The Relation between Implied and Realized Volatility of S&P 500 Index

This paper studies the relation between implied and realized volatility by using daily S&P 500 index option price over the period between January 1995 and December 1999. In particular, we want to test the how different measurement errors affect the stability of this relationship. Two sources of measurement errors are considered. The first one is the measurement error in realized volatility. Four different estimators of computing realized volatility are tested. They are the standard deviation of daily return; the Parkinson (1980) extreme value volatility estimator, the Yang & Zhang (2000) range estimator, and the square root of intraday return squares (Andersen, 2000). The second source of error comes from model specification. The implied volatility computed from Black Scholes model is compared with that from calibrated Heston (1993) stochastic volatility optionpricing model. We find the improvement of the measurement of realized volatility can significantly improve the forecast ability of implied volatility, with the realized volatility estimated from intraday return data is most predictable. However, there is no significant difference in forecasting realized volatility using implied volatility either from BlackScholes model or from Heston model. When both implied volatility and historical volatility are used to forecast realized volatility, we find implied volatility outperforms historical volatility and even subsumes information of historical volatility. This result h olds for all measurements of realized volatility and implied volatility.