  The aim of this chapter is to show how option prices in jump-diffusion models
can be computed using meshless methods based on Radial Basis Function (RBF)
interpolation. The RBF technique is demonstrated by solving the partial
integro-differential equation (PIDE) in one-dimension for the American put and
the European vanilla call/put options on dividend-paying stocks in the Merton
and Kou jump-diffusion models. The radial basis function we select is the Cubic
Spline. We also propose a simple numerical algorithm for finding a finite
computational range of an improper integral term in the PIDE so that the
accuracy of approximation of the integral can be improved. Moreover, the
solution functions of the PIDE are approximated explicitly by RBFs which have
exact forms so we can easily compute the global integral by any kind of
numerical quadrature. Finally, we will not only show numerically that our
scheme is second order accurate in both spatial and time variables in a
European case but also second order accurate in spatial variables and first
order accurate in time variables in an American case.
