# Stock Returns, Implied Volatility Innovations, and the Asymmetric Volatility Phenomenon

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JOURNAL OF FINANCIAL AND OUANTITATIVE ANALYSIS VOL. 41, NO. 2, JUNE 2006 COPYRIGHT 2006, SCHOOL OF BUSINESS ADMINISTRATION, UNIVERSITY OF WASHINGTON, SEATTLE, WA 98195 Stock Returns, Implied Volatility Innovations, and the Asymmetric Volatility Phenomenon Patrick Dennis, Stewart Mayhew, and Chris Stivers* Abstract We study the dynamic relation between daily stock retums and daily innovations in option- derived implied volatilities. By simultaneously analyzing innovations in index- and firm- level implied volatilities, we distinguish between innovations in systematic and idiosyn- cratic volatility in an effort to better understand the asymmetric volatility phenomenon. Our results indicate that the relation between stock retums and innovations in systematic volatility (idiosyncratic volatility) is substantially negative (near zero). These results sug- gest that asymmetric volatility is primarily attributed to systematic market-wide factors rather than aggregated firm-level effects. We also present evidence that supports our as- sumption that innovations in implied volatility are good proxies for innovations in expected stocic volatility. I. Introduction Understanding the relation between stock returns and innovations in ex- pected volatility is a fundamental issue in understanding financial markets. This relation has long been studied by financial economists (see, e.g., Cox and Ross (1976), Black (1976), and Christie (1982)), and it is of practical importance for areas such as risk management, option pricing, and event studies. Asymmetric volatility is a well documented empirical regularity in this area. The asymmetric volatility phenomenon (AVP) refers to the stylized fact that neg- ative retum shocks tend to imply higher future volatility than do positive retum shocks of the same magnitude (see, e.g., Wu (2001), Bekaert and Wu (2000)). The AVP has also been described as a negative correlation between stock retums * Dennis, pjd9v@virginia.edu, Mclntire School of Commerce, University of Virginia, Char- lottesville, VA 22904; Mayhew, mayhews@sec,gov. Office of Economic Analysis, U.S, Securities and Exchange Commission, Washington, DC 20549; and Stivers, cstivers@terry,uga.edu. Department of Banking and Finance, Terry College of Business, University of Georgia, Athens, GA 30602. We thank Don Chance, Bob Connolly, Jennifer Conrad, Rob Engle, Bin Gao, Bill Lastrapes, Lars Norden, Guojun Wu (the referee), and seminar participants at the 2003 European Finance Association confer- ence, the University of Georgia, and the Federal Reserve Bank of Atlanta for helpful comments. The U.S. Securities and Exchange Commission disclaims responsibility for any private publication or statement of any SEC employee or Commissioner. This study expresses the authors' views and does not necessarily reflect those of the Commission, the Commissioners, or other members of the staff. Stivers is also a Visiting Scholar at the Federal Reserve Bank of Atlanta. The views expressed in this article do not necessarily reflect the position of the Federal Reserve Bank of Atlanta or Ihe Federal Reserve System. 381

382 Journal of Financial and Quantitative Anaiysis and innovations in expected volatility.' The AVP literature has debated whether the AVP may be more attributable to firm-level effects such as leverage (a nega- tive return shock causes increased volatility) or systematic market-wide influences such as volatility feedback (an increase in expected market-level volatility causes a negative return).^ In this paper, we study the time series of stock returns and option-derived implied volatilities for the S&P 100 index and 50 large U.S. firms. By simultane- ously analyzing daily innovations in both index- and firm-level implied volatili- ties, we distinguish between innovations in systematic and idiosyncratic volatility. This decomposition allows us to contribute to the asymmetric volatility literature by distinguishing between systematic market-wide explanations and firm-level explanations for the phenomenon. Our findings also bear on other related lit- erature where the relation between stock returns and volatility innovations is an important consideration. Our empirical investigation relies on the following intuition. Just as a finn's stock return can be decomposed into systematic and idiosyncratic components, innovations to a stock's volatility may be attributed to changes in systematic or idiosyncratic volatility. Systematic volatility shocks may result from macro events such as an interest rate shock or international financial crisis, while idiosyncratic volatility shocks may originate from firm-specific events such as product introduc- tions and patent events. If the AVP is attributed more to systematic, market-level factors, then the negative relation between firm-level stock returns and volatility innovations should be primarily evident in the relation between the stock returns and market-level volatility innovations. In contrast, if the AVP is attributed more to firm-level effects, then the negative relation between firm-level stock returns and volatility innovations should be similar for both market-level volatility inno- vations and idiosyncratic volatility innovations. For the implied volatility in our empirical work, we use a standardized im- plied volatility (IV) that may be interpreted as an average IV from at-the-money call and put options with one month to expiration. Specifically, our standardized IV is a weighted average of eight different IVs, where our weighting method fol- lows the Chicago Board Options Exchange's procedure for calculating their im- plied volatility index (VIX) over our sample period (see Whaley (1993)). Use of this standardized IV both reduces measurement errors and mitigates the concern that daily IV changes are linked to variations in the option's moneyness and time to expiration. Our empirical approach is to assume that the daily change in our standardized IV is an observable proxy for innovations in the expected volatility of stock returns. We present new evidence that supports this assumption. 'The strong form of the AVP, as termed by Bekaert and Wu (2000), suggests a simple negative correlation between stock returns and volatility innovations where negative (positive) return shocks imply an increase (decrease) in expected volatility. By contrast, the weak form of the AVP suggests a negative correlation between stock returns and innovations in expected volatility, after controlling for the relation between the absolute return shock and the volatility innovation, ^By referring to the leverage effect as a firm-level effect, we do not mean that changes in leverage could not induce changes in expected market-level volatility. For example, a decrease in expected future market-wide cash flows (or an increase in systematic risk aversion) is likely to devalue eq- uity more than corporate debt and thus increase market-level leverage, which could induce a higher expected market-level volatility. However, the leverage effect applies for each firm in isolation, re- gardless of whether changes in leverage are related across firms.

Dennis, Mayhew, and Stivers 383 Our empirical contributions may be broken down into the following five ar- eas, discussed in order of prominence for our study. First, we present new ev- idence about the dynamic relation between stock retums and innovations in ex- pected volatility, as proxied for by the daily change in IV. Wefindthat the relation between index retums and index-level volatility innovations is substantially more negative (with a correlation of —0.679) than the relation between individual stock retums and the respective firm-level volatility innovation (with a median correla- tion of —O.t65). Further, the negative relation between individual stock retums and index-level volatility innovations (with a median correlation of —0.339) is notably stronger than the negative relation between individual stock retums and their respective firm-level volatility innovation. Second, we use a market-model variance decomposition to obtain an im- plied idiosyncratic volatility for each firm. We find that the relation between the individual stock retums and their respective innovation in implied idiosyncratic volatility is only marginally negative (with a median correlation of -0.046). This relation is not statistically different than zero for 37 of the 50 firms. These first two empirical contributions suggest that the AVP is more related to systematic market-wide influences, rather than an aggregation of firm-level effects. Third, we present new evidence supporting the assumption that the daily IV changes are good proxies for the innovation in expected volatility. Many studies have evaluated the infonnation in the IV level for the future volatility of realized stock retums and recent findings indicate that the IV level impounds nearly all information about future realized volatility. However, the critical issue for our study is whether the daily changes in IV contain reliable incremental information about future retum volatility beyond the previous IV level. We are unaware of any other studies that have evaluated the incremental volatility information in the daily IV change. Controlling for the IV level from period f - 2, we find that the IV innovation over period t - I contains reliable incremental information for stock volatility in period t. Further, we find that daily firm-level IV innovations contain more information about the respective firm-level volatility innovation than does the daily index-level IV innovation. These findings support the assumption that daily IV changes are a good proxy for the daily innovation in expected volatility. Fourth, we reexamine the AVP in the traditional way by evaluating the re- lation between conditional volatility and lagged retums shocks in a time-series model. Our new wrinkle is to distinguish between firm- and market-level re- tum shocks when examining the AVP at the firm level. This exploration is moti- vated by our prior findings, which suggest that the asymmetry in the relation be- tween conditional firm-level volatility and lagged retum shocks will be greater for lagged market-level retum shocks than for lagged firm-level retum shocks. Con- sistent with earlier evidence (see, e.g., Andersen, Bollerslev, Diebold, and Ebens (2001)), we find that the AVP is stronger for index retums than for firm-level stock returns. Consistent with our first two empirical contributions discussed above, we also find that AVP behavior in firm-level conditional volatility is stronger and more reliable when relating firm-level conditional volatility to lagged market- level retum shocks rather than to lagged own-firm retum shocks. Fifth, studying the relation between stock retums and volatility innovations may also provide insight into other related areas of the literature, such as the IV

384 Journai of Financiai and Quantitative Analysis smile, the bias in IV, and return skewness. Accordingly, we perfonn additional data analysis related to these topics and then discuss what our collective findings suggest for these areas. One key finding is that "index versus firm" differences in the relation between stock returns and volatility innovations are consistent with "index versus firm" differences in the slope ofthe IV smiles. However, even when controlling for the movements in IV suggested by last period's IV smile and the day's stock price change, the relation between stock returns and the residual IV innovations remains reliably and sizably negative. Overall, our additional analysis supports conclusions in other studies that indicate there are important differences between individual stocks and indices in the areas of option pricing and stock re- turn behavior (see, e.g., Bakshi and Kapadia (2003a), Bollen and Whaley (2004), and Andersen et al. (2001)). The remainder of this study is organized as follows. Section II describes our data and variable construction. Section III presents our primary findings on the dynamic relation between stock returns and innovations in expected volatility. Section IV presents new supporting evidence on the infonnation content in IV with respect to the future volatility of spot stock returns. Section V reexamines the AVP in the traditional approach that focuses only on stock returns and presents evidence that complements our primary findings in Section III, Finally, Section VI discusses other analysis and potential implications of our findings, and Section VII concludes. II. Data and Variable Construction A. Stock Returns and Standardized Implied Volatility We analyze 50 individual stocks with options traded on the Chicago Board Options Exchange (CBOE). Daily dividend-adjusted returns were obtained from CRSP for the period January 4, 1988, through December 31, 1995, The sample contains the 50 firms that had the highest total option trading volume over the period 1988 to 1995 and that met the following additional screens. First, the stock had to have the complete set of daily returns from CRSP and bave the same ticker over our sample period. Second, each stock had to have adequate option data so that we could construct our standardized IV for at least 1,800 of the 2,022 daily periods.^ We also examine two different stock indices in our empirical work. The primary index is the S&P 100 since it is the underlying asset for the index options that we examine. Total returns for the S&P 100 are obtained from Datastream International, For comparison, we also analyze returns of the value-weighted NYSE/AMEX/NASDAQ stock index from CRSP in order to evaluate a different. ^We choose an option-volume screen because of evidence in Mayhew and Stivers (2003) that the quality of volatility infonnation in IV degrades appreciably for firms with thinly traded options. The Berkeley Options Data base lists 150 firms that have option volume listed in every year of our sample. The average trading volume for the largest 10 option-volume firms is about nine times the average for option-volume firms 41-50, 33 times the average for option-volume firms 91-100, and 323 times the average for option-volume firms 141-150, Thus, we feel that 50 firms is a good compromise between sample size (number of firms) and information quality in the data. The additional screens for return availability and standardized IV availability removed only eight ofthe top 58 option-volume firms.

Dennis, Mayhev*/, and Stivers 385 broader tnarkef index. In our empirical work, when we refer to the market-level stock retum, we mean this broad CRSP index. For the index IV in our empirical work, we use the Volatility Index (VIX) from the CBOE. This index represents the IV of an at-the-money option on the S&P 100 index with 22 trading days (30 calendar days) to expiration. It is con- structed by taking a weighted average of the IVs for eight options, including a call and a put at the two strike prices closest to the money and the nearest two expirations (excluding options within one week of expiration). Each of the eight component IVs is calculated from the bid-ask quote midpoint using a binomial tree that accounts for early exercise and dividends. Note that each of the eight component IVs is a single observation of the traditional IV that features the as- sumptions of the Black-Scholes framework, except that the binomial model must be used to allow for early exercise. This procedure was designed to reduce noise and mitigate measurement errors (see details in Whaley (1993) and Fleming, Ost- diek, and Whaley (1995)). For each firm, we constmct daily standardized IVs using eight individual stock options, where our weighting method follows the CBOE's procedure for calculating VIX over our sample period. We believe this standardized IV is a good choice for our empirical work for the following reasons. First, when model- ing retum volatility, this type of IV has been shown to subsume nearly all of the information from the historical time series of retums, at least for widely traded options (see, e.g., Christensen and Prabhala (1998), Fleming (1998), Blair, Poon, and Taylor (2001), and Mayhew and Stivers (2003)). Second, we are interested in the relation between daily retums and daily changes in IV in the time series. Errors in absolute pricing with the Black-Scholes framework should be less im- portant for analysis that relies on IV innovations rather than the IV level. In other words, we are not assuming that the Black-Scholes framework gives the correct option price, but that it is a good enough model to empirically control for changes in moneyness, the riskless interest rate, and time to expiration so that we can use market prices of options to investigate changes in the volatility implied from op- tion prices. Third, our procedure focuses on near at-the-money options, which are the most widely traded and typically yield IVs that are relatively less biased. Focusing on near at-the-money options also serves to standardize the daily IV observations since the IV may vary with the moneyness of the options (the well- known IV smile). Fourth, this standardization procedure mitigates measurement errors and avoids IV biases that may occur if one only examines calls (or puts). Fifth, we present additional evidence in Section IV that supports our assumption that the daily changes in our standardized IV are good proxies for the daily inno- vation in the expected volatility of spot stock retums. Finally, our use of this IV follows from many previous studies."* We use option price data from the Berkeley Options Data Base, realized dividends from CRSP, and T-bill rates from Datastream. For each stock on every trading day, we calculate the eight designated IVs using midpoints of the final "•Other option pricing models could conceivably be used to back out an alternate tV. For example, the Heston (1993) option pricing model incorporates stochastic volatility. However, the Heston model requires the estimation of seven unobserved parameters to specify the dynamics of the underlying asset and this model was not in existence for over half of our sample.

386 Journal of Financial and Quantitative Analysis quote of the day, matched with a contemporaneous observation of the underlying stock price. The eight estimates are then aggregated using the VIX weighting procedure. For a few days, one or more of the eight options was unavailable or else a reported option price was below the lower arbitrage bound. We treat these days as missing observations for our standardized IV. ^ In our subsequent empirical testing, we treat the missing standardized IVs as follows. First, for the testing that requires an IV level, we sitnply use the previous day's IV estimate. Using stale IV data in this case will create a small bias that suggests weaker explanatory power for the IV when trying to explain subsequent spot volatility. Second, when calculating correlations between daily stock retums and standardized IV changes (Table 2), we throw out the observation whenever there is a missing standardized IV observation that prevents us from calculating a daily IV change. Third, for the maximum likelihood estimation in Tables 3 through 8, we fill in missing values with the prior day's value of IV. For these models, this means the "daily IV change" is set equal to zero for days that originally had missing IV values. This treatment means that our analysis will tend to slightly understate the comovement between retums and IV changes for these models.^ Table 1 reports summary statistics for our sample of 50 firms including the stocks' market capitalization (size), volatility of spot stock retums, IV level, daily IV variability, option trading volume, and the number of available standardized IV observations per firm. On average, over 92% of the days have valid observa- tions for the standardized IV. We analyze the days with missing standardized IVs and do not find any apparent clustering where missing values tend to occur on the same day across firms, so we do not believe there is any systematic relation in the missing values. We also note that the number of standardized IV observations varies little with firm size. This suggests that any differences in our results re- lated to firm size are unlikely to be related to variation in the number of available standardized IV observations. Table 1 also reports summary statistics for the largest and smallest size-based quintiies of the 50 firms, based on the finns' average size over our sample period. First, note that the median size of the largest 10 firms is about 20 times that of the smallest 10 firms. Second, note that the retum standard deviation, the IV level, and the daily variability in IV are all appreciably larger for the smaller firms as compared to the larger firms. Third, note that the median option trading volume ^This procedure of averaging the call and put IVs when calculating the standardized IV reduces the estimation error for two reasons. First, to the extent that the noise due to microstructure is independent across options, it can be reduced by averaging multiple observations. Second, errors induced by using an incorrect dividend or interest rate will bias call and put estimates of IV in opposite directions. So, averaging call and put IVs mitigates such errors. For these reasons and because we felt it was more conservative, we only calculated a standardized IV when all eight individual IVs were available. *An alternate way to handle missing standardized IVs is to interpolate and assume that a missing observation of the standardized IV is equal to the average of the preceding period's value and the subsequent period's value. With this approximation, we could analyze the data set as if there are no missing values. Using this alternate interpolation method to handle missing values, we re-estimate our primary results in Section III and find essentially identical results (the correlations with this alter- nate approach are within 0.01 of the summary correlations reported in Table 2). Since both methods for handling missing values give the same answer, we present our tabular results using the method described in the main text since this method seems more conservative.

Dennis, Mayhevi/, and Stivers 387 TABLE 1 Sample Description Table 1 reports descriptive statistics for our sample of 50 large firms: GE, AT&T, IBM, Wal-Mart, Coca-Cola, Merck, Bristol Myers, GM, Johnson & Johnson, Mobil, Amoco, Pepsico, Ford, Bell Atlantic, 3M, Hewiett-Paci

388 Journal of Financial and Quantitative Anaiysis the implied idiosyncratic return variance to a standard deviation and then calcu- late the daily change in the implied standard deviation of the idiosyncratic return component {AlV\'^i°). This procedure results in a few firm-days where the estimate of implied id- iosyncratic variance is negative. Across the 50 firms, the median (mean) number of days for a firm where the estimate of the implied idiosyncratic variance is neg- ative is only two (25), Seventeen (32) of the 50 firms have no days (five days or less) where the estimate of the implied idiosyncratic variance is negative. Neg- ative values are implausible and we cannot convert the variance into a standard deviation. Accordingly, we set the implied idiosyncratic variance to zero for these few firm-days where its value is negative,^ The estimates of the implied idiosyncratic volatility seem reasonable. The average (median) value of the implied idiosyncratic volatility across the 50 firms is 1,45% (1,41%) per day. We also compute the ratio of a firm's implied idiosyn- cratic variance divided by its total implied variance for each firm for each day. Across the 50 firms, the median (average) of this ratio is 0,755 (0,736), which is consistent with the R^ obtained from estimating the market model, equation (1), and also indicates that the substantial majority of a firm's volatility is idiosyn- cratic. III. The Dynamic Relation between Stock Returns and Expected Volatility Innovations Here, we evaluate the dynamic relation between spot stock returns and inno- vations in expected volatility. In this section, we assume that the daily change in our standardized IV is a good proxy for the innovation in expected return volatility, an assumption that is supported by findings in Section IV, We ana- lyze innovations in expected index-level volatility, firm-level total volatility, and firm-level idiosyncratic volatility. Since the AVP implies a negative correlation between stock returns and IV innovations, our analysis should have implications for understanding the nature of the AVP, 'This simple method to estimate the daily change in a firm's implied idiosyncratic volatility is open to other criticisms. First, it assumes a constant beta over the entire sample period. For robust- ness evaluation, we have recomputed the time series of the daily changes in idiosyncratic volatility using time-varying betas, calculated from rolling regressions on the preceding 500 daily observations. Across the 50 firms, the average correlation (median correlation) between the "daily change in id- iosyncratic volatility using time-varying betas" and the "daily change in idiosyncratic volatility using a fixed beta" is 0,988 (0,992), Thus, over our sample, the dynamic behavior of the daily change in idiosyncratic volatility is nearly identical, whether one uses a constant beta or time-varying beta. Sec- ond, since we need both the market VIX and the firm's IV to estimate a day's implied idiosyncratic variance, the method is subject to noise in both these implied volatilities. Further, to calculate the daily change in implied idiosyncratic volatility, we need both the VIX and firm IV in consecutive trading days. Since we have some missing values in our daily time series of our standardized IV, this means the average number of observations for the daily change in idiosyncratic volatility is limited to 1,744 (out of 2,022 possible days) across the 50 firms. Despite these criticisms, we feel that this measure of idiosyncratic volatility is useful because of the clear intuition of the measure, the consistency of the results across our sample of 50 firms, the consistency of the results across subperiods, and the results in Section IV that indicate the "change in our implied idiosyncratic volatility" contains reliable incremental information for a firm's future idiosyncratic return volatility.

Dennis, Mayhev^/, and Stivers 389 A. Correlation Analysis We begin by computing the simple correlations between tbe daily stock re- tums and innovations in expected volatility. Results are reported in Table 2. There are a number of notable findings. First, we find that the correlation between the S&P 100 index retum (CRSP index retum) and tbe index volatility innovation is negative and large at -0.679 (-0.663). Poteshman (2000), Benzoni (2002), and Pan (2002) use different methods to estimate this correlation, yet each of these studies comes up with correlation estimates tbat are close to our point estimate. ^ TABLE 2 Correlations between Stock Returns and Implied Volatiiity Innovations Table 2 reports correlations between the daily time series of spot stock returns and the daily changes in our standardized implied volatility. We report on our sample of 50 large U.S. stocks, the S&P 100 index, and the CRSP vaiue-weighted index over the 1988 to 1995 period. For the individuai stocks, we report means and medians for each correiation for the group of stocks in column one. In the tabie, R;, is the daiiy stock return, / i l V , , is the daiiy change in the respective f stocks stanaaraizea iv, zivix, is tne daily change in the CBOEs VIX, and AWy" IS the daily change in a firm's implied idiosyncratic volatility based on a market-model variance decomposition. Correlation of «;,,,4IV,,, R,.,,ziVIX ANi i.AVKt Ali 50 firms Mean: -0.163 -0.330 -0.047 0.172 Median : -0.165 -0.339 -0,046 0.156 Largest 10 firms Mean: -0.291 -0.421 -0.078 0.298 Median : -0.280 -0.417 -0.049 0.317 Smallest 10 firms Mean: -0.042 -0.208 0.002 0.107 Median : -0.040 -0.270 0.002 0.123 S&P 100 n/a -0.679 n/a n/a CRSP Index n/a -0.663 n/a n/a Second, in contrast to tbe sizable negative correlation for tbe index, the av- erage (median) correlation between individual stock retums and the respective own-firm volatility innovation is only -0.163 (-0.165). Thus, the correlation at the index level is over four times tbe average value for the individual stocks. Third, we find that the mean (median) correlation between the individual stock retums and the index volatility innovation is —0.330 (-0.339) across tbe 50 stocks. Tbe correlation between the individual stock retums and tbe index volatility innovation is more negative than the correlation between tbe individ- ual stock retums and the respective own-firm volatility innovation for 46 of the 50 firms. In our view, these results in Table 2 suggest tbat the AVP is more re- lated to systematic, market-level factors rather tban firm-level influences because: i) tbe correlation between index retums and index volatility innovations is much ^Poteshman (2000) uses S&P 500 option prices over June 1988 to August 1997 and estimates the correlation to be —0.61 by minimizing the sum of squared option pricing errors. Benzoni (2002) uses a two-stage approach where he first estimates the structural parameters of the dynamic process from daily S&P 500 index prices by using a simulated method of moments estimation. In the second stage, he uses the estimated price dynamics to examine S&P 500 index options and estimates the risk adjustment necessary for option pricing. His estimate of the correlation between the spot price incre- ment and the variance increment is -0.58. Finally, Pan (2002) uses an "implied-state" generalized method of moments estimation that uses spot prices and option prices jointly. She examines weekly observations over the January 1989 to December 1996 period and estimates a correlation of -0.53 for the case that allows a risk premia for stochastic volatility but no jump dynamics.

390 Journal of Financial and Quantitative Analysis more negative than the correlation between firm retums and own-firm volatility innovations, and ii) the correlation between firm retums and index volatility inno- vations is appreciably more negative than the correlation between firm retums and own-firm volatility innovations. Though our results do not allow us to pinpoint the market-level factors behind the AVP, some plausible explanations might be the volatility feedback effect (Bekaert and Wu (2000)) or herding on the part of traders (Avramov, Chordia, and Goyal (2006)). Fourth, the notion that the AVP is primarily related to market-level volatility innovations suggests that the correlation between the individual stock retums and innovations in their expected idiosyncratic volatility should be near zero. We calculate the correlation between the individual stock retums and the chatige in their respective implied idiosyncratic volatility (calculated per Section II.B). As reported in column 3 of Table 2, the mean (mediati) of these correlations is near zero at only -0.047 (-0.046). Finally, as reported in the final columti in Table 2, note that the correlations between the firm IV innovations and the VIX innovations are modest. This is consistent with the fact that most of a firm's volatility is firm specific and miti- gates any multicollinearity concems about the firm IV innovations and the VIX innovations. B. Multivariate Analysis Next, we investigate the dynamic relation between stock returns and ex- pected volatility innovations in a multivariate framework that allows for condi- tional heteroskedasticity in the retum residuals. We point out that our specifica- tions here are not meant to imply economic causality between stock retums and innovations in the standardized IVs. Rather, the specifications are meant to ex- amine the reliability of the statistical relation between stock retums and volatility innovations, which is important for practical applications such as risk manage- ment, option pricing, and event studies. To begin with, we investigate the relation between stock returns and volatility innovations in a specification that includes own-firm IV innovations and index IV innovations simultaneously. We estimate the following system, (3) Ri,, - V (4) hi,, = V _ , , , l where /?,,, is the daily retum of individual stock or stock index /, e,^ is the retum residual, /J, , is the conditional variance of e,,,, IV?, is the standardized implied variance of stock i (VIX? for the S&P 100 and CRSP index), ZilV,,, is the daily change in the individual firm's implied volatility, AYIX, is the daily change in the CBOE's VIX, Dr^_| is a dummy variable that equals one if £,-,,_i is nega- tive, and the ^s are coefficients to be estimated for each stock or index. The implied volatility variables are in "annualized standard deviation" units for the mean equation and are in "daily implied variance" units for the variance equation. The coefficients of interest are tpi and ip2 in (3). We choose the variance equation based oti results in May hew and Stivers (2003).

Dennis, Mayhew, and Stivers 391 For this system and all our subsequent specifications that specify both a con- ditional mean and conditional variance, the system is estimated simultaneously by maximum likelihood estimation using the conditional normal density. Inference about estimated coefficients for our maximum likelihood estimation in this paper is based on robust quasi-maximum likelihood standard errors, in accordance with Bollerstev and Wooldridge (1992). Table 3, Panel A reports the firm-level results. We find that the estimated coefficients on the ZiVIX, term (i/'2) are negative and statistically significant at a 1% /?-value for 48 of the 50 firms. The relation between the firm retums and the own-firm IV innovations also tends to be negative, but the relation is less reliable. For 28 of the 50 firms, the estimated coefficients on the /ilV,,, term (?/)i) are negative and statistically significant at a 1% p-value. For 48 of the 50 finns, the V'2 estimates are more reliably negative than the ipi estimates. In Table 3, Panel A we also report the R^ from an ordinary least-squares (OLS) estimation of different variations of the mean equation (3). For the OLS estimation with both ZiVIX, and ZilV,,, as explanatory variables, the average of the R^s is 13.5% across the 50 firms. The sizable R^ also suggests a substantial TABLE 3 The Dynamic Relation between Stock Returns and Implied Volatiiity Innovations Table 3 reports on estimating variations of the following model on each individual stock and index. where R,^, is the daily return of individual stook or index /, zilV, , is the daiiy change in the individual fi.rrri's IV, ZiVIX, is the daily change in the CBOE's VIX (^IV terms are all in "anriualized standard deviation" units), ^ I V ) * is the daily change in the individual firm's implied idiosyncratic voiatiiity based on a market-modei variance decomposition, e, , is the residual, h , , is the conditional variance of E, ,, IV?, is the standardized daiiy impiied variance of stock / (Vixf for'the S&P 100 and CRSP index), and the i/;s are estimated coefficients for each stock or index. For each size-based stock grouping in column one, we report the mean and median of the coefficients of interest. The rightmost columns report fl^s from OLS estimations of the mean equation with the noted restrictions. Panel A reports firm-level results using only the 4VIX, and ANj, explanatory terms in the mean equation (i/.g = 0). Panel B reports firm-level results using only the AN'y'l' explanatory term in the mean equation (t/;,, i/'2 = 0). Panel C reports stock index results. For Panels A and B, the numbers in brackets indicate the number cf firms where the respective coeffioient is positive/negative and signifioant at a 1% p-level, (-statistics are reported in parentheses for Panel C. The sample period is 1988 through 1995. Panel A. Firm-Levet Results with VtX Innovations and Firm-Levet tV Innovations (xpg = 0) V-i i'2 -/-a ^ 6 - 0 V'i .i>B=o Mean/Median Mean/Median Mean Mean fvlean 1988 to 1995 Period All 50 firms -0.067/-0.066 -0.444/-0.447 0.135 0.034 0.119 [2/28] [1/48] Largest 10 firms -0.121/-0.102 -0.464 / -0.455 0.208 0.077 0.178 [0/9] [0/10] Smallest 10 firms -0.010/0.002 -0.336 / -0.449 0.071 0.007 0.065 [1/2] [1/8] 1988 to 1991 Subperiod All 50 firms -0.052 / -0.040 -0.456/-0.463 0.189 0.041 0.175 [1/18] [0/48] 1992 to 1995 Subperiod All 50 firms -0.108/-0.101 -0.419/-0.394 0.078 0.032 0.055 [0/29] [1/48] (continued on next page)

392 Journal of Financial and Quantitative Analysis TABLE 3 (continued) The Dynamic Relation between Stock Returns and Implied Volatility Innovations Panel B. Firm-Level Results with Innovations in the Firm's Implied Idiosyncratic Volatility Only (ip-f,ip2 = '^) Mean R^ Mean/Median "At. 1*2 = 0 1988 to 1995 Period All 50 firms -0.021 / -0.013 0.0066 [2/13] Largest 10 firms - 0 . 0 3 5 / - 0 013 0.0019 [0/2] Smallesf 10 firms 0.001/0.006 0.0110 (1/2J 198810 1991 Subperiod All 50 firms -0.002/0.003 0.0064 [2/5] 1992 to 1995 Subperiod All 50 firms -0.082/-0.059 0.0182 [0/24] Panel C. Stock Index Results CVi, i>6 -= 0) Index i>2 1988 to 1995 Period S&P 100 -0.512 0.469 (-32.57) CRSP -0.414 0.438 (-35.37) 1988 to 1991 Subperiod S&P 100 -0.515 0.493 (-25.10) CRSP -0.400 0.466 (-22.14) 1991 to 1995 Subperiod S&P 100 -0.499 0.361 (-18.33) CRSP -0.431 0.369 (-20.40) negative relation between stock retums and volatility innovations. Next, in a vari- ation of (3) where the own-firm IV innovation is the only explanatory variable, the average R^ across the 50 firms is only 3.4%. In contrast, in a variation of (3) where the VIX innovation is the only explanatory variable, the average R ^ across the 50 firms is over three times as large at 11.9%. Thus, both in terms of statistical significance of the coefficients and of ^^ evaluation, the relation between the firm retums and index volatility innovations is stronger than the relation between the firm retums and own-firm volatility innovations. We also estimate the model over the 1988-1991 and 1992-1995 subperiods and report summary results in Table 3. The subperiod results are qualitatively similar to the overall sample results. Next, we isolate the relation between individual stock retums and the inno- vations in their respective implied idiosyncratic volatility by estimating the fol- lowing system, (5) Ri,, = Vo + V-e^IV;.*"-!-£,•,„ (6) hi,, = ^3 + ^4lVf_,_ I + Vse

Dennis, Mayhew, and Stivers 393 where AYVfl" is the daily change in the individual firm's implied idiosyncratic volatility (based on the market-model variance decomposition described in Sec- tion II.B) and the other terms are as defined for (3) and (4). Table 3, Panel B, reports the results. We find that the estimated coefficients on the Aiyfj° term (V'e) are negative and significant for only 13 of the 50 firms witb a modest mean (median) value of -0.021 (—0.013). Further, the average R ^ for an OLS estimation of (5) is only 0.66%, which is quite small compared to the R^ values in Panel A. Subperiod results are consistent. Finally, in Table 3, Panel C, we report index-level results for the S&P 100 and CRSP indices. We find a very reliable negative relation between the index retum and the index volatility innovation. The R^s are very substantial at 46.9% for the S&P 100 and 43.8% for the CRSP index, as compared to the firm-level results. Again, subperiod results are consistent. Thus, the evidence in Table 3 reinforces the findings in Table 2, again indicating that the negative relation between stock retums and expected volatility innovations is primarily related to tbe market-wide component of expected volatility. In Table 3, we do not report the estimated coefficients for the variance equa- tions for brevity. Tbe coefficients are consistent with findings in Mayhew and Stivers (2003). The ip4 estimates on the IV?,_| term are positive and significant ata 1% p-value for all 50 firms and both indices. TbelV?,_j term captures nearly all of the volatility information.' C. Cross-Sectional Variation in Results We do not attempt a formal analysis of cross-sectional variation in our re- sults because we have only 50 individual firms in our sample, because we employ reduced form empirical models, and because we have no obvious theoretical mo- tivation. Further, since options are widely traded for only a modest proportion of publicly traded firms, a broad cross-sectional analysis is not possible. However, as we noted in Section II, there £ire appreciable differences across the size quintiles of firms in terms of the stock's spot retum volatility, the IV level, the daily variability in IV, and the option trading volume. These size-related variations motivate our choice to differentiate our results across size quintiles in each table. For all five quintiles, the correlation between stock returns and index IV in- novations is appreciably more negative than the correlation between stock retums and own-firm IV innovations. This consistency is important when considering what our evidence suggests about the AVP. We do note a few apparent size-related differences. We note that the negative relation between stock retums and IV innovations in Tables 2 and 3 is stronger for the larger firms, both in the relation between stock retums and own-firm volatility innovations and in the relation between stock retums and index volatility inno- vations. The relation between stock retums and own-firm volatility innovations 'The estimated tps coefficients on the lagged retum residuals are only positive and significant for about half the firms (26 firms at a 5% p-value), and the estimated tp^ coefficients that allow for the sign asymmetry are only positive and significant for three firms. For the indices, only i/)5 is positive and significant and only for the CRSP index (at a 0.04 p-value).

394 Journal of Financial and Quantitative Analysis is especially prominently weaker for the smallest quintile of firms, which may be due to poorer quality infonnation in IV for the smaller firms with appreciably lower option trading volume. Consistent with this conjecture, we also find in later analysis that the volatility infonnation from the IV innovation appears somewhat weaker for the smallest quintile of firms (see Table 4). We perform additional cross-sectional analysis regarding the relation be- tween VIX innovations and stock retums. Specifically, we evaluate the correlation between daily VIX innovations and stock retums of 10 size-based stock portfolios from CRSP (formed from NYSE/AMEX stocks). We find that the con-elations be- tween the daily VIX innovations and the size-based portfolio retums are reliably negative for all 10 size-based decile portfolios. The magnitude of the correlation declines monotonically with size, with correlations of -0.671, -0.601, -0.567, -0.533, -0.520, -0.507, -0.470, -0.450, -0.361, and -0.238 for decile-10 (the largest firms) through decile-1 (the smallest firms). This additional evidence suggests that the sizable negative correlations between stock retums and VIX in- novations extend beyond large firms (such as the large firms in our sample). IV. Daily Changes in Implied Volatility and the Future Volatility of Spot Stock Returns Our findings in Section III have clear implications for understanding the AVP, provided that the daily IV change is a good proxy for the daily innova- tion in expected stock retum volatility. In this section, we present new evidence that supports this key assumption. The fundamental question is whether the daily IV innovation over period t - 1 (ZiIV,-,,_ i) contains reliable incremental informa- tion for the conditional volatility of period t, beyond the information contained in the IV level at; - 2. If daily changes in individual stock IVs reflect primarily measurement error and noise, then these IV changes may contain relatively little incremental information about future stock retum volatility. For example, results in Bollen and Whaley (2004) suggest that supply and demand imbalances may have a material influence on day to day IV changes. If so, it is not clear that daily IV changes will contain substantial incremental infonnation about future stock volatility. Thus, from a practical perspective, the signal-to-noise ratio in daily IV innovations is an empirical issue. Another concem is that the daily VIX innovation might contain more in- cremental information about future firm-level volatility than does the respective firm-level IV innovation, perhaps due to a higher signal-to-noise ratio in the more widely traded index options. If so, this could help explain why we find larger comovements between individual stock retums and index IV innovations than be- tween individual stock retums and own-firm IV innovations. First, we investigate the incremental volatility information in the lagged own- firm IV innovations by estimating the following system, (7) Ri,i = 70 (8) k, = where /?,,, is the daily retum of individual stock /, /?M,I is the market-level stock retum, e,^, is the retum residual, /i,-,, is the conditional variance of e,-,,, IV?,_2 is

Dennis, Mayhew, and Stivers 395 the implied daily variance of stock i at the end of period t-2, AYV ?,_, is the one- day change in the stock option's implied variance from the end of period / - 2 to the end of period t - I, and the 7s are estimated coefficients. '^ The coefficient of interest is 75 on the ZiIV?,_, term. We report results in Table 4. We find that the /iivf,_, term provides reliable incremental infonnation about future stock volatility. For 37 of the 50 firms, the estimated coefficient on the Z\IV^,_, term (75) is positive and statistically signif- icant at a 5% level or better. Further, the mean (median) 75 is 0.766 (0.746), as compared to a mean (median) 74 of 0.912 (0.895) for the IVf_2 term, which sug- gests a high signal-to-noise ratio in the AW f,_, term. These results are consistent across the size quintiles in Table 4, although the information in the IV innovation for the larger firms (with higher option trading volume) appears to be of somewhat higher quality. TABLE 4 Conditional Volatility Information from the Daily IV Change for Individual Stocks Table 4 reports on Ihe information content of the daily IV change for the conditional volatility of individuai stock returns. We estimate the foilowing model, ''/,( = 70 + 7iW|,(-l hi,, = 73+74

396 Journal of Financial and Quantitative Anaiysis where AVlX^_^ is the one-day change in the implied variance of the S&P 100 index from the end of period f - 2 to the end of period t — 1 and the other terms are as defined for (7) and (8), Here, the coefficient of interest is 76 on the Z\VIXf_, term. The results are reported in Table 5, We find that the daily VIX innovation provides very reliable incremental information about future volatility for both the S&P 100 and CRSP stock index. At the firm level, the daily VIX innovation also tends to provide incremental volatility information. Across the 50 firms, the mean (median) ofthe 76 estimates is 0,866 (0,714), which is close to the 76 estimated for the S&P 100 return. The 76 estimates are positive (negative) and statistically significant for 14 (1) of the 50 firms at the 5% level. For the individual stocks, the number of 76 estimates that are statistically significant seems modest, which likely refiects the high proportion of idiosyncratic volatility in firm returns and our use of the sizable quasi-maximum likelihood robust standard errors. TABLE 5 Conditional Volatility Information from the Daily Index-Level IV Change Table 5 reports on the information content of the daily VIX change for the conditional voiatiiity of individual stock returns and index returns. We estimate the foilowing model, "/,( = 70 + T l W i . l - i hij = 73 + 74ivf,,_ where /?,•, is stock or index /'s daily return, Rf^j is the market-level stock return, e, , is the return residuai, h, , is the conditional variance of e, ,, iV?, is the daily impiied variance for stock / (ViX^ fcr the indices), 4VIX^_, is the one-day change in the implied variance of the S&P 100 index from the end of period ( - 2 to the end of period (— 1, and the 7S are coefficients to be estimated for each stock or index. The 72 *erm is omitted for the indices. For each size-based grouping of individual stocks in coiumn one, we report the mean and median coefficients. The numbers in brackets indicate the number of firms where the respective coefficient is positive/negative and significant at a 5% p-vaiue. For the stock index results, /-statistics are reported in parentheses. The sample period is 1988 through 1995. 7) 72 74 76 Lk. Function Value All 50 firms Mean: 0.001 0.054 0.867 0.866 7265.11 Median; 0.001 0.049 0.856 0.714 [5/5] [15/6] [50/0] [14/1] Largest 10 firms Mean: 0.013 -0.073 0.900 0.903 7614.51 Median: 0.028 -0.075 0.875 0.939 [0/0] [1/4] [10/0] [5/0] Smallest 10 firms Mean: -0.015 0.166 0.823 1.757 6864.20 Median: -0.025 0.180 0.796 1.377 [1/31 [5/0] [10/0] [4/1] S&P 100 0.004 0.769 0.817 (0,18) (9.19) (2.92) CRSP 0.122 0.461 0.550 (5.17) (7.61) (2.94) For 49 of the 50 firms, we find that the likelihood function value for the system of equations (7) and (9) is less than the comparable system of (7) and (8), which indicates there is more incremental volatility information from own-firm IV changes than from VIX changes. Overall, the results in Tables 4 and 5 support our use of the daily IV innovation as a proxy for the change in the expected return volatility at both the firm and market level. Finally, our interpretation ofthe implied idiosyncratic volatility from Section III requires that this measure be informative about the conditional volatility of the

Dennis, Mayhew, and Stivers 397 idiosyncratic component of firm-level stock retums. We evaluate this issue with the following model, (10) /?,,, = 4>o (11) l ^ ° =

398 Journal ot Financial and Quantitative Analysis the own-firm (index) retum shocks. Instead of estimating a traditional asymmetric GARCH model (such as in Glosten et al. (1993)), we use the IV from period t — 2to control for volatility information from t — 2 and older. This specification allows us to isolate the AVP behavior related to the period / — 1 retum shock and to incorporate empirical evidence that the volatility information in IV largely subsumes volatility infonnation from older retum shocks. A. Asymmetric Volatility with Lagged Own-Stock Return Shocks We estimate the following system to evaluate the traditional univariate AVP in a given firm's (or index's) retums, (12) Ri^i = 9o +diRi,i-]+02RM,I-\ + £i,i, (13) hi,, = 9^+64l%_2 + e,el,_,+e(,D-,_,el,_,, where /?,-,, is the daily retum of individual stock or index i, R^, is the market-level retum, e,^, is the retum residual,ft,,,is the conditional variance of e,,,, IV?,_2 is the implied daily variance of stock / (or VIX^ for the indices) at the end of period t — 2, D~i_^ is a dummy variable that equals one if £,-,(_i is negative and zero otherwise, and the ^s are coefficients to be estimated. The 62 term is omitted for the indices. The primary coefficient of interest is de since it allows for the volatility asymmetry. Table 6, Panel A reports results for the indices. We find that the AVP is reli- ably evident as indicated by the size and statistical significance of the 6^ estimates. The relation between the conditional variance and the positive retum shocks is -0.067 (-0.063) for the CRSP (S&P 100) index. Conversely, the relation be- tween the conditional v£iriance and the negative market retum shocks (^5 + ^6) is 0.117 (0.045) for the CRSP (S&P 100) index. Note that this is the strong form of AVP behavior for the index retums, where positive (negative) retum shocks imply a lower (higher) future volatility. Subperiod results are consistent. Table 6, Panel B reports the comparable results for the individual stock re- tums. For the 50 individual stocks, the AVP related to the lagged own-firm's retum shocks is much weaker than that for the indices. The mean (median) 9 (, es- timate across the 50 firms is only 0.033 (0.024) and the 9(, estimates are positive and statistically significant for only seven of the 50 firms. '^ Further, in contrast to the strong form AVP in the index retums, both positive and negative retum shocks are positively related to future volatility for the individual stock retums, with neg- ative retum shocks implying only a marginally higher volatility. There is little variation in this AVP across the different size quintiies, except for the smallest quintile where the AVP is essentially non-existent. '^Also note that the significance level denoted for Tables 6 and 7 is lessened to 10% (rather than the 1% and 5% in Tables 3 through 5). We make this choice because the statistical relations are weaker for these return-based AVP models. Thus, we feel this significance notation is more informative since it qualifies more individual coefficients.

Dennis, Mayhew, and Stivers 399 TABLE 6 Asymmetric Volatility Behavior with Lagged Own-Stock Return Shocks Table 6 reexamines the asymmetric volatility phenomenon in daily stock returns at the index and firm level. We report on the following model, where the focus is on the relation between the return shook in period ( — 1 and the conditional voiatiiity for period t, where R,-, is the daily return of individual stock or index i, RM,t-^ is the iagged market-level return, E, , is the return residual, h,_, is the conditional variance of e ; , , IV? j is the daily irhplied variance of stock / (or ViX^ for the in'dices), D~|_ ^ is a dummy variable that equals one if £/,f_i is negative and zero otherwise, and the (?s are coefficients to be estimated for eaoh stock or index. The 62 term is omitted for the indices. Panel A reports stock index results, and Panel i3 reports on firm-level results. For the S&P 100 and CRSP index returns, (-statistios are in parentfieses. For Panel B, for each size-based grouping cf individual stocks in coiumn one, we report the mean and median for the coeffioients of interest. The numbers in braol

400 Journal of Financial and Quantitative Analysis TABLE 7 Asymmetric Volatility Behavior with Lagged Market-Level Return Shocks Table 7 reports on the following asymmetric volatility model for daily firm-level stock returns where the conditional variance is a function of lagged market-level return shocks. where fl, , is the daily return for stock (, RM,:-: is the lagged market-level return, e,, is the return residual, h, , is the conditional variance of £,_,, IV?, is the daily' implied variance of stock /, D^ , _ , is a dummy variable that equals one if RM,!-\ is negative and zero'otherwise, and the 9s are estimated coefficients for each stock. For each size-based grouping of individual stocks in column one, we report the mean and median for the coefficients of interest. The numbers in brackets report the number of firms where the estimated coefficient is positive/negative and statistically significant at a 10% p-value or better. The sample period is 1988 through 1995. fvlean/Median fvlean/Median Mean/Median 1988 to 1995 Period All 50 firms 0.833/0.822 0.124/0.011 0.188/0.178 [50/0] [2/8] [20/0] Largest 10 firms 0.862/0.824 -0.082/-0.101 0.259 / 0.284 [10/0] [0/3] [9/0] Smallest 10 firms 0.795/0.801 0.259/0.149 0.078/0.075 [10/0] [1/11 [2/0] 198S to 1991 Subperiod All 50 firms 0.852/0.885 0.233/0.005 0.068/0.159 [49/0] [3/6] [17/0] 1992 to 199S Subperiod All 50 firms 0.685 / 0.683 -0.011/-0.066 0.409/0.167 [39/0] [2/11] 18/1] By contrast, the lagged positive return shocks have little relation to future firm-level volatility. The median of the 9^ estimates is only 0,011, and the esti- mates display little statistical significance, Subperiod results are consistent; espe- cially comparing the median of the Og estimates for the overall period (at 0,178) and the two subperiods (at 0,159 and 0,167, respectively). In regard to cross-sectional variation, we note that the largest quintile of firms has noticeably stronger asymmetric volatility behavior with respect to the lagged market return shocks in the Table 7 model. On the other hand, there is little apparent variation across firm size in the univariate AVP in Table 6, If the AVP is more of a systematic market-level phenomenon, this result seems consistent with the observation that these large firm returns also have a more negative relation with the index volatility innovations (as reported in Section III), To conclude, our examination of the asymmetric relation between condi- tional volatility and lagged return shocks yields two observations. First, asym- metric volatility is more reliably evident in the index returns than in the firm-level returns, which is consistent with prior findings that have contrasted index versus firm behavior (see, e,g,, Andersen et al, (2001)), Second, we contribute with a new finding which indicates that asymmetric volatility is more sizable and reli- ably evident in firm-level volatility when relating a finn's conditional volatility to the lagged market-level return shock (rather than the own-firm return shock). Both observations above seem consistent with our primary findings in Section III,

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