A computationally efficient numerical approach for multi-asset option pricing

ABSTRACT

Numerical solution of the multi-dimensional partial differential equations arising in the modelling of option pricing is a challenging problem. Mesh-free methods using global radial basis functions (RBFs) have been successfully applied to several types of such problems. However, due to the dense linear systems that need to be solved, the computational cost grows rapidly with dimension. In this paper, we propose a numerical scheme to solve the Black–Scholes equation for valuation of options prices on several underlying assets. We use the derivatives of linear combinations of multiquadric RBFs to approximate the spatial derivatives and a straightforward finite difference to approximate the time derivative. The advantages of the scheme are that it does not require solving a full matrix at each time step and the algorithm is easy to implement. The accuracy of our scheme is demonstrated on a test problem.

KEYWORDS:

• Multiquadric radial basis functions
• multi-asset option pricing
• mesh-free method
• sparse
• finite difference

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M70
• 65D05
• 65D25
• 65N06
• 65F50

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Definition. A real symmetric matrix $\mathbf{A}=[A{]}_{ij}=[{\phi }_j\left(x_i\right){]}_{ij}$ is called conditionally positive semi-definite of order one if its associate quadratic form is non-negative, i.e. $\sum _{i,j=1}^N{\lambda }_i{\lambda }_j{A}_{ij}\ge 0;$ for all $\mathrm{\Lambda }=[{\lambda }_1,\dots ,{\lambda }_N{]}^{\mathbf{T}}\in {\mathbb{R}}^N$ which satisfy $\sum _{j=0}^N{\lambda }_j=0,$ if $\mathrm{\Lambda }\ne 0$ implies strict above-mentioned inequality, then $\mathbf{A}$ is called a conditionally positive definite matrix of order k.