  We prove the strong consistency and the asymptotic normality of the maximum
likelihood estimator of the parameters of a general conditionally
heteroscedastic model with $\alpha$-stable innovations. Then, we relax the
assumptions and only suppose that the innovation process converges in
distribution toward a stable process. Using a pseudo maximum likelihood
estimator with a stable density, we also obtain the strong consistency and the
asymptotic normality of the estimator. This framework seems relevant for
financial data exhibiting heavy tails. We apply this method to several
financial index and compute stable Value-at-Risk.
