
π’
π‘0
πΌ (π , π’) ππ ππ’,
Var [βπ1π π‘0,π‘1
] = π2
1β«
π‘1
π‘0
(β«
π‘1
π
π (π , π’) ππ’)
2
ππ .
(45)
By using the formula
πΈ [exp (βπ1π π‘0,π‘1
)]
= exp(πΈ [βπ1π π‘0,π‘1
] +1
2Var [βπ
1π π‘0,π‘1
]) .
(46)
And substituting (45),(46) into (47), we can obtain (43).Formula (43) can be rewritten asπΊ2(π1, π2; π‘0, π‘1, π‘2)
= πΈ [exp (βπ1π π‘0,π‘1
β π2π π‘1,π‘2
)]
= πΈ [exp (βπ1π π‘0,π‘1
) πΈπ‘1
[exp (βπ2π π‘1,π‘2
)]]
= πΈ [exp (βπ1π π‘0,π‘1
) πΏ1(π2; π‘1, π‘2)] .
(47)
Similar to the computation of (42), (43) can be obtained, andwe omit it here.
Since the length π of settlement period for reference assetπΆ is 0; namely, if firm π΅ has no default, π΅ pays compensationfor πΆβs loss immediately once πΆ defaults, thus (18) becomes
π = (πΈ[
π
β
π=1
π·(ππ) πΆπ1{ππ΅β§ππΆ>ππ}]
βπΈ[
π
β
π=1
π·(ππ) πΏππΏ (π
πβ1, ππβ1) 1{ππ΅β§ππΆ>ππ}])
Γ(πΈ[
π
β
π=1
π·(ππ) πΏπ1
{ππ΅β§ππΆ>ππ}])
β1
β
πΈ [π· (ππΆ) 1{ππΆβ€π}1{ππ΅>ππΆ}]
πΈ [βπ
π=1π·(ππ)π1{ππ΅β§ππΆ>ππ}]
.
(48)
The price π of TRS is given by the theorem below.
8 Abstract and Applied Analysis
Theorem 6. With default intensities (38) and HJMmodel (9),the price π of TRS is given by
π =βπ
π=1πΆπexp (β (π
0+ π0) ππ) β πΊ1((1 + π + π) ; 0, π
π)
βπ
π=1πΏπ exp (β (π
0+ π0) ππ) β πΊ1((1 + π + π) ; 0, π
π)
β (
π
β
π=1
M (πΊ2(1 + π + π, π + π; 0, π
πβ1, ππ, π‘2)
βπΊ1(1 + π + π; 0, π
π)))
Γ (
π
β
π=1
πΏπ exp (β (π0+π0) ππ) β πΊ1((1+π+ π) ; 0, π
π))
β1
β
(π/ (1 + π + π)) (1 β πβ(π+π)π
β πΊ1(1 + π + π; 0, π))
βπ
π=1πΏπ exp (β (π
0+ π0) ππ) β πΊ1((1 + π + π) ; 0, π
π)
β((π0βπ (π0+π0)
1+π+π)Γβ«
T
0
πβ(π0+π0)Vβ πΊ1(1+π +π; 0, V) πV)
Γ (
π
β
π=1
πΏπ exp (β (π0+π0) ππ) β πΊ1((1+π+π) ; 0, π
π))
β1
.
(49)
Proof. Consider
π = (πΈ[
π
β
π=1
π·(ππ) πΆπ1{ππ΅β§ππΆ>ππ}]
β πΈ[
π
β
π=1
π·(ππ) πΏππΏ (π
πβ1, ππβ1) 1{ππ΅β§ππΆ>ππ}])
Γ (πΈ[
π
β
π=1
π·(ππ) πΏπ1
{ππ΅β§ππΆ>ππ}])
β1
β
πΈ [π· (ππΆ) 1{ππΆβ€π}1{ππ΅>ππΆ}]
πΈ [βπ
π=1π·(ππ)π1{ππ΅β§ππΆ>ππ}]
.
(50)
To compute (48), we need to only deduce the following threeexpectation values:
οΏ½ΜοΏ½1:=
π
β
π=1
πΈ [π· (ππ) 1{ππ΅β§ππΆ>ππ}] ,
οΏ½ΜοΏ½2:=
π
β
π=1
πΈ [π· (ππ) πΏ (ππβ1, ππβ1) 1{ππ΅β§ππΆ>ππ}] ,
οΏ½ΜοΏ½3:= πΈ [π· (π
πΆ) 1{ππΆβ€π}1{ππ΅>ππΆ}] ,
(51)
where
οΏ½ΜοΏ½1=
π
β
π=1
πΈ [π· (ππ) 1{ππ΅β§ππΆ>ππ}]
=
π
β
π=1
πΈ [πΈ [π· (ππ) 1{ππ΅β§ππΆ>ππ}| Fπ
πβ]]
=
π
β
π=1
πΈ [πβπ 0,ππ β πΈ [1
{ππ΅β§ππΆ>ππ}| Fπ
πβ]]
=
π
β
π=1
πΈ [exp (β (π0+ π0) ππ) β exp (β (1 + π + π) π
0,ππ
)]
=
π
β
π=1
exp (β (π0+ π0) ππ) β πΊ1((1 + π + π) ; 0, π
π) ,
οΏ½ΜοΏ½2=
π
β
π=1
πΈ [π· (ππ) πΏ (ππβ1, ππβ1) 1{ππ΅β§ππΆ>ππ}]
=
π
β
π=1
πΈ [πΈ [π· (ππ) πΏ (ππβ1, ππβ1) 1{ππ΅β§ππΆ>ππ}| Fπ
πβ]] .
(52)
Substituting the expression (14) of Libor interest rate πΏ(π‘, π)into the formula above, we have
οΏ½ΜοΏ½2=
π
β
π=1
πΈ [πΈ [πβπ 0,ππ
1
πΏ(ππ ππβ1,ππβ1+πΏ β 1) 1
{ππ΅β§ππΆ>ππ}| Fπ
πβ]]
=
π
β
π=1
πΈ [1
πΏ(πβπ 0,ππβ1 β π
βπ 0,ππ ) πΈ [1
{ππ΅β§ππΆ>ππ}| Fπ
πβ]]
=
π
β
π=1
1
πΏ(πΊ2(1 + π + π, π + π; 0, π
πβ1, ππ)
βπΊ1(1 + π + π; 0, π
π)) ,
(53)
οΏ½ΜοΏ½3= πΈ [π· (π
πΆ) 1{ππΆβ€π}1{ππ΅>ππΆ}]
= πΈ [πΈ [π· (ππΆ) 1{ππΆβ€π}1{ππ΅>ππΆ}| Fπ
πβ]]
= πΈ [β«
π
0
β«
β
Vπβπ 0,Vπ (π’, V | Fπ
πβ) ππ’ πV]
= πΈ[β«
π
0
(π0+ ππV) π
β(π0βπ1)Vβ(1+π)π
0,V
Γ β«
β
V(π0+ π1+ πππ’) πβ(π0+π1)π’βππ
0,π’ππ’ πV]
Abstract and Applied Analysis 9
= πΈ[(π0βπ (π0+ π0)
1 + π + π)
Γβ«
π
0
exp (β (π0+π0) V) exp (β (1+π+π) π
0,V) πV
+π
1 + π + π
Γ ( (1 β exp (β (π0+ π0) π β (1 + π + π) π
0,π)) ]
=π
1 + π + π(1 β exp (β (π
0+ π0) π)
β πΊ1(1 + π + π; 0, π))
+ (π0βπ (π0+ π0)
1 + π + π)
Γ β«
π
0
exp (β (π0+ π0) V) β πΊ
1(1 + π + π; 0, V) πV.
(54)
Thus, substituting (52)β(54) into (48), we obtain
π =βπ
π=1πΆπexp (β (π
0+ π0) ππ) β πΊ1((1 + π + π) ; 0, π
π)
βπ
π=1πΏπ exp (β (π
0+ π0) ππ) β πΊ1((1 + π + π) ; 0, π
π)
β (
π
β
π=1
π(πΊ2(1 + π + π, π + π; 0, π
πβ1, ππ)
βπΊ1(1 + π + π; 0, π
π)))
Γ(
π
β
π=1
πΏπ exp (β (π0+π0) ππ) β πΊ1((1+π+π) ; 0, π
π))
β1
β
(π/ (1 + π + π)) (1 β πβ(π+π)π
β πΊ1(1 + b + π; 0, π))
βπ
π=1πΏπ exp (β (π
0+ π0) ππ) β πΊ1((1 + π + π) ; 0, π
π)
β ((π0βπ (π0+ π0)
1 + π + π)
Γ β«
π
0
πβ(π0+π0)Vβ πΊ1(1 + π + π; 0, V) πV)
Γ(
π
β
π=1
πΏπ exp (β(π0+π0) ππ) β πΊ1((1+π+π) ; 0, π
π))
β1
.
(55)
The proof is complete.
Remark 7. From the expression (49), the price π of TRS isrelated to the default free interest rate risk (because there are πand π). 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 π΅and πΆ are described as follows:
ππ΅
π‘= π0+ πππ‘+
π2
π1(π‘ β ππΆ) + 1
1{ππΆβ€π‘},
ππΆ
π‘= π0+ πππ‘+
π2
π1(π‘ β ππ΅) + 1
1{ππ΅β€π‘},
(56)
where π0, π0, π, π, π
1, and π
1are nonnegative, satisfying π
0+π2>
0 and π0+π2> 0. When π
1= π1= 0, model is simplified to the
looping default model (38).To obtain the analytic solution of π , we consider the
following simplified model:
ππ΅
π‘= π0+ πππ‘β
π1
π1(π‘ β ππΆ) + 1
1{ππΆβ€π‘},
ππΆ
π‘= π0+ πππ‘β
π1
π1(π‘ β ππ΅) + 1
1{ππ΅β€π‘}.
(57)
By the result in [27], the joint conditional survival probabilityof (ππ΅, ππΆ) under the model (57) is given by
π (ππ΅> π‘1, ππΆ> π‘2| Fπ
πβ)
= (π1(π‘2β π‘1) + 1) π
β(π0π‘2+ππ 0,π‘2
)
+ πβ(π0+π0)π‘2 exp (β (π + π) π
0,π‘2
)
β π1πβπ0π‘2 β«
π‘2
π‘1
πβπ0(π βπ‘1) exp (βππ
0,π‘2
β ππ π‘1,π ) ππ
β πβπ0(π‘2βπ‘1)βπ0π‘2 exp (βππ
0,π‘2
β ππ π‘1,π‘2
)
for π‘1β€ π‘2β€ π,
π (ππ΅> π‘1, ππΆ> π‘2| Fπ
πβ)
= (π1(π‘1β π‘2) + 1) π
β(π0π‘1+ππ 0,π‘1
)
+ πβ(π0+π0)π‘1 exp (β (π + π) π
0,π‘1
)
β π1πβπ0π‘1 β«
π‘1
π‘2
πβπ0(π βπ‘2) exp (βππ
0,π‘1
β ππ π‘2,π ) ππ
β πβπ0(π‘1βπ‘2)π0π‘1 exp (βππ
0,π‘1
β ππ π‘2,π‘1
)
for π‘2β€ π‘1β€ π,
(58)
10 Abstract and Applied Analysis
and the joint conditional density function is given by
π (π‘1, π‘2| Fπ
πβ)
=
{{{{{{
{{{{{{
{
(π0+ πππ‘1
) [(π0+ πππ‘2
) (π1(π‘2β π‘1) + 1) β π
1]
Γπβπ0π‘1βπ0π‘2βππ 0,π‘1
βππ 0,π‘2 , π‘
1β€ π‘2
(c0+ πππ‘2
) [(π0+ πππ‘1
) (π1(π‘1β π‘2) + 1) β π
1]
Γπβπ0π‘1βπ0π‘2βππ 0,π‘1
βππ 0,π‘2 . π‘
2β€ π‘1.
(59)
Under the model (57), the price formula of π is given by
π = (πΈ[
π
β
π=1
π·(ππ) πΆπ1{ππ΅β§ππΆ>ππ}]
β πΈ[
π
β
π=1
π·(ππ) πΏππΏ (π
πβ1, ππβ1) 1{ππ΅β§ππΆ>ππ}])
Γ (πΈ[
π
β
π=1
π·(ππ) πΏπ1
{ππ΅β§ππΆ>ππ}])
β1
β
πΈ [π· (ππΆ) 1{ππΆβ€π}1{ππ΅>ππΆ}]
πΈ [
π
β
π=1
π·(ππ) πΏπ1
{ππ΅β§ππΆ>ππ}]
.
(60)
Combining with Lemma 5, we can obtain the price π ofTRS, and the result is the following theorem.
Theorem 8. With default intensities (57) and HJM interestrate model (9), the price π of TRS is given by
π =βπ
π=1πΆπexp (β (π
0+ π0) ππ) β πΊ1((1 + π + π) ; 0, π
π)
βπ
π=1πΏπ exp (β (π
0+ π0) ππ) β πΊ1((1 + π + π) ; 0, π
π)
β (
π
β
π=1
π(πΊ2(1 + π + π, π + π; 0, π
πβ1, ππ)
β πΊ1(1 + π + π; 0, π
π)))
Γ (
π
β
π=1
πΏπ exp (β (π0+π0) ππ) β πΊ1((1+π +π) ; 0, π
π))
β1
β
(π/ (1 + π + π)) (1 β πβ(π+π)π
β πΊ1(1 + π + π; 0, π))
βπ
π=1πΏπ exp (β (π
0+ π0) ππ) β πΊ1((1 + π + π) ; 0, π
π)
β ((π0βπ (π0+ π0)
1 + π + π)
Γ β«
π
0
πβ(π0+π0)Vβ πΊ1(1 + π + π; 0, V) πV)
Γ(
π
β
π=1
πΏπ exp (β(π0+π0) ππ) β πΊ1((1+π+π) ; 0, π
π))
β1
.
(61)
Proof. From (60), to obtain the analytic solution of π , we needto only compute three expectation values:
πΎ1:=
π
β
π=1
πΈ [π· (ππ) 1{ππ΅β§ππΆ>ππ}] ,
πΎ2:=
π
β
π=1
πΈ [π· (ππ) πΏ (ππβ1, ππβ1) 1{ππ΅β§ππΆ>ππ}] ,
πΎ3:= πΈ [π· (π
πΆ) 1{ππΆβ€π}1{ππ΅>ππΆ}] ,
(62)
where
πΎ1=
π
β
π=1
πΈ [π· (ππ) 1{ππ΅β§ππΆ>ππ}]
=
π
β
π=1
πΈ [πΈ [π· (ππ) 1{ππ΅β§ππΆ>ππ}| Fπ
πβ]]
=
π
β
π=1
πΈ [πβπ 0,ππ β πΈ [1
{ππ΅β§ππΆ>ππ}| Fπ
πβ]]
=
π
β
π=1
exp (β (π0+ π0) ππ) β πΊ1((1 + π + π) ; 0, π
π) ,
πΎ2=
π
β
π=1
πΈ [π· (ππ) πΏ (ππβ1, ππβ1) 1{ππ΅β§ππΆ>ππ}]
=
π
β
π=1
πΈ [πΈ [1
πΏπβπ 0,ππ (ππ ππβ1,ππβ1+πΏ β 1) 1
{ππ΅β§ππΆ>ππ}| Fπ
πβ]]
=
π
β
π=1
1
πΏ(πΊ2(1 + π + π, π + π; 0, π
πβ1, ππ)
β πΊ1(1 + π + π; 0, π
π)) ,
πΎ3= πΈ [π· (π
πΆ) 1{ππΆβ€π}1{ππ΅>ππΆ}]
= πΈ [πΈ [π· (ππΆ) 1{ππΆβ€π}1{ππ΅>ππΆ}| Fπ
πβ]]
= πΈ[(π0βπ (π0+ π0)
1 + π + π)
Γ β«
π
0
exp (β (π0+ π0) V)
Abstract and Applied Analysis 11
Γ exp (β (1+π +π) π 0,V) πV
+π
1 + π + π
Γ (1 β exp (β (π0+ π0) π β (1 + π + π) π
0,π) ]
= (π0βπ (π0+ π0)
1 + π + π)
Γ β«
π
0
exp (β (π0+ π0) V) β πΊ
1(1 + π + π; 0, V) πV
+π
1 + π + π(1 β exp (β (π
0+ π0) π)
β πΊ1(1 + π + π; 0, π)) .
(63)
Substituting expressions (63) of πΎ1, πΎ2, and πΎ
3into (60), we
can obtain (61). We complete the proof.
Remark 9. From formulas (49) and (61), we conclude thatif the length π of the reference assetβs settlement period is0, the price π 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).
References
[1] R. C. Merton, βOn the pricing of corporate debt: the riskstructure of interest rates,β The Journal of Finance, vol. 29, no.2, pp. 449β470, 1974.
[2] F. Black and J. C. Cox, βValuing corporate securities: someeffects of bond indenture provisions,β The Journal of Finance,vol. 31, no. 2, pp. 351β367, 2007.
[3] R. Geske, βThe valuation of corporate liabilities as compoundoptions,βThe Journal of Financial andQuantitative Analysis, vol.12, no. 4, pp. 541β552, 1977.
[4] P. Artzner and F. Delbaen, βDefault risk insurance and incom-plete markets,βMathematical Finance, vol. 5, no. 3, pp. 187β195,1995.
[5] D. Duffie, M. Schroder, and C. Skiadas, βRecursive valuationof defaultable securities and the timing of resolution of uncer-tainty,βTheAnnals of Applied Probability, vol. 6, no. 4, pp. 1075β1090, 1996.
[6] R. A. Jarrow and S. M. Turnbull, βPricing options on financialsecurities subject to default risk,βThe Journal of Finance, vol. 50,no. 1, pp. 53β86, 1995.
[7] D. B. Madan and H. Unal, βPricing the risks of default,β Reviewof Derivatives Research, vol. 2, no. 2-3, pp. 121β160, 1998.
[8] D. Duffie and D. Lando, βTerm structures of credit spreads withincomplete accounting information,β Econometrica, vol. 69, no.3, pp. 633β664, 2001.
[9] R. A. Jarrow, D. Lando, and S. M. Turnbull, βA Markov modelfor the term structure of credit risk spreads,β The Review ofFinancial Studies, vol. 10, no. 2, pp. 481β523, 1997.
[10] R. B. Litterman and T. Iben, βCorporate Bond Valuation andTerm Structure of Credit Spreads, Financial Anal,βThe Journalof Portfolio Management, vol. 17, no. 3, pp. 52β64, 1991.
[11] D. Duffie and K. J. Singleton, βModeling term structures ofdefaultable bonds,β Review of Financial Studies, vol. 12, no. 4,pp. 687β720, 1999.
[12] D. Lando, βOn cox processes and credit risky securities,β Reviewof Derivatives Research, vol. 2, no. 2-3, pp. 99β120, 1998.
[13] P. Collin-Dufresne and B. Solnik, βOn the term structure ofdefault premia in the swap and LIBOR markets,β The Journalof Finance, vol. 56, no. 3, pp. 1095β1115, 2001.
[14] G. R. Duffee, βEstimating the price of default risk,β Review ofFinancial Studies, vol. 12, no. 1, pp. 197β226, 1999.
[15] R. A. Jarrow and F. Yu, βCounterparty risk and the pricing ofdefaultable securities,βThe Journal of Finance, vol. 56, no. 5, pp.1765β1799, 2001.
[16] R. J. Elliott, M. Jeanblanc, and M. Yor, βOn models of defaultrisk,βMathematical Finance, vol. 10, no. 2, pp. 179β195, 2000.
[17] F. Jamshidian,Valuation of Credit Default Swaps and Swaptions,NIB Capital Bank, The Hague, The Netherlands, 2003.
[18] Z. X. Ye andR.X. Zhuang, βPricing of total return swap,βChineseJournal of Applied Probability and Statistics, vol. 28, no. 1, pp. 79β86, 2012.
[19] S. R. Das and R. K. Sundaram, βAn integrated model for hybridsecurities,β Management Science, vol. 53, no. 9, pp. 1439β1451,2007.
[20] J. M. Harrison and D. M. Kreps, βMartingales and arbitragein multiperiod securities markets,β Journal of Economic Theory,vol. 20, no. 3, pp. 381β408, 1979.
[21] J.M.Harrison and S. R. Pliska, βMartingales and stochastic inte-grals in the theory of continuous trading,β Stochastic Processesand Their Applications, vol. 11, no. 3, pp. 215β260, 1981.
[22] D. Heath, R. Jarrow, andA.Morton, βBond pricing and the termstructure of interest rates: a new methodology for contingentclaims valuation,β Econometrica, vol. 60, no. 1, pp. 77β105, 1992.
[23] S. E. Shreve, Stochastic Calculus for Finance: Continuous-TimeModels, Springer, New York, NY, USA, 2004.
12 Abstract and Applied Analysis
[24] S. Y. Leung and Y. K. Kwok, βCredit default swap valuation withcounterparty risk,βTheKyoto Economic Review, vol. 74, no. 1, pp.25β45, 2005.
[25] Y.-F. Bai, X.-H. Hu, and Z.-X. Ye, βA model for dependentdefault with hyperbolic attenuation effect and valuation ofcredit default swap,β Applied Mathematics and Mechanics, vol.28, no. 12, pp. 1643β1649, 2007.
[26] A.-J. Wang and Z.-X. Ye, βThe pricing of credit risky securitiesunder stochastic interest rate model with default correlation,βApplications of Mathematics, vol. 58, no. 6, pp. 703β727, 2013.
[27] R.-L. Hao and Z.-X. Ye, βThe intensity model for pricingcredit securities with jump diffusion and counterparty risk,βMathematical Problems in Engineering, vol. 2011, Article ID412565, 16 pages, 2011.
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Stochastic AnalysisInternational Journal of
Top Related