A behavioral explanation for the negative asymmetric return–volatility relation☆
Introduction
Empirical evidence shows a negative relation between realized daily and weekly market returns and volatility. More specifically, negative (positive) innovations to return are correlated with positive (negative) innovations to volatility, with a greater asymmetric effect when returns decline/volatility increases. Two documented theories attempt to explain this negative relation. Black (1976) postulates that negative shocks to returns increase financial leverage, making stocks riskier and therefore subsequently driving up volatility, labeled the leverage hypothesis. Poterba and Summers, 1986, Campbell and Hentschel, 1992 present the volatility feedback hypothesis, where any innovations to volatility (especially positive ones) lead to a decrease in returns. The leverage hypothesis has few supporters (see e.g. Low, 2004), while the volatility feedback hypothesis involves a complicated economic process that passes through expectations and dividends to validate the negative relation and only (weakly) explains the longer-term return–volatility relation. More recently, Low (2004) suggests that a behavioral explanation could be the cause of the asymmetric effect of losses being associated with larger volatility changes than are gains, but he does not relate his results to behavioral concepts and only examines the leverage effect to test the overall relation.
We investigate the relation between daily and intraday changes using the new CBOE Volatility Index (the VIX) and the returns on the S&P 500 index, as well as the corresponding Nasdaq volatility (VXN) and index return. We focus on the short-term dynamics of the return–volatility relation, contrary to the majority of past studies that employ weekly and monthly data on realized volatility to examine this relation. Our aim is to provide a detailed analysis of the short-term relation between market returns and implied volatility in order to identify the characteristics of the strong negative and asymmetric correlation between these variables.
We add to the literature by providing intraday results for the return–volatility relation, determining the factors affecting the relation, comparing five different forms of the model, and linking specific behavioral explanations with the observed daily and intraday results. In particular, we show that the negative and asymmetric association of return to changes in implied volatility is consistent with behavioral explanations of this phenomenon, while the leverage and volatility feedback models do not explain our results. We also examine: (1) return quintiles to show how implied volatility reactions are associated with the size of return innovations and (2) different measures of implied volatility to investigate the influence of the implied volatility skew, as well as determining the importance of realized volatility.
The empirical aspects of our study include four major differences from previous research. First, we use both the new VIX and the new VXN to measure implied volatility, with the new measures being better metrics of market expectations since they include the entire strike price range of implied volatilities. Second, we compare results using the VIX (VXN) to those of the near-the-money implied volatility, as well as including 5-min realized volatility as an independent variable. This allows us to disentangle the effects of the implied volatility skew from near-the-money implied volatility to examine the characteristics of the return–volatility relation and to distinguish the importance of implied volatility from current volatility. Third, we quantify the volatility response to the magnitude of return innovations, unlike other studies that only test for the presence of an asymmetric response. Fourth, in addition to using daily data, we investigate the relation at the intraday frequency using data sampled at 30-min and 5-min intervals, which allows us to solidify our behavioral explanation.
Our main empirical findings can be summarized along three dimensions. First, consistent with earlier studies, we find a significant negative and asymmetric correlation between innovations in return and (implied) volatility for stock indexes. However, by using regression models similar to those of Bollerslev and Zhou (2006), the results are consistent with behavioral explanations of the relation, but not the leverage or volatility feedback explanations. The results also show the superiority of employing the new VIX (VXN) to examine the return–volatility relation compared to either the near-the-money implied volatility or the contemporaneous realized volatility.
Our second contribution is a detailed analysis of the relation between return and implied volatility through time, as well as for quintiles of returns and their associated volatility innovations. We find that the individual years show a consistently strong relation over the different periods, unlike the sample inconsistency reported by others. Moreover, the quintiles of return results show that the strongest support for the negative and asymmetric relation is associated with the extreme changes in returns and volatility. The main implication of this finding is that “tail” events are important determinants of the return–volatility relation, which subsequently relates to the shape of the return distribution.
Third, by comparing the results of two implied volatility measures, the new VIX that employs all strike prices and the near-the-money implied volatility of the market, we show the importance of the implied volatility skew in explaining the return–volatility asymmetry. This supports the inferences of Dennis et al. (2006), who suggest that the magnitude of the asymmetry might be related to the slope of the implied volatility function, although they do not calculate the IVF.
Taken as a whole, our research shows that there is more to the return–volatility relation than suggested by the established hypotheses. In particular, we show the lack of support for established leverage and volatility feedback theories concerning this relation, while the results are consistent with behavioral explanations. In addition, we examine the characteristics of the relation using different models, across samples, for different measures of volatility, and for the sign and size of the return innovations.
Section snippets
The leverage and volatility feedback hypotheses
The negative relation between returns and volatility is widely documented in the literature. As pointed out by Bollerslev et al. (2007), most studies show a negative correlation between current return shocks and future volatility, with some studies illustrating that negative news is associated with a larger increase in volatility than positive news. The two popular theories associated with the negative return–volatility relation are the leverage hypothesis and the volatility feedback
Data and variable description
This analysis employs data from three sources. We obtain the daily values for the S&P 500 stock index, the Nasdaq 100 index, the new VIX and the new VXN from the Chicago Board Options Exchange (CBOE) Master Data Retrieval (MDR) file. The near-the-money implied volatilities on the S&P 500 options and the Nasdaq 100 options are provided by Historical Options Data. We obtain 5-min returns to calculate realized volatility and the 30- and 5-min VIX and return data for the intraday analysis of the
Methodology
Use of the VIX provides several advantages for examining the return–volatility relation compared to realized volatility. First, the VIX is based on market determined bid and ask option prices, which allows us to examine how traders and option dealers react to the return dynamics of the market. Second, use of the VIX avoids statistical estimation problems associated with realized measures of volatility.
Daily summary measures
Table 1 provides summary mean and standard deviation statistics for each variable for the daily data using the overall sample period (yearly results are available upon request). Over the sample period the change in the VIX is well behaved, with a mean of −0.005 and a standard deviation of 1.301. Both the mean of the VXN (31.037) and its standard deviation (14.977) are much larger than the corresponding VIX values, which is expected due to the greater volatility of the returns in the Nasdaq
Analysis using intraday data
Summary statistics for 5- and 30-min interval data on the VIX/VXN are omitted for space considerations. Both the means and standard deviations are stable over the sample period and are approximately equal to one-tenth of the equivalent daily values for the 5-min data. The correlation matrix for the intraday variables are consistent with the daily data, although the large sample size causes some correlations to be significant at very low correlation values. Surprisingly, a highly significantly
Return–volatility asymmetry
Panels A and B of Table 5 show the results of the five regression models for the VIX when returns are segregated into positive and negative daily changes (the VXN results are similar in nature but possess lower R2s and lower significance values for the variables than the VIX results). Comparison of the Panel A and Panel B R2 values shows that a better fit is achieved in the negative return partition for both markets. The coefficients for the contemporaneous return and some of the lagged return
Summary and conclusions
This paper takes a different approach to investigating the negative asymmetric risk–return relation by using the new VIX implied volatility measure, comparing different models of implied volatility, analyzing intraday and Nasdaq results, and examining the effect of quintile rankings of returns. Our results imply that the leverage hypothesis and the volatility feedback hypothesis are not the primary explanations of the return–implied volatility relation. We propose a behavioral approach that is
Acknowledgements
We thank Naresh Bansal for extensive comments on an earlier draft and participants of the 2007 Eastern Finance and Southwestern Finance meetings for their suggestions. We also thank Hersh Shefrin and John Nofsinger for their ideas concerning the behavioral finance elements used in this paper and Giorgio Szego, editor, for his help and support. We are indebted to two anonymous referees for their valuable comments.
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This paper was reviewed and accepted while Prof. Giorgio Szego was the Managing Editor of The Journal of Banking and Finance and by the past Editorial Board.