Research On Pricing Defaultable Option And Bond Based On Defaultable Price Processes

With the development of financial innovation, financial derivatives have been more and more in fiancial markets. These financial derivatives provide important tools for investors to avoid financial risks, and also provide tremendous investment opportunities for investors. And same time, these derivatives also give huge financial risks. The global financial crisis which occurred in 2008 has been caused by financial derivatives, such as the mortgage bonds. The pricing of default derivatives has been caused widespread concern for scholars, and has been a hot problem in modern finance.How to characterize the risk of options and bonds for investors has become the urgent question which must solve. In this article, we use the mean variance hedging methodology to study the default risk, and give the pricing formula for defaultable options and bonds. Firstly, we aim at the defaultable options based on different price processes, such as defaultable jump diffusion processes and defaultable semimartingales with general jumps. Then we also aim at defaultable bond in complete market and in incomplete market conditions; give the pricing formula for defaultable bonds. In the last, we careful study the optimal-variance martingale measures, and give an explicit construction of the optimal-variance martingale measures.In chapter two we derive a complete solution for pricing defaultable options based on defaultable jump-difusion process. The approach consists of three steps:First, we set up a defaultable martingale representation theorem for defaultable jump-difusion process. Then, we construct two backward stochastic differential equations for hedging as a stochastic control process and the option. Finally, we find out the optimal investment strategy through minimizing the cost function, which leads to the pricing equation for defaultable options.In chapter three we propose a methodology for pricing defaultable option for semimartingales with general jumps in incomplete markets. First of all, a default process martingale representation theorem based on semimartingale is proved. Then two backward semimartingale differential equations (BSDEs) about control process and option are constructed, and their solutions are proved existent and unique. Finally a mean-variance hedging methodology is used to derive the pricing formula of defaultable options for semimartingales with general jumps.In chapter four, we treat defaultable zero coupon bond as a contingent claim in incomplete markets. By setting up an investment portfolio with risky asset, we develop the bond pricing equation by finding the optimal investment strategy with minimum risk through linear-quadratic hedging (LQ) method.In chapter five, we assume the short-term interest rate of bonds is a diffuse process; in this chapter addresses the pricing process of defaultable zero-coupon bonds and their pricing method. By employing the equivalent martingale measure and the principle of affine term structure of interest rate, an explicit pricing equation of defaultable zero-coupon bonds is derived under the assumptions of Vasicek model.In chapter six, we proposed a local martingale representation theorem for exponential defaultabale jump-diffusion processes based on the defalutable jump-diffusion price processes. And we construct an optimal problem for optimal-variance martingale measures under the martingale condition, we use Euler-Lagrange equation, give an explicit construction of the optimal-variance martingale measures.