Group Classification of a Generalized Black--Scholes--Merton Equation

TL;DR

This paper performs a comprehensive symmetry-based classification of a generalized Black-Scholes-Merton equation, deriving invariant solutions for various nonlinear cases relevant to financial boundary conditions.

Contribution

It introduces a complete group classification for a generalized model and provides explicit invariant solutions under key financial boundary conditions.

Findings

Classified nonlinear cases of the generalized model.

Derived invariant solutions for terminal and barrier options.

Enhanced understanding of symmetries in financial PDEs.

Abstract

The complete group classification of a generalization of the Black-Scholes-Merton model is carried out by making use of the underlying equivalence and additional equivalence transformations. For each non linear case obtained through this classification, invariant solutions are given. To that end, two boundary conditions of financial interest are considered, the terminal and the barrier option conditions.