Use of radial basis functions for meshless numerical solutions applied to financial engineering barrier options

Abstract

A large number of financial engineering problems involve non-linear equations with non-linear or time-dependent boundary conditions. Despite available analytical solutions, many classical and modified forms of the well-known Black-Scholes (BS) equation require fast and accurate numerical solutions. This work introduces the radial basis function (RBF) method as applied to the solution of the BS equation with non-linear boundary conditions, related to path-dependent barrier options. Furthermore, the diffusional method for solving advective-diffusive equations is explored as to its effectiveness to solve BS equations. Cubic and Thin-Plate Spline (TPS) radial basis functions were employed and evaluated as to their effectiveness to solve barrier option problems. The numerical results, when compared against analytical solutions, allow affirming that the RBF method is very accurate and easy to be implemented. When the RBF method is applied, the diffusional method leads to the same results as those obtained from the classical formulation of Black-Scholes equation.