Radial Basis Function generated Finite Differences for option pricing problems

Abstract

In this paper we present a numerical method to price options based on Radial Basis Function generated Finite Differences (RBF-FD) in space and the Backward Differentiation Formula of order 2 (BDF-2) in time. We use Gaussian RBFs that depend on a shape parameter ε. The choice of this parameter is crucial for the performance of the method. We chose ε as const⋅h−1 and we derive suitable values of the constant for different stencil sizes in 1D and 2D. This constant is independent of the problem parameters such as the volatilities of the underlying assets and the interest rate in the market. In the literature on option pricing with RBF-FD, a constant value of the shape parameter is used. We show that this always leads to ill-conditioning for decreasing h, whereas our proposed method avoids such ill-conditioning. We present numerical results for problems in 1D, 2D, and 3D demonstrating the useful features of our method such as discretization sparsity, flexibility in node placement, and easy dimensional extendability, which provide high computational efficiency and accuracy.