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

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 .


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 reducedform 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 lognormal 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
The filtration F is generated by the state variables and the default processes of n companies as follows where Denote that

2.3
We assume the default time τ i i 1, . . ., n possesses a strictly positive F X T * ∨ H −i T * -adapted intensity process λ i t satisfying t 0 λ i s ds < ∞, P -a.s. for all t ∈ 0, T * .The intensity process λ i t 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 λ i t Δt.These N i , 1 ≤ i ≤ n generate the defaults of n companies.Their intensity processes λ i , 1 ≤ i ≤ n depend on state variables and the default states of all other companies.Due to the counterparty risk, {τ i } n i 1 may no longer be assumed independent conditionally on F X .

Primary-Secondary Framework
We divide n 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 variable X t may contain the risk-free spot rate r t 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, {F t } T * t 0 , P , we add several independent unit exponential random variables {E i , 1 ≤ i ≤ l} which are independent of X under probability measure P .The default times of l primary firms can be defined as where λ i t is adapted to F X t .Then, we add another series independent unit exponential random variables {E j , l 1 ≤ j ≤ n} which are independent of X and τ i , 1 ≤ i ≤ l.The default times of n − l secondary firms can be defined as where Assumption 3 the default probability .Because E i 1 ≤ i ≤ l is independent of state variables X, the conditional and unconditional survival probability distributions of primary firm i are given by Similarly, since E j 1 l ≤ j ≤ n is independent of X and τ i , 1 ≤ i ≤ l, we have the conditional and unconditional survival probability distributions of secondary firm j

2.8
Assumption 4 the default intensity .Because the Primary firms' default processes only depend on macrovariables, we denote their default intensities by In addition, secondary firms' default processes depend on the macrovariables and the default processes of the primary firms.We denote the intensities by where Λ j k,t is adapted to F X T * for all k.Λ i 0,t and Λ j 0,t can be constants or stochastic processes which are correlated with the state variables.
Assumption 5 the risk-free interest rate .The risk-free interest rate r t 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.

The Pricing of Credit Securities
In this section, we price the defaultable bonds and credit default swap CDS in the primarysecondary framework satisfying Assumption 1 to Assumption 5. To obtain some explicit results, we give another specific assumptions.We assume that the state variable X t only contains the risk-free spot rate r t 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 where W t is a standard Brownian motion on the probability space Ω, F, P and Y t is a Possion process under P with intensity μ. q t is a deterministic function and α, σ, K are constants.We assume W t and Y t are mutually independent.
Remark 3.1.In fact, from Park 17 , we know 3.1 has the explicit solution as follows: Moreover, in accordance with the properties of W s and Y s , we can check that r t is a F r -Markov process, which plays an important role in the following.

Defaultable Bonds' Pricing
We first give some general pricing formulas for bonds in the primary-secondary framework described in Section 2.2.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 where E t • represents the conditional expectation with respect to F t , β i 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 r s λ i s ds , t ≤ T.

3.5
In the following, we only consider the case with two firms.Firm A 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: We price the bonds issued by firm A 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 r 0 e −αu , then for u ≥ t, 3.2 can be expressed as q v e α v−u dY v .

3.9
Now, we present an important theorem in the pricing process of credit securities.
Theorem 3.4.Suppose that r t follows 3.1 and R t,T T t r s ds be the cumulative interest from time t to T. Let E t e −aR t,T g a, t, T for all a ∈ R, then one obtains

3.18
where c T v is given by 3.11 .Moreover, h 3 t, T follows the normal distribution with mean 0 and variance σ 2 a 2 T t c 2 T v dv.Therefore, by the independent increments of the diffusion process,

Mathematical Problems in Engineering
So

3.20
In addition, using the results in Park 17 , based on independent increments for the jump process, E e h 4 t,T exp μ T t e −aq v c T v − 1 dv .
Thus, from Lemma 3.2 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 firm A and B have the same maturity date T and recovery rate β A β B 0. If the intensity processes λ A t and λ B t 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 and the time-t price of bond issued by secondary firm B is

3.23
where for for all k, v, u ∈ 0, T , c v u is defined as 3.11 e αu e −αv − e −αk , 3.24

3.26
Mathematical Problems in Engineering 9 Proof.Firstly, from Lemma 3.2 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 By 2.6 , the property of conditional expectation and the law of integration by parts, we check that

3.29
where 3.29 involves the interchange of the expectation and the integral.Further, using the law of iterated conditional expectations, we have

3.32
Again, by 3.9 , we have

3.34
We easily check that where d s, T, u is given by 3.24 .Moreover, using F übini's theorem, we obtain

3.36
Therefore, we give a different expression below:

3.37
Then, from 3.12 and 3.37 , we find that

3.38
In addition, applying 3.19 and the results in Park 17 , we have e −q u m 1 c s u m 2 d s,T,u − 1 du .

3.39
Case 1.The defaults of firm A and firm B are mutually independent conditional on the riskfree interest rate.
Case 2. Firm A 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. Firm B is the primary party and the firm A is the secondary party.
Case 4. The defaults of firm A, and firm B are mutually contagious looping default .Now, we make use of the results in previous sections to price the CDS in Case 2. 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 r t , let the default times of firm A and B be τ A with the intensity λ A and τ B with the intensity λ B , respectively.Theorem 3.6.Suppose the risk-free interest rate r t satisfies 3.1 and the intensities λ A and λ B satisfy 3.6 and 3.7 , respectively.Then, the swap rate c has the following expression: where R 0,s is defined as Theorem 3.4.Secondly, the time-0 market value of firm B's promised payoff in case of firm A's default is Thus, in accordance with the arbitrage-free principle, we obtain 0 E e −R 0,s ds .

3.44
Further, we can use the properties of conditional expectation to simplify 3.44 as follows: where the last one is obtained by 3.4 .Note that V B 0, T 2 can be obtained by 3.23 and E e −R 0,s g 1, 0, s by Theorem 3.4.We substitute 3.7 into the above expectation term

3.46
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 Case 1 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 λ A t and λ B t satisfy the below relations: 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 Lemma 3.2 because we can get p t, T g 1, t, T from Theorem 3.4.We omit the process.

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.

1 m 1 and 1 b B 1 m 2 .
Then, from Theorem 3.4, we show that where g •, •, • and V B 0, T 2 are given by Theorems 3.4 and 3.5, respectively.Proof.Firstly, the time-0 market value of buyer C's payments to seller B is

1 ,
and b are positive constants, then the swap ratec g 1 b B 1 , 0, T 2 e −b B 0 T 2 − e − b B and b are positive constants.Remark 3.3.The interest rate r t 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.
A ≤t} , 0}, we will discuss it in other paper.

11
Proof.The proving ideas are similar to Jarrow and Yu 12 and Park 17 .Since r t follows 3.1 , it has an explicit expression as 3.8 .Then, we have t e −aR t,T E e −aR t,T | r t .3.17Hence, we mainly need to obtain E e h 3 t,T | r t and E e h 4 t,T | r t .By F übini's theorem, 3.15 and 3.16 become