Author(s)

E. Kita & Y. Goto

Abstract

This paper describes the application of a radial basis function approximation for evaluating European and American stock options. The European stock option is evaluated with the Black-Sholes equation and the boundary condition related to the exercise price. The Black-Sholes equation is solved with the Crank-Nicholson scheme and the radial basis function approximation. Numerical results are compared with the theoretical solutions to discuss the validity of the formulation. Finally, the extension of the present formulation for evaluating the American option is also explained. 1 Introduction Recently, the financial derivatives are widely dealt and the importance is expanded. The importance of the derivative transaction is increasing for the adequate sharing of the financial risk. The option transaction is one of the most important financial derivatives and therefore, several schemes have been presented by many researchers for the pricing of the options [1, 2]. Several types of the financial options have been developed; European option, American option, Look-Back option, Exotic option and so on. In this study, we will consider the pricing of the European and the American options. The price of the European option can be evaluated as the solution of the Black-Sholes differential equation by taking the payoff condition in maturity day. The Black-Sholes equation is discretized according to the Crank-Nicholson scheme on the time axis and the option price is approximated with Radial Bases Function with unknown parameters at each time step. The initial values of the parameters are determined from the payoff condition in maturity day. Then, the parameters at the pricing day

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