  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.
