A radial basis function — Hermite finite difference approach to tackle cash-or-nothing and asset-or-nothing options

Abstract

A novel local meshfree method is proposed for simulating cash-or-nothing and asset-or-nothing options, at which the initial condition is discontinuous. The novelty of the scheme is in the use of Hermite radial basis function (RBF) interpolation on set of local nodes. We also construct the RBF generated Hermite finite difference (HFD) formulas and subsequently obtain their associated differentiation matrices (DMs). The developed accuracy equipped with a θ-shift algorithm for smoothing the discontinuous initial conditions are illustrated with a series of financial tests.