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

TL;DR

This study compares 15 financial density forecasting schemes over 21 years, finding risk-neutral densities generally outperform historical ones, with specific models like Variance Gamma and GJR-FHS showing superior predictive performance.

Contribution

It provides a comprehensive, multi-criteria evaluation of popular density forecasting models, introducing the Integrated Forecast Score (IFS) for better assessment.

Findings

Risk-neutral densities outperform historical densities in information content.

Variance Gamma model has the highest out-of-sample likelihood and lowest errors.

GJR-FHS provides the most consistent forecasts across the density range.

Abstract

We investigate the forecasting ability of the most commonly used benchmarks in financial economics. We approach the usual 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. Using a new composite 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 Breeden-Litzenberger formula yield biased predictions and are rejected in statistical tests.