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.
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.
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Black-Scholes-Merton model
Black--Scholes--Merton Model
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\%
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the Orstein-Uhlenbeck
an Orstein--Uhlenbeck
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Black-Scholes-Merton equation
Black--Scholes--Merton Equation
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Stein-Stein
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