Research On The Pricing Model Of American Option Based On Tree Graph Method And Finite Difference Method

Finite difference method(FDM) and the tree diagram method(binomial methods) is the most important and most commonly used numerical method for American option pricing, each of them plays an important role in the study of American options pricing issues. Based on the previous studies, the paper gives the idea of making widely linkage relationship between the different option prices, and based on this, the FDM the tree diagram method has been studied theoretically and empirically.The paper firstly differencing the Black-Scholes PDE and get the general form of FDM(Explicit model, Implicit model and Crank-Nicolson model) that including more high-dimensional differential price changes. Then proving the compatibility of the three models by calculating the truncation error of them, and discussing the stability and convergence of the three models according to the Fourier transform, von Neumann norms and Lax equivalence theorem. Finally, creating a multi-stage iterative model of the general form of FDM by establishing a flat grid and matrix model, and verify that the general form of FDM are improved compared with the conventional method in convergence speed and accuracy after numerical simulation and empirical analysis. The second part of this paper is the study of trinomial methods, according to geometric Brownian motion and the risk- neutral principles, we deduce the single period trinomial methods. Since the changes in asset prices follow geometric Brownian motion that is normally distributed, therefore, by fitting the probability of trinomial methods coefficients with the probability function of Poisson distribution which is the alternative of normal distribution we can get the best undetermined parameter of the model, and then get the specific form of trinomial methods. At last, according to the numerical simulation of pricing the European option by trinomial methods to analyze the convergence properties of the Solutions, then through the empirical analysis of pricing American option to validate the result of trinomial methods in practical applications. At the end of the two parts, It also proposed a "generalized finite difference model" and higher dimensional tree model conjecture, attempting to explore the theory to be deeper and propose some ideas for the future research related.