Density Forecast Comparisons for Stock Prices, Obtained from High-Frequency Returns and Daily Option Prices

This paper presents the first comparison of the accuracy of density forecasts for stock prices. Six sets of forecasts are evaluated for DJIA stocks, across four forecast horizons. Two forecasts are risk-neutral densities implied by the Black-Scholes and Heston models. The third set are historical lognormal densities with dispersion determined by forecasts of realized variances obtained from 5-minute returns. Three further sets are defined by transforming risk-neutral and historical densities into real-world densities. The most accurate method applies the risk transformation to the Black-Scholes densities. This method outperforms all others for 87% of the comparisons made using the likelihood criterion