Finite Difference Method For Solving American Lookback Put Option Under The Black-Scholes Model

In this paper, we mainly study the numerical method for valuing American look-back put options under the Black-Scholes model. It is well known that the American lookback option satisfies the two-dimension nonlinear parabolic problem, which is hard to use the numerical method to solve directly. Based on the analysis of the is-sues on solving the problem, this paper introduces an approach to settle it. First of all, we transform the problem into the one-dimension form applying by the numeri-cal technique, whereafter, using the Landau’s transformation to normalize the defined domain. Due to the nonlinear feature, applying the finite difference method, we ob-tain the discretization form of the Black-Scholes equation, which is used to solve the option value, and we get the optimal exercise boundary by Newton’s method. Solving this problem in turn, we get the option price and the optimal exercise boundary simul-taneously. We also prove the numerical solution is nonnegative under some appropri-ate assumptions. In addition, we verify the validity and correctness of the proposed methods by comparing with the binomial tree method, which provides a theoretical basis for the application. Compared with the binomial method, our method not only can guarantee the numerical accuracy, but also can reduce the time cost, which is an efficient method of quickly pricing the financial products.