Numerical Methods For American Options Pricing

Option is one of the core tools of financial derivatives, which plays an important role in the effective management of risk and speculation. Risk management is depended on the right evaluation of option in a certain extent .The critical thing is how to value its fair price. A wide variety of the options traded in exchanges are American options. It is thus important to find appropriate ways to price American options. Thus, unlike European options, no explicit closed-form formulas have been found for American options, approximation methods have to be used in practice. Generally the classic Black-Scholes model for an American option leads to a free boundary problem with a degenerate partial differential operator.The part of this text introduction has done the reviewing of generality to the financial derivative and pricing theory. At the second chapter, the article expatiates the instauration of the Black-Scholes Differential Equation. At the third chapter of the article, nonlinearity error has been sharply reduced by grafting one or more small sections of fine high-resolution lattice onto the traditional Binomial Tree and Trinomial Tree with coarser time and price steps on the proper area. In the fourth chapter, the traditional differential scheme, based on the B-S differential equation, has been improved by changing some parameters presented, and collecting their both advantages. The following chapter presents the procedure for deriving canonical forms of European options of multi-assets is introduced. Then, the MFS based on the convection-diffusion fundamental solution is proposed to solve the resulted canonical forms. The MFS method has been expanded to the American option of single asset after considering the characteristics of MFS and American option. Numerical example were showed the efficiency and practicability of the algorithm at the end of the chapter 3, 4 & 5.