Crash Option Pricing Under The Jump Diffusion Process

In the financial investment transactions, the drawdown of the asset is an important market risk in the trading market, which is also a key criteria to choose asset for trading. To avoid the market crash for investors, a variety of financial derivatives to hedge the thus trading risks are introduced. Among them, as a fundamental hedging derivative, the Crash option with the drawdown risk, has become the main consideration of financial derivatives.Recently a large proportion of references and research pay more attention to the pricing problem of the Crash option with the geometric Brown motion asset dynamics. Hence it does not incorporate jump risk which is also crucial in the pricing issue. In this paper, we are concerned with the Crash option pricing based on maximum drawdown when the underlying asset dynamics is described as a jump diffusion process. We prove that the price function corresponding to the Crash option with jump-diffusion is a unique solution to is a partial integro-differential equation. Further, we deduce a semi-analytical solution of the price of the Crash option under three different jump distributions. Finally, the numerical analysis of Crash option price and the related properties are presented using the finite difference numerical scheme.