Pricing options in illiquid markets: optimal systems, symmetry reductions and exact solutions

TL;DR

This paper analyzes nonlinear option pricing models in illiquid markets, using symmetry methods to find exact solutions and understand the impact of nonlinear demand functions on pricing dynamics.

Contribution

It introduces a novel analysis of nonlinear demand in option pricing models and applies Lie group techniques to derive exact solutions and reductions.

Findings

Derived families of exact solutions for nonlinear PDEs

Identified symmetry properties and reductions of the models

Provided insights into the effects of nonlinear demand on pricing

Abstract

We study a class of nonlinear pricing models which involves the feedback effect from the dynamic hedging strategies on the price of asset introduced by Sircar and Papanicolaou. We are first to study the case of a nonlinear demand function involved in the model. Using a Lie group analysis we investigate the symmetry properties of these nonlinear diffusion equations. We provide the optimal systems of subalgebras and the complete set of non-equivalent reductions of studied PDEs to ODEs. In most cases we obtain families of exact solutions or derive particular solutions to the equations.