Analytical shape functions and derivatives approximation formulas in local radial point interpolation methods with applications to financial option pricing problems

Abstract

The radial point interpolation method is increasingly being applied for the numerical solution of partial differential equations in different fields. Most implementations in the literature for obtaining the matrix coefficients make use of numerical approximations of the shape functions and their derivatives. To avoid the solution of linear systems required for computation of derivative approximations, this work derives analytical shape functions for three and five-node support domains in a local radial point interpolation method (LRPIM) with multiquadrics as basis functions. A weak form algorithm for the Black–Scholes equation using three-node analytical shape functions is developed and its unconditional stability and convergence are theoretically established. LRPIM finite-difference (FD) formulas are derived and applied to the solution of one and two-asset financial options. A five-node LRPIM-FD method in one-dimension is shown to yield fourth-order accuracy and applications to two-asset problems also yield accurate prices for options on a minimum of two risky assets and exchange options.