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We study a three-firm contagion model with counterparty risk and apply this model to price defaultable bonds and credit default swap (CDS). This model assumes that default intensities are driven by external common factors as well as other defaults in the system. Using the “total hazard” approach, default times can be generated and the joint density function is obtained. We represent the pricing method of defaultable bonds and obtain the closed-form pricing formulas. By the approach of “change of measure,” analytical solutions of CDS swap rate (swap premuim) are derived in the continuous time framework and the discrete time framework, respectively.

The corporate bonds and their credit derivatives are typical financial tools in the markets which undertake and avoid the credit risk of the companies. There are two basic approaches to modeling the pricing of defaultable securities: the value-of-the-firm (or structural) approach and the intensity-based approach. The structural model is based on the work of Merton [

Nevertheless, the value of the firm assets is not observable, which brings difficulties to the pricing of credit derivatives. Reduced-form approach for credit risks avoids the disadvantage of structural approach which models the firm's value directly. They use risk-neutral pricing principle of contingent claims and take the time of default or other credit events as an exogenous random variable.

Reduced-form models are developed by Artzner and Delbaen [

In this paper, we mainly discuss the pricing of the defaultable zero-coupon bonds and CDS based on the intensity model with correlated default. The structure of this paper is organized as follows: in Section

We consider an uncertain economy with a time horizon of

On this probability space, there is an

According to the information contained in the state variables and the default processes, the filtration is

Let

According to the filtration

Let

According to the Doob-Meyer decomposition, we have that

Under the above characterization, the conditional survival probability of firm

The unconditional survival probability of firm

Now, we give the recursive construction of default time as Yu in his paper [

The total hazard accumulated by firm

Define the inverse functions

Let

Given that Step

Norros [

In this subsection, we explore the three-firms contagion model with an interaction term. Consider the case where the default intensity of one firm is affected by the default of other two firms, so that when one firm defaults the default probabilities of other two firms will jump. In the three-firms contagion model, the interdependent structure between firm

Recall Leung and Kwok's three-firms model:

Nevertheless, Leung and Kwok have not allow the effect of two parties' simultaneous default on the third party, namely, there is not an interaction term in their model. Thus, if three firms are copartners, then the default risk of each firm may be overestimated and the asset value may be underestimated because there exists the case in which the default events might overlap. If they are competitors, then the case is contrary.

For the above reason, we allow the following three-firms contagion model:

Nextly, we employ the three-firms model specified by (

A defaultable claim maturing at

The dividend process

The exdividend price process

By Definitions

The exdividend price of the defaultable claim

From Lemma

For the default-free zero-coupon bond which pays one dollar, the dividend process is

If the dividend process is

For the defaultable zero-coupon bond which pays one dollar if not default and pays

If the dividend process

We assume that there are three firms

We adopt the change of measure introduced by Collin-Dufresne et al. [

Applying the result of Jarrow and Yu [

Because of the symmetry of default intensities, we need only to compute one firm's value of the three firms. In the remainder of this subsection, we will derive the closed-form pricing formula of firm

For firm

Let intensity processes

As one of the important credit derivatives, CDS is a contract agreement which allows the transfer of credit risk of a risky asset (basket of risky assets) from one party to the other. A financial institution may use a CDS to transfer credit risk of a risky asset while continues to retain the legal ownership of the asset. To determine a fair swap rate of a CDS in the presence of counterparty risks, the interdependent default risk structures between these parties must be considered simultaneously.

On CDS valuation, there have been numerous works in recent years. Based on the reduced-form approach with correlated market and credit risks, the closed-form valuation formula for the swap rate of a CDS is obtained by Jarrow and Yildirim [

We assume that party

Similar to the description in [

The structure of CDS.

We employ the three-firms model specified by (

In this section, we will analyze the effect of correlated risks between three parties in a CDS using a similar contagion model as in Leung and Kwok's model [

To price CDS swap rate

Assume that

With the total hazard method (

The Jacobi determinant of

The density function

In this subsection, we employ the three-firms model specified by (

In this framework, the value of the contingent leg at time 0 is equal to

We can derive

Let the intensity processes

According to the arbitrage-free principle, we set the present value of protection buyer's payment equal to the present value of the compensation payment made at

Since it takes no cost to enter a CDS, the value of

Recall that the change of measure is defined by

Thus, by (

The right side of (

By (

From (

From (

In the discrete time framework, let

The value of the contingent leg at time 0 is given by

The value of the fee leg at time 0 is given by

Let the intensity processes

Similar to the discussion in the continuous time, since it takes no cost to enter a CDS, the value of the swap rate

We can obtain

As analyzed in the continuous time, the expression for the swap premium

In this paper, we present a three-firms contagion model with an interaction term which is an improvement in the model of Leung and Kwok [

Our model has its actual background. For example, before and during the global financial crisis, as default risk of the reference asset issuer increased, the protection seller collected higher CDS swap premiums. Thus, default risk of the protection buyer increased since more CDS swap premiums were payed. On the other hand, the protection seller compensated more and more for the loss of reference asset (if it defaulted). When the protection seller (such as a monoline insurer) had no ability to compensate for the loss of reference asset, it went bankrupt. All of these could be important reasons for the financial crisis. So our model is of some significance.

This paper was supported by the National Basic Research Program of China (973 program 2007CB814903).