  We introduce the notion of continuously invertible volatility models that
relies on some Lyapunov condition and some regularity condition. We show that
it is almost equivalent to the ability of the volatilities forecasting using
the parametric inference approach based on the SRE given in [16]. Under very
weak assumptions, we prove the strong consistency and the asymptotic normality
of the parametric inference. Based on this parametric estimation, a natural
strongly consistent forecast of the volatility is given. We apply successfully
this approach to recover known results on univariate and multivariate GARCH
type models and to the EGARCH(1,1) model. We prove the strong consistency of
the forecasting as soon as the model is invertible and the asymptotic normality
of the parametric inference as soon as the limiting variance exists. Finally,
we give some encouraging empirical results of our approach on simulations and
real data.
