Pricing Default Bond In Financial Market Driven By Fractional Brownian Motion And Jump Process

Vulnerable option is a financial derivative product, because it has th e characteristics of credit risk, it can be called with credit risk options. In t he domestic financial market, institutional imperfections and low creditw orthiness of investors, making China's financial market is facing serious c redit risk, affecting the stability and stability of the market and security an d so on. At present, many researchers at home and abroad are strengtheni ng the research on credit risk options, which is helpful to establish long-te rm stable financial market environment, and also provide more theoretical and time guidance for institutional investors and individual investors, wh ich is also conducive to reducing the credit risk to their losses. In the past,many researches on option pricing were based on the general geometric Brownian motion of asset prices. However, the capital market was changi ng rapidly, and some policies and important information were changed en ough to make the asset prices jump irregularly or discontinuously. In fact,whether in practice or in the theoretical study of scholars, we find that th e price of various assets in the stock market has a long-term memory, that is, the price of a certain time may have an effect on the later price after a long time , which means that these prices do not follow the geometric Bro wnian motion. However, the Brownian motion of these features is precis ely the fractional Brownian motion, so it is more appropriate to use the fr actional Brown motion to model the asset price process. Based on this, thi s paper describes the change of asset price by Fractional Brownian motio n, which making the established model more in line with the actual situati on. In addition, to describe the discontinuity of asset prices, this paper stu dies the pricing of fragile European options under jump-diffusion. This pa per mainly includes the following two parts. The first part, we set up the p ricing model of the default bond driven by the fractional Brownian motio n. We use the theory of stochastic analysis about fBM to derive the pricin g formulas of the defaultable bonds and its analytical solution in the speci al case. In addition, we analyze the influence of counterparty risk and rec overy rate on the value of bonds by numerical analysis. The second part,we studied the pricing of the fragile European option under the jump-diff usion process, in addition to considering the relevant credit risk, we also u se the jump-diffusion process to simulate the process of the change of the related asset price. By using the Ito formula and the Taylor expansion met hod, we obtain a fragile European option pricing formula under the jump-diffusion process. And some special cases under the model are given.