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Numerical solution using radial basis functions for multidimensional fractional partial differential equations of type Black-Scholes

A. Torres-Hernandez, F. Brambila-Paz, C. A. Torres-Martínez

TL;DR

This work tackles numerical solution of space-time-fractional Black-Scholes-type PDEs by employing a meshless radial basis function (RBF) framework. It combines Caputo time discretization with a radial interpolant to discretize the interior, and uses a QR-based preconditioning strategy to mitigate matrix ill-conditioning. Key contributions include demonstrating node-type flexibility (e.g., Chebyshev, Halton, Cartesian), reducing discretization conditioning, and validating the approach with one- and two-dimensional examples and RMSE assessments. The method offers a practical, dimension-agnostic alternative to traditional mesh-based schemes for multidimensional fractional PDEs in finance, with performance tied to node placement and kernel choice.

Abstract

The aim of this paper is to solve numerically, using the meshless method via radial basis functions, time-space-fractional partial differential equations of type Black-Scholes. The time-fractional partial differential equation appears in several diffusion problems used in physics and engineering applications, and models subdiffusive and superdiffusive behavior of the prices at the stock market. This work shows the flexibility of the radial basis function scheme to solve multidimensional problems with several types of nodes and it also shows how to reduce the condition number of the matrices involved.

Numerical solution using radial basis functions for multidimensional fractional partial differential equations of type Black-Scholes

TL;DR

This work tackles numerical solution of space-time-fractional Black-Scholes-type PDEs by employing a meshless radial basis function (RBF) framework. It combines Caputo time discretization with a radial interpolant to discretize the interior, and uses a QR-based preconditioning strategy to mitigate matrix ill-conditioning. Key contributions include demonstrating node-type flexibility (e.g., Chebyshev, Halton, Cartesian), reducing discretization conditioning, and validating the approach with one- and two-dimensional examples and RMSE assessments. The method offers a practical, dimension-agnostic alternative to traditional mesh-based schemes for multidimensional fractional PDEs in finance, with performance tied to node placement and kernel choice.

Abstract

The aim of this paper is to solve numerically, using the meshless method via radial basis functions, time-space-fractional partial differential equations of type Black-Scholes. The time-fractional partial differential equation appears in several diffusion problems used in physics and engineering applications, and models subdiffusive and superdiffusive behavior of the prices at the stock market. This work shows the flexibility of the radial basis function scheme to solve multidimensional problems with several types of nodes and it also shows how to reduce the condition number of the matrices involved.

Paper Structure

This paper contains 10 sections, 3 theorems, 85 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Basic Definitions of the Fractional Derivative
  3. Introduction to the Definition of Riemann-Liouville
  4. Introduction to the Definition of Caputo
  5. Discretization of the Fractional Derivative of Caputo
  6. Meshless Methods
  7. Interpolation with Radial Basis Functions
  8. Solution of Differential Equations with Radial Basis Functions
  9. Examples
  10. Conclusions

Key Result

Proposition 2.2

Let $f$ be a function. If $f \in L_{loc} ^ 1 (a, \infty)$, then the fractional integrals of $f$ fulfill that

Figures (5)

  • Figure 1: Different numbers of Chebyshev nodes used.
  • Figure 2: The analytical solution and the numerical solutions with respect to space for different moments in time are presented. The $RMSE$ is presented with respect to time for the different numerical solutions.
  • Figure 3: The analytical solution and the numerical solutions with respect to space for the final time step are presented. The $RMSE$ is presented with respect to time for the different numerical solutions.
  • Figure 4: Different numbers of nodes used.
  • Figure 5: The analytical solution and the numerical solutions with respect to space, with $y=x$, for different moments in time are presented.

Theorems & Definitions (9)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Example 3.3
  • Example 3.4