Implied Volatilities as Forecasts of Future Volatility, the Market Risk Premia, and Returns Variability

The unbiasedness tests of implied volatility as a forecast of future realized volatility have found implied volatility to be a biased predictor. We explain this puzzle by recognizing that option prices contain a market risk premium not only on the asset itself, but also on its volatility. We show using a stochastic volatility model, that a call option price can be represented as an expected value of the Black-Scholes formula evaluated at the average integrated volatility. If we allow volatility risk to be priced, this expectation should be taken under the risk-neutral probability measure, and can be decomposed into the expectation with respect to the physical measure and the risk-premium term. This term is just a linear function of the unobservable spot volatility. The decomposition explains the bias documented in the empirical literature and shows that the realized and historical volatility, which are used in the tests, are in fact the estimates of the unobserved quadratic variation and spot volatility of the stock-return generating process. Therefore, the use of these estimates generates the error-in-the-variables problem. We provide an empirical example based on the S&P 100 returns and the VIX index. We find, that when we take into an account the risk-premium and use efficient methods to estimate volatility, the unbiasedness hypothesis can not be rejected, and the point estimate of the slope in the traditional regression is exactly equal to 1. JEL classification: