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

Volatility Puzzles: A Uni(cid:12)ed Framework for (cid:3) Gauging Return-Volatility Regressions y Tim Bollerslev and z Hao Zhou August 20, 2003 Abstract This paper provides a simple uni(cid:12)ed framework for assessing the empirical linkages between returns and realized and implied volatilities. First, we show that whereas the volatility feedback e(cid:11)ect 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 unambiguouslypositiveforallreasonableparametercon(cid:12)gurations. Second,thelagged return-volatility asymmetry, or the leverage e(cid:11)ect, 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 (cid:12)ndings 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. JEL Classi(cid:12)cation: G12, C51, C22. Keywords: Leverage Asymmetry, Volatility Feedback, Implied Volatility Forecast, Realized Volatility, Stochastic Volatility Model, Model Misspeci(cid:12)cation, Estimation Bias. (cid:3)The work of Bollerslev was supported by a grant from the NSF to the NBER. We have bene(cid:12)ted from discussionswithJimClouse andMike Gibson. Matthew Chesnes providedexcellentresearchassistance. We would also like to thank George Jiang, Neil Shephard, Rossen Valkanov, along with seminar participants in the University of Arizona, and the Symposium of New Frontiers in Financial Volatility Modeling (Florence, Italy) for their helpful comments and suggestions. The views presented here are solely those of the authors and do not necessarily represent those of the Federal Reserve Board or its sta(cid:11). yDepartment of Economics, Duke University, Post O(cid:14)ce Box 90097, Durham NC 27708 USA, Email boller@econ.duke.edu, Phone 919-660-1846,Fax 919-684-8974. zDivision of Research and Statistics, Federal Reserve Board, Mail Stop 91, Washington DC 20551 USA, E-mail hao.zhou@frb.gov,Phone 202-452-3360,Fax 202-728-5889. 1

1 Introduction Following the realization in the late eighties that (cid:12)nancial market volatility is both timevarying and predictable, empirical investigations into the temporal linkages between aggregate stock market volatility and returns have (cid:12)gured very prominently in the literature. Of course, volatility per se is not directly observable, and several di(cid:11)erent 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-returnrelationships di(cid:11)eringsigni(cid:12)cantly acrosscompeting studiesandprocedures. The present paper provides a uni(cid:12)ed theoretical framework for reconciling these conflicting empirical (cid:12)ndings. Speci(cid:12)cally, by postulating a relatively simple 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 e(cid:11)ect, (ii) the lagged returns and the current market-based volatilities, which we refer to as the leverage e(cid:11)ect, and (iii) the two di(cid:11)erent 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 e(cid:11)ect 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; Ghysels et al., 2002) have reported consistently positive and signi(cid:12)cant 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) document negative values, unstable signs, or otherwise insigni(cid:12)cant estimates. Moreover, the contemporaneous 2

risk-return tradeo(cid:11) appears sensitive to the use of ARCH as opposed to stochastic volatility formulations (Koopman and Uspensky, 1999), 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 andKang,2002). As we show below, these conflicting results are not necessarily inconsistent with the basic ICAPM model, in that the risk-return tradeo(cid:11) relationship depends importantly on the particular volatility measure employed in the empirical investigations.1 The leverage e(cid:11)ect, which predicts a negative correlation between current returns and futurevolatilities, was(cid:12)rstdiscussedbyBlack(1976)andChristie(1982). Thee(cid:11)ect(andthe 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 volatilityofthereturntoequity.2 Empiricalevidencealongtheselinesgenerallycon(cid:12)rmsthat aggregate market volatility responds asymmetrically to negative and positive returns, but the economic magnitude is often small and not always statistically signi(cid:12)cant (e.g., Schwert, 1989; Nelson, 1991; Gallant et al., 1992; Glosten et al., 1993; Engle and Ng, 1993; Du(cid:11)ee, 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, 2003) 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 coe(cid:14)cients less than unity in regressionbased unbiasedness tests (see, e.g., Canina and Figlewski, 1993; Christensen and Prabhala, 1998;DayandM.Lewis,1992;Flemingetal.,1995;Fleming,1998;LamoureuxandLastrapes, 1993, along with the recent survey in Poon and Granger, 2002). As formally shown in the 1More generalmulti-factor models also complicate the risk-returntradeo(cid:11)relationship, as the projection ofthe returnsonthe volatilitymustnowcontrolforotherstate variables(see,e.g.,Abel,1988;Tauchenand Hussey, 1991; Backus and Gregory,1993; Scruggs, 1998). 2Note, the volatility feedback e(cid:11)ect, along with the well-documented persistent volatility dynamics, also implies an observationallyequivalentnegative correlationbetween currentreturns and future volatility,as a shock to the volatility will require an immediate return adjustment to compensate for the increased future risk. We follow the convention in the literature of referring to the negative correlation between future volatility and current returns as the leverage e(cid:11)ect. 3

recent studies by Bates (2002), Chernov (2002), and Pan (2002), 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 modelpopularizedbyHeston(1993). Thisallows ustoutilizevariousclosedformexpressions fortheconditionalmoments previously derived byAndersen etal.(2002b)andBollerslev and Zhou (2002). However, the same basic idea could in principle be generalized to other more complicated model structures, including multiple volatility factorsand jumps, at the expense of notational and computational complexity (see, e.g. Andersen et al., 2002a; Eraker et al., 2003;Chernov et al.,2003). Nonetheless, therelatively simple one-factora(cid:14)neHeston model is rich enough to explain our empirical (cid:12)ndings in regards to the monthly return-volatility regressions for the Standard & Poor’s aggregate market index. 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 e(cid:11)ect, the leverage e(cid:11)ect, and the implied volatility forecasting bias, respectively. Section 3 provides con(cid:12)rmatory empirical evidence based ona (cid:12)fteen-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. 2 Theoretical Model Structure Let p denote the time-t logarithmic price of the risky asset, or portfolio. The one-factor t continuous-time a(cid:14)ne stochastic volatility model of Heston (1993) then postulates the following dynamics for the instantaneous returns, p dp t = ((cid:22)+(cid:21) s V t )dt+ p V t dB t ; dV = (cid:20)((cid:18)−V )dt+(cid:27) V dW ; (1) t t t t corr(dB ;dW ) = (cid:26); t t where the latent stochastic volatility, V , is assumed to follow a square-root process. A t negative instantaneous correlation between the two separate Brownian motions driving the price and volatility processes, or (cid:26) < 0, is directly associated with the leverage e(cid:11)ect in the raw returns; i.e., the tendency for contemporaneous returns and volatility to be negatively correlated. Similarly, the volatility feedback e(cid:11)ect is captured directly by the risk-return 4

trade-o(cid:11) parameter, (cid:21) > 0. Other more complicated model structures, including multiple s latent volatility factors along with jumps in the price and/or volatility, could in principle be analyzed by similar means. However, for expositional purposes we restrict our analysis to the relatively simple model in equation (1). Given this dynamics for the underlying price process, standard pricing arguments imply the existence of the following equivalent Martingale measure, or \risk-neutralized" distribution, p dp t = (r t (cid:3) −d t )dt+ V tp dB t (cid:3); dV = (cid:20)(cid:3)((cid:18)(cid:3) −V )dt+(cid:27) V dW(cid:3); (2) t t t t corr(dB(cid:3);dW(cid:3)) = (cid:26); t t where d refers to the dividend payout rate and r(cid:3) denotes the risk-neutral interest rate. t t The value of any contingent claim written on the underlying asset is now readily evaluated by calculating the expected payo(cid:11) in this risk-neutral distribution.3 We will refer to this expectation by the superscript (cid:3), as in E(cid:3)((cid:1)). The values of the risk-neutral parameters in (2) are directly related to the parameters of the actual price process in equation (1) by the functional relationships, (cid:20)(cid:3) = (cid:20)+(cid:21) and (cid:18)(cid:3) = (cid:20)(cid:18)=((cid:20)+(cid:21) ). The (cid:21) parameter refers to the v v v stochastic volatility risk premium, which is generally estimated to be negative. Hence, the degree of mean reversion for the risk-neutralized volatility process, as determined by (cid:20)(cid:3), is therefore slower (possibly even explosive) than the mean reversion for the actual volatility, as determined by (cid:20) (for a more detailed discussion of the connection between the objective and the risk-neutral distributions, see also Benzoni, 2001; Chernov, 2002; Wu, 2001; Pan, 2002). We next turn to our discussion of the corresponding model-based implications for the di(cid:11)erent return-volatility regressions, starting with the volatility feedback e(cid:11)ect. 2.1 Volatility Feedback E(cid:11)ect Empirical assessments of the relationship between returns and contemporaneous volatility have typically found the volatility feedback e(cid:11)ect to be statistically insigni(cid:12)cant, and sometimes even negative. These results may appear at odds with the ICAPM and the corresponding one-factor model in equation (1). Thus, as discussed in the introduction, several studies have resorted to more complicated multi-factor representations as a way to resolve 3Notice, that in the presence of stochastic volatility it is generallynot possible to perfectly hedge contingent claims payo(cid:11), and options are therefore no longer redundant assets. 5

this apparent empirical puzzle (see, e.g., Scruggs, 1998, and the discussion therein). Meanwhile, consider the continuously compounded returns from time t to t+(cid:1) implied by the simple model in (1), Z Z q t+(cid:1) t+(cid:1) R = p −p = (cid:22)(cid:1)+(cid:21) V du+ V dB : (3) t;t+(cid:1) t+(cid:1) t s u u u t t R p Although the \residual" de(cid:12)ned by t+(cid:1) V dB is heteroskedastic, the population ret u u gression of the returns on a constant and the integrated volatility would correctly uncover the volatility feedback e(cid:11)ect ((cid:21) > 0), provided that the orthogonality condition s (cid:16)R p R (cid:17) E t+(cid:1) V dB (cid:2) t+(cid:1)V du = 0 holds true. However, with a non-zero \leverage" eft u u t u fect, or (cid:26) < 0, the residual and the integrated volatility will be correlated, resulting in a biased estimate for (cid:21) . s Speci(cid:12)cally, consider the population regression, Z t+(cid:1) R = (cid:11)+(cid:12) V du+e : (4) t;t+(cid:1) u t;t+(cid:1) t Then as formally shown below, unless (cid:26) = 0, the population feedback coe(cid:14)cient (cid:12) will di(cid:11)er from the true feedback coe(cid:14)cient (cid:21) . Of course, the integrated volatility is not directly obs servable, so the sample counterpart to the population regression in (4) isn’t actually feasible. However, the integrated volatility may in theory be approximated arbitrarily well by the corresponding realized volatility constructed by the summation of su(cid:14)ciently (cid:12)nely sampled high-frequency squared returns (see, e.g., Andersen et al., 2003). This approach, which is now routinely employed in the literature, also underlies our empirical analysis in Section 3 below. Alternatively, consider the corresponding implied volatility-return regression, ! Z t+(cid:1) R = (cid:11)(cid:3) +(cid:12)(cid:3)E(cid:3) V du +e(cid:3) ; (5) t;t+(cid:1) t u t;t+(cid:1) t where the risk neutral expectation is taken under the distribution in (2). In this situation, unless the stochastic volatility risk-premium equals zero, or (cid:21) = 0, the population feedback v coe(cid:14)cient will again di(cid:11)er from the true feedback coe(cid:14)cient in equation (3), that is (cid:12)(cid:3) 6= (cid:21) . Hence, to correctly uncover the volatility feedback parameter from a contemporaneous s return-volatility type regression, either the leverage e(cid:11)ect must be zero if the regression is based on a realized volatility proxy, or the stochastic volatility risk premium must be zero 6

when using options implied volatilities. Of course neither case is likely to hold empirically. Proposition 1 characterizes the exact form of the resulting biases.4 Proposition 1 Assume that the parameters in (1) and (2) adhere to the standard sign restrictions, (cid:20) > 0, (cid:18) > 0, (cid:27) > 0, (cid:26) < 0, (cid:21) < 0, (cid:21) > 0, and that (cid:22) 6= 0. The population v s feedback coe(cid:14)cient in the integrated volatility regression in equation (4) is then given by, (cid:26)(cid:20) (cid:12) = (cid:21) + < (cid:21) : (6) s (cid:27) s Let a = (1−e−(cid:20)(cid:1))=(cid:20) and a(cid:3) = (1−e−(cid:20)(cid:3)(cid:1))=(cid:20)(cid:3). The population feedback coe(cid:14)cient in the (cid:1) (cid:1) implied volatility regression in (5) may then be expressed as, a (cid:12)(cid:3) = (cid:21) (cid:1) < (cid:21) : (7) sa(cid:3) s (cid:1) Moreover, assuming that 0 < (cid:21) < −(cid:26)(cid:20) , the two slope parameters are related by, s (cid:27) (cid:12) < 0 < (cid:12)(cid:3) < (cid:21) ; (8) s while for 0 < −(cid:26)(cid:20) < (cid:21) < a(cid:3) (cid:1) (cid:26)(cid:20) , we have (cid:27) s a −a(cid:3) (cid:27) (cid:1) (cid:1) 0 < (cid:12) < (cid:12)(cid:3) < (cid:21) : (9) s The proof of the proposition is given in the technical Appendix A. Theimplicationsofthepropositionforempiricalstudies designedtouncover thevolatility feedback e(cid:11)ect are immediate. First, regression-based procedures utilizing realized volatility proxies will invariably result in a downward biased slope estimate, with the sign and magnitude depending on the underlying structural parameters. This, of course, is entirely consistent with the extant literature discussed above reporting inconclusive and sometimes even negative estimates for (cid:12). Only if the leverage, or asymmetry, e(cid:11)ect is zero ((cid:26) = 0) will the regression be unbiased for estimating (cid:21) . Second, regression estimates based on implied s volatility will generally show less of a downward bias and remain positive under the most general parameter setting. However, only if the stochastic volatility risk premium equals zero ((cid:21) = 0) will the bias completely disappear. Again, this is directly in line with the v existing literature discussed above, as well as the new empirical results reported in Section 3 below. 4As shown in the appendix, the population intercepts (cid:11) and (cid:11)(cid:3) will also generally di(cid:11)er from the true drift in equation (3), that is (cid:22)(cid:1). However, we will focus our discussion on the slope coe(cid:14)cients which are typically associated with the volatility feedback e(cid:11)ect. 7

This result also helps explain why various versions of (cid:12)ltered volatility (obtained by projecting on lagged historical squared and/or absolute returns) may produce less biased or even positive (cid:12) estimates. Speci(cid:12)cally, instead of the realized return - realized volatility trade-o(cid:11)regression in(4), consider therealizedreturn- expectedvolatility trade-o(cid:11)regression ! Z t+(cid:1) R = (cid:11)~ +(cid:12)~E V du +e : t;t+(cid:1) t u t;t+(cid:1) t This regression explicitly purges the simultaneous correlation between the return and volatility innovations. As such, this regression corresponds more closely to the implied returnvolatility trade-o(cid:11) regression in (5) that obtain by replacing the expected integrated volatil- (cid:16) (cid:17) (cid:16) (cid:17) R R ity, E t+(cid:1)V du , with itsrisk neutral equivalent, E(cid:3) t+(cid:1)V du . Of course, theexpected t t u t t u integrated volatility will generally depend upon the underlying structural model, but may be approximated empirically through the use of instrumental variables procedures. However, as previously noted, the resulting estimates for the risk-return trade-o(cid:11) relationship are often very sensitive to theparticular ad hoc choice ofinstruments employed inthe estimation (Harvey, 2001; Brandt and Kang, 2002). We shall return to this issue in the empirical Section 3.1 below. At a more general level Proposition 1 clearly highlights the importance of the volatility proxy used in the estimation of the risk-return trade-o(cid:11) relationship, and as such indirectly explains the instability in the estimates reported in the extant literature in regards to the model choice, instrument control, and return horizon. Similar issues arise in the empirical estimation of the leverage e(cid:11)ect, to which we turn next. 2.2 Leverage E(cid:11)ect Several di(cid:11)erent parametric volatility models and volatility-return regressions have been employed inthe literature forempirically assessing the leverage e(cid:11)ect (see e.g., the discussion in Bekaert and Wu (2000), along with the surveys of the ARCH literature in Bollerslev et al. (1992) and Bollerslev et al. (1994)). Although most estimates support the hypothesis that aggregate stock market volatility responds asymmetrically to past negative and positive returns, as discussed in the introduction, the magnitude and the statistical signi(cid:12)cance of the estimated e(cid:11)ect is quite sensitive to the return horizon andthe particular volatility proxy employed in the estimation. At the most basic level the leverage e(cid:11)ect is generally associated with a negative correlation between current volatility and lagged returns. To formally quantify this correlation, 8

consider the corresponding population regressions for the integrated volatility,5 Z t+(cid:1) V du = γ +(cid:14)R +e ; (10) u t−(cid:1);t t;t+(cid:1) t and the option implied volatility, ! Z t+(cid:1) E(cid:3) V du = γ(cid:3) +(cid:14)(cid:3)R +e(cid:3) ; (11) t u t−(cid:1);t t−(cid:1);t t where the expectation in equation (11) is again taken with respect to the risk-neutral distribution. Of course, the slope parameters in the simpli(cid:12)ed asymmetry regressions in (10) and (11) do not correspond directly to the leverage, or asymmetry, parameter (cid:26) determining the correlation between the two Brownian motions in (1). However, as the following proposition makes clear, the population regression parameters may be expressed as explicit nonlinear functions of the underlying structural parameters in (1) and (2). These functional relationships in turn explain the stronger asymmetry observed empirically between implied volatility and lagged-returns. Proposition 2 Assume that the parameters in (1) and (2) adhere to the standard sign restrictions, (cid:20) > 0, (cid:18) > 0, (cid:27) > 0, (cid:26) < 0, (cid:21) < 0, and (cid:21) > 0. Let a = (1 − e−(cid:20)(cid:1))=(cid:20), v s (cid:1) a(cid:3) = (1−e−(cid:20)(cid:3)(cid:1))=(cid:20)(cid:3), and c = (e−(cid:20)(cid:1)+(cid:20)(cid:1)−1)=(cid:20). The population slope parameters in (10) (cid:1) (cid:1) and (11) may then be expressed as, (cid:21) (cid:18)(cid:27)2a2 +(cid:26)(cid:27)(cid:18)a2 (cid:14) = s 2(cid:20) (cid:1)(cid:16) (cid:1)(cid:17) ; (12) (cid:18)(cid:1)+ (cid:21)s(cid:27)(cid:18) (cid:21)s(cid:27) +(cid:26) c (cid:20) (cid:20) (cid:1) and (cid:26)(cid:27)(cid:18)a(cid:3) a (cid:14)(cid:3) = (cid:16)(cid:1) (cid:1) (cid:17) : (13) (cid:18)(cid:1)+ (cid:21)s(cid:27)(cid:18) (cid:21)s(cid:27) +(cid:26) c (cid:20) (cid:20) (cid:1) Moreover, assuming 0 < (cid:21) < −2(cid:26)(cid:20) it follows that, s (cid:27) (cid:14)(cid:3) < (cid:14) < 0; (14) while for 0 < −2(cid:26)(cid:20) < (cid:21) , (cid:27) s (cid:14)(cid:3) < 0 < (cid:14): (15) 5Intheempiricalsectionwealsoreporttheresultsfromalongerregressioninwhichweincludethelagged volatility along with di(cid:11)erent response coe(cid:14)cients for positive and negative returns. 9

The proof of the proposition is given in Appendix B. It is noteworthy that for the integrated volatility regression, the \leverage" coe(cid:14)cient, (cid:14), depends critically on both the volatility feedback parameter (cid:21) (positively), as well as s the \structural" leverage coe(cid:14)cient (cid:26) (negatively). Thus, although most empirical studies report strong realized volatility asymmetry for the aggregate market portfolio ((cid:14) < 0), this may help explain the lack of statistical signi(cid:12)cance, and sometimes even reverse asymmetry reported occasionally. For the implied volatility regression, the \leverage" coe(cid:14)cient (cid:14)(cid:3) depends directly on (cid:26) (negatively), and the stochastic volatility risk premium (cid:21) through a(cid:3) v (cid:1) (magnitude). In contrast to the integrated volatility regression, the coe(cid:14)cient in the implied volatility regression is unambiguously negative provided that(cid:26) < 0. Moreover, provided that the volatility feedback e(cid:11)ect is positive ((cid:21) > 0), and the stochastic volatility risk premium is s negative ((cid:21) < 0), as it is commonly assumed in the literature, the magnitude of the implied v volatility asymmetry always exceeds that of the integrated volatility, that is (cid:14)(cid:3) < (cid:14). Similar considerations help toexplain thedownward biasin theimplied-realized volatility forecasting regressions, to which we now turn. 2.3 Implied Volatility Forecasting Bias The two previous subsections demonstrate how the use of realized or implied volatility proxies can result in quite di(cid:11)erent population parameters in the contemporaneous and lagged return-volatility regressions. A closely related question, concerns the extent to which implied volatilities provide unbiased forecasts of the corresponding future realized volatilities. The most common approach employed in the literature for assessing the forecasting bias is based on regressing the ex-post realized volatility over some time period, say [t;t+(cid:1)], on a constant and the time t implied volatility for an option maturing at t + (cid:1) (for a recent survey of this extensive empirical literature see Poon and Granger, 2002). In population, ! Z Z t+(cid:1) t+(cid:1) V du = (cid:30) +(cid:30) E(cid:3) V du +e : (16) u 0 1 t u t;t+(cid:1) t t Obviously, for the implied volatility to provide unbiased forecasts, the two projection coef- (cid:12)cients should equal (cid:30) = 0 and (cid:30) = 1, respectively. Meanwhile, most empirical studies 0 1 report statistically signi(cid:12)cant biases in the direction of (cid:30) > 0 and (cid:30) < 1. These empiri- 0 1 cal biases have in part been explained by a standard errors-in-variables type problem arising fromtheuseof(cid:12)nite-sampleequivalentstothepopulationregressionin(16)(Christensenand 10

Prabhala, 1998), along with very persistent volatility dynamics rendering standard statistical inference unreliable (Bandi and Perron, 2003). However, these statistical considerations aside, it follows that if the stochastic volatility risk premium, (cid:21) , di(cid:11)ers from zero, the two v population regression coe(cid:14)cients in (16) implied by the structural model in (1) and (2) will not equal zero and unity, respectively. Proposition 3 Assume that the parameters in (1) and (2) adhere to the standard sign restrictions, (cid:20) > 0, (cid:18) > 0, (cid:27) > 0, (cid:26) < 0, (cid:21) < 0, and (cid:21) > 0. The population parameters in v s the regression in (16) are then given by, a a (cid:30) = b − (cid:1)b(cid:3) and (cid:30) = (cid:1) < 1; (17) 0 (cid:1) a(cid:3) (cid:1) 1 a(cid:3) (cid:1) (cid:1) where a = (1−e−(cid:20)(cid:1))=(cid:20), a(cid:3) = (1−e−(cid:20)(cid:3)(cid:1))=(cid:20)(cid:3), b = (cid:18)((cid:1)−a ), and b(cid:3) = (cid:18)(cid:3)((cid:1)−a(cid:3) ). (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) The proof of the proposition is given in Appendix C. The proposition immediately explains the typical (cid:12)nding of a downward bias in the estimated slope coe(cid:14)cient. Intuitively, for (cid:21) < 0, the stochastic volatility risk premium v reduces the degree of mean reversion in the risk-neutral volatility process relative to that of the actualvolatility process ((cid:20)(cid:3) < (cid:20)), in turn resulting in the ratioa =a(cid:3) becoming less than (cid:1) (cid:1) one.6 Thus, any estimate of (cid:30) should be gauged against this population bias. Of course, 1 the true structural model parameters, (cid:20) and (cid:20)(cid:3), are generally unknown and would have to be estimated. Conversely, the population return-volatility regressions in Propositions 1-3 (coupled with additional moment restrictions along the lines of Bollerslev and Zhou, 2002) could in principle be employed as a system of equations in estimating the structural model parameters underlying the actual and risk-neutral distributions in (1) and (2). We shall not pursue this approach any further here. Instead, we next turn to a concrete empirical implementation of the (cid:12)ve regression equations based on aggregate U.S. stock market volatility and returns for directly illustrating the various biases implied by Propositions 1-3. 6Closely related results, along with a more detailed analysis of the impact of jumps, have recently been derived in concurrent work by Bates (2002) and Chernov (2002). 11

3 Empirical Illustration Our empirical analysis is based on monthly returns and volatilities for the S&P composite index spanning the period from January 1986 through February 2002.7 The monthly continuously compounded percentage returns are constructed from the daily S&P500 closing prices supplied by the Wall Street Journal. Normalizing the monthly time interval to unity, we will refer to the return over the t+1’th month as R . t;t+1 The corresponding realized volatilities are based on the summation of the (cid:12)ve-minute squaredreturnswithinthemonth. Thehigh-frequencydatafortheS&P500indexisprovided by the Institute of Financial Markets. Speci(cid:12)cally, with n trading days in month t+1, t+1 78X(cid:1)n t+1 RV (cid:17) (logP −logP ) 2; (18) t;t+1 t+i=78(cid:1)n t+(i−1)=78(cid:1)n t+1 t+1 i=1 where the 78 (cid:12)ve-minute subintervals represents the normal trading hours from 9:30am to 4:00pm,includingtheclose-to-open(cid:12)ve-minuteinterval. Asdiscussedintheprevioussection, the realized volatility, RV , is readily interpreted as a consistent (for increasing sampling t;t+1 R frequency) estimate of the corresponding integrated volatility, IV (cid:17) t+1V ds. t;t+1 t s The monthly implied volatility (variance) is formally de(cid:12)ned by, (cid:20)Z (cid:21) t+1 IV (cid:3) (cid:17) E(cid:3) V dsjF ; (19) t;t+1 s t t where E(cid:3) refers to the risk-adjusted expectation of the one-month ahead integrated volatility, IV .8 The most actively traded equity index options are written on the S&P100 t;t+1 index. Hence for liquidity reasons, we rely on the corresponding option implied volatilities (cid:3) (VIX) provide by the CBOE in empirically quantifying IV . The VIX index (reported t;t+1 in annualized percentage standard deviation form) is based on a weighted average of the one-month ahead volatilities inverted from eight near-the-money puts and calls, and is now widely regarded among market participants as the \implied volatility index" (see Fleming 7The start date ofJanuary1986reflects the availability of both the S&P high-frequency return data and VIX implied volatility index. However,all of the regressionresults reportedbelow are materially una(cid:11)ected by excluding the October 1987 stock market crash and starting the sample in January 1988. These results are available upon request. 8Thismeasureisalsosometimesreferredtoasthe\risk-neutralintegratedvolatility." Inempiricalstudies the implied volatilityconceptis usedalmostexclusivelyin referenceto the volatilitythat equatesthe Black- Scholes price with the actual price of an option. This generally provides an accurate approximation for short-lived at-the-money options; see, e.g., Ledoit et al. (2002). 12

et al., 1995; Fleming, 1998, for a precise de(cid:12)nition and further discussion of the VIX index).9 To facilitate the theoretical derivations, all of the volatility regressions analyzed in the previous section were cast intheformofvariances corresponding to theempirical RV and t;t+1 (cid:3) IV measures de(cid:12)ned above. However, for robustness reasons previous empirical studies t;t+1 have often been implemented in the form of standard deviations. Hence, we augment the 1=2 (cid:3)1=2 variance regressions with the analogous regressions based on RV and IV . t;t+1 t;t+1 Summary statistics for all of the variables are reported in Table 1. For comparison purposes the standard deviations and the variances are converted to percentage and squared percentage points, respectively. From the (cid:12)rst column, the average annualized return on the market was about ten percent, with a sample standard deviation of around sixteen percent. The returns are negatively skewed with much fatter tails than the normal distribution. The implied volatilities are systematically higher than the realized volatilities, and their unconditional distributions also deviate more from the normal. The returns are approximately serially uncorrelated, while the volatility series (both in standard deviation and variance forms) exhibit pronounced own temporal dependencies. In fact, the (cid:12)rst ten autocorrelations reported in the bottom part of the table are all highly signi(cid:12)cant with the gradual, but very slow, decay suggestive of long-memory type features. This is also evident from the time series plots for each of the (cid:12)ve series given in Figure 1. 3.1 Volatility Feedback E(cid:11)ect Our estimates of the volatility feedback e(cid:11)ect are based on the empirical equivalents to the two population regressions in equations (4) and (5), R = (cid:11)+(cid:12)RV +u ; (20) t;t+1 t;t+1 t+1 R = (cid:11)(cid:3) +(cid:12)(cid:3) IV (cid:3) +u(cid:3) ; (21) t;t+1 t;t+1 t+1 along with the corresponding robust regressions in standard deviation form, R = (cid:11)+(cid:12)RV 1=2 +u ; (22) t;t+1 t;t+1 t+1 R = (cid:11)(cid:3) +(cid:12)(cid:3) IV (cid:3)1=2 +u(cid:3) ; (23) t;t+1 t;t+1 t+1 9Ideally, we would want realized and implied volatilities and returns for the same index. However, reliable high-frequency data, required for the construction of the realized volatilities, are only available for the S&P500index, while liquidity considerations in the options market dictates the use of the S&P100VIX index. Of course, the returns on the two indexes are very close, with a monthly sample correlationof 0.988. 13

where the residuals from the regressions are generically denoted by u and u(cid:3) , respect+1 t+1 tively. The coe(cid:14)cient estimates, along with their asymptotic standard errors based on a Newey-West covariance matrix estimator allowing for a two-month lag, are reported in Table 2. Interestingly, for the regressions involving the implied volatility, the intercept and slope coe(cid:14)cients are all insigni(cid:12)cant, and both of the R-squares are very close to zero. In contrast, the two realized volatility regressions both result in highly signi(cid:12)cant estimates for the intercepts (positive) and slopes (negative). These contemporaneous return-volatility regressions also explainabouttenandthirteenpercent ofthevariabilityintheex-post monthly returns, respectively. Although the empirical (cid:12)nding of a signi(cid:12)cant negative relationship between aggregated stock market returns and realized volatility may appear counter intuitive, the result is, of course, entirely consistent with Proposition 1 and the ranking of the corresponding population parameters, (cid:12) < 0 < (cid:12)(cid:3), given that the condition 0 < (cid:21) < −(cid:26)(cid:20) is s (cid:27) satis(cid:12)ed by the underlying structural model parameters.10 It is worth noting, that while all of the regressions above involve a trade-o(cid:11) between monthly returns and volatilities over the identical time horizon, the one-month implied volatility is determined at time t, whereas the realized volatility is not observable until t+1. As such, the results in Table 2 are also consistent with the recent empirical (cid:12)ndings by Brandt and Kang (2002), who report a (puzzling) negative contemporaneous relation between the conditional mean and the conditional variance of the market returns, along with a more conventional positive tradeo(cid:11) for the one-month lagged volatility. Similarly, Ghysels et al. (2002) report a signi(cid:12)cant positive trade-o(cid:11) relationship when the squared returns 20-50 days in the past are weighted most heavily in their realized volatility constructs. In order to further illustrate this point, the last columns in each of the panels in Table 2 report the results from a standard instrumental variables procedure in which we rely on the absolute lagged returns as instruments for the realized volatilities in the two regressions in (20) and (22). Although the slope coe(cid:14)cient estimates for both the standard deviation and the variance formulation remain negative, they are clearly much closer to zero, and the regression R-squares drop from 10-13% to about 5-6%. Moreover, the corresponding standard errors are also much larger so that the estimates are no longer statistically signi(cid:12)cant. It is also noteworthy that on using the lagged raw returns as instruments, the estimates for the 10Recall that the positive return volatility trade-o(cid:11) observed in some empirical studies, 0<(cid:12) < (cid:12)(cid:3), may similarly be justi(cid:12)ed by the secondcase of Proposition1, when the underlying structuralparameterssatisfy the condition 0<−(cid:26) (cid:27) (cid:20) <(cid:21) s < a(cid:1) a − (cid:3) (cid:1) a(cid:3) (cid:1) (cid:26) (cid:27) (cid:20) . 14

two slopecoe(cid:14)cients (availableuponrequest) arebothpositive, albeitevencloser tozerostatistically.11 As such, these results further highlight the sensitivity to the particular volatility proxy and instrument choice employed in the reduced form volatility feedback regressions, despite the fact that some instruments may produce conventional positive feedback e(cid:11)ect. 3.2 Leverage E(cid:11)ect The empirical equivalents to the simple leverage regressions analyzed in Section 2.2 above take the form, RV = γ +(cid:14)R +u ; (24) t;t+1 t−1;t t+1 IV (cid:3) = γ(cid:3) +(cid:14)(cid:3) R +u(cid:3); (25) t;t+1 t−1;t t with the theoretical prediction from Proposition 2 that in population (cid:14)(cid:3) < (cid:14) < 0; i.e., the implied volatility is more responsive to the lagged return than the realized volatility. Again, for robustness reasons, the asymmetry implications may alternatively be tested in standard deviation form, RV 1=2 = γ +(cid:14)R +u ; (26) t;t+1 t−1;t t+1 IV (cid:3)1=2 = γ(cid:3) +(cid:14)(cid:3) R +u(cid:3); (27) t;t+1 t−1;t t with the similar predictions in regards to the sign and ordering of the slope coe(cid:14)cients. Moreover, to account for the strong own temporal dependencies in the volatility and to allow for di(cid:11)erent impacts from past negative and positive returns, a slightly longer asymmetry regression is often estimated empirically, RV = γ +(cid:12)RV +(cid:11)(R ) 2 −(cid:14)(R ) 2I +u ; (28) t;t+1 t−1;t t−1;t t−1;t (R (cid:20)0) t+1 t−1;t IV (cid:3) = γ(cid:3) +(cid:12)(cid:3) IV (cid:3) +(cid:11)(cid:3) (R ) 2 −(cid:14)(cid:3) (R ) 2I +u(cid:3): (29) t;t+1 t−1;t t−1;t t−1;t (R (cid:20)0) t t−1;t In these longer regressions, weak asymmetry would again be implied by negative (cid:14)’s, while strong asymmetry would have the (cid:11)’s be negative as well. Similarly, if the implied volatility responds more asymmetrically to past returns than do the realized volatility, we would 11We also experimented with a number of other instrumental variables, including the lagged squared returns and the lagged volatilities in standard deviation and variance forms. Although not statistically signi(cid:12)cant, only the results based on the absolute lagged returns were generally consistent in terms of their signs across di(cid:11)erent subsamples. 15

expect to (cid:12)nd that the estimates for the (cid:14)’s satisfy the relation (cid:14)(cid:3) < (cid:14) < 0. These same considerations apply to the pair of robust standard deviation regressions, RV 1=2 = γ +(cid:12)RV 1=2 +(cid:11)jR j−(cid:14)jR jI +u ; (30) t;t+1 t−1;t t−1;t t−1;t (R (cid:20)0) t+1 t−1;t IV (cid:3)1=2 = γ(cid:3) +(cid:12)(cid:3) IV (cid:3)1=2 +(cid:11)(cid:3)jR j−(cid:14)(cid:3)jR jI +u(cid:3): (31) t;t+1 t−1;t t−1;t t−1;t (R (cid:20)0) t t−1;t Note that for (cid:12) = 0 and (cid:14) = 2(cid:11) the long regression in equation (30) collapses to the short regressioninequation(26). Likewise, for(cid:12)(cid:3) = 0and(cid:14)(cid:3) = 2(cid:11)(cid:3) thetworisk-neutralregressions in equations (31) and (27) coincide. The actual S&P estimation results for the leverage regressions are reported in Table 3. The intercepts and slope coe(cid:14)cients for the short regressions in the (cid:12)rst panel of the table are all highly signi(cid:12)cant. The R-squares for the realized volatility regressions are systematically lower than the corresponding R-squares for the implied volatilities. Importantly, the estimates of the (cid:14)’s from the variance regression, (cid:14)(cid:3) = −3:31 and (cid:14) = −0:82, also adhere exactly to the theoretical predictions from Proposition 2 of negative and more pronounced asymmetry for the implied volatility. Similarly, the short regressions in standard deviation form results in estimates of (cid:14)(cid:3) = −0:17 and (cid:14) = −0:08, both of which are signi(cid:12)cantly less than zero. Turning to the longer regressions, it is noteworthy that the asymmetry in the realized volatility areno longerstatistically signi(cid:12)cant, while the two estimates of(cid:14)(cid:3) fromthe implied volatility regressions are both overwhelmingly signi(cid:12)cant, and economically large. Again, this is directly in line with the implications from Proposition 2 of strong asymmetry for the implied volatility along with weak, and possibly even reverse, asymmetry for the realized volatility.12 The well-documented strong own temporal dependencies in the volatility also result in fairly large and highly signi(cid:12)cant estimates for the (cid:12)’s, along with much higher R-squares for the long volatility regressions (84-88% for the implied and 38-55% for the realized volatilities). All-in-all, the at (cid:12)rst somewhat puzzling empirical (cid:12)ndings for the di(cid:11)erent regressions reported in Table 3 again highlight the importance of properly interpreting the estimated asymmetry in lieu of the theoretical implications for the di(cid:11)erent volatility proxies detailed in Proposition 2. In this regard, the results in Table 3 are also consistent with previous empirical evidence in the literature related to the signi(cid:12)cance, or the lack thereof, of the 12Indeed, subsample analysis excluding 1986-1987 and 1998-2002, result in reverse asymmetry for the realized volatility, while the implied volatility asymmetry remains very strong. This is therefore consistent with the second case of Proposition 2, i.e., (cid:14)(cid:3) < 0 < (cid:14). Further details concerning these subsample results are again available upon request. 16

volatility asymmetry e(cid:11)ect for other markets and time periods (see, e.g., Schwert, 1989; Nelson, 1991; Gallant et al., 1992; Engle and Ng, 1993; Du(cid:11)ee, 1995; Bekaert and Wu, 2000; Wu, 2001, among others). We next turn to discussion of the related empirical evidence concerning the unbiasedness regressions directly linking the implied and realized volatility. 3.3 Implied Volatility Forecasting Bias The question of whether implied volatilities provide unbiased and informationally e(cid:14)cient forecasts of the corresponding future realized volatilities have been studied extensively in the empirical (cid:12)nance literature. The typical regression employed in the literature takes the form, RV = (cid:30) +(cid:30) IV (cid:3) +u ; (32) t;t+1 0 1 t;t+1 t+1 or in terms of standard deviations, RV 1=2 = (cid:30) +(cid:30) IV (cid:3)1=2 +u ; (33) t;t+1 0 1 t;t+1 t+1 where unbiasedness would be associated with (cid:30) = 0 and (cid:30) = 1. Of course, the theoretical 0 1 results in Proposition 3 implies that these are not the values to be expected empirically if the stochastic volatility risk is priced.13 The actual estimation results reported in Table 4 also do not support the unbiasedness hypothesis. Both of the estimates for (cid:30) are signi(cid:12)cantly less than unity, and the estimates 1 for (cid:30) are greater than zero, albeit not signi(cid:12)cantly so. The regression in standard deviation 0 formresults inafairlyhighR-squareof0.51, while theR-squarefromthelessrobust variance regression equals 0.34. These (cid:12)ndings of a downward bias in the implied volatility forecasts along with fairly high explanatory power when judged by the high-frequence based realized volatility measures are directly in line with recent empirical results in the literature (see, e.g., Martens and Zein, 2002; Neely, 2003, and the survey by Poon and Granger, 2002). Moreover, the direction of the estimated biases are exactly as expected from Proposition 3, and as such do not necessarily suggest any ine(cid:14)ciencies. 13The existence of a non-zero stochastic volatility risk premium for explaining estimates of (cid:30) 0 > 0 and (cid:30) 1 < 1, along the lines of Proposition 3, has previously been discussed by Bates (2002), Benzoni (2001), Chernov (2002), and Pan (2002), among others. Meanwhile, as in Fleming (1998), di(cid:11)erent subsamples result in estimates of (cid:30) 0 (cid:20) 0, which is still consistent with Proposition 3. These results are available upon request. 17

4 Conclusion The simple uni(cid:12)ed continuous-time framework derived in this paper for assessing the linkages between discretely observed returns and realized and implied volatilities help explain a number of puzzling (cid:12)ndings in the extant empirical literature. In particular, we show that whereas the sign of the correlation between return and implied volatility is unambiguously positive, the correlation between contemporaneous return and realized volatility is generally undetermined. Similarly, the lagged return-volatility asymmetry is always stronger for implied than realized volatility. Also, implied volatilities generally provide downward biased forecasts of subsequent realized volatilities. It would be interesting to extend the empirical analysis for the aggregate S&P market index presented here to other markets. In particular, the volatility feedback and asymmetry e(cid:11)ects may not be as important for other markets, and consequently result in qualitatively di(cid:11)erent return-volatility linkages. Thetheoreticalanalysisofmorecomplicatedmodelstructures allowing for jumps in the volatility and/or returns along with multiple volatility factors may also give rise to additional new insights. The regression-based implications derived here could also be used in directly estimating the underlying objective and risk-neutral dynamics, including the stochastic volatility risk premium, by appropriately matching the sample and population moments for the realized and implied volatilities. We leave further work along these lines for future research. 18

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A Proof of Proposition 1 To simplify the exposition, de(cid:12)ne a = (1−e−(cid:20)(cid:1))=(cid:20), a(cid:3) = (1−e−(cid:20)(cid:3)(cid:1))=(cid:20)(cid:3), b = (cid:18)((cid:1)−a ), (cid:1) (cid:1) (cid:1) (cid:1) b(cid:3) = (cid:18)(cid:3)((cid:1)−a(cid:3) ), and c = (e−(cid:20)(cid:1) +(cid:20)(cid:1)−1)=(cid:20). The proof consists of three steps. (cid:1) (cid:1) (cid:1) To determine the projection coe(cid:14)cients in the realized volatility-return trade-o(cid:11) relationship, note that from Andersen et al. (2002b), the variance term may be written as, ! Z t+(cid:1) (cid:18)(cid:27)2 (cid:16) (cid:17) (cid:18)(cid:27)2 VAR V du = e−(cid:20)(cid:1) +(cid:20)(cid:1)−1 = c : u (cid:20)3 (cid:20)2 (cid:1) t Similarly, the covariance term takes the form, (cid:16) R (cid:17) COV R ; t+(cid:1)V du (cid:16) t;t+(cid:1) t R u R p R (cid:17) = COV (cid:22)(cid:1)+(cid:21) t+(cid:1)V du+ t+(cid:1) V dB ; t+(cid:1)V du (cid:16)R s t (cid:17) u (cid:16)t R pu u t R u (cid:17) = (cid:21) VAR t+(cid:1)V du +COV t+(cid:1) V dB ; t+(cid:1)V du : s t u t u u t u Rearranging and integrating by parts, the second term becomes, (cid:16)R p R (cid:17) COV t+(cid:1) V dB ; t+(cid:1)V du (cid:16) R tp u R u t (cid:17)u = E t+(cid:1) V dB t+(cid:1)V du hR t (cid:16)R up u t (cid:17) u R R p i = E t+(cid:1) u V dB V du+ t+(cid:1) ( uV ds) V dB hR t (cid:16)R t p s s(cid:17) u i t t s u u = E t+(cid:1) u V dB V du hR t (cid:16)R t p s s(cid:17)(cid:16)u R R p (cid:17) i = E t+(cid:1) u V dB V + u(cid:20)((cid:18)−V )ds+ u(cid:27) V dW du hR t t (cid:16)R sp s R t t (cid:17)i shR (cid:16)R t p s s R p (cid:17) i = E t+(cid:1)−(cid:20) u V dB uV ds +E t+(cid:1) u V dB u(cid:27) V dW du R t h R t(cid:16)R ps s t (cid:17)s i t(cid:16)R t R s s t (cid:17) s s = E t+(cid:1) −(cid:20) u s V dB V ds du+E t+(cid:1) u(cid:26)(cid:27)V dsdu : t t t (cid:28) (cid:28) s t t s R Notice that the recursive structure within the Riemann integral t+(cid:1) ((cid:1))du must hold over t any time interval (cid:1), so that in particular, Z (cid:18)Z q (cid:19) Z Z (cid:18)Z q (cid:19) t+(cid:1) u t+(cid:1) u s E V dB V du = −(cid:20) E V dB V dsdu s s u (cid:28) (cid:28) s t t t t t Z (cid:18)Z (cid:19) t+(cid:1) u + (cid:26)(cid:27)(cid:18)ds du; t t which gives rise to the linear (cid:12)rst order ordinary di(cid:11)erential equation, (cid:16)R p (cid:17) dE u V dB V (cid:18)Z q (cid:19) t s s u u = −(cid:20)E V dB V +(cid:26)(cid:27)(cid:18): (A1) du s s u t Solving this equation yields, (cid:18)Z u q (cid:19) (cid:18)Z t q (cid:19) (cid:26)(cid:27)(cid:18) (cid:16) (cid:17) E V dB V = e−(cid:20)(u−t)E V dB V + 1−e−(cid:20)(u−t) : (A2) s s u s s t (cid:20) t t 24

Realizing that the (cid:12)rst term on the right-hand-side equals zero, and completing the outside integration operator now yields, Z t+(cid:1) (cid:18)Z u q (cid:19) (cid:26)(cid:27)(cid:18) (cid:16) (cid:17) (cid:26)(cid:27)(cid:18) V dB V du = e−(cid:20)(cid:1) +(cid:20)(cid:1)−1 = c : (A3) s s u (cid:20)2 (cid:20) (cid:1) t t Therefore, (cid:16) R (cid:17) COV R ; t+(cid:1)V du (cid:26)(cid:20) t;t+(cid:1) t u (cid:12) = VAR (cid:16)R t+(cid:1)V du (cid:17) = (cid:21) s + (cid:27) ; (A4) t u which is less than zero provided that 0 < (cid:21) < −(cid:26)(cid:20) and great than zero if 0 < −(cid:26)(cid:20) < (cid:21) . In s (cid:27) (cid:27) s addition, the intercept may be written as, ! Z t+(cid:1) (cid:26)(cid:20)(cid:18) (cid:11) = E(R )−(cid:12)E V du = (cid:22)(cid:1)+(cid:21) (cid:18)(cid:1)−(cid:12)(cid:18)(cid:1) = (cid:22)(cid:1)− (cid:1); (A5) t;t+(cid:1) u s (cid:27) t which is larger than (cid:22)(cid:1) under the usual parameter restrictions stipulated in the Proposition. The determination of the projection coe(cid:14)cients in the implied volatility-return regression proceed by analogous arguments. First, the variance term may be written as, " !# Z t+(cid:1) (cid:27)2(cid:18) VAR E(cid:3) V du = VAR(a(cid:3) V +b(cid:3) ) = a(cid:3) a(cid:3) : t t u (cid:1) t (cid:1) 2(cid:20) (cid:1) (cid:1) Similarly, for the covariance term h (cid:16) (cid:17)i R COV R ;E(cid:3) t+(cid:1)V du h t;t+(cid:1) R t t u R p i = COV (cid:22)(cid:1)+(cid:21) t+(cid:1)V du+ t+(cid:1) V dB ;a(cid:3) V +b(cid:3) hR s t u i t u(cid:16)R u p(cid:1) t (cid:1) (cid:17) = (cid:21) COV t+(cid:1)V du;a(cid:3) V +a(cid:3) COV t+(cid:1) V dB ;V ; s t u (cid:1) t (cid:1) t u u t where h (cid:16) (cid:17)i R R COV t+(cid:1)V du;E(cid:3) t+(cid:1)V du h(cid:16)R t u(cid:17) (cid:16)t R t u (cid:17)i (cid:16)R (cid:17) h (cid:16)R (cid:17)i = E t+(cid:1)V du E(cid:3) t+(cid:1)V du −E t+(cid:1)V du E E(cid:3) t+(cid:1)V du h t(cid:16) u (cid:17)t t(cid:16) u (cid:17)i th (cid:16)u t(cid:17)i t h (cid:16)u (cid:17)i R R R R = E E t+(cid:1)V du E(cid:3) t+(cid:1)V du −E E t+(cid:1)V du E E(cid:3) t+(cid:1)V du th t (cid:16)R u t(cid:17) t (cid:16)R u (cid:17)i t t u t t u = COV E t+(cid:1)V du ;E(cid:3) t+(cid:1)V du t t u t t u = (cid:27)2(cid:18)a a(cid:3) ; 2(cid:20) (cid:1) (cid:1) while the second term equals zero, (cid:16) R p (cid:17) h(cid:16) R p (cid:17) i h (cid:16) R p (cid:17) i COV t+(cid:1) V dB ;V = E t+(cid:1) V dB V = E E t+(cid:1) V dB V = 0: t u u t t u u t t t u u t Hence, hR (cid:16)R (cid:17)i COV t+(cid:1)V du;E(cid:3) t+(cid:1)V du (cid:27)2(cid:18)a a(cid:3) a (cid:12)(cid:3) = VA t R h E u (cid:3) (cid:16)R t+ t (cid:1)V t du (cid:17)i u = (cid:21) s(cid:27) 2 2 (cid:20) (cid:18)a (cid:1) (cid:3) a (cid:1) (cid:3) = (cid:21) sa (cid:1) (cid:3) ; (A6) t t u 2(cid:20) (cid:1) (cid:1) (cid:1) 25

which is less than (cid:21) but larger than zero, provided that (cid:21) < 0 and (cid:21) > 0. Also, s v s " !# Z t+(cid:1) a (cid:11)(cid:3) = E(R )−(cid:12)(cid:3)E E(cid:3) V du = (cid:22)(cid:1)+(cid:21) (cid:18)(cid:1)−(cid:21) (cid:1)(cid:18)(cid:1); (A7) t;t+(cid:1) t u s sa(cid:3) t (cid:1) which is larger than (cid:22)(cid:1) given (cid:21) < 0 and (cid:21) > 0. v s Finally, combining the results above, it follows readily that if 0 < (cid:21) < −(cid:26)(cid:20) we have s (cid:27) (cid:12) < 0 < (cid:12)(cid:3); (A8) while for 0 < −(cid:26)(cid:20) < (cid:21) < a(cid:3) (cid:1) (cid:26)(cid:20) we have (cid:27) s a −a(cid:3) (cid:27) (cid:1) (cid:1) 0 < (cid:12) < (cid:12)(cid:3): (A9) B Proof of Proposition 2 By de(cid:12)nition (cid:16)R (cid:17) COV t+(cid:1)V du;R t u t−(cid:1);t (cid:14) = : VAR(R ) t−(cid:1);t The denominator may be rewritten as, VAR(R ) (cid:16)t− R (cid:1);t (cid:17) (cid:16)R (cid:17) (cid:16)R R p (cid:17) = (cid:21)2VAR t V du +E t V du +(cid:21) COV t V du; t V dB s (cid:16) t−(cid:1) u (cid:17) t−(cid:1) u (cid:16) s t−(cid:1)(cid:17)u t−(cid:1) u u = (cid:21)2(cid:18)(cid:27)2 e−(cid:20)(cid:1) +(cid:20)(cid:1)−1 +(cid:18)(cid:1)+(cid:21) (cid:26)(cid:27)(cid:18) e−(cid:20)(cid:1) +(cid:20)(cid:1)−1 s (cid:20)3 (cid:16) (cid:17) s (cid:20)2 = (cid:18)(cid:1)+ (cid:21)s(cid:27)(cid:18) (cid:21)s(cid:27) +(cid:26) c ; (cid:20) (cid:20) (cid:1) where the second equality follows from the results in Andersen et al. (2002b), Bollerslev and Zhou (2002), and Proposition 1 above. The numerator may be expressed as, (cid:16)R (cid:17) COV t+(cid:1)V du;R t(cid:16) R u t R −(cid:1);t (cid:17) (cid:16) R R p (cid:17) = (cid:21) COV t+(cid:1)V du; t V du +COV t+(cid:1)V du; t V dB s t u t−(cid:1) u t u t−(cid:1) u u (cid:16) (cid:17) (cid:16) (cid:17) 2 2 = (cid:21) (cid:18)(cid:27)2 1−e−(cid:20)(cid:1) + (cid:26)(cid:27)(cid:18) 1−e−(cid:20)(cid:1) s2(cid:20)3 (cid:20)2 = (cid:21) (cid:18)(cid:27)2a2 +(cid:26)(cid:27)(cid:18)a2 ; s 2(cid:20) (cid:1) (cid:1) where the (cid:12)rst term uses the result from Andersen et al. (2002b), and the second term uses the result from the proof of Proposition 1, (cid:16)R R p (cid:17) (cid:16)R R p (cid:17) COV t+(cid:1)V du; t V dB = E t+(cid:1)V du t V dB t u t−(cid:1) u u h t (cid:16)R u t−(cid:1)(cid:17)R u pu i = E E t+(cid:1)V du t V dB (cid:16) t t R pu t−(cid:1)(cid:17) u u = E a V t V dB (cid:1) t t−(cid:1) u u = (cid:26)(cid:27)(cid:18)a2 : (cid:1) 26

Combining the two equations it follows therefore that (cid:21) (cid:18)(cid:27)2a2 +(cid:26)(cid:27)(cid:18)a2 (cid:14) = s 2(cid:20) (cid:1)(cid:16) (cid:1)(cid:17) : (A10) (cid:18)(cid:1)+ (cid:21)s(cid:27)(cid:18) (cid:21)s(cid:27) +(cid:26) c (cid:20) (cid:20) (cid:1) The intercept in the realized volatility asymmetry regression can be easily shown as γ = (cid:18)(cid:1)−(cid:14)((cid:22)(cid:1)+(cid:21) (cid:18)(cid:1)): (A11) s The coe(cid:14)cient of the implied volatility asymmetry is similarly de(cid:12)ned by, h (cid:16)R (cid:17) i COV E(cid:3) t+(cid:1)V du ;R (cid:14)(cid:3) = t t u t−(cid:1);t : VAR(R ) t−(cid:1);t The numerator may be rewritten as, h (cid:16)R (cid:17) i COV E(cid:3) t+(cid:1)V du ;R t t u t−(cid:1);t = COV [a(cid:3) V +b(cid:3) ;R ] (cid:1) t(cid:16) R(cid:1) t−(cid:1);t (cid:17) (cid:16) R p (cid:17) = a(cid:3) (cid:21) COV V ; t V du +a(cid:3) COV V ; t V dB (cid:1) s t t−(cid:16)(cid:1) u (cid:17) (cid:1) t t−(cid:1) u u = −a(cid:3) (cid:21) 0+a(cid:3) (cid:26)(cid:27)(cid:18) 1−e−(cid:20)(cid:1) (cid:1) s (cid:1) (cid:20) = (cid:26)(cid:27)(cid:18)a(cid:3) a ; (cid:1) (cid:1) (cid:16) (cid:17) where a(cid:3) = 1 1−e−(cid:20)(cid:3)(cid:1) and b(cid:3) = (cid:18)(cid:3)((cid:1)−a(cid:3) ), and the last line of the proof utilizes the (cid:1) (cid:20)(cid:3) (cid:1) (cid:1) results from the proof of Proposition 1 that (cid:16) R (cid:17) COV V ; t V du (cid:16) t t−(cid:1) (cid:17)u R = E V t V du −(cid:18)2(cid:1) (cid:16) t t− R (cid:1) u (cid:17) (cid:16) R R (cid:17) = E V t V du +E (cid:20)(cid:18) t du t V du (cid:16)t−(cid:1) R t−(cid:1) u R (cid:17)t−(cid:1) (cid:16) t− R (cid:1) up R (cid:17) +E −(cid:20) t V du t V du +E (cid:27) t V dW t V du −(cid:18)2(cid:1) t−(cid:1) u t−(cid:1) u h t−(cid:1) (cid:16)R u u t−(cid:17)(cid:1) u i = E[V (a V +b )]+(cid:20)(cid:18)2(cid:1)2 −(cid:20) VAR t V du +(cid:18)2(cid:1)2 t−(cid:16)(cid:1) (cid:1) t−(cid:1) (cid:1)(cid:17) t−(cid:1) u + (cid:27)2(cid:18) e−(cid:20)(cid:1) +(cid:20)(cid:1)−1 −(cid:18)2(cid:1) (cid:20)2 (cid:16) (cid:17) (cid:16) (cid:17) = (cid:18)2(cid:1)− (cid:18)(cid:27)2 e−(cid:20)(cid:1) +(cid:20)(cid:1)−1 + (cid:27)2(cid:18) e−(cid:20)(cid:1) +(cid:20)(cid:1)−1 −(cid:18)2(cid:1) (cid:20)2 (cid:20)2 = 0; and, (cid:16) R p (cid:17) COV V ; t V dB (cid:16) R t pt−(cid:1) u(cid:17) u = E V t V dB (cid:16)t t−(cid:1) (cid:17)u u = (cid:26)(cid:27)(cid:18) 1−e−(cid:20)(cid:1) (cid:20) = (cid:26)(cid:27)(cid:18)a : (cid:1) 27

Now combing the di(cid:11)erent equations it follows that (cid:26)(cid:27)(cid:18)a(cid:3) a (cid:14)(cid:3) = (cid:16)(cid:1) (cid:1) (cid:17) : (A12) (cid:18)(cid:1)+ (cid:21)s(cid:27)(cid:18) (cid:21)s(cid:27) +(cid:26) c (cid:20) (cid:20) (cid:1) The intercept in the implied volatility asymmetry regression can be easily shown as γ(cid:3) = (a(cid:3) (cid:18)+b(cid:3) )−(cid:14)(cid:3) ((cid:22)(cid:1)+(cid:21) (cid:18)(cid:1)): (A13) (cid:1) (cid:1) s Lastly, note that the common denominator of (cid:14) and (cid:14)(cid:3) (corresponding to a variance) is always positive. The requirement that (cid:14)(cid:3) < (cid:14), or (cid:18)(cid:27)2 (cid:26)(cid:27)(cid:18)a(cid:3) a < (cid:21) a2 +(cid:26)(cid:27)(cid:18)a2 ; (cid:1) (cid:1) s 2(cid:20) (cid:1) (cid:1) is equivalent to (cid:18)(cid:27)2 (cid:21) a2 > 0 > (cid:26)(cid:27)(cid:18)a (a(cid:3) −a ): s 2(cid:20) (cid:1) (cid:1) (cid:1) (cid:1) The assumption that (cid:21) < 0 ensures that a(cid:3) − a > 0. Thus, given the usual parameter v (cid:1) (cid:1) restrictions in the Proposition, the assumptions that (cid:21) > 0 and (cid:26) < 0 are su(cid:14)cient to s guarantee the ordering of the (cid:14)’s. C Proof of Proposition 3 From the proof of Proposition 1, hR (cid:16)R (cid:17)i (cid:16) (cid:17) COV t+(cid:1)V du;E(cid:3) t+(cid:1)V du (cid:27)2(cid:18)a a(cid:3) a 1−e−(cid:20)(cid:1) (cid:20)(cid:3) (cid:30) 1 = VA t R h E u (cid:3) (cid:16)R t+ t (cid:1)V t du (cid:17)i u = (cid:27) 2 2 (cid:20) (cid:18)a (cid:1) (cid:3) a (cid:1) (cid:3) = a (cid:1) (cid:3) = (1−e−(cid:20)(cid:3)(cid:1))(cid:20) < 1; t t u 2(cid:20) (cid:1) (cid:1) (cid:1) where the last inequality follows directly by the assumption that (cid:20)(cid:3) = (cid:20)+(cid:21) < (cid:20). Similarly, v the intercept may be evaluated as, ! " !# Z Z t+(cid:1) t+(cid:1) a a (cid:30) = E V du −(cid:30) E E(cid:3) V du = (cid:18)(cid:1)− (cid:1) (a(cid:3) (cid:18)+b(cid:3) ) = b − (cid:1)b(cid:3) ; 0 u 1 t u a(cid:3) (cid:1) (cid:1) (cid:1) a(cid:3) (cid:1) t t (cid:1) (cid:1) which can generally not be signed. 28

D Tables and Figures Table 1: Summary Statistics for Monthly Returns and Volatilities Return Standard Deviation Variance Implied Realized Implied Realized Mean 0.857 6.074 3.363 41.190 13.558 Std. Dev. 4.595 2.078 1.504 33.673 13.907 Skewness -1.213 1.634 1.476 3.967 2.916 Kurtosis 7.611 8.602 5.637 27.516 14.883 Minimum -24.543 2.835 1.366 8.036 1.866 5% Qntl. -6.634 3.465 1.733 12.003 3.004 25% Qntl. -1.914 4.615 2.261 21.300 5.113 50% Qntl. 1.232 5.828 2.881 33.970 8.301 75% Qntl. 3.863 7.137 4.093 50.934 16.751 95% Qntl. 7.074 9.724 6.287 94.565 39.523 Maximum 12.378 17.728 10.284 314.266 105.767 (cid:26) -0.011 0.773 0.738 0.642 0.616 1 (cid:26) -0.077 0.627 0.588 0.438 0.413 2 (cid:26) -0.034 0.523 0.519 0.321 0.343 3 (cid:26) -0.117 0.485 0.458 0.275 0.273 4 (cid:26) 0.065 0.483 0.487 0.272 0.319 5 (cid:26) 0.012 0.485 0.512 0.289 0.373 6 (cid:26) 0.066 0.460 0.472 0.251 0.307 7 (cid:26) -0.032 0.447 0.462 0.244 0.300 8 (cid:26) -0.028 0.441 0.457 0.243 0.293 9 (cid:26) 0.122 0.422 0.441 0.233 0.291 10 29

Table 2: Volatility Feedback E(cid:11)ect Standard Deviation Variance Implied Realized Expected Implied Realized Expected (cid:11)(cid:3) = 0.637 (cid:11) = 4.156 (cid:11)~ = 1.717 (cid:11)(cid:3) = 0.828 (cid:11) = 2.486 (cid:11)~ = 1.231 (0.932) (1.171) (1.838) (0.489) (0.552) (12.91) (cid:12)(cid:3) = 0.036 (cid:12)= -0.981 (cid:12)~ = -0.264 (cid:12)(cid:3) = 0.001 (cid:12) = -0.120 (cid:12)~ = -0.029 (0.170) (0.390) (0.594) (0.013) (0.047) (0.965) R2 = 0.000 R2 = 0.103 R2 = 0.047 R2 =0.000 R2 = 0.132 R2 = 0.057 Note: The "Expected" volatility regressions refer to the instrumental variables regressions using the absolute lagged returns as instruments for the realized volatilities. Table 3: Leverage E(cid:11)ect Short Regression Standard Deviation Variance Implied Realized Implied Realized γ(cid:3) = 6.225 γ = 3.438 γ(cid:3) = 44.087 γ = 14.317 (0.238) (0.168) (3.891) (1.471) (cid:14)(cid:3) = -0.170 (cid:14) = -0.081 (cid:14)(cid:3) = -3.305 (cid:14) = -0.824 (0.059) (0.028) (1.384) (0.283) R2 = 0.141 R2 = 0.062 R2 = 0.204 R2 = 0.074 Long Regression Standard Deviation Variance Implied Realized Implied Realized γ(cid:3) = 1.218 γ = 0.900 γ(cid:3) = 12.110 γ = 4.967 (0.279) (0.195) (2.494) (1.183) (cid:12)(cid:3) = 0.689 (cid:12) = 0.723 (cid:12)(cid:3) = 0.580 (cid:12) = 0.654 (0.059) (0.054) (0.067) (0.068) (cid:11)(cid:3) = 0.072 (cid:11) = 0.007 (cid:11)(cid:3) = 0.032 (cid:11) = -0.003 (0.037) (0.036) (0.052) (0.044) (cid:14)(cid:3) = -0.309 (cid:14) = -0.009 (cid:14)(cid:3) = -0.444 (cid:14) = 0.015 (0.033) (0.050) (0.062) (0.058) R2 = 0.844 R2 = 0.547 R2 = 0.884 R2 = 0.383 30

Table 4: Implied Volatility Forecasting Bias Standard Deviation Variance (cid:30) = 0.231 (cid:30) = 3.705 0 0 (0.392) (2.643) (cid:30) = 0.516 (cid:30) = 0.239 1 1 (0.069) (0.066) R2 = 0.507 R2 = 0.336 31

Monthly S&P 500 Return 10 5 0 −5 −10 1987 1990 1992 1995 1997 2000 Monthly S&P 100 Implied Volatility (Standard Deviation) 15 10 5 0 1987 1990 1992 1995 1997 2000 Monthly S&P 500 Realized Volatility (Standard Deviation) 15 10 5 0 1987 1990 1992 1995 1997 2000 Monthly S&P 100 Implied Volatility (Variance) 150 100 50 0 1987 1990 1992 1995 1997 2000 Monthly S&P 500 Realized Volatility (Variance) 150 100 50 0 1987 1990 1992 1995 1997 2000 Figure 1: Time Series Plot of Returns and Volatilities. 32