Pricing Barrier Option Under Jump-Diffusion Model

Option pricing is one of the core question in financial mathematics.Since Black & Scholes publish their well known models,there are a great many test in the financial market,but the statistic data press indicate that Black-Scholes Model have some systemic difference with the real financial market.There are two distinct phenomenons:The first is the distribution of risk-free return's Leptokurtic is much less than the real.Moreover,there is a Implied volatility smile,that is the distribution of underlying asset is no a exact a Normal distribution.In order to improve the Black-Scholes model,many scholar give many methods,there publish a great many models,and the Jump-Diffusion is one of the most models. In this article we assume the underlying asset fellow a Jump-Diffusion model,then pricing a barrier option. A barrier option is a financial derivative contract whose payoff depends on whether the price path of the underlying asset (could be a stock,an index,an exchange rate,etc) has reached a certain predetermined level(called a barrier).These barrier option can be classified as either knock-out options or know-in options.Most models assume continuous monitoring of the barrier,while in practice, most contacts with barrier features specify fixed times for monitoring of the barrier-typically, daily closings,which called discrete barrier options.There are following results:1, we assume the underlying fellow a Constant Jump-Diffusion model,then get the solution of the stochastic differential equation.2, With risk-free pricing option pricing theorem,we get the price of Up-and-In put option V(H),then get the discrete option with similar method.3, Compare the distribution of X,τand X,τ′, then give the relation of V(H) and V_m(H).4,Use Laplace Transform and Monto Carlo method to simulate the continuous and discrete ones respectively.