Option Pricing with Lie Symmetry Analysis and Similarity Reduction Method

TL;DR

This paper introduces a novel analytical method for option pricing under state-dependent volatility using Lie symmetry analysis and similarity reduction, simplifying complex PDEs to find explicit solutions.

Contribution

It applies Lie symmetry and similarity reduction to derive analytical option pricing formulas for specific volatility functions, advancing mathematical finance techniques.

Findings

Derived explicit solutions for certain volatility functions

Reduced PDE complexity through symmetry analysis

Validated method with case studies producing analytical formulas

Abstract

With some transformations, we convert the problem of option pricing under state-dependent volatility into an initial value problem of the Fokker-Planck equation with a certain potential. By using the Lie symmetry analysis and similarity reduction method, we are able to reduce the dimensions of the partial differential equation and find some of its particular solutions of the equation. A few case studies demonstrate that our new method can be used to produce analytical option pricing formulas for certain volatility functions.