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A RADIAL BASIS FUNCTION METHOD FOR SOLVING OPTIONS PRICING MODEL Yiu-Chung Hon 1 Xian-Zhong Mao 2

ABSTRACT: This paper applies the global radial basis functions as a

spatial collocation scheme for solving the Options Pricing model. Dierent numerical time integration schemes are employed for the time derivative of the model. In the case of the European options, it is shown that the major numerical error is from the time integration instead of the spatial approximation by comparing with the analytical solution. The numerical results for the American options indicate that this proposed scheme oers a highly accurate approximation compared with existing numerical methods. Since the basis functions are in nitely dierentiable, the numerical approximation of the derivatives of the options price can be computed directly without using extra interpolation techniques. The numerical approximation of the optimal exercise boundary in the case of American options can also be obtained eectively by using the Newton's iterative scheme. Key words: Black-Scholes; American Options; Radial Basis Functions. Yiu-Chung Hon, Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong. 2 Xian-Zhong Mao, Zhejiang Provincial Institute of Estuarine and Coastal Engineering Research, China. 1

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I. INTRODUCTION

Black and Scholes (1973) rstly proposed an analytical formula for evaluating European call options. By assuming that the asset price is risk-neutral, Black and Scholes showed that the European call options value satis es a lognormal diusion partial dierential equation which is now known as the celebrated Black-Scholes equation. It is well known that the American options can be treated as a free boundary problem in which no analytical formula is available. Until recently there are several numerical methods for the American options valuation out of which the nite dierence method (Geske and Shastri 1985) is most commonly employed. To list some of the other numerical methods for the American Options: binomial method by Cox, Ross and Rubinstein (1979); projected SOR method by Wilmott, Dewynne and Howison (1993); Front- xing nite dierence method by Wu and Kwok (1997); Monte Carlo simulation by Grant, Vora and Weeks (1996); integral equation method by Huang, Subrahmanyam and Yu (1996); and more recently linear programming technique by Dempster and Hutton (1997). A comparison of some of these numerical methods can be found in Broadie and Detemple's (1996) review paper. In this paper a new numerical scheme is devised by applying the global radial basis functions (RBF), particularly Hardy's multiquadric (MQ), as a spatial approximation for the numerical solution of the options value and its derivatives in the Black-Scholes equation. This transforms the Black-Scholes equation to a system of rst order equations in time and the American options can then be approximated by using a high order backward time integration scheme like fourth order Runge Kutta method. Numerical results indicate that this RBF method oers a highly accurate spatial approximation to the solution. The method does not require the generation of a grid as in the nite dierence method and the computational domain is composed of scattered collocation points. Since the radial basis functions are in nitely continuously dierentiable, the higher order partial derivatives of the options value can directly be computed by using the derivatives of the basis functions. 2

Hardy (1971) rstly developed this MQ to approximate two-dimensional geographical surfaces. In Franke's (1982) review paper, the MQ was rated one of the best methods among 29 scattered data interpolation schemes based on their accuracy, stability, eciency, memory requirement, and ease of implementation. Stead (1984) showed that the high order partial derivatives obtained from the MQ interpolant are much more accurate than most conventional techniques like nite dierence quotient. Recently, Kansa (1990) successfully modi ed the MQ for solving partial dierential equations (PDEs) of elliptic, parabolic, and hyperbolic types. Hon et al. (1997) further extended this MQ to solve various nonlinear initial and boundary-value problems including Burgers' equation with shock wave and biphasic mixture model with solid and uid phases. Golberg and Chen (1994) improved the numerical accuracy by using the MQ in their dual reciprocity method for approximating the particular solution of partial dierential equations. However, The performance of this MQ depends on the choice of a user-speci ed parameter c, which is often referred to shape parameter. Golberg, Chen, and Karur (1996) and Hickernell and Hon (1997) applied the technique of cross validation to obtain an optimal value of the shape parameter. In this paper, the value for the shape parameter suggested by Hardy (1971) is used in the computation. The organization of this paper is as follows. In Section II we introduce the radial basis functions method for solving the Black-Scholes equation. Numerical comparison with the analytical formula in the European options case are given in Section III. In Section IV the algorithm for American options valuation is given. The optimal exercise boundary is also computed by using the ecient Newton's iteration method. Numerical comparisons with the binomial method are also given. Conclusion is presented in the last Section V. Furthermore, this numerical scheme proposed here is of a general nature and can be used for solving nonlinear PDEs arising in other areas.

II. RADIAL BASIS FUNCTIONS FOR SOLVING BLACK-SCHOLES EQUATION 3

To illustrate how to apply the radial basis functions as a spatial collocation scheme for options pricing, we consider the values of the European options which satis es the following Black-Scholes equation: @V + 1 2S 2 @ 2V + rS @V ? rV = 0 (1) @t 2 @S 2 @S where r is the risk-free interest rate,  is the volatility of the stock price S , and V (S; t) is the option value at time t and stock price S . The initial condition is given by the terminal payo valuation: 8