Volatility Puzzles: A Unified Framework for Gauging Return-Volatility Regressions

This paper provides a simple unified framework for assessing the empirical linkages between returns and realized and implied volatilities. First, we show that whereas the volatility feedback effect as measured by the sign of the correlation between contemporaneous return and realized volatility depends importantly on the underlying structural model parameters, the correlation between return and implied volatility is unambiguously positive for all reasonable parameter configurations. Second, the lagged return-volatility asymmetry, or the leverage effect, is always stronger for implied than realized volatility. Third, implied volatilities generally provide downward biased forecasts of subsequent realized volatilities. Our results help explain previous findings reported in the extant empirical literature, and is further corroborated by new estimation results for a sample of monthly returns and implied and realized volatilities for the aggregate S&P market index.

Introduction
Following the realization in the late eighties that financial market volatility is both time-varying and predictable, empirical investigations into the temporal linkages between aggregate stock market volatility and returns have figured very prominently in the literature. Of course, volatility per se is not directly observable, and several different volatility proxies have been employed in empirically assessing the linkages, including (i) model-based procedures that explicitly parameterize the volatility process as an ARCH or stochastic volatility model, (ii) direct market-based realized volatilities constructed by the summation of intra-period higher-frequency squared returns, and (iii) forward looking market-based implied volatilities inferred from options prices (see Andersen et al., 2003, for further discussion of the various volatility concepts and procedures). Meanwhile, a cursory read of the burgeoning volatility literature reveals a perplexing set of results, with the sign and the size of the reported volatility-return relationships differing signicantly across competing studies and procedures.

Author: Tim Bollerslev and Hao Zhou

Source: Lancaster University Management School

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