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

A. Paliathanasis, K. Krishnakumar, K. M. Tamizhmani, P. G. L. Leach

Published 2015-08-27Version 1

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 the 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 on two independent variables, the value of the underlying asset $S$ and a new variable, $y$, which follow from the Orstein-Uhlenbeck process. We find that for arbitrary functional form of the volatility, $\sigma(y)$, the (1+2) evolution equation admits always two Lie symmetries, plus the linear symmetry and the infinity number of solution symmetries. However when $\sigma(y)=\sigma_{0}$ and since the price of the option depends on 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 symmetries, plus the linear symmetry and the infinity number of solution symmetries. Furthermore we apply the zero-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.