Volatility puzzles: a simple framework for gauging return-volatility regressions

Abstract

This paper provides a simple theoretical 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 asymmetric response of current volatility to lagged negative and positive returns, typically referred to as 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 S&P500 aggregate market index.

Introduction

Following the realization in the late 1980s that financial 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., 2004, 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 significantly across competing studies and procedures.

Building on the popular Heston (1993) one-factor stochastic volatility model, the present paper provides a simple theoretical framework for reconciling these conflicting empirical findings. Specifically, by postulating a parametric volatility model for the dynamic dependencies in the underlying returns, we show how the sign and the magnitude of the linear relationships between (i) the contemporaneous returns and the market-based volatilities, which we refer to as the volatility feedback effect, (ii) the lagged returns and the current market-based volatilities, which we refer to as the leverage effect, and (iii) the two different market-based volatilities, which we refer to as the implied volatility forecasting bias, all depend importantly on the parameters of the underlying structural model and the stochastic volatility risk premium.

The classical Intertemporal CAPM (ICAPM) model of Merton (1980) implies that the excess return on the aggregate market portfolio should be positively and directly proportionally related to the volatility of the market (see also Pindyck, 1984). This volatility feedback effect also underlies the ARCH-M model originally developed by Engle et al. (1987). However, empirical applications of the ARCH-M, and related stochastic volatility models, have met with mixed success. Some studies (see e.g., French et al., 1987, Chou, 1988, Campbell and Hentschel, 1992, Bali and Peng, 2003, Guo and Whitelaw, 2003, Ghysels et al., 2004) have reported consistently positive and significant estimates of the risk premium, while others (see, e.g., Campbell, 1987, Turner et al., 1989, Breen et al., 1989, Chou et al., 1992, Glosten et al., 1993, Lettau and Ludvigson, 2004) document negative values, unstable signs, or otherwise insignificant estimates. Moreover, the contemporaneous risk-return tradeoff appears sensitive to the use of ARCH as opposed to stochastic volatility formulations (Koopman and Uspensky, 2002), the length of the return horizon (Harrison and Zhang, 1999), along with the instruments and conditioning information used in empirically estimating the relationship (Harvey, 2001, Brandt and Kang, 2004). As we show below, these conflicting results are not necessarily inconsistent with the basic ICAPM model, in that the risk-return tradeoff relationship depends importantly on the particular volatility measure employed in the empirical investigations.¹

The so-called leverage effect, which predicts a negative correlation between current returns and future volatilities, was first discussed by Black (1976) and Christie (1982). The effect (and the name) may (in part) be attributed to a chain of events according to which a negative return causes an increase in the debt-to-equity ratio, in turn resulting in an increase in the future volatility of the return to equity.² Empirical evidence along these lines generally confirms that aggregate market volatility responds asymmetrically to negative and positive returns, but the economic magnitude is often small and not always statistically significant (e.g., Schwert, 1990, Nelson, 1991, Gallant et al., 1992, Glosten et al., 1993, Engle and Ng, 1993, Duffee, 1995, Bekaert and Wu, 2000). Moreover, the evidence tends to be weaker for individual stocks (e.g., Tauchen et al., 1996, Andersen et al., 2001). Importantly, the magnitude also depends on the volatility proxy employed in the estimation, with options implied volatilities generally exhibiting much more pronounced asymmetry (e.g., Bates, 2000, Wu and Xiao, 2002, Eraker, 2004).

A closely related issue concerns the bias in options implied volatilities as forecasts of the corresponding future realized volatilities. An extensive literature has documented that the market-based expectations embedded in options prices generally exceed the realized volatilities resulting in positive intercepts and slope coefficients less than unity in regression-based unbiasedness tests (see, e.g., Canina and Figlewski, 1993, Christensen and Prabhala, 1998, Day and Lewis, 1992, Fleming et al., 1995, Fleming, 1998, Lamoureux and Lastrapes, 1993, along with the recent survey in Poon and Granger, 2003). As formally shown in the recent studies by Chernov (2002), Pan (2002), and Bates (2003), this bias is intimately related to the market price of volatility risk, and some of our theoretical results in regards to the implied volatility forecasting bias parallel the developments in these concurrent studies.

Our theoretical results are based on the one-factor continuous-time stochastic volatility model popularized by Heston (1993), which explicitly assumes that the stochastic volatility premium is linear. This in turn allows us to utilize various closed form expressions for the conditional moments previously derived by Andersen et al. (2004) and Bollerslev and Zhou (2002). Although the exact relationship and implication derived in this paper may not hold for other more complicated model structures, the basic idea could in principle be generalized to cases of multiple volatility factors and jumps and/or non-linear volatility premia, albeit at the expense of considerable notational and computational complexity (see e.g., Andersen et al., 2002, Eraker et al., 2003, Chernov et al., 2003). Interestingly however, the theoretical results for the relatively simple one-factor affine Heston model turn out to be rich enough to explain the apparent conflicting empirical findings in regards to the monthly return-volatility regressions for the S&P500 aggregate market index and corresponding realized and implied volatilities.

The plan for the rest of the paper is as follows. Section 2 starts out by a discussion of the basic model structure, followed by the theoretical predictions related to the volatility feedback effect, the leverage effect, and the implied volatility forecasting bias, respectively. Section 3 provides confirmatory empirical evidence based on a thirteen-year sample of monthly returns, and high-frequency-based realized and implied volatilities for the Standard & Poor's composite index. Section 4 concludes. All of the derivations are given in a technical Appendix.