Vulnerable Option Pricing In Bi-Fractional Brownian Motion Environment

Vulnerable option is an option that has credit risk when trading in an overthe-counter market.Due to the existence of credit risk,there is a possibility of default when both parties entered the option transaction,so that the option can not be implemented.In the past,people assumed that the stock price satisfy the geometric Brownian motion or fractional Brownian motion.As the financial market changes rapidly,a lot of financial evidence shows that the differential equations that existing stock prices satisfy are difficult to adapt the market demand.In recent years,some scholars proposed the bi-fractional Brownian motion,which is a more generalized Gaussian process and random than the fractional Brownian motion,the bi-fractional Brownian motion can be described more general financial phenomena.This paper mainly discusses the pricing problem of the vulnerable option in the bi-fractional Brownian environment.The main research results are as follows:(1)Assume that the stock price,company value and corporate debt both satisfies the stochastic differential equation driven by bi-fractional Brownian motion,assume that the interest rate is a constant,the corresponding market model is established by using the stochastic analytical theory of the bi-fractional Brownian motion,applying the actuarial approach,the pricing formula of vulnerable option in bi-fractional Brownian motion environment is obtained.(2)Assume that stock price satisfy the stochastic differential equation driven by the bifractional Brownian motion and jump process,both the firm value and the corporate debt satisfies stochastic differential equation driven by bi-fractional Brownian motion.The financial mathematical model in bi-fractional jump-diffusion process is built by the stochastic analysis theory of the bi-fractional Brownian motion and jump process.Using the actuarial approach,the pricing formula of vulnerable option in bi-fractional jump-diffusion environment is obtained.(3)Introduce the bi-fractional Vasicek interest rate model to discuss the pricing problem of the vulnerable option.Suppose that stock price,company value and corporate debt satisfies the stochastic differential equation driven by bi-fractional Brownian motion,establishes the corresponding market model,using the actuarial approach,the pricing formula of vulnerable option in bi-fractional Vasicek rate environment is obtained.