Inferring the Forward Looking Equity Risk Premium from Derivative Prices

Abstract

This paper considers the measurement of the equity risk premium in financial markets from a new perspective that picks up on a suggestion from Merton (1980) to use implied volatility of options on a market portfolio as a direct ex-ante estimate for market variance, and hence the risk premium. Here the time variation of the unobserved risk premium is modelled by a system of stochastic differential equations connected by arbitrage arguments between the spot equity market, the index futures and options on index futures. We motivate and analyse a mean-reverting form for the dynamics of the risk premium. Since the risk premium is not directly observable, information about its time varying conditional distribution is extracted using an unobserved component state space formulation of the system and Kalman filtering methodology. In order to cater for the time variation of volatility we use the option implied volatility in the dynamic equations for the index and its derivatives. This quantity is in a sense treated as a signal that impounds the markets ex-ante, forward looking, view on the equity risk premium. The results using monthly U.S. market data over the period January 1995 to June 2003 are presented and the model fit is found to be statistically significant using a number of measures. Comparisons with ex-post returns indicate that such historical measures may be understating the market risk premium.