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 10 classical and multifractal volatility models. Forecasting performance is evaluated out-of-sample based on the empirical MSE, MAE, and QLIKE criteria 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 the best model throughout under the MAE criterion, i.e., for all indices and forecast horizons between one day and 100 days, with uniform results for forecast evaluations against both realized volatilities and squared returns or during tranquil/turbulent market periods. Based on the evaluation of MSE and QLIKE forecast errors, the RV-MRW, RV-BMSM, and RV-ARFIMA provide the most accurate forecasts during our tranquil sample from 2016–2018, where we can observe a transition from RV-MRW dominating long-term forecasts to RV-BMSM and RV-ARFIMA dominating in the short term. The new RV-BMSM takes the lead in phases of more turbulent market dynamics (sample 2010–2012), when it appears throughout in the 90% model confidence set at horizons $\le 10$ days and for 13 out of 14 indices at 20 days. These results are very promising if we consider that this is the first empirical application of the RV-MRW and RV-BMSM. Moreover, whereas RV-ARFIMA forecasts are often a time-consuming task, the RV-MRW stands out due to its fast execution and straightforward implementation.