The most relevant stylized facts of time-varying financial volatility are the asymmetric response to return shocks and the long memory property. These have largely been modeled in isolation. To capture the asymmetry or/and the long memory in conditional variance, three models are employed FIGARCH,LSTGARCH and LSTFIGARCH. The estimation results on daily returns of five major stock indices of the G7 countries and 6 selected emerging market reveal that for most series there is strong evidence of long memory and asymmetry in their conditional variance. Both emerging and developed markets can be reasonably well modeled using STFIGARCH model except CAC and FTSII which can be much better characterized by FIGARCH. Our findings show that STFIGARCH is the best model and it provide more accurate out of sample forecasts than the LSTGARCH model but the FIGARCH stays as a competing model. We note in particular that the transition between two outer regimes is faster for stock markets of emerging economies than for developed economics where in these markets the changes occur in a smooth manner. This corroborates the usual observation that emerging stock markets may collapse much more suddenly and recover more slowly than the developed stock markets.