Financial density forecasts: A comprehensive comparison of risk-neutral and historical schemes

We investigate the forecasting ability of the most commonly used benchmarks in financial economics. We approach the main methodological caveats of probabilistic forecasts studies –small samples, limited models and non-holistic validations– by performing a comprehensive comparison of 15 predictive schemes during a time period of over 21 years. All densities are evaluated in terms of their statistical consistency, local accuracy and forecasting errors. Through the development of a new indicator, the Integrated Forecast Score (IFS), we show that risk-neutral densities outperform historical-based predictions in terms of information content. 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. In contrast, lognormal densities, the Heston model or the non-parametric Breeden-Litzenberger formula yield biased predictions and are rejected in statistical tests.