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

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.

Published version, 14pages, 4 figures

References in corpus (4)

• Lie point symmetries of a general class of PDEs: The heat equation
• Group Classification of a Generalized Black--Scholes--Merton Equation
• Group Classification of a generalization of the Heath Equation
• The algebraic properties of the space- and time-dependent one-factor model of commodities

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