Numerical Partial Differential Solution of the Black-Scholes Equation

Meshfree radial basis functions (RBF) is an interpolation technique for constructing an unknown function from scattered data. We apply the RBF method in evaluating the price of standard American options. The analytical solution of the European option exists and can be obtained by the Black-Scholes formula. There is no exact solution of the American option problem due to the existence of an early exercise constraint which leads to a free boundary condition. We evaluate the American Option by adding a small continuous nonlinear penalty term to the Black-Scholes model to remove the free boundary condition. The application of RBFs leads to a system ordinary differential equations which are solved by a time integration scheme known as the -method. The option price is approximated with RBF with unknown parameters at each time step. We compare the accuracy, efficiency and computational cost of three RBFs Gaussian, Multiquadric and the Inverse-multiquadric. Finally a comparison is made between the three RBFs and the solution obtained by finite difference approximations.