A High Order Method for Pricing of Financial Derivatives using Radial Basis Function generated Finite Differences

TL;DR

This paper introduces a high-order numerical method using Radial Basis Function generated Finite Differences for efficiently pricing 2D European options, achieving high accuracy with fewer nodes and potential for higher-dimensional problems.

Contribution

The paper develops a fourth-order spatial discretization method for option pricing that handles nonuniform nodes and improves computational efficiency over standard techniques.

Findings

Achieves fourth-order accuracy in space.

High accuracy with fewer nodes in nonuniform layouts.

Potential for higher-dimensional derivative pricing.

Abstract

In this paper, we consider the numerical pricing of financial derivatives using Radial Basis Function generated Finite Differences in space. Such discretization methods have the advantage of not requiring Cartesian grids. Instead, the nodes can be placed with higher density in areas where there is a need for higher accuracy. Still, the discretization matrix is fairly sparse. As a model problem, we consider the pricing of European options in 2D. Since such options have a discontinuity in the first derivative of the payoff function which prohibits high order convergence, we smooth this function using an established technique for Cartesian grids. Numerical experiments show that we acquire a fourth order scheme in space, both for the uniform and the nonuniform node layouts that we use. The high order method with the nonuniform node layout achieves very high accuracy with relatively few nodes. This renders the potential for solving pricing problems in higher spatial dimensions since the computational memory and time demand become much smaller with this method compared to standard techniques.