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This paper mainly discusses the pricing of credit default swap (CDS) in the fractional dimension environment. We assume that the default intensity of a firm depends on the default states of counterparty firms and the term structure of interest rates, but the contagious impact of the counterparty firm is decreasing over time, until disappears. The interest rate risk is reflected by the fractional Vasicek interest rate model. We model the firm’s default intensity in the looping default framework and derive the pricing formulas of risky bonds and credit default swap.

Credit default swap was one of the most important derivatives in the financial market, which was created by JP Morgan in 1995 to manage credit risk. Credit default swap (CDS) is a kind of bilateral agreements. Because it was easy to implement standardization which was firstly founded by the ISDA (the international swaps and derivatives association) in 1998, the credit default swap market had the rapid expansion. However, some con- cealed contradictions exposed gradually, such as the United States subprime crisis and the European sovereign debt crisis. They make people realize that credit derivatives bring convenience and contain huge risk at the same time, especially contagious risk. Therefore, the pricing problem of credit default swap became a hot research topic in recent years.

Until now, there have been mainly two basic CDS’s pricing models: the structural model and the reduced- form model. Structured model was firstly built by [

Reduced model contains intensity model and non-intensity model. The intensity model was pioneered by by [

[

Recently, [

As the fractional Brownian motion has the properties of self-similarity and long-range dependence and many phenomena in financial market show these properties in some certain, the fractional Brownian motion becomes a very suitable tool in different applications such as mathematical finance. The fractal Brownian motion was introduced by Kolmogorov in Hilbert space. This paper also consider the Hurst index

Definition 1. Let

where

Definition 2. (Fractional Brownian motion) Let

1)

2)

Definition 3. (Quasi-conditional expectation) Let

where

Definition 4. (Quasi-martingale) Suppose that

From the above definitions, it is easy to prove the following theorems:

Theorem 1. ( [

1)

2) Let

3) Let

The interest rate has an important influence on pricing credit derivatives, especially after the fixed interest rate is replaced by the floating interest rate, the impact will be more important. From the point of time, the interest rate also has the characteristics of the fractional Brownian motion. Therefore, [

where

where

To make the formula simple, we suppose that the face value of bond is 1 dollar. The default-free bond’s price was obtained in [

Theorem 2. ( [

where a, b and

where

The above conclusions were all obtained by using the classical theory of the fractional Brownian motion in ref. [

Suppose that

As

where

Denote

This paper consider that the interest rate is the only state variable and

where

and

In the following, we consider the simple case with two firms: firm A and firm B. Their defaults are mutually influenced and both correlated with the market interest rate. We assume that their default intensity satisfy the below relations respectively:

where

We will give the defaultable bond’s price without the proof (see [

Lemma 1. ( [

Now, we calculate the conditionally marginal distributions of default time

Under the new measure

Thus, the default model can be simplified and the calculation of the default probabilities and bonds’ prices will be relatively easy.

Theorem 3. Let

defaults occur up to time t, then the joint conditional distribution of

when

The proof can be found in the Appendix.

Corollary 1. Let

Proof. We can obtain the corollary from Theorem 3, so omit the process.

Now, we apply the above results to price the bonds issued by firm A and B in the looping default framework. We firstly give the other form of pricing formula for the bond. Later, we will price the bonds based on this formula.

Lemma 2. ( [

In this paper, we will not consider the risk from the recovery rate. Therefore, without loss of generality, we suppose that the recovery rates

Theorem 4. Assume the interest rate

where

where

The proof can be found in the Appendix.

In this section, we apply the results in section 3 to price CDS related to the zero coupon bond issued by firm A. Firm C holds a bond issued by the reference firm A with the maturity date T. To seek protection against the possible loss, firm C buys a default swap with the maturity date

In the following, we discuss a simple situation which only contains the default risk from reference firm A and the CDS’s seller 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 notional is 1 dollar. In the event of firm A’s default, firm B compensates firm C for 1 dollar if he doesn’t default, otherwise 0 dollar.

Now, we give some notations. Denoted the swap rate by a constant c and interest rate by

the time-0 market value of firm B’s promised payoff in case of firm A’s default is

Then, in accordance with the arbitrage-free principle, we obtain

Theorem 5. Suppose the interest rate

where

and

Proof.

To derive the swap rate of CDS in the looping default framework, we define a firm-specific probability measure

then

Substituting the quasi-conditional expectation into the above formula of the swap rate C, we deduce (40).

This paper studies the pricing of the defaultable bonds and credit default swap when contagious risk has the attenuation effect in the fractional dimension environment. We consider that the default intensity is correlated with the counterparty’s default and the interest rate following fractional Vasicek model. Moreover, we mainly discuss the CDS’s pricing that the default of the firms has an impact on each other and the default intensity has linear correlation with short-term market interest rates. In fact, we can also consider other more complex cases, such as:

Case 1: The default intensity has nonlinear correlation with short-term market interest rate;

Case 2: We can consider other economic state variables than short-term market interest rate;

Case 3: In our model, we only study two counterparts, however, there are many counterparts in the financial market and we can discuss the case of three counterparts or more in further studies.

We thank the editor and the referee for their comments. Research of W.J. Gu et al. is funded by the Innovation Program of Shanghai Municipal Education Commission (No.: 13YZ125); funding scheme for training young teachers in Shanghai Colleges (ZZshjr12010). This support is greatly appreciated.

WenjingGu,YinglinLiu,RuiliHao, (2016) Attenuated Model of Pricing Credit Default Swap under the Fractional Brownian Motion Environment. Journal of Mathematical Finance,06,247-259. doi: 10.4236/jmf.2016.62021

1) Proof of Theorem 3

Proof. Let

as

so

then

and then

and when

and when

2) Proof of Theorem 4

Proof. Firstly, according to the pricing formula on fractional quasi-martingale in [

where

According to Corollary 1, and then

And we find that the key step is to calculate the three quasi-conditional expectation

As shown in the definition above, we deduce

Let

So

where

Let

is quasi-martingale, so

and, we can deduce

So we can get