Pricing Vulnerable Options With Stochastic Interest Rates Under Jump-diffusion Process

Option is a very important financial derivative,which has the function of hedging financial risk.Options traded on exchanges generally do not have default risk because of third-party supervision,but options traded on the over-the-counter market are different.Owing to the absence of strict and controllable regulations in the OTC market and the absence of institutions like central clearing houses to maintain the order,defaults often occur and option holders face greater credit risks.Johnson and Stulz(1987)first defined options with credit risk as vulnerable options,and studied such options.Klein uses the structured approach and assumes that the credit risk is related to the underlying asset price to obtain a vulnerable option pricing model.In recent years,the over-the-counter market has been developing and expanding,so it is of great practical significance to study the pricing of vulnerable options.This paper mainly extends the Klein model.Firstly,considering the assumption that stock price and company value are subject to geometric Brownian motion in the Klein model,we can not accurately describe the jump-type changes in the process of real market price changes.Therefore,we assume that both stock price and company value follow jump-diffusion model.Its continuous part to characterize market price and the jump part depicts the price jump caused by unexpected events.In this paper,we introduce a special updating process more general than Poisson process to represent the jumping part,which makes the model more extensive.Under jump-difiusion model,we derive the pricing formula of vulnerable options with fixed interest rates.Then,considering the volatility of market interest rate,we assume that the risk-free interest rate in the model is a Possible stochastic short-term interest rate.We further extend the model and obtain the pricing formula of fragile options with stochastic interest rate in jump-diffusion process,which can be simplified to Klein model under certain conditions.Finally,the convergence and validity of the proposed model are verified by numerical analysis.Compared with the classical B-S model and Klein model,the effects of various parameters of the model on the price of vulnerable options are studied by numerical examples,with the assumption that the stochastic interest rate follows the Vasicek model.The impact of market risk,default risk,jump risk and interest rate risk on the price of call options is considered comprehensively.