Option pricing models and longer-term value-at-risk (VaR) models generally require volatility forecasts over horizons considerably longer than the data frequency. The typical recursive procedure for generating longer-term forecasts keeps the relative weights of recent and older observations the same for all forecast horizons. In contrast, we find that older observations are relatively more important in forecasting at longer horizons. We find that the Ederington and Guan (2005) model and a modified EGARCH (exponential generalized autoregressive conditional heteroskedastic) model in which parameter values vary with the forecast horizon forecast better out-of-sample than the GARCH (generalized autoregressive conditional heteroskedastic), EGARCH, and Glosten, Jagannathan, and Runkle (GJR) models across a wide variety of markets and forecast horizons.

This paper estimates how the shape of the implied volatility smile and the size of the variance risk premium relate to parameters of GARCH-type time-series models measuring how conditional volatility responds to return shocks. Markets in which return shocks lead to large increases in conditional volatility tend to have larger variance risk premia than markets in which the impact on conditional volatility is slight. Markets in which negative (positive) return shocks lead to larger increases in future volatility than positive (negative) return shocks tend to have downward (upward) sloping implied volatility smiles. Also, differences in how volatility responds to return shocks as measured by GARCH-type models explain much, but not all, of the variations in excess kurtosis and multi-period skewness across different markets.