Encompassing a very broad family of ARCH-GARCH models, we show that the AT-GARCH (1,1) model, where volatility rises more in response to bad newsthan to good news, and where news are considered bad only below a certain level, is a remarkably robust representation of worldwide stock market returns. The residual structure is then captured by extending ATGARCH (1,1) to an hysteresis model, HGARCH, where we modelstructured memory effects from past innovations. Obviously, this feature relates to the psychology of the markets and the way traders process information. For the French stock market we show that votalitity is affected differently, depending on the recent past being characterized by returns all above or below a certain level. In the same way a longer term trend may also influence volatility. It is found that bad news are discounted very quickly in volatility, this effect being reinforced when it comes after a negative trend in the stock index. On the opposite, good news have a very small impact on volatility except when they are clustered over a few days, which in this case reduces volatility.