Study Of American Option Pricing Based On Singularity Separation Method

In 1973, based on non-dividend-paying stocks option, Fischer Black and Myron Scholes derived Black-Scholes option pricing formula and gave a precise analytical solution for European option. However, due to the possibility of early exercise of American option, we could only find the approximated analytical solution or, mostly the numerical solution in stead of analytical one. Finite difference method which solves discrete equation on a finite area, is one of the popular methods in American option pricing. The irregularity of the difference area may lead to relatively big truncation error. By making use of the relation between the American option and the corresponding European option and introducing some appropriate variable transformations, in this paper, we simplify the pricing model for American option which involves a parabolic equation with free boundary condition. These simplifications convert the original area where to take difference into a regular shape and eliminate the singularity in the initial condition. Moreover, the separation of singularity decreases the truncation error of numerical computation.B-S equation is the pillar of the option pricing. In this paper, assuming that the price of the stock follows the geometric Brownian motion, we deduce the B-S equation in details. Based on this equation, we study the pricing model for American option which essentially is an obstacle problem. Take the American put option as an example, for any particular time before expiration, there exists a corresponding price for which the value of the American put option equals the corresponding payoff function if the real price is less than the fixed price otherwise the value of the put option is greater than the payoff function. This particular price depends on time, which leads to a so called free boundary. Due to the uncertainty of this boundary, it is hard to find the analytical solution of the American option. Therefore, numerical methods become the main approaches to study the pricing of American option. The fundamental idea of singularity separation method is discussed in details. Because the Europe option price could be obtained precisely, we consider the difference of the price of American and the corresponding Europe option instead of the price of the American option itself, which could decrease the relative error. Moreover, the bound of the equation of the American option pricing is not smooth enough. Therefore, we introduce some appropriate variable transformation in order to control the truncation error. The binomial method and Projected SOR method are two normal method in American option pricing, we compare the singularity separation method with the these two and find that singularity separation method efficiently improves the accuracy and the speed of computation.