A Credit Structural Model With Jump Risks And Related Problems

Credit risk is the possibility that either trader fails to fulfill obligations in a financial contract.In recent years,along with the default events and emergencies in financial market rising frequently,credit risk has become an indispensable part of finance risk.So,it is urgent and meaningful for credit risk to establish mathematic models and develop further discussions.Mathematic models of studying credit risk obtain two fundamental models:the structural one and the reduced one.However,the two models are almost based on the geometric Brown motion and cannot simulate emergencies in the financial market.Thus,the urgent affair is to improve the traditional models.Based on the multidimensional affine jumps diffusion process,this dissertation studies credit risk problems and establish the credit structural model with jump risks.The kernel of the model is that a default occurs whenever the value of assets falls below debts.Compared with the tradi-tional one,it considers the stochastic volatility,the stochastic interest rate,and their jumps,in the changing process of the total assets value.It overcomes the imperfection that emergencies cannot be simulated by the traditional model.The major contents of this dissertation are as follows:To begin with,for the default probability,when the default may come in the maturity only,the closed explicit formula of the conditional default probability is obtained,according to the affine structure,by the semi-martingale Ito formula,the Feynman-Kac theorem,the Fourier transform,and the characteristic function method;when the default may come in any time,the probability density function of the first passage time is obtained by the curve fitting method.Furthermore,applying these conclusions,by the equilibrium method,forward measure tech-nique,and Fourier inversion transform,for the securities correlated with credit,are obtained,the defaultable zero-coupon bonds value,the risk interest rate of the defaultable coupon bonds,and the credit spread of the zero-coupon bonds;for the credit derivatives,are obtained,the value of the defaultable zero-coupon bond options and the credit spread options,along with the expectation of the discounted compensation and the year premium,of the credit default swaps.Finally,by numerical simulation,the term structures of the conditional default probability,the defaultable securities value and the credit derivatives value are obtained,and the impacts of the key parameters in this model for them are analyzed.Numerical results show:all kinds of jumps have more remarkable effects on the default probability,the defaultable securities value,and the credit derivatives value.For example,it can make the default probability of companies and the credit derivatives value rise remarkably,increasing the simultaneous correlated jumps intensity,among the assets value,the volatility and the interest rate.