Before revision:  We find that the Variance Gamma model generates the highest out-of-sample likelihood of observed prices and the lowest predictive errors, whereas the ARCH-based GRJ-FHS delivers the most consistent forecasts across the entire density range.
After revision:  We find that the Variance Gamma model generates the highest out-of-sample likelihood of observed prices and the lowest predictive errors, whereas the ARCH-based GJR-FHS delivers the most consistent forecasts across the entire density range.