This paper studies the impact of modelling time‐varying variances of stock returns in terms of risk measurement and extreme risk spillover. Using a general class of regime‐dependent models, we find that volatility can be disaggregated into distinct components: a persistent stable process with low sensitivity to shocks and a high volatility process capturing rather short‐lived rare events. Out‐of‐sample forecasts show that, once regime shifts are accounted for, accuracy is improved compared to the standard generalized autoregressive conditional heteroscedasticity or the historical volatility model. Volatility plays an important role in controlling and monitoring financial risks. Therefore, by means of a risk management application, we illustrate the economic value and the practical implications of risk control ability of the models in terms of value at risk. Finally, tests for predictability in co‐movements in the tails of stock index returns suggest that large losses are strongly correlated, supporting asymmetric transmission processes for financial contagion in the left tail of return distributions, whereas contagion in reverse direction (gains) is weak.