Models of self-financing hedging strategies in illiquid markets: symmetry reductions and exact solutions

TL;DR

This paper analyzes a nonlinear PDE modeling self-financing hedging strategies in illiquid markets, employing Lie symmetry methods to find reductions and explicit solutions, including a new special case with practical applications.

Contribution

It provides a comprehensive symmetry analysis of the PDE model, classifies functions g(alpha) with extended symmetries, and derives explicit solutions for a new special case.

Findings

Classification of functions g(alpha) with extended Lie symmetries

Complete reductions of the PDE to ODEs using symmetry methods

Explicit solutions for the new special model, including power derivative descriptions

Abstract

We study the general model of self-financing trading strategies in illiquid markets introduced by Schoenbucher and Wilmott, 2000. A hedging strategy in the framework of this model satisfies a nonlinear partial differential equation (PDE) which contains some function g(alpha). This function is deep connected to an utility function. We describe the Lie symmetry algebra of this PDE and provide a complete set of reductions of the PDE to ordinary differential equations (ODEs). In addition we are able to describe all types of functions g(alpha) for which the PDE admits an extended Lie group. Two of three special type functions lead to models introduced before by different authors, one is new. We clarify the connection between these three special models and the general model for trading strategies in illiquid markets. We study with the Lie group analysis the new special case of the PDE describing the self-financing strategies. In both, the general model and the new special model, we provide the optimal systems of subalgebras and study the complete set of reductions of the PDEs to different ODEs. In all cases we are able to provide explicit solutions to the new special model. In one of the cases the solutions describe power derivative products.