Realized Volatility Forecasting: Continuous versus Discrete Time Models

Forecasting realized volatility (RV) is central to financial econometrics, with important implications for risk management, asset allocation, and derivative pricing. Motivated by the ongoing debate on volatility modeling, this paper provides a comprehensive empirical comparison of many alternative models. We evaluate leading continuous time models estimated using state-of-the-art methods from the rough volatility literature, together with both standard long-memory autoregressive fractionally integrated moving average (ARFIMA) models and their rough-volatility extensions, as well as several variants of the heterogeneous autoregressive (HAR) model and their logarithmic counterparts. The models are applied to a large panel of equities and cryptocurrencies, with performance assessed using both statistical and economic criteria. Our results show that for equities, continuous time models consistently outperform discrete time alternatives across all evaluation criteria and forecasting horizons. The fractional Brownian motion model for log RV performs best at short horizons, while the fractional Ornstein Uhlenbeck model for log RV dominates in the long run. For cryptocurrencies, a mild divergence emerges between economic and statistical performance: based on realized utility, the quarticity-augmented heterogeneous autoregressive (HARQ) model for RV leads in the short term and the Brownian semistationary models prevail at longer horizons, whereas the HAR-type models for log RV deliver superior statistical accuracy.