Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility

TL;DR

This paper classifies Lie symmetries of a stochastic volatility extension of the Black-Scholes-Merton model for European options, revealing symmetry structures and reductions to simpler equations, with applications to Heston and Stein–Stein models.

Contribution

It provides a comprehensive symmetry classification for the stochastic volatility model, identifying conditions for additional symmetries and reducing complex PDEs to ODEs, extending prior analyses of option pricing models.

Findings

For arbitrary volatility functions, the model admits two Lie point symmetries.

When volatility is constant, the model admits five Lie point symmetries.

The PDE reduces to a linear second-order ODE using Lie invariants.

Abstract

We perform a classification of the Lie point symmetries for the Black--Scholes--Merton Model for European options with stochastic volatility, $\sigma$, in which the last is defined by a stochastic differential equation with an Orstein--Uhlenbeck term. In this model, the value of the option is given by a linear (1 + 2) evolution partial differential equation in which the price of the option depends upon two independent variables, the value of the underlying asset, $S$, and a new variable, $y$. We find that for arbitrary functional form of the volatility, $\sigma(y)$, the (1 + 2) evolution equation always admits two Lie point symmetries in addition to the automatic linear symmetry and the infinite number of solution symmetries. However, when $\sigma(y)=\sigma_{0}$ and as the price of the option depends upon the second Brownian motion in which the volatility is defined, the (1 + 2) evolution is not reduced to the Black--Scholes--Merton Equation, the model admits five Lie point symmetries in addition to the linear symmetry and the infinite number of solution symmetries. We apply the zeroth-order invariants of the Lie symmetries and we reduce the (1 + 2) evolution equation to a linear second-order ordinary differential equation. Finally, we study two models of special interest, the Heston model and the Stein--Stein model.