Pricing The Credit Derivatives Under The Stochastic Volatility Jump Diffusion Model

In recent years, how to effectively manage and defense credit risk has been the main concern in both academia and industry. There are mainly two kinds of methods on the study of credit risk. One is a structured method, represented by classical B-S model, known as the Merton method and the other is a reduced method, pioneered by Jarrow and Turnbul. Based on the capital structure within the enterprise, the structured model provides a series of necessary assumptions, such as constant interest rate, constant fluctuation rate, to adjust the enterprise dynamic assets process to meet the demands of the B-S model. It is the assumptions that makes the structured model difficult to capture the sharp peak and thick tail phenomenon of financial variables in actual market. And the model’s assumptions make extensive models more complex and difficult to estimate and correct. In addition, as it being difficult to accurately and timely acquire the relevant financial information of enterprise’s assets, liabilities, the structured model is rarely useful in practice. The reduced model does not consider the company’s capital structure and thinks that the corporate defaults, being considered as an external variable, have no obvious relationship with the enterprise value. It directly builds the company default model to research the company defaults. The reduced models based on intensity include the credit rate conversion model, the period structure model, and the affine intensity model. The affine intensity model assumes that the state variables determining the probability of default follow an affine jump process, which introduces the process of affine jump features to the default correlation and thereby introduces sudden analysis to default process. Due to the applicability of the affine process and the maturity of the development of the theory, credit risk research under the framework of affine has become one of the hot pots.The thesis studied the credit default swap and its derivatives pricing by using the affine strength model under the situation of volatility and interest rates with random jump risk considered. Main work of the thesis are as follows:Firstly, the default bond pricing was studied. The model assumed that the default intensity was a linear function of the economic variables that can be directly ob-served by the company and considered the volatility and interest rates as random with jump risk. Then, by using relevant knowledge of characteristic function and stochastic differential equations, pricing formula of default bond is given based on this model. Secondly, based on the default bond pricing, credit default swap pricing of having counterparty risk and having no counterparty risk are studied, respectively. The corresponding credit default swap pricing formula are given. Thirdly, based on the study of the second section, credit default swap share option pricing was studied by using the methods of stochastic differential equations, characteristic function, and Fourier trans-form. Then the expression of European bullish credit default swap share option was obtained. Fourthly, numerical calculation was conducted to analyze the influence of some parameters in the model on credit default swap difference and credit default swap share option price.The results are as follows:(1) Credit default swap difference decreases when recovery rate increases. Price of credit default swap share option in the short term increases rapidly when time goes by while in long term is stable relatively.(2) The correlation coefficients of diffusion fluctuation pand diffusion bounce rhoj both have a positive impact on credit default swap difference and credit default swap share option price although the impact degree of ρ and ρ J are different, where the diffusion fluctuation grows slowly while bounce fluctuation increases slowly first and then dramatically.(3)Of four types of classical models (SV, SVJD, SCIJ and SVCJ), change and performance of the credit default swap difference along with the change of the default intensity and change and performance of the credit default swap option price along with the change of credit default swap are compared respectively.The result shows that the jump risk in default intensity or default volatility has a significant impact on credit default swaps difference and credit default swap option price. The correlation coefficient of diffusion volatility and bounce has positive impact on credit default swaps spreads and credit default swap option price. However, the impact degree is different, where the diffu-sion volatility grows slowly while bounce fluctuations increases slowly first and then dramatically. Bounce risk existed in the default intensity or volatility has a significant impact on credit default swap option price. Bounce risk existed in the credit default swap spreads or volatility has a signif-icant effects on credit default swaps option price as well. Furthermore, bounce together of the two has much larger influence on the option price than that of bounce independently.