Meshless collocation method for option pricing by variance gamma model

Based on the use of radial basis functions (RBFs), we present in this paper a meshless collocation method to compute both European and American option prices by solving the variance gamma (VG) model. The valuation of the financial derivatives is performed by solving a corresponding partial integro-differential equation (PIDE). In the case of European option, numerical comparison with the analytical solution shows that the proposed scheme achieves a higher accurate approximation than most existing numerical methods. When analytical solution is not available in the case of American option, we use a dividend process to obtain an alternative characterization of the American option so that solution to the PIDE can be achieved in the entire computational region. Since the RBFs used in this paper are infinitely differentiable, the approximation of the derivatives of option prices can be obtained at no extra interpolation cost. In addition, the leave-one-out cross validation (LOOCV) algorithm is generalized for obtaining a local optimal choice of the shape parameter contained in the RBFs for superior convergence. Several numerical examples are given to verify the efficiency and stability of the proposed method. © 2013 World Scientific Publishing Company.