# The Intensity Model for Pricing Credit Securities with Jump Diffusion and Counterparty Risk

Authors: Ruili Hao; Zhongxing Ye
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

We present an intensity-based model with counterparty risk. We assume
the default intensity of firm depends on the stochastic interest rate driven by the jump-diffusion
process and the default states of counterparty firms. Furthermore, we make use
of the techniques in Park (2008) to compute the conditional distribution of default times
and derive the explicit prices of bond and CDS. These are extensions of the models in
Jarrow and Yu (2001).


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## Body

## 1. Introduction

As credit securities are actively traded and the financial market becomes complex, the valuation of credit securities has called for more effective models according to the real market. Until now, there have been mainly two basic models: the structural model and the reduced-form model. In the first model, the firm's default is governed by the value of its assets and debts, while the default in the reduced-form model is governed by the exogenous factor.The structural approach was pioneered by Merton [1], then extended by Black and Cox [2] and Longstaff and Schwartz [3], assuming the default before the maturity date and others. In the above models, the asset process was all driven by the Brownian motion. Since the asset value may suffer a sudden drop for the reason of some events in the economy, Zhou [4] provided a jump-diffusion model with credit risk in which jump amplitude followed a log-normal distribution and valuated defaultable securities. In his model, Zhou gave the explicit expressions of defaultable securities' prices when the default occurred at the maturity date T, but only gave a tractably simulating approach when the firm defaulted before time T. For the first-passage-time models of credit risk with jump-diffusion process, Steve and Amir [5] and Zhang and Melnik [6] used the approach of Brownian bridge to estimate the jump-diffusion process and priced barrier options. Kou and Wang [7] and Kou et al. [8] made use of the Laplace transform to valuate the credit risk after estimating the jump-diffusion process with an infinitesimal generator. For the problem of the valuation of credit derivatives involving jump-diffusion process, it is still difficult to get explicit results in the event of defaulting before the maturity date, despite using the above approaches. However, it is more convenient to use reduced-form approach for valuating the credit risk in such situation.Comparing with the structural approach, the reduced-form approach is flexible and tractable in the real market. It is pioneered by Jarrow et al. [9] and Duffie and Singleton [10]. They introduced exogenous mechanism of firm's default. Their models considered the default as a random event which is controlled by a exogenous intensity process.Davis and Lo [11] firstly proposed the model of credit contagion to account for concentration risk in large portfolios of defaultable securities (DL Model). Later, motivated by a series of events such as the South Korean banking crisis, Long Term Capital Management's potential default and so on, Jarrow and Yu [12] thought the traditionally structural and reduced-form models were full of problems because they all ignored the firm's specific source of credit risk. They made use of the Davis's contagious model and introduced the concept of counterparty risk which is from the default of firm's counterparties. In their models, they paid more attention to the primary-secondary framework in which the intensity of default was influenced by the economy-wide state variables and the default state of the counterparty. Later, there are also other similar applications such as Leung and Kwork [13], Bai et al. [14] and so on. In recent years, some authors applied this into portfolio credit securities such as Yu [15] and Leung and Kwok [16]. Nevertheless, the stochastic interest rate in the above models still was driven by diffusion processes.At present, aggregate credit risk is still one of the most pervasive threats in the financial markets, which is from the contagious risk caused by business counterparties. In this paper, we mainly discuss the pricing of defaultable securities in primary-secondary framework, extending the models in Jarrow and Yu [12]. We consider that the macroeconomic variables contain the risk-free interest rate which shows the interaction between credit risk and market risk. However, the interest rate may drop suddenly due to some events in the modern economy. Therefore, we allow the stochastic interest rate to follow a jump-diffusion process rather than the continuous diffusion process in Jarrow and Yu [12]. Thus, our model not only reflects the real market much better, but more precisely to identify the impact of counterparty risk on the valuation of credit securities. Moreover, we apply the techniques in Park [17] to the pricing of bond and CDS, so that we avoid solving the PDEs.


## 2. Model

### 2.1. The Setting of Model

Let(Ω,ℱ,{ℱt}t=0T*,P) be the filtered probability space satisfying the usual conditions, where ℱ=ℱT*, T* is large enough but finite and P is an equivalent martingale measure under which discounted securities' prices are martingales.On(Ω,ℱ,{ℱt}t=0T*,P), there is an ℛd-valued process X=(X1,…,Xd), where {Xi}i=1d are Markov processes and represent d economy-wide state variables. Beside these, there are n companies with n point processes {Ni}i=1n(N0i=0) which represent the default processes of n companies, respectively. When Ni first jumps from 0 to 1, we call the company i defaults and denote τi be the default time of company i. Thus, Nti=1{τi≤t}, where 1{·} is the indicator function.The filtrationℱ is generated by the state variables and the default processes of n companies as follows (2.1)Ft=FtX∨Ht1∨⋯∨Htn,
where (2.2)FtX=σ(Xs,0≤s≤t),Hti=σ(Nsi,0≤s≤t).
Denote that (2.3)Ht=Ht1∨⋯∨Htn,Ht-i=Ht1∨⋯∨Hti-1∨Hti+1∨⋯∨Htn.
We assume the default time τi(i=1,…,n) possesses a strictly positive ℱT*X∨ℋT*-i-adapted intensity process λti satisfying ∫0tλsids<∞, P-a.s. for all t∈[0,T*]. The intensity process λti shows the local default probability in the sense that the default probability of company i over a small interval (t,t+Δt) is equal to λtiΔt. These Ni, 1≤i≤n generate the defaults of n companies. Their intensity processes λi, 1≤i≤n depend on state variables and the default states of all other companies. Due to the counterparty risk, {τi}i=1n may no longer be assumed independent conditionally on ℱX.


### 2.2. Primary-Secondary Framework

We dividen firms into two mutually exclusive types: l primary firms and n-l secondary firms. Primary firms' default processes only depend on state variables, while secondary firms' default processes depend on the state variables and the default states of the primary firms. This model was proposed by Jarrow and Yu [12]. Now, we provide some assumptions of the model.Assumption 1 (economy-wide state variables).
The state variableXt may contain the risk-free spot rate rt or other economical variables in the economy environment which may impact on the default probability of the companies.Assumption 2 (the default times).
On(Ω,ℱ,{ℱt}t=0T*,P), we add several independent unit exponential random variables {Ei,1≤i≤l} which are independent of X under probability measure P. The default times of l primary firms can be defined as
(2.4)τi=inf⁡{t:∫0tλsids≥Ei},1≤i≤l,
where λti is adapted to ℱtX. Then, we add another series independent unit exponential random variables {Ej,l+1≤j≤n} which are independent of X and τi, 1≤i≤l. The default times of n-l secondary firms can be defined as
(2.5)τj=inf⁡{t:∫0tλsjds≥Ej},1+l≤j≤n,
where λtj is adapted to ℱtX∨ℋt1∨⋯∨ℋtl.Assumption 3 (the default probability).
BecauseEi(1≤i≤l) is independent of state variables X, the conditional and unconditional survival probability distributions of primary firm i are given by
(2.6)P(τi>t∣FT*X)=exp⁡(-∫0tλsids),(2.7)P(τi>t)=E[exp⁡(-∫0tλsids)],t∈[0,T*].
Similarly, since Ej(1+l≤j≤n) is independent of X and τi, 1≤i≤l, we have the conditional and unconditional survival probability distributions of secondary firm j(2.8)P(τj>t∣FT*X∨HT*1∨⋯∨HT*l)=exp⁡(-∫0tλsjds),P(τj>t)=E[exp⁡(-∫0tλsjds)],t∈[0,T*].Assumption 4 (the default intensity).
Because the Primary firms' default processes only depend on macrovariables, we denote their default intensities by(2.9)λti=Λ0,ti,1≤i≤l.
In addition, secondary firms' default processes depend on the macrovariables and the default processes of the primary firms. We denote the intensities by
(2.10)λtj=Λ0,tj+Σk=1lΛk,tj1{τk≤t},1+l≤j≤n,
where Λk,tj is adapted to ℱT*X for all k. Λ0,ti and Λ0,tj can be constants or stochastic processes which are correlated with the state variables.Assumption 5 (the risk-free interest rate).
The risk-free interest ratert in this framework is stochastic which may follow CIR model, HJM model, Vasicek model or their extensions. It has effect on the defaults of n companies.


## 2.1. The Setting of Model

Let(Ω,ℱ,{ℱt}t=0T*,P) be the filtered probability space satisfying the usual conditions, where ℱ=ℱT*, T* is large enough but finite and P is an equivalent martingale measure under which discounted securities' prices are martingales.On(Ω,ℱ,{ℱt}t=0T*,P), there is an ℛd-valued process X=(X1,…,Xd), where {Xi}i=1d are Markov processes and represent d economy-wide state variables. Beside these, there are n companies with n point processes {Ni}i=1n(N0i=0) which represent the default processes of n companies, respectively. When Ni first jumps from 0 to 1, we call the company i defaults and denote τi be the default time of company i. Thus, Nti=1{τi≤t}, where 1{·} is the indicator function.The filtrationℱ is generated by the state variables and the default processes of n companies as follows (2.1)Ft=FtX∨Ht1∨⋯∨Htn,
where (2.2)FtX=σ(Xs,0≤s≤t),Hti=σ(Nsi,0≤s≤t).
Denote that (2.3)Ht=Ht1∨⋯∨Htn,Ht-i=Ht1∨⋯∨Hti-1∨Hti+1∨⋯∨Htn.
We assume the default time τi(i=1,…,n) possesses a strictly positive ℱT*X∨ℋT*-i-adapted intensity process λti satisfying ∫0tλsids<∞, P-a.s. for all t∈[0,T*]. The intensity process λti shows the local default probability in the sense that the default probability of company i over a small interval (t,t+Δt) is equal to λtiΔt. These Ni, 1≤i≤n generate the defaults of n companies. Their intensity processes λi, 1≤i≤n depend on state variables and the default states of all other companies. Due to the counterparty risk, {τi}i=1n may no longer be assumed independent conditionally on ℱX.


## 2.2. Primary-Secondary Framework

We dividen firms into two mutually exclusive types: l primary firms and n-l secondary firms. Primary firms' default processes only depend on state variables, while secondary firms' default processes depend on the state variables and the default states of the primary firms. This model was proposed by Jarrow and Yu [12]. Now, we provide some assumptions of the model.Assumption 1 (economy-wide state variables).
The state variableXt may contain the risk-free spot rate rt or other economical variables in the economy environment which may impact on the default probability of the companies.Assumption 2 (the default times).
On(Ω,ℱ,{ℱt}t=0T*,P), we add several independent unit exponential random variables {Ei,1≤i≤l} which are independent of X under probability measure P. The default times of l primary firms can be defined as
(2.4)τi=inf⁡{t:∫0tλsids≥Ei},1≤i≤l,
where λti is adapted to ℱtX. Then, we add another series independent unit exponential random variables {Ej,l+1≤j≤n} which are independent of X and τi, 1≤i≤l. The default times of n-l secondary firms can be defined as
(2.5)τj=inf⁡{t:∫0tλsjds≥Ej},1+l≤j≤n,
where λtj is adapted to ℱtX∨ℋt1∨⋯∨ℋtl.Assumption 3 (the default probability).
BecauseEi(1≤i≤l) is independent of state variables X, the conditional and unconditional survival probability distributions of primary firm i are given by
(2.6)P(τi>t∣FT*X)=exp⁡(-∫0tλsids),(2.7)P(τi>t)=E[exp⁡(-∫0tλsids)],t∈[0,T*].
Similarly, since Ej(1+l≤j≤n) is independent of X and τi, 1≤i≤l, we have the conditional and unconditional survival probability distributions of secondary firm j(2.8)P(τj>t∣FT*X∨HT*1∨⋯∨HT*l)=exp⁡(-∫0tλsjds),P(τj>t)=E[exp⁡(-∫0tλsjds)],t∈[0,T*].Assumption 4 (the default intensity).
Because the Primary firms' default processes only depend on macrovariables, we denote their default intensities by(2.9)λti=Λ0,ti,1≤i≤l.
In addition, secondary firms' default processes depend on the macrovariables and the default processes of the primary firms. We denote the intensities by
(2.10)λtj=Λ0,tj+Σk=1lΛk,tj1{τk≤t},1+l≤j≤n,
where Λk,tj is adapted to ℱT*X for all k. Λ0,ti and Λ0,tj can be constants or stochastic processes which are correlated with the state variables.Assumption 5 (the risk-free interest rate).
The risk-free interest ratert in this framework is stochastic which may follow CIR model, HJM model, Vasicek model or their extensions. It has effect on the defaults of n companies.


## 3. The Pricing of Credit Securities

In this section, we price the defaultable bonds and credit default swap (CDS) in the primary-secondary framework satisfying Assumption1 to Assumption 5. To obtain some explicit results, we give another specific assumptions. We assume that the state variable Xt only contains the risk-free spot rate rt and the default of a firm is correlated with the default-free term structure. Namely, we will present a one-factor model for credit risk. Furthermore, we mainly consider single counterparty. There are one primary firm and one secondary firm in our pricing model. Counterparty risk may occur when secondary firm holds large amounts of debt issued by the primary firm.We suppose that the risk-free interest rate follows the jump-diffusion process(3.1)drt=α(K-rt)dt+σdWt+qtdYt,
where Wt is a standard Brownian motion on the probability space (Ω,ℱ,P) and Yt is a Possion process under P with intensity μ. qt is a deterministic function and α, σ, K are constants. We assume Wt and Yt are mutually independent.Remark 3.1.
In fact, from Park [17], we know (3.1) has the explicit solution as follows:
(3.2)rt=r0e-αt+αK∫0te-α(t-s)ds+σ∫0te-α(t-s)dWs+∫0tqse-α(t-s)dYs.
Moreover, in accordance with the properties of Ws and Ys, we can check that rt is a ℱr-Markov process, which plays an important role in the following.


### 3.1. Defaultable Bonds' Pricing

We first give some general pricing formulas for bonds in the primary-secondary framework described in Section2.2. Suppose that the face value of bond i(i=1,…,n) is 1 dollar. Under the equivalent martingale measure P, the default-free and defaultable bond's prices are, respectively, given by(3.3)p(t,T)=Et[exp⁡(-∫tTrsds)],(3.4)Vi(t,T)=Et[e-∫tTrsds(βi1{τi≤T}+1{τi>T})],
where Et[·] represents the conditional expectation with respect to ℱt, βi is the recovery rate of defaultable bond i, and T(<T*) is the maturity date.Lemma 3.2 (see [12]).
The defaultable bond price can also be expressed as(3.5)Vi(t,T)=βip(t,T)+1{τi>t}(1-βi)Et[exp⁡(-∫tT(rs+λsi)ds)],t≤T.In the following, we only consider the case with two firms. FirmA is the primary firm whose default is independent of the default risk of secondary firm B but depends on the interest rate r, while firm B's default is correlated with the state of firm A and the risk-free interest rate. One assumes their intensity processes, respectively, satisfy some linear relations below:(3.6)λtA=b0A+b1Art,(3.7)λtB=b0B+b1Brt+b1{τA≤t},
where b0A, b1A, b0B, b1B, and b are positive constants.Remark 3.3.
The interest ratert in our model is an extension of Vasicek model. It may cause negative intensity. We use the similar method in Jarrow and Yu [12] to avoid this case. We can assume λtA=max{b0A+b1Art,0}, λtB=max{b0B+b1Brt+b1{τA≤t},0}, we will discuss it in other paper.We price the bonds issued by firmA and firm B. To be convenient, we use time-t forward interest rate instead of time-0 forward interest rate in (3.2). Let f(0,u)=r0e-αu, then for u≥t, (3.2) can be expressed as(3.8)ru=f(0,u)+(∫0t+∫tu)αKeα(v-u)dv+(∫0t+∫tu)σeα(v-u)dWv+(∫0t+∫tu)qveα(v-u)dYv=f(t,u)+∫tuαKeα(v-u)dv+∫tuσeα(v-u)dWv+∫tuqveα(v-u)dYv,
where(3.9)f(t,u)=f(0,u)+∫0tαKeα(v-u)dv+∫0tσeα(v-u)dWv+∫0tqveα(v-u)dYv.
Now, we present an important theorem in the pricing process of credit securities.Theorem 3.4.
Suppose thatrt follows (3.1) and Rt,T=∫tTrsds be the cumulative interest from time t to T. Let Et[e-aRt,T]=g(a,t,T) for all a∈ℛ, then one obtains
(3.10)g(a,t,T)=exp⁡(∫tT[-af(t,u)+12σ2a2cT2(u)+μ(e-aqucT(u)-1)]du-aK(T-t)+aKcT(t)),
where
(3.11)cv(u)=-1α(eα(u-v)-1),0≤v,u≤T.Proof.
The proving ideas are similar to Jarrow and Yu [12] and Park [17]. Since rt follows (3.1), it has an explicit expression as (3.8). Then, we have
(3.12)-a∫tTrudu=-∫tTaf(t,u)du-∫tTdu∫tuaαKeα(v-u)dv-∫tTdu∫tuσaeα(v-u)dWv-∫tTdu∫tuqvaeα(v-u)dYv≜h1(t,T)+h2(t,T)+h3(t,T)+h4(t,T),
where a∈ℛ and
(3.13)h1(t,T)=-a∫tTf(t,u)du,(3.14)h2(t,T)=-∫tTdu∫tuaαKeα(v-u)dv=-aK(T-t)-aKα(eα(t-T)-1),(3.15)h3(t,T)=-∫tTdu∫tuσaeα(v-u)dWv,(3.16)h4(t,T)=-∫tTdu∫tuqvaeα(v-u)dYv.
By the Markov property of r, we have
(3.17)Et[e-aRt,T]=E[e-aRt,T∣rt].
Hence, we mainly need to obtain E[eh3(t,T)∣rt] and E[eh4(t,T)∣rt]. By Fübini's theorem, (3.15) and (3.16) become
(3.18)h3(t,T)=-∫tTdWv∫vTσaeα(v-u)du=-∫tTσacT(v)dWv,h4(t,T)=-∫tTdYv∫vTqvaeα(v-u)du=-∫tTaqvcT(v)dYv,
where cT(v) is given by (3.11). Moreover, h3(t,T) follows the normal distribution with mean 0 and variance σ2a2∫tTcT2(v)dv. Therefore, by the independent increments of the diffusion process,
(3.19)E[exp⁡(A+B)]=exp⁡((Var⁡[A]+Var⁡[B])2).
So
(3.20)E[eh3(t,T)]=exp⁡(12σ2a2∫tTcT2(v)dv).
In addition, using the results in Park [17], based on independent increments for the jump process,
(3.21)E[eh4(t,T)]=exp⁡(μ∫tT(e-aqvcT(v)-1)dv).
We substitute (3.13), (3.14), (3.20) and (3.21) into Et[e-aRt,T] and deduce (3.10).Thus, from Lemma3.2 and Theorem 3.4, we can derive the pricing formulas of defaultable bonds.Theorem 3.5.
In the primary-secondary framework described as above, the bonds issued by firmA and B have the same maturity date T and recovery rate βA=βB=0. If the intensity processes λtA and λtB satisfy (3.6) and (3.7) and no defaults occur up to time t, then the time-t price of bond issued by primary firm A is
(3.22)VA(t,T)=g(1+b1A,t,T)exp⁡(-b0A(T-t))
and the time-t price of bond issued by secondary firm B is
(3.23)VB(t,T)=g(1+b1B,t,T)e-(b0B+b)(T-t)+be-(K+Kb1B+b0B+b)T+(1+b0B+b0A+b1B+b1A)t⋅∫tTe-(1+b1B+b1A)∫tsf(t,u)du+(b-b1AK-b0A)s+(1+b1B+b1A)Kcs(t)+(1+b1B)(K-r0)d(s,T,0)-f1(t,s)+M(s)ds,
where for for all k,v,u∈[0,T], cv(u) is defined as (3.11)
(3.24)d(k,v,u)=-1αeαu(e-αv-e-αk),(3.25)f1(t,s)=∫0tσ(1+b1B)d(s,T,u)dWu+∫0t(1+b1B)qud(s,T,u)dYu,(3.26)M(s)=∫sT[12σ2(1+b1B)2cT2(u)+μ(e-(1+b1B)qucT(u)-1)]du+∫ts12σ2[(1+b1B+b1A)cs(u)+(1+b1B)d(s,T,u)]2du+∫tsμ[e-qu((1+b1B+b1A)cs(u)+(1+b1B)d(s,T,u))-1]du.Proof.
Firstly, from Lemma3.2 and Theorem 3.4, we can easily show that (3.22) holds. Secondly, according to Lemma 3.2, (3.7), and the properties of conditional expectation, we obtain the price of bond issued by firm B at time t(3.27)VB(t,T)=Et[exp⁡(-∫tT(rs+λsB)ds)]=Et[exp⁡(-b0B(T-t)-(1+b1B)Rt,T-b(T-τA)1{τA≤T})]=Et[exp⁡(-b0B(T-t)-(1+b1B)Rt,T)⋅E[exp⁡(-b(T-τA)1{τA≤T})∣Ft∨FT*r]].
By (2.6), the property of conditional expectation and the law of integration by parts, we check that
(3.28)E[exp⁡(-b(T-τA)1{τA≤T})∣Ft∨FT*r]=(∫tT+∫T∞)e-b(T-s)1{s≤T}d(1-e-b0A(s-t)-b1ARt,s)=e-b(T-t)(1+b∫tTe-(b0A-b)(s-t)-b1ARt,sds).
Hence,
(3.29)VB(t,T)=e-(b0B+b)(T-t)Et[e-(1+b1B)Rt,T]+be-(b0B+b)T+(b0B+b0A)t⋅∫tTe-(b0A-b)sEt[e-(1+b1B+b1A)Rt,s-(1+b1B)Rs,T]ds,
where (3.29) involves the interchange of the expectation and the integral. Further, using the law of iterated conditional expectations, we have
(3.30)Et[e-(1+b1B+b1A)Rt,s⋅e-(1+b1B)Rs,T]=Et[e-(1+b1B+b1A)Rt,s⋅Es[e-(1+b1B)Rs,T]]=Et[e-(1+b1B+b1A)Rt,s⋅g(1+b1B,s,T)]≜Et[eI].
Denote 1+b1B+b1A=m1 and 1+b1B=m2. Then, from Theorem 3.4, we show that
(3.31)I≜-m1∫tsrudu-m2∫sTf(s,u)du-m2K(T-s)+Km2cT(s)+∫sT[12σ2m22cT2(u)+μ(e-m2qucT(u)-1)]du≜I1+I2,
where
(3.32)I1=-m1∫tsrudu-m2∫sTf(s,u)du,I2=∫sT[12σ2m22cT2(u)+μ(e-m2qucT(u)-1)]du-m2K(T-s)+Km2cT(s).
Again, by (3.9), we have
(3.33)-m2∫sTf(s,u)du=-∫sTm2f(0,u)du-∫sTdu∫0sm2αKeα(v-u)dv-∫sTdu∫0sσm2eα(v-u)dWv-∫sTdu∫0sm2qveα(v-u)dYv≜h1′(s,T)+h2′(s,T)+h3′(s,T)+h4′(s,T),
where
(3.34)h1′(s,T)=-m2∫sTf(0,u)du,h2′(s,T)=-∫sTdu∫0sm2αKeα(v-u)dv,h3′(s,T)=-∫sTdu∫0sσm2eα(v-u)dWv,h4′(s,T)=-∫sTdu∫0sm2qveα(v-u)dYv.
We easily check that
(3.35)h1′(s,T)=-m2r0d(s,T,0),h2′(s,T)=-m2KcT(s)+m2Kd(s,T,0),
where d(s,T,u) is given by (3.24). Moreover, using Fübini's theorem, we obtain
(3.36)h3′(s,T)=-∫0sσm2d(s,T,u)dWu,h4′(s,T)=-∫0sm2qud(s,T,u)dYu.
Therefore, we give a different expression below:
(3.37)-m2∫sTf(s,u)du=-m2r0d(s,T,0)-m2KcT(s)+m2Kd(s,T,0)-∫0sσm2d(s,T,u)dWu-∫0sm2qud(s,T,u)dYu.
Then, from (3.12) and (3.37), we find that
(3.38)I1=-m1∫tsrudu-m2∫sTf(s,u)du=-m1∫tsf(t,u)du-m1K(s-t)+m1Kcs(t)+m2(K-r0)d(s,T,0)-m2KcT(s)-∫0tσm2d(s,T,u)dWu-∫0tm2qud(s,T,u)dYu-∫tsσ(m1cs(u)+m2d(s,T,u))dWu-∫tsqu(m1cs(u)+m2d(s,T,u))dYu.
In addition, applying (3.19) and the results in Park [17], we have
(3.39)E[e-∫tsσ(m1cs(u)+m2d(s,T,u))dWu]=exp⁡(12σ2∫ts(m1cs(u)+m2d(s,T,u))2du),E[e-∫tsqu(m1cs(u)+m2d(s,T,u))dYu]=exp⁡(μ∫ts[e-qu(m1cs(u)+m2d(s,T,u))-1]du).
Therefore, combining (3.39) and (3.31), we obtain
(3.40)Et[eI]=eI2Et[eI1]=(i)exp⁡(-(1+b1B+b1A)∫tsf(t,u)du-b1AKs+(1+b1B+b1A)Kt-(1+b1B)KT)⋅exp⁡[(1+b1B+b1A)Kcs(t)+(1+b1B)(K-r0)d(s,T,0)-f1(t,s)]⋅Et[e-∫tsσ[(1+b1B+b1A)cs(u)+(1+b1B)d(s,T,u)]dWu]⋅Et[e-∫tsqu[(1+b1B+b1A)cs(u)+(1+b1B)d(s,T,u)]dYu]=(ii)exp⁡(-(1+b1B+b1A)∫tsf(t,u)du-b1AKs+(1+b1B+b1A)Kt-(1+b1B)KT)⋅exp⁡[(1+b1B+b1A)Kcs(t)+(1+b1B)(K-r0)d(s,T,0)-f1(t,s)]⋅exp⁡[12σ2∫ts[(1+b1B+b1A)cs(u)+(1+b1B)d(s,T,u)]2du]⋅exp⁡[μ∫ts[e-qu((1+b1B+b1A)cs(u)+(1+b1B)d(s,T,u))-1]du]⋅exp⁡[∫sT[12σ2(1+b1B)2cT2(u)+μ(e-(1+b1B)qucT(u)-1)]du],
where (i) follows from the independence of Wu and Yu and the property of conditional expectation and (ii) holds from the Markov property and the independent increment property of Wu and Yu. Finally, we substitute Et[eI] into (3.29) and obtain (3.23). The proof is completed.


### 3.2. CDS's Pricing

FirmC holds a bond issued by the reference firm A with the maturity date T1. To decrease the possible loss, firm C buys protection with the maturity date T2(T2≤T1) from firm B on condition that firm C gives the payments to firm B at a fixed swap rate in time while firm B promises to make up firm C for the loss caused by the default of firm A at a certain rate. Each party has the obligation to make payments until its own default. The source of credit risk may be from three parties: the issuer of bond, the buyer of CDS and the seller of CDS.In the following, we consider a simple situation which only contains the risk from reference firmA and firm B. At the same time, to make the calculation convenient, we suppose the recovery rate of the bond issued by firm A is zero and the face value is 1 dollar. In the event of firm A's default, firm B compensates firm C for 1 dollar if he does not default, otherwise 0 dollar. There are four cases for the defaults of firm A and firm B.Case 1.
The  defaults of firmA and firm B are mutually independent conditional on the risk-free interest rate.Case 2.
FirmA is the primary party whose default only depends on the risk-free interest rate (the only economy state variable) and the firm B is the secondary party whose default depends on the risk-free interest rate and the default state of firm A.Case 3.
FirmB is the primary party and the firm A is the secondary party.Case 4.
The  defaults of firmA, and firm B are mutually contagious (looping default).Now, we make use of the results in previous sections to price the CDS in Case2. We assume firm A is the primary party and the firm B is the secondary party. Denoted the swap rate by a constant c and interest rate by rt, let the default times of firm A and B be τA with the intensity λA and τB with the intensity λB, respectively.Theorem 3.6.
Suppose the risk-free interest ratert satisfies (3.1) and the intensities λA and λB satisfy (3.6) and (3.7), respectively. Then, the swap rate c has the following expression:
(3.41)c=VB(0,T2)-e-(b0B+b0A)T2g(1+b1B+b1A,0,T2)∫0T2g(1,0,s)ds,
where g(·,·,·) and VB(0,T2) are given by Theorems 3.4 and 3.5, respectively.Proof.
Firstly, the time-0 market value of buyerC's payments to seller B is
(3.42)E[∫0T2ce-∫0srududs]=c∫0T2E[e-R0,s]ds,
where R0,s is defined as Theorem 3.4.
Secondly, the time-0 market value of firmB's promised payoff in case of firm A's default is(3.43)E[1{τA≤T2}e-∫0T2rudu1{τB>T2}].
Thus, in accordance with the arbitrage-free principle, we obtain
(3.44)c=E[1{τA≤T2}e-∫0T2rudu1{τB>T2}]∫0T2E[e-R0,s]ds.
Further, we can use the properties of conditional expectation to simplify (3.44) as follows:
(3.45)c=E[E[1{τA≤T2}∣FT*r∨HT*B]e-∫0T2rudu1{τB>T2}]∫0T2E[e-R0,s]ds=E[1{τB>T2}e-∫0T2rudu]-E[1{τA>T2}e-∫0T2rudu1{τB>T2}]∫0T2E[e-R0,s]ds=VB(0,T2)-E[1{τA>T2}e-∫0T2(ru+λuB)du]∫0T2E[e-R0,s]ds,
where the last one is obtained by (3.4). Note that VB(0,T2) can be obtained by (3.23) and E[e-R0,s]=g(1,0,s) by Theorem 3.4. We substitute (3.7) into the above expectation term
(3.46)E[1{τA>T2}exp⁡(-∫0T2(ru+λuB)du)]=E[1{τA>T2}exp⁡(-∫0T2(b0B+(1+b1B)ru+b1{τA≤u})du)]=E[1{τA>T2}exp⁡(-b0BT2-(1+b1B)∫0T2rudu-b(T2-τA)1{τA≤T2})]=E[1{τA>T2}e-b0BT2-(1+b1B)R0,T2]=(i)E[E[1{τA>T2}∣FT*r]e-b0BT2-(1+b1B)R0,T2]=(ii)E[e-b0BT2-(1+b1B)R0,T2-∫0T2λuAdu]=(iii)e-(b0B+b0A)T2E[e-(1+b1B+b1A)∫0T2rudu]=(iv)e-(b0B+b0A)T2g(1+b1B+b1A,0,T2),
where (i) involves the property of conditional expectation, (ii) follows from (2.6), and (iv) follows from Theorem 3.4. Now, substituting these results into (3.45), we show (3.41) holds. The proof is complete.Remark 3.7.
The model in Case1 can be considered a special case of primary-secondary model and the price of CDS can be derived by the similar method. The pricing of CDS in Case 4 will be discussed in another paper. In Case 3, if λtA and λtB satisfy the below relations:
(3.47)λtA=b0A+b1Art+b1{τB≤t},λtB=b0B+b1Brt,
where b0A, b1A, b0B, b1B, and b are positive constants, then the swap rate
(3.48)c=g(1+b1B,0,T2)e-b0BT2-e-(b0B+b0A)T2g(1+b1B+b1A,0,T2)∫0T2g(1,0,s)ds,
where g(·,·,·) are given by Theorem 3.4. The deriving process is similar to Theorem 3.6, so we omit it.Remark 3.8.
In our models, to make the expressions comparatively simple, we all assume that the recovery rates are zero. When the relevant recovery rates are nonzero constant, the pricing formulas are still easily obtained from Lemma3.2 because we can get p(t,T)=g(1,t,T) from Theorem 3.4. We omit the process.


## 3.1. Defaultable Bonds' Pricing

We first give some general pricing formulas for bonds in the primary-secondary framework described in Section2.2. Suppose that the face value of bond i(i=1,…,n) is 1 dollar. Under the equivalent martingale measure P, the default-free and defaultable bond's prices are, respectively, given by(3.3)p(t,T)=Et[exp⁡(-∫tTrsds)],(3.4)Vi(t,T)=Et[e-∫tTrsds(βi1{τi≤T}+1{τi>T})],
where Et[·] represents the conditional expectation with respect to ℱt, βi is the recovery rate of defaultable bond i, and T(<T*) is the maturity date.Lemma 3.2 (see [12]).
The defaultable bond price can also be expressed as(3.5)Vi(t,T)=βip(t,T)+1{τi>t}(1-βi)Et[exp⁡(-∫tT(rs+λsi)ds)],t≤T.In the following, we only consider the case with two firms. FirmA is the primary firm whose default is independent of the default risk of secondary firm B but depends on the interest rate r, while firm B's default is correlated with the state of firm A and the risk-free interest rate. One assumes their intensity processes, respectively, satisfy some linear relations below:(3.6)λtA=b0A+b1Art,(3.7)λtB=b0B+b1Brt+b1{τA≤t},
where b0A, b1A, b0B, b1B, and b are positive constants.Remark 3.3.
The interest ratert in our model is an extension of Vasicek model. It may cause negative intensity. We use the similar method in Jarrow and Yu [12] to avoid this case. We can assume λtA=max{b0A+b1Art,0}, λtB=max{b0B+b1Brt+b1{τA≤t},0}, we will discuss it in other paper.We price the bonds issued by firmA and firm B. To be convenient, we use time-t forward interest rate instead of time-0 forward interest rate in (3.2). Let f(0,u)=r0e-αu, then for u≥t, (3.2) can be expressed as(3.8)ru=f(0,u)+(∫0t+∫tu)αKeα(v-u)dv+(∫0t+∫tu)σeα(v-u)dWv+(∫0t+∫tu)qveα(v-u)dYv=f(t,u)+∫tuαKeα(v-u)dv+∫tuσeα(v-u)dWv+∫tuqveα(v-u)dYv,
where(3.9)f(t,u)=f(0,u)+∫0tαKeα(v-u)dv+∫0tσeα(v-u)dWv+∫0tqveα(v-u)dYv.
Now, we present an important theorem in the pricing process of credit securities.Theorem 3.4.
Suppose thatrt follows (3.1) and Rt,T=∫tTrsds be the cumulative interest from time t to T. Let Et[e-aRt,T]=g(a,t,T) for all a∈ℛ, then one obtains
(3.10)g(a,t,T)=exp⁡(∫tT[-af(t,u)+12σ2a2cT2(u)+μ(e-aqucT(u)-1)]du-aK(T-t)+aKcT(t)),
where
(3.11)cv(u)=-1α(eα(u-v)-1),0≤v,u≤T.Proof.
The proving ideas are similar to Jarrow and Yu [12] and Park [17]. Since rt follows (3.1), it has an explicit expression as (3.8). Then, we have
(3.12)-a∫tTrudu=-∫tTaf(t,u)du-∫tTdu∫tuaαKeα(v-u)dv-∫tTdu∫tuσaeα(v-u)dWv-∫tTdu∫tuqvaeα(v-u)dYv≜h1(t,T)+h2(t,T)+h3(t,T)+h4(t,T),
where a∈ℛ and
(3.13)h1(t,T)=-a∫tTf(t,u)du,(3.14)h2(t,T)=-∫tTdu∫tuaαKeα(v-u)dv=-aK(T-t)-aKα(eα(t-T)-1),(3.15)h3(t,T)=-∫tTdu∫tuσaeα(v-u)dWv,(3.16)h4(t,T)=-∫tTdu∫tuqvaeα(v-u)dYv.
By the Markov property of r, we have
(3.17)Et[e-aRt,T]=E[e-aRt,T∣rt].
Hence, we mainly need to obtain E[eh3(t,T)∣rt] and E[eh4(t,T)∣rt]. By Fübini's theorem, (3.15) and (3.16) become
(3.18)h3(t,T)=-∫tTdWv∫vTσaeα(v-u)du=-∫tTσacT(v)dWv,h4(t,T)=-∫tTdYv∫vTqvaeα(v-u)du=-∫tTaqvcT(v)dYv,
where cT(v) is given by (3.11). Moreover, h3(t,T) follows the normal distribution with mean 0 and variance σ2a2∫tTcT2(v)dv. Therefore, by the independent increments of the diffusion process,
(3.19)E[exp⁡(A+B)]=exp⁡((Var⁡[A]+Var⁡[B])2).
So
(3.20)E[eh3(t,T)]=exp⁡(12σ2a2∫tTcT2(v)dv).
In addition, using the results in Park [17], based on independent increments for the jump process,
(3.21)E[eh4(t,T)]=exp⁡(μ∫tT(e-aqvcT(v)-1)dv).
We substitute (3.13), (3.14), (3.20) and (3.21) into Et[e-aRt,T] and deduce (3.10).Thus, from Lemma3.2 and Theorem 3.4, we can derive the pricing formulas of defaultable bonds.Theorem 3.5.
In the primary-secondary framework described as above, the bonds issued by firmA and B have the same maturity date T and recovery rate βA=βB=0. If the intensity processes λtA and λtB satisfy (3.6) and (3.7) and no defaults occur up to time t, then the time-t price of bond issued by primary firm A is
(3.22)VA(t,T)=g(1+b1A,t,T)exp⁡(-b0A(T-t))
and the time-t price of bond issued by secondary firm B is
(3.23)VB(t,T)=g(1+b1B,t,T)e-(b0B+b)(T-t)+be-(K+Kb1B+b0B+b)T+(1+b0B+b0A+b1B+b1A)t⋅∫tTe-(1+b1B+b1A)∫tsf(t,u)du+(b-b1AK-b0A)s+(1+b1B+b1A)Kcs(t)+(1+b1B)(K-r0)d(s,T,0)-f1(t,s)+M(s)ds,
where for for all k,v,u∈[0,T], cv(u) is defined as (3.11)
(3.24)d(k,v,u)=-1αeαu(e-αv-e-αk),(3.25)f1(t,s)=∫0tσ(1+b1B)d(s,T,u)dWu+∫0t(1+b1B)qud(s,T,u)dYu,(3.26)M(s)=∫sT[12σ2(1+b1B)2cT2(u)+μ(e-(1+b1B)qucT(u)-1)]du+∫ts12σ2[(1+b1B+b1A)cs(u)+(1+b1B)d(s,T,u)]2du+∫tsμ[e-qu((1+b1B+b1A)cs(u)+(1+b1B)d(s,T,u))-1]du.Proof.
Firstly, from Lemma3.2 and Theorem 3.4, we can easily show that (3.22) holds. Secondly, according to Lemma 3.2, (3.7), and the properties of conditional expectation, we obtain the price of bond issued by firm B at time t(3.27)VB(t,T)=Et[exp⁡(-∫tT(rs+λsB)ds)]=Et[exp⁡(-b0B(T-t)-(1+b1B)Rt,T-b(T-τA)1{τA≤T})]=Et[exp⁡(-b0B(T-t)-(1+b1B)Rt,T)⋅E[exp⁡(-b(T-τA)1{τA≤T})∣Ft∨FT*r]].
By (2.6), the property of conditional expectation and the law of integration by parts, we check that
(3.28)E[exp⁡(-b(T-τA)1{τA≤T})∣Ft∨FT*r]=(∫tT+∫T∞)e-b(T-s)1{s≤T}d(1-e-b0A(s-t)-b1ARt,s)=e-b(T-t)(1+b∫tTe-(b0A-b)(s-t)-b1ARt,sds).
Hence,
(3.29)VB(t,T)=e-(b0B+b)(T-t)Et[e-(1+b1B)Rt,T]+be-(b0B+b)T+(b0B+b0A)t⋅∫tTe-(b0A-b)sEt[e-(1+b1B+b1A)Rt,s-(1+b1B)Rs,T]ds,
where (3.29) involves the interchange of the expectation and the integral. Further, using the law of iterated conditional expectations, we have
(3.30)Et[e-(1+b1B+b1A)Rt,s⋅e-(1+b1B)Rs,T]=Et[e-(1+b1B+b1A)Rt,s⋅Es[e-(1+b1B)Rs,T]]=Et[e-(1+b1B+b1A)Rt,s⋅g(1+b1B,s,T)]≜Et[eI].
Denote 1+b1B+b1A=m1 and 1+b1B=m2. Then, from Theorem 3.4, we show that
(3.31)I≜-m1∫tsrudu-m2∫sTf(s,u)du-m2K(T-s)+Km2cT(s)+∫sT[12σ2m22cT2(u)+μ(e-m2qucT(u)-1)]du≜I1+I2,
where
(3.32)I1=-m1∫tsrudu-m2∫sTf(s,u)du,I2=∫sT[12σ2m22cT2(u)+μ(e-m2qucT(u)-1)]du-m2K(T-s)+Km2cT(s).
Again, by (3.9), we have
(3.33)-m2∫sTf(s,u)du=-∫sTm2f(0,u)du-∫sTdu∫0sm2αKeα(v-u)dv-∫sTdu∫0sσm2eα(v-u)dWv-∫sTdu∫0sm2qveα(v-u)dYv≜h1′(s,T)+h2′(s,T)+h3′(s,T)+h4′(s,T),
where
(3.34)h1′(s,T)=-m2∫sTf(0,u)du,h2′(s,T)=-∫sTdu∫0sm2αKeα(v-u)dv,h3′(s,T)=-∫sTdu∫0sσm2eα(v-u)dWv,h4′(s,T)=-∫sTdu∫0sm2qveα(v-u)dYv.
We easily check that
(3.35)h1′(s,T)=-m2r0d(s,T,0),h2′(s,T)=-m2KcT(s)+m2Kd(s,T,0),
where d(s,T,u) is given by (3.24). Moreover, using Fübini's theorem, we obtain
(3.36)h3′(s,T)=-∫0sσm2d(s,T,u)dWu,h4′(s,T)=-∫0sm2qud(s,T,u)dYu.
Therefore, we give a different expression below:
(3.37)-m2∫sTf(s,u)du=-m2r0d(s,T,0)-m2KcT(s)+m2Kd(s,T,0)-∫0sσm2d(s,T,u)dWu-∫0sm2qud(s,T,u)dYu.
Then, from (3.12) and (3.37), we find that
(3.38)I1=-m1∫tsrudu-m2∫sTf(s,u)du=-m1∫tsf(t,u)du-m1K(s-t)+m1Kcs(t)+m2(K-r0)d(s,T,0)-m2KcT(s)-∫0tσm2d(s,T,u)dWu-∫0tm2qud(s,T,u)dYu-∫tsσ(m1cs(u)+m2d(s,T,u))dWu-∫tsqu(m1cs(u)+m2d(s,T,u))dYu.
In addition, applying (3.19) and the results in Park [17], we have
(3.39)E[e-∫tsσ(m1cs(u)+m2d(s,T,u))dWu]=exp⁡(12σ2∫ts(m1cs(u)+m2d(s,T,u))2du),E[e-∫tsqu(m1cs(u)+m2d(s,T,u))dYu]=exp⁡(μ∫ts[e-qu(m1cs(u)+m2d(s,T,u))-1]du).
Therefore, combining (3.39) and (3.31), we obtain
(3.40)Et[eI]=eI2Et[eI1]=(i)exp⁡(-(1+b1B+b1A)∫tsf(t,u)du-b1AKs+(1+b1B+b1A)Kt-(1+b1B)KT)⋅exp⁡[(1+b1B+b1A)Kcs(t)+(1+b1B)(K-r0)d(s,T,0)-f1(t,s)]⋅Et[e-∫tsσ[(1+b1B+b1A)cs(u)+(1+b1B)d(s,T,u)]dWu]⋅Et[e-∫tsqu[(1+b1B+b1A)cs(u)+(1+b1B)d(s,T,u)]dYu]=(ii)exp⁡(-(1+b1B+b1A)∫tsf(t,u)du-b1AKs+(1+b1B+b1A)Kt-(1+b1B)KT)⋅exp⁡[(1+b1B+b1A)Kcs(t)+(1+b1B)(K-r0)d(s,T,0)-f1(t,s)]⋅exp⁡[12σ2∫ts[(1+b1B+b1A)cs(u)+(1+b1B)d(s,T,u)]2du]⋅exp⁡[μ∫ts[e-qu((1+b1B+b1A)cs(u)+(1+b1B)d(s,T,u))-1]du]⋅exp⁡[∫sT[12σ2(1+b1B)2cT2(u)+μ(e-(1+b1B)qucT(u)-1)]du],
where (i) follows from the independence of Wu and Yu and the property of conditional expectation and (ii) holds from the Markov property and the independent increment property of Wu and Yu. Finally, we substitute Et[eI] into (3.29) and obtain (3.23). The proof is completed.


## 3.2. CDS's Pricing

FirmC holds a bond issued by the reference firm A with the maturity date T1. To decrease the possible loss, firm C buys protection with the maturity date T2(T2≤T1) from firm B on condition that firm C gives the payments to firm B at a fixed swap rate in time while firm B promises to make up firm C for the loss caused by the default of firm A at a certain rate. Each party has the obligation to make payments until its own default. The source of credit risk may be from three parties: the issuer of bond, the buyer of CDS and the seller of CDS.In the following, we consider a simple situation which only contains the risk from reference firmA and firm B. At the same time, to make the calculation convenient, we suppose the recovery rate of the bond issued by firm A is zero and the face value is 1 dollar. In the event of firm A's default, firm B compensates firm C for 1 dollar if he does not default, otherwise 0 dollar. There are four cases for the defaults of firm A and firm B.Case 1.
The  defaults of firmA and firm B are mutually independent conditional on the risk-free interest rate.Case 2.
FirmA is the primary party whose default only depends on the risk-free interest rate (the only economy state variable) and the firm B is the secondary party whose default depends on the risk-free interest rate and the default state of firm A.Case 3.
FirmB is the primary party and the firm A is the secondary party.Case 4.
The  defaults of firmA, and firm B are mutually contagious (looping default).Now, we make use of the results in previous sections to price the CDS in Case2. We assume firm A is the primary party and the firm B is the secondary party. Denoted the swap rate by a constant c and interest rate by rt, let the default times of firm A and B be τA with the intensity λA and τB with the intensity λB, respectively.Theorem 3.6.
Suppose the risk-free interest ratert satisfies (3.1) and the intensities λA and λB satisfy (3.6) and (3.7), respectively. Then, the swap rate c has the following expression:
(3.41)c=VB(0,T2)-e-(b0B+b0A)T2g(1+b1B+b1A,0,T2)∫0T2g(1,0,s)ds,
where g(·,·,·) and VB(0,T2) are given by Theorems 3.4 and 3.5, respectively.Proof.
Firstly, the time-0 market value of buyerC's payments to seller B is
(3.42)E[∫0T2ce-∫0srududs]=c∫0T2E[e-R0,s]ds,
where R0,s is defined as Theorem 3.4.
Secondly, the time-0 market value of firmB's promised payoff in case of firm A's default is(3.43)E[1{τA≤T2}e-∫0T2rudu1{τB>T2}].
Thus, in accordance with the arbitrage-free principle, we obtain
(3.44)c=E[1{τA≤T2}e-∫0T2rudu1{τB>T2}]∫0T2E[e-R0,s]ds.
Further, we can use the properties of conditional expectation to simplify (3.44) as follows:
(3.45)c=E[E[1{τA≤T2}∣FT*r∨HT*B]e-∫0T2rudu1{τB>T2}]∫0T2E[e-R0,s]ds=E[1{τB>T2}e-∫0T2rudu]-E[1{τA>T2}e-∫0T2rudu1{τB>T2}]∫0T2E[e-R0,s]ds=VB(0,T2)-E[1{τA>T2}e-∫0T2(ru+λuB)du]∫0T2E[e-R0,s]ds,
where the last one is obtained by (3.4). Note that VB(0,T2) can be obtained by (3.23) and E[e-R0,s]=g(1,0,s) by Theorem 3.4. We substitute (3.7) into the above expectation term
(3.46)E[1{τA>T2}exp⁡(-∫0T2(ru+λuB)du)]=E[1{τA>T2}exp⁡(-∫0T2(b0B+(1+b1B)ru+b1{τA≤u})du)]=E[1{τA>T2}exp⁡(-b0BT2-(1+b1B)∫0T2rudu-b(T2-τA)1{τA≤T2})]=E[1{τA>T2}e-b0BT2-(1+b1B)R0,T2]=(i)E[E[1{τA>T2}∣FT*r]e-b0BT2-(1+b1B)R0,T2]=(ii)E[e-b0BT2-(1+b1B)R0,T2-∫0T2λuAdu]=(iii)e-(b0B+b0A)T2E[e-(1+b1B+b1A)∫0T2rudu]=(iv)e-(b0B+b0A)T2g(1+b1B+b1A,0,T2),
where (i) involves the property of conditional expectation, (ii) follows from (2.6), and (iv) follows from Theorem 3.4. Now, substituting these results into (3.45), we show (3.41) holds. The proof is complete.Remark 3.7.
The model in Case1 can be considered a special case of primary-secondary model and the price of CDS can be derived by the similar method. The pricing of CDS in Case 4 will be discussed in another paper. In Case 3, if λtA and λtB satisfy the below relations:
(3.47)λtA=b0A+b1Art+b1{τB≤t},λtB=b0B+b1Brt,
where b0A, b1A, b0B, b1B, and b are positive constants, then the swap rate
(3.48)c=g(1+b1B,0,T2)e-b0BT2-e-(b0B+b0A)T2g(1+b1B+b1A,0,T2)∫0T2g(1,0,s)ds,
where g(·,·,·) are given by Theorem 3.4. The deriving process is similar to Theorem 3.6, so we omit it.Remark 3.8.
In our models, to make the expressions comparatively simple, we all assume that the recovery rates are zero. When the relevant recovery rates are nonzero constant, the pricing formulas are still easily obtained from Lemma3.2 because we can get p(t,T)=g(1,t,T) from Theorem 3.4. We omit the process.


## 4. Conclusion

This paper gives the pricing formulas of defaultable bonds and CDSs. In our model, we consider the case that the default intensity is correlated with the risk-free interest rate following jump-diffusion process and the counterparty's default, which is more realistic. We involve the jump risk of risk-free interest rate in the pricing, generalizing the contagious model in Jarrow and Yu [12].In fact, we only consider the comparatively simple situation. We can further study the more general model. For example, we consider the case that the relevant recovery rates are stochastic and the interest rate satisfies more general jump-diffusion process. Moreover, the model in this paper is actually one-factor model with one state variable, while we can discuss multifactor models in which there are several state variables. In a word, the contagious model of credit security with counterparty risk is very necessary to be further discussed in the future.



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*Source: 412565-2011-04-26.xml*