Pricing Of The Perpetual Bermudan Option And Partial Differential Equation

We consider pricing problem of the perpetual Bermudan option. We make use of the Binomial Tree Method(BTM) and partial differential equation method(PDE) to value the price of the perpetual Bermudan option in the discrete and continuous case respectively, constructing the corresponding the discrete model (the algorithm of BTM) and the continuous mathematical model (the value problem of the parabolic differential equations). The existence and uniqueness of the periodic solution of the perpetual Bermudan option (discrete and continuous) is proved by using the constraction mapping theorem. Furthermore the existence and uniqueness of the optimal exercised boundary is also proved by using the property of the perpetual Bermudan option (discrete and continuous) with respect to the underlying asset and the pricing formulas of the perpetual Bermudan option(discrete and continuous) in the form of series and the corresponding the nonlinear equation which the optimal exercise boundary in the exercise date satisfies are also given by iterative process. In the meanwhile, due to considering exhibiting some biases in Black-Scholes model, we also value the perpetual Bermudan option with jump-diffusion by PDE. The mathematical model of it is given. Furthermore the corresponding the pricing formula of it and the nonlinear equation which the optimal exercise boundary in the exercise date satisfies are present by iterative process.