Filtering Equity Risk Premia from Derivative Prices

This paper considers the measurement of the equity risk premium in financial markets. While there exists a vast amount of research into its behaviour, particularly in US markets, this is largely based on regression based techniques which do not capture well the dynamic and forward looking nature of the risk premium. In this paper 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. Although various processes for the dynamics of the risk premium may be considered, we motivate and analyse a mean-reverting form. Since the risk premium is not directly observable, information on it 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 market’s forward looking view on the equity risk premium. The results using monthly Australian and U.S. market data over a period of five years are presented. The model fit is found to be statistically significant for both markets. The time series of the mean and standard deviation of the risk premia generated by the Kalman filter are compared with risk premia computed from ex-post returns. It is found that the ex-post risk premia have a general tendency to lie within a two standard deviations band around the filtered mean. However there are frequent movements outside the band, particularly on the downside, indicating that the ex-post measure may be understating the risk premium.