Computing stable numerical solutions for multidimensional American option pricing problems: a semi-discretization approach

TL;DR

This paper introduces a stable semi-discretization numerical scheme for multi-asset American option pricing, reducing computational costs and ensuring stability, positivity, and boundedness of solutions, validated through numerical examples.

Contribution

It presents the first stability analysis for a semi-discretization approach in multi-asset American option pricing, incorporating a variable transformation and exponential time differencing.

Findings

Stability conditions on step sizes are derived.

The method ensures positivity and boundedness of solutions.

Numerical examples confirm stability and competitiveness.

Abstract

The matter of the stability for multi-asset American option pricing problems is a present remaining challenge. In this paper a general transformation of variables allows to remove cross derivative terms reducing the stencil of the proposed numerical scheme and underlying computational cost. Solution of a such problem is constructed by starting with a semi-discretization approach followed by a full discretization using exponential time differencing and matrix quadrature rules. To the best of our knowledge the stability of the numerical solution is treated in this paper for the first time. Analysis of the time variation of the numerical solution with respect to previous time level together with the use of logarithmic norm of matrices are the basis of the stability result. Sufficient stability conditions on step sizes, that also guarantee positivity and boundedness of the solution, are found. Numerical examples for two and three asset problems justify the stability conditions and prove its competitiveness with other relevant methods.