We introduce a new stochastic volatility model that postulates a general correlation structure between the shocks of the measurement and log volatility equations at different temporal lags. The resulting specification is able to better characterize the leverage effect and propagation in financial time series. Furthermore, it nests other asymmetric volatility models and can be used for testing and diagnostics. We derive the simulated maximum likelihood and quasi maximum likelihood estimators and investigate their finite sample performance in a simulation study. An empirical illustration shows that the postulated correlation structure improves the fit of the leverage propagation and leads to more precise volatility predictions.