Financial Density Forecasts: A Comprehensive Comparison of Risk-Neutral and Historical Schemes

Abstract

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, log-normal densities, the Heston model or the non-parametric Breeden-Litzenberger formula yield biased predictions and are rejected in statistical tests.