Lie symmetries of (1+2) nonautonomous evolution equations in Financial Mathematics

TL;DR

This paper applies Lie symmetry analysis to (1+2) evolution equations in Financial Mathematics, revealing symmetry properties and reductions to classical heat equations for both autonomous and nonautonomous cases.

Contribution

It provides a detailed symmetry analysis of two key financial PDEs, showing how they can be reduced to simpler heat equations, especially highlighting differences between autonomous and nonautonomous forms.

Findings

Autonomous Black-Scholes equation has maximal symmetry and reduces to classical heat equation.

Nonautonomous equations have submaximal symmetry, indicating less reducibility.

Reduced equations in both cases are of maximal symmetry and equivalent to classical heat equations.

Abstract

We analyse two classes of $(1+2)$ evolution equations which are of special interest in Financial Mathematics, namely the Two-dimensional Black-Scholes Equation and the equation for the Two-factor Commodities Problem. Our approach is that of Lie Symmetry Analysis. We study these equations for the case in which they are autonomous and for the case in which the parameters of the equations are unspecified functions of time. For the autonomous Black-Scholes Equation we find that the symmetry is maximal and so the equation is reducible to the $(1+2)$ Classical Heat Equation. This is not the case for the nonautonomous equation for which the number of symmetries is submaximal. In the case of the two-factor equation the number of symmetries is submaximal in both autonomous and nonautonomous cases. When the solution symmetries are used to reduce each equation to a $(1+1)$ equation, the resulting equation is of maximal symmetry and so equivalent to the $(1+1)$ Classical Heat Equation.