Symmetry reductions of a nonlinear option pricing model

TL;DR

This paper applies Lie group theory to a nonlinear Black-Scholes model, deriving invariant solutions that account for feedback effects in large trader hedging strategies.

Contribution

It introduces a symmetry-based reduction method for a nonlinear option pricing PDE, providing explicit invariant solutions for specific parameter values.

Findings

Derived invariant solutions for the nonlinear PDE.

Reduced the PDE to ODEs using Lie symmetries.

Identified special cases with explicit solutions.

Abstract

The studied model was suggested to design a perfect hedging strategy for a large trader. In this case the implementation of a hedging strategy affects the price of the underlying security. The feedback-effect leads to a nonlinear version of the Black-Scholes partial differential equation. Using the Lie group theory we reduce the partial differential equation in special cases to ordinary differential equations. The found Lie group of the model equation gives rise to invariant solutions. Families of exact invariant solutions for special values of parameters are described.