Multiple Jumps-Diffusion Model and Vulnerable European Option Pricing

A mixed diffusion process involving various sources of jumps is introduced to characterize both the price of underlying asset and the ratio of firm's assets to liabilities. Continuous component is modeled as geometric Brownian motion to describe their ``normal'' revolution, and discontinuous component is modeled as jumps with several Poisson arrival processes in conjunction with corresponding random jump size to characterize their sudden increase or drop in a surprising manner instantaneously. This may be due in part to the impact of rare events and new information, such as technological innovation, regulatory effects, catastrophic rare events and so on \ldots These jumps are assumed independent of each other, with each type having a log-normally distributed jump size, we also supposed that all jumps risk is diversifiable and hence not priced in equilibrium. By applying It\^o lemma and equivalent martingale measure transformation within the framework of our model, we derived a closed form of analytic solution for vulnerable European option, and therefore generalized classical formula for vulnerable European option with jump and quantified the works by Zhou\,(2001) and Lobo\,(1999).