Pricing vulnerable European options under a Markov-modulated jump diffusion process

In this paper, we investigate the pricing of vulnerable European options under a Markov-modulated jump diffusion process. The states of market economy which are described by a two-state continuous time Markov-chain are explained as a stable state and a high volatility state. The dynamic of the risky asset is described by a Markov-modulated geometry Brownian motion when the market state is stable, otherwise, it follows a Markov-modulated jump diffusion process. We consider two types of models to describe default risk: one is the structural model, the other is the reduced form model. By utilizing techniques of measure changes, some analytic formulas for pricing vulnerable European options are derived under these models.