Lie symmetry analysis of the Black-Scholes-Merton model for European options with stochastic volatility (Q515438)

Summary: 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 Ornstein-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.