Pricing Financial Derivatives using Radial Basis Function generated Finite Differences with Polyharmonic Splines on Smoothly Varying Node Layouts

TL;DR

This paper introduces an improved RBF-FD method using polyharmonic splines and smoothly varying node layouts, demonstrating enhanced accuracy and efficiency in pricing complex financial derivatives across multiple models.

Contribution

The paper develops a significantly improved RBF-FD scheme with polyharmonic splines and variable node densities, advancing numerical methods for multidimensional financial PDEs.

Findings

Enhanced convergence and stability of the RBF-FD method.

Superior performance in pricing European and American options.

Better conditioning of the discrete systems compared to previous methods.

Abstract

In this paper, we study the benefits of using polyharmonic splines and node layouts with smoothly varying density for developing robust and efficient radial basis function generated finite difference (RBF-FD) methods for pricing of financial derivatives. We present a significantly improved RBF-FD scheme and successfully apply it to two types of multidimensional partial differential equations in finance: a two-asset European call basket option under the Black--Scholes--Merton model, and a European call option under the Heston model. We also show that the performance of the improved method is equally high when it comes to pricing American options. By studying convergence, computational performance, and conditioning of the discrete systems, we show the superiority of the introduced approaches over previously used versions of the RBF-FD method in financial applications.