Forecasting the Variability of Stock Index Returns with the Multifractal Random Walk Model for Realized Volatilities

We adapt the multifractal random walk model by Bacry et al. (2001) to realized volatilities (denoted RV-MRW) and take stock of recent theoretical insights on this model in Duchon et al. (2012) to derive forecasts of financial volatility. Moreover, we propose a new extension of the binomial Markov-switching multifractal (BMSM) model by Calvet and Fisher (2001) to the RV framework. We compare the predictive ability of the two against seven classical and multifractal volatility models. Forecasting performance is evaluated out-of-sample based on the empirical MSE and MAE as well as using model confidence sets following the methodology of Hansen et al. (2011). Overall, our empirical study for 14 international stock market indices has a clear message: The RV-MRW is throughout the best model for all forecast horizons under the MAE criterium as well as for large forecast horizons h=50 and 100 days under the MSE criterion. Moreover, the RV-MRW provides most accurate 20-day ahead forecasts in terms of MSE for the great majority of indices, followed by RV-ARFIMA, the latter dominating the competition at the 5-day-horizon. These results are very promising if we consider that this is the first empirical application of the RV-MRW. Moreover, whereas RV-ARFIMA forecasts are often a time consuming task, the RV-MRW stands out due to its fast execution and straightforward implementation. The new RV-BMSM appears to be specialized in short term forecasting, the model providing most accurate one-day ahead forecasts in terms of MSE for the same number of cases as RV-ARFIMA.