Spherical Harmonics Applied to Differential and Integro-Differential Equations Arising in Mathematical Finance

Abstract

This paper is devoted to extend the spherical harmonics technique to the solution of parabolic differential equations and to integro-differential equations. The heat equation and the Black-Scholes equation are solved by using the method of spherical harmonics. We also discuss the Black-Scholes equation in annular domains, and generalized Black-Scholes equations. Finally we solve some integro-differential equation arising in financial models with jumps by using the method of spherical harmonics.

Operator splitting kernel based numerical method for a generalized Leland’s model

“…We will implement the described above kernel-based splitting for parabolic sub-problem (4). Assuming that A(·) is known, for example from old time level or old iteration (as will be in our case), solving (4) in 2D case is equivalent of solving first (12), then (11) …”

“…Parabolic sub-problem (4). Now instead of (4), we approximate two sub-problems of type (11) and (12) for…”

“…We consider the generalized Hoggard-Whalley-Wilmott, Leland's model for higher dimensions by SenGupta and Mariani [4] in the case of European Call option…”