A class of Asymmetric GARCH models is presented. It shares the same unconditional variance and volatility forecast formula as the standard GARCH(P,Q) model under the assumption of a symmetric conditional distribution for innovations. use three models of this class to assess their ability to forecast S&P 500 market volatility and to make better decisions for the purpose of risk management and investment. Subsequently, a comparison is made with respect to competing models (GARCH, EGARCH, GJR). It was found that for the in-sample evaluation, the best model is obtained from the Stochastic Unit GARCH (SUGARCH) model where leverage effects are introduced through the GARCH (i.e) parameter. For the out-of-sample evaluation (QLIKE loss function), it is better to use the SUGARCH class where the asymmetry appears on the ARCH (i.e ) parameter.