Abstracts

The past decade has seen an increased interest in the use of Radial Basis Function (RBF) methodology for the numerical solution of the pricing equations of Finance. RBF interpolation seeks to interpolate an arbitrary function by a linear combination of translates of a given radially symmetric basis function or RBF, on a given set of interpolation points. This interpolation problem is uniquely solvable for a large class of basis functions which can be completely characterized. RBF interpolation also provides good approximations to the function which is being interpolated. The quality of the approximation depends on the RBF as well as on interpolation technique used (stationary or non-stationary) and may require basis functions which grow at infinity (and as such are the opposite end of the well-localized functions used in, for example, Finite Element methods). RBF interpolation can be used as the basis for a numerical scheme (a variant of the method of lines) to solve the parabolic Partial Integral-Differential Equations encountered in Finance. In the first part of this talk we report on results obtained when applying a stationary RBF scheme to pricing European and American options in exponential Lévy models of the CGMY - KoBoL class (Brummelhuis and Chan, Applied Math. Finance, to appear). We show how the scheme can deal with arbitrary singularities of the Lévy measure in 0, without introducing further approximations. In numerical experiments the scheme is found to be second order convergent in the log-price variable, including for infinite activity and unbounded variation Lévy processes. Since the scheme does not discretise the operator, it circumvents some of the complications encountered with Finite Difference (FD) methods, and leads to a better convergence rate in the unbounded variation case, thereby providing a viable alternative to FD. On the theoretical side, we rigorously determine the order of convergence of the scheme on a regular grid as the grid-size tends to 0. We show that in certain cases, the convergence is only apparent, saturating when the grid-size reaches a certain minimal level, but can nevertheless lead to a very good approximation. This is an illustration of the 'approximate approximation' phenomenon first noted by Mazj'a and Powell in the '90's.