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We perform a classification of the Lie point symmetries for theBlack--Scholes--Merton Model for European options with stochastic volatility,$\sigma$, in which the last is defined by a stochastic differential equationwith an Orstein--Uhlenbeck term. In this model, the value of the option isgiven by a linear (1 + 2) evolution partial differential equation in which theprice of the option depends upon two independent variables, the value of theunderlying asset, $S$, and a new variable, $y$. We find that for arbitraryfunctional form of the volatility, $\sigma(y)$, the (1 + 2) evolution equationalways admits two Lie point symmetries in addition to the automatic linearsymmetry and the infinite number of solution symmetries. However, when$\sigma(y)=\sigma_{0}$ and as the price of the option depends upon the secondBrownian motion in which the volatility is defined, the (1 + 2) evolution isnot reduced to the Black--Scholes--Merton Equation, the model admits five Liepoint symmetries in addition to the linear symmetry and the infinite number ofsolution symmetries. We apply the zeroth-order invariants of the Lie symmetriesand we reduce the (1 + 2) evolution equation to a linear second-order ordinarydifferential equation. Finally, we study two models of special interest, theHeston model and the Stein--Stein model.

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