  In this work we study the averaging principle for non-autonomous slow-fast
systems of stochastic differential equations. In particular in the first part
we prove the averaging principle assuming the sublinearity, the Lipschitzianity
and the Holder's continuity in time of the coefficients, an ergodic hypothesis
and an $\mathcal{L}^2$-bound of the fast component. In this setting we prove
the weak convergence of the slow component to the solution of the averaged
equation. Moreover we provide a suitable dissipativity condition under which
the ergodic hypothesis and the $\mathcal{L}^2$-bound of the fast component,
which are implicit conditions, are satisfied.
  In the second part we propose a financial application of this result: we
apply the theory developed to a slow-fast local stochastic volatility model.
First we prove the weak convergence of the model to a local volatility one.
Then under a risk neutral measure we show that the prices of the derivatives,
possibly path-dependent, converge to the ones calculated using the limit model.
