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Under the native-born model of default and the circular model of default, we take the price of credit derivatives into account. It’s supposed that the short-term market interest rates are based on Vasicek model in this article. Firstly, we calculate the price of default-free bonds in zero-coupon bond. Then, we give the default-intensity expressions under the two models. We calculate the prices of default-free bonds under the two default models. For different situations, we estimate the parameters by maximum likelihood estimation method and calculate the default probability of the company. From the analysis of the result, we find that the result conforms to reality. So the models of default intensity we suppose in the bond pricing are reasonable.

In the end of the 20th century, credit derivatives are developed as a way to transfer the credit risk of financial products, their value is developed from the bonds and other financial assets. The value of credit dervatives is to restructure credit risk. At the earliest, it appeared in the form of credit default swaps which were derived from corporate bonds. Credit default swap is a kind of insurance contract. Investors do not need to pay to the dealing party that is equal to the contract of credit default swaps in cash. In this way we can separate the credit risk from the cash flow. With the development of market economy, the forms of credit derivatives are enriched.

Because of the market requirement, the variety of credit derivatives becomes enormous rapidly. So its pricing becomes a very popular object of study. As usual, asset pricing is aimed at single case or default assets. When talking about the bonding pricing, we can consider it in the case of recovery rate, default time and market risk-free interest rate. The researching of recovery rate can refer to Brennan & Schwartz [

Bai Yun-fen [

In this article, the credit derivatives pricing is based on the Vasicek model. Because of this, we first introduce the Vasicek model in short. The Vasicek model is proposed by Vasicek (1977), to set a price of bond in the equilibrium model which is mono-factorial. We measure the market risk with Brownian motion. Because the model has explicit solutions, it is very convenient in the process of practical application. So this article we use the interest rate model to study the credit risk pricing model.

Definition 1 The rate in the Vasicek model satisfies the following stochastic differential equation:

where σ is the standard deviation, we use it to judge the random fluctuation.

Solve the differential equation above, we get that

Because ITO integral expectation is zero, so

To take the limit (3), we can get

where parameter b is the long-term average of interest rate.

interest rate r deviate from b and back to b get faster, the mean square deviation of interest rate gets smaller.

This article is based on Vasicek model. We do some research on counterparty risk model under native-default framework of default and annular framework of default. We apply this interest rate model to bond pricing.

Definition 2 assume that probability space

gious limitary number. p is the unique equivalent martingale measure in Hariso and Pliska.

In order to be easy, we just consider two companies once time. The epidemic model could be complicated with many company situation and it’s not easy to export pricing formula. However, we couldn’t get the default parameter estimation. Assume that firm A and firm B each has zero coupon bond at the value of one dollar, expiration date is T, recovery rate is

Assume that the firm A and firm B’s default process is

When comes to default-able bond, the pay should be

The first part in Equation (6) shows that when credit event happens, the debtor pay to the creditors. The second part in Equation (6) shows credit event doesn’t happen, the debtor pay to the creditors. So the price of default-able zero bond at time t equal to the discount price of the pay at time T. That is:

Equation (7) can translate into the following form:

Assume that short-term market interest rate is the unique macro state variable and the interest rate meet the Vasicek model

where

Because P is equivalent martingale measure, we use ITO formula. From that we get

Boundary condition is whatever the r is, there should be F(T,r) = 1 and has solution like

In the following part, we use shreve’s method to calculate the equation.

We can get

Take Equation (11) into above we can easily get

Theorem 1 Assume the short-term market interest rate

1) Native Cluster Framework

Assume that the default intensity of firm A and firm B meet the following relations:

where

Theorem 2 Assume that short-term market interest rate submits to Vasicek model, the default intensity of firm A and firm B obeys Equation (12) and Equation (13). If firm A and firm B don’t break a contract at time t, then, native firm A’s bond price at time t:

Firm B’s default-able bond price at time t:

where

Proof:

There is no relation between breach of contract and interest rate, so we can get that from Equation (8):

We take

In a similar way, we know the default-able bond price of firm B at time t:

In Equation (17),

Take Equation (18) into Equation (17) we can just get the default-able bond price of firm B.

2) Annular Framework of Default

Assume that the default intensity of firm A and firm B meets the following relations:

where

Theorem 3 Assume that short-term market interest rate submits to Vasicek model, the default intensity of firm A and firm B obeys Equation (19) and Equation (20). If firm A and firm B don’t break a contract at time t, then, native firm A’s bond price at time t:

Default-able bond price of firm B at time t:

where

Proof:

By the bond pricing formula

When the default intensity of firm A and firm B obeys Equation (19) and Equation (20),

Random

Now we need to calculate:

Take Equation (25) into Equation (23) we can get firm A’s bond price:

Because the firm A and firm B are a ring of default, so the calculation process is similar to the firm A’s.

This article explores the market short-term interest rates which conform to the Vasicek model. According to the existing literature, we want to get the default-free bond prices. Because it is a markov process, there is an explicit solution by using Shreve’s method. We suppose the form of default intensity with different framework of default. Under the native cluster model, we use the bond pricing formula to get the companies’ default-free bond prices. Under annular framework of default, in order to calculate the price of default-free bond, firstly, we get the two companies’ joint probability density and marginal probability density; then, we take it to the bong pricing formula; finally, we can easily get the default-free bond price under annular framework of default. In the last section, we estimate the probability of default by maximum likelihood estimation method. Through analysis and discussion, the parameters that we estimate correspond to reality. Of course, there are still many deficiencies in this article. For example, we should get further research through actual data. Calibration is carried out to the parameter estimation. If the parameter estimation is not so accurate through empirical analysis, it would produce deviation to the probability of default.

I would like to thank teacher Zhao. He helps me a lot in thinking and writing this article.

HanghangZhou,DianliZhao, (2015) The Pricing of Credit Derivatives and Estimation of Default Probability. Journal of Mathematical Finance,05,243-248. doi: 10.4236/jmf.2015.53022