Predicting the Volatility of the S&P 500 Equity Index
Posted: 16 Mar 2007
Date Written: January 2007
This paper presents empirical tests of seven alternative estimators for the volatility of the S&P 500 equity index. Two of the estimators are the implied volatilities derived from the option models of Geske (1979) and Black-Scholes (1973) and using market prices for options. The Geske volatility estimator is stochastic, and this paper presents its first empirical tests. The Black-Scholes estimator in theory is deterministic, but herein when computed daily it is ad hoc allowed to change randomly with changes in the index level. The other five estimators are empirical, based on sample data. Four of the empirical estimators are GARCH approaches (GARCH11, EGARCH, TGARCH, and Heston-Nandi GARCH) which accommodate stochastic volatilities, and the fifth empirical estimator is simply the sample variance viewed as an historical estimate. All seven estimators are used to predict the actual realized volatility over the life of each option. If the market processes information efficiently, when the actual realized volatility over the option life is regressed on each estimator, the slope coefficient and intercept should be 1.0 and 0.0, respectively. In the case of the implied volatility estimators, this can be considered a test of these models. Our results show that in all cases the slope coefficients and intercepts of the implied volatilities are much closer to 1.0, more highly significant, have intercepts closer to 0.0 which are insignificant, and exhibit higher R2 than the empirical estimators. Furthermore, the Geske estimator appears better in these respects than this ad hoc Black-Scholes estimator. Encompassing regressions are run paring the Geske estimator separately with each of the five empirical estimators. In every encompassing regression test the slope coefficients and intercepts of the Geske estimator remain much closer to 1.0 and 0.0, respectively, while the competing estimators slope coefficients are much reduced from 1.0 toward 0.0, and are often insignificant. Thus, the Geske stochastic volatility estimator appears to capture a very significant portion but not all of the information relevant for predicting the volatility of the S&P 500 equity index.
Keywords: Derivatives, Volatility Forecast, Stochastic Stock Volatility, Leverage
JEL Classification: G12
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