A Multi-Horizon Comparison of Density Forecasts for the S&P 500 Using Index Returns and Option Prices

Abstract

We compare density forecasts of the S&P 500 index from 1991 to 2004, obtained from option prices and daily and five-minute index returns. Risk-neutral densities are given by using option prices to estimate diffusion and jump-diffusion processes, that incorporate stochastic volatility, and then three transformations are used to obtain real-world densities. These densities are compared with historical densities defined by ARCH models. The best forecasts are obtained from risk-transformations of the risk-neutral densities, for horizons of two and four weeks, while the historical forecasts are superior for the one-day horizon; our ranking criterion is the out-of-sample likelihood of observed index levels. Mixtures of the real-world and historical densities have higher likelihoods than both components for short forecast horizons.

Keywords: ARCH Models, Density Forecasts, Risk-Neutral Densities, Risk Transformations

JEL Classification: C14, C22, C52, C53, G13