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 timedependent 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.Keywords: financial engineering; radial basis functions; diffusional method; barrier options. ResumoMuitos problemas de engenharia financeira envolvem equações não-lineares com condições de contorno não-lineares ou dependentes do tempo. Apesar de soluções analíticas disponíveis, várias formas clássicas e modificadas da conhecida equação de Black-Scholes (BS) requerem soluções numéricas rápidas e acuradas. Este trabalho introduz o método de função de base radial (RBF) aplicado à solução da equação BS com condições de contorno não-lineares relacionadas a opções de barreira dependentes da trajetória. Além disso, explora-se o método difusional para solucionar equações advectivo-difusivas quanto à sua efetividade para solucionar equações BS. Utilizam-se funções de base radial Cúbica e Thin-Plate Spline (TPS), aplicadas à solução de problemas de opções de barreiras. Os resultados numéricos, quando comparados com as soluções analíticas, permitem afirmar que o método RBF é muito acurado e fácil de ser implementado. O método difusional associado ao método RBF leva aos mesmos resultados obtidos pela formulação clássica da equação de Black-Scholes.

Técnica De Funções De Base Radial Acurada Para Soluções Competitivas E Eficientes De Equações Não Lineares De Black-Scholes

“…According to Santos et al (2009) most practical problems involve complex nonlinear problems, for which there are no analytical solutions. Therefore, these problems should be solved by means of numerical methods.…”

Convective Drying Analysis of a Single Wheat Kernel Based on an Irreversible Thermodynamics Model

“…The H-weighted method allowed discretizing the mass balance equation, Eq. (19) [38,54] ; the resulting expressions for spherical and axisymmetric cylindrical coordinates are, respectively: @Mðr;z; tÞ @t % ð1 À HÞ Á f ðM; T; r; z;tÞ þ H Á f ðM;T;r;z;t þ DtÞ…”

“…Additionally, the RBF method is a meshless method, easily applicable to multidimensional problems. [34][35][36][37][38] As mentioned by Sarra, [34] RBF methods for time-dependent PDEs have enjoyed large advantages in accuracy over other flexible but low-order methods, such as finite differences, finite volumes, and finite elements. RBF methods have shared the ease of implementation and flexibility of these lowerorder methods.…”

“…[34][35][36][37][38] However, in the reviewed literature there is no work on the application of the RBF method to solve the highly nonlinear system of partial differential equations associated with the theoretical models of grain or food drying.…”