Pricing Defaultable bonds and CDS with PDE methods

The market for credit derivatives is growing rapidly. The credit derivative market’s global size was estimated to be $100 billion to $200 billion in 1996. The British Bankers Association estimated that the size was $1.6 trillion in 2001. Now the size is about $62 trillion [10]. The demand is strong because credit derivatives provide varieties that can fit different clients. The fundamental credit derivative is the defaultable bond. When pricing defaultable bonds, we need to consider not only the face value and the coupon of the bonds but also the default risk. Bonds with high default risk should be cheaper than bonds with low default risk. In this paper, we derive PDE models to price credit derivatives. In Chapter 2, we will discuss the reduced form approach, which is the popular method used in pricing a credit derivative. The reduced form approach models the default probability without considering the value of the firm [1]. The main idea is the yield spread, i.e. the difference between the yield of a defaultable bond and the yield of a default-free bond. Under the approach Li [13] concludes that the yield spread is attributed to two components, the default probability and the recovery rate, i.e. the fraction of the bond’s face value paid in case of default. The reduced form approach is easy to calibrate because we can obtain the data of the yield spread from the market easily. However the yield spread is not only caused by these two components in the real world. Longstaff [15] mentions that the yield spread also may be caused by the liquidity. In Chapter 3 we introduce the structural approach to price of defaultable bonds. The structural approach models default risk by modeling the value of a firm directly [1]. Under the structural approach, the pricing PDE is a 1-D PDE with a moving boundary. We assume that the interest rate is a constant in Chapter 3. But interest rates are an important factor when pricing bonds, so we introduce stochastic interest rate models in Chapter 4. In Chapter 5, we review the papers that price defaultable bonds with a stochastic interest rate. One derives the fundamental PDE. There are analytic solutions for the PDEs, if the model of the interest rate is simple enough. If the model of the interest rate is complicated, however one has to use numerical methods in solving the PDEs. We introduce a new method to price the most popular credit derivative in the market: the credit default swap (CDS) in Chapter 6. The CDS is a kind of insurance that protects the buyer of the CDS when a default event occurs. As a traditional insurance, the protection buyer makes regular premium payments quarterly or semiannually. When the default event occurs, the protection seller pays par value of the bond to the