Mathematical Financial Modeling: A radial basis functions method for Financial Barrier Options

In recent decades, the topics of Mathematical Finance(MF) has been widely used in universities around the world. Some part of discussion of Mathematical Finance consists of Stochastic Differential Equation(SDE), Partial Differential Equation(PDE), Economic and Numerical Analysis. In recent years, In our country, this subject has been studied in some universities. This thesis introduces a specific field of financial mathematics related to barrier options contracts. This note 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. Cubic and Thin-Plate Spline (TPS) radial basis functions were employed and evaluated as to their effectiveness to solve Up-and-Out (UAO) and Down-and-Out (DAO) barrier problems. The numerical results, when compared against analytical solutions, allow affirming that the RBF method is very accurate and easy to be implemented.