Aplicação de um método adaptativo temporal de funções de base radial à solução da equação de Black-Scholes

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 compared one against the other 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. Furthermore, a time adaptive scheme was implemented, based on available predictive and corrective algorithms associated with Bixlers (1989) time adaptive ordinary differential equation solver. The time adaptive methodology showed itself to be highly efficient, when efficiency is defined both in terms of computing speed (number of time steps required to reach solutions at a desired simulation time) and in terms of accuracy or precision. Actually, the use of adaptiveness associated to numerical truncation errors of the order of 10-7, in the case of call options, and 10-5 to 10-4 when barrier options were considered, led to excellent results. This work shows a series of graphs reflecting the dependence of the numerical error with the integration method, initial time step, underlying asset value, specified truncation error and maximum stock value for implementing boundary conditions. In the case of call option simulations, the Cubic RBF method was more efficient than the TPS one, while in the case of barrier options, both methods led to essentially equivalent results. The time adaptive technique applied to the solution of Black-Scholes equation allows considerable computer processing efficiency; indeed, the number of time steps required to reach a final desired simulation time, under a given required accuracy, can be 500 times smaller than when no adaptiveness is used; the economy is problem dependent.