An Adaptive Finite Difference Method Using Far-Field Boundary Conditions for the Black-Scholes Equation

Abstract

Abstract. We present an accurate and efficient numerical method for solving the Black-Scholes equation. The method uses an adaptive grid technique which is based on a far-field boundary position and the Peclet condition. We present the algorithm for the automatic adaptive grid generation: First, we determine a priori suitable far-field boundary location using the mathematical model parameters. Second, generate the uniform fine grid around the non-smooth point of the payoff and a non-uniform grid in the remaining regions. Numerical tests are presented to demonstrate the accuracy and efficiency of the proposed method. The results show that the computational time is reduced substantially with the accuracy being maintained.

Accurate and Efficient Computations of the Greeks for Options Near Expiry Using the Black-Scholes Equations

“…Λ(s) is the payoff function at maturity . There are three classical techniques which are the finite difference method (FDM) [3][4][5][6][7][8][9][10][11][12][13], the finite element method [14], and the finite volume method [15] for the numerical solutions of the BS PDE.…”

Path Averaged Option Value Criteria for Selecting Better Options

“…Beginning in 1973, it was described that a mathematical framework for finding the fair price of a European option by Black and Scholes [1,2], several numerical methods have been presented for the cases where analytic solutions are neither available nor easily computable. See more details about numerical methods such as finite difference method (FDM) [3,4,5,6,7,8,9,10,11,12,13], finite element method [14,15,16], finite volume method [17,18,19], and a fast Fourier transform [20,21,22,23,24]. For convenience, we use the closed-form of the Black-Scholes equation in this work.…”

Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations

“…Therefore we need to use a numerical approximation. To obtain an approximation of the option value, one can compute a solution of the BS equations (1) and (2) using a finite difference method (FDM) [2][3][4][5][6][7][8], finite element method [9][10][11], finite volume method [12][13][14], a fast Fourier transform [15][16][17], and also their optimal BC [18].…”