Optimal systems of subalgebras for a nonlinear Black-Scholes equation

TL;DR

This paper classifies the symmetry structures of a nonlinear Black-Scholes model with market feedback effects, providing a foundation for deriving all invariant solutions through optimal subalgebra systems.

Contribution

It offers a classification of a parametric family of Lie algebras related to the nonlinear Black-Scholes equation and constructs optimal subalgebra systems for solution analysis.

Findings

Classification of a one-parametric family of Lie algebras.

Description of optimal systems of subalgebras for the symmetry algebra.

Facilitation of deriving complete invariant solutions.

Abstract

The main object of our study is a four dimensional Lie algebra which describes the symmetry properties of a nonlinear Black-Scholes model. This model implements a feedback effect which is typical for an illiquid market. The structure of the Lie algebra depends on one parameter, i.e. we have to do with a one-parametric family of algebras. We provide a classification of these algebras using Patera--Winternitz method. Optimal systems of one-, two- and three- dimensional subalgebras are described for the family of symmetry algebras of the nonlinear Black-Scholes equation. The optimal systems give us the possibility to describe a complete set of invariant solutions to the equation.