Interest Rate Swap Pricing With Default Risk Under Stochastic Interest Rate

Being one of the most popular and highly liquid instruments of the international cap-ital market, interest rate swap is in possession of functions such as price discovery, risk aversion and asset allocation. Since the first trade in the year of 1981, the year-on-year average growth of interest rate swap trading was over 30%. Interest rate swap has been popular all along and its influences on a corporation's capital management are becoming more and more significant. The Chinese interest rate swap market also goes into a fast growing stage from the opening in 2006. The purchase of appropriate pricing mode of the burgeoning interest rate swap market became more urgency.Based on these reasons, this paper corporate the developing the credit default mea-surement theory, providing three stepwise interest rate swap pricing model with default risk. Duffie & Huang introduce a reduce model by merging the risk factors into risk free discount rate in 1996. With counterparties of different default risk, the promised cash flows of a swap are discounted by a switching discount rate that, at any given state and time, is equal to the discount rate of the counterparty of which the swap is currently out of the money (that is, a liability). They showed that defaultable securities can be priced in a way similar to the standard risk neutral pricing of default-free securities with the effective discount rate being used instead of the usual risk free short rate.The first two models presented in this paper is based by the reduced model of Duffie & Huang (1996), assuming the dynamics of firm assets following the Geometrical Brown motion process and the Exponential Variance Gamma process respectively. The risk free discount rate is adjusted by the default density after it is calculated by structure model. Then the interest rate swap is discounted by the risk adjustment discount rate to get the final rates.As a process with jump item, EVG process possesses some advantages, such as it can solve the problem of volatility smile and the fat-tail phenomena. The second model in this paper is an advanced model than the first one theoretically, because the EVG model is a more proper process in describing dynamics assets price.Both of the two models'numerical results show that, a one hundred basis point of credit spread (bond spread) cause only 0.088 basis point of swap spread averagely. It seems that the advance of assets dynamics price model don't behave better in interest rate swap pricing. It's because of the special swap clauses of without exchange the principle and the netting, causing the little sensitivity of swap to default risk. And by another side, the swap is not always assets or liability to a counterparty, and swap is suffered from the counter-party's default risk only when the swap is a liability to this counterparty. Additionally, our paramenters in this paper are all let the log return under the EVG model is larger than under the Geometrical Brown Motion, and this positive difference of log return counteracts the risk caused by the jump under EVG model.The first two models both calculate the swap rate by two steps, first step calculate the default density and the second step solve the PDE, which will cause double error. In order to solve this problem, we present another interest rate swap pricing model, which we consider the interest rate swap as a contingent claim of three stochastic variables:assets price, risk free interest rate, and time t, under the risk neutral measurement. We suppose the firm assets price follows the EVG process, and then deduce the PIDE which the value of the interest rate swap satisfies under the stochastic interest rate. The numerical results show that, a one hundred basis point of bond spread results in a 0.111 basis point of swap spread. Although the result seems increasing in more than 25% corresponding to the former two models, but the absolute difference is still very small too. It seems that the double error caused by the first two models is not so obviously in swap pricing, and all the three models are suitable pricing model of swap.We apply the three models into bond pricing with small modification since bond pric-ing is relative to swap pricing. The numerical results show that the behave of the three models are very different in bond pricing. Under the same parameters, the difference of bond rate can be more than 1000 basis point which is very significant. As a result, as- sets as bond, which are sensitive to default risk, the choice of a proper pricing model is very important. And theoretically, the third model use the advanced assets dynamics price model-EVG process and solve the problem of double error, we consider the third model is the relative proper model in bond pricing.