Implied volatility sentiment: a tale of two tails

Abstract

We propose a sentiment measure jointly derived from out-of-the-money index puts and single stock calls: implied volatility (IV-) sentiment. In contrast to implied correlations, our measure uses information from the tails of the risk-neutral densities from these two markets rather than across their entire moneyness structures. We find that IV-sentiment measure adds value over and above traditional factors in predicting the equity risk premium out-of-sample. Forecasting results are superior when constrained ensemble models are used vis-à-vis unregularized machine learning techniques. In a mean-reversion strategy, our IV-sentiment measure delivers economically significant results, with limited exposure to a set of cross-sectional equity factors, including Fama and French's five factors, the momentum factor and the low-volatility factor, and seems valuable in preventing momentum crashes. Our novel measure reflects overweight of tail events, which we interpret as a behavioral bias. However, we cannot rule out a risk-compensation rationale.

Keywords:

• Sentiment
• Implied volatility
• Equity-risk premium
• Reversals
• Predictability
• Machine learning

JEL classification:

• G12
• G14
• G17

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

† It is important to disentangle the (equity) hedging behavior of institutional investor to their overall trading activity. Studies, such as Frijns et al. (2018), provide some evidence that institutional investors price stocks rationally, supporting the idea that the argued behavioral bias might be confined to institutional investors' portfolio insurance decisions.

† Reversals in the context of this paper are not to be confused with the, so-called, reversal (cross-sectional) strategy, i.e. a strategy that buys (sells) stocks with low (high) total returns over the past month. We focus on the overall equity market, rather than investigating single stocks.

‡ The literature on the link between the skew and the overall stock market is still incipient. Doran et al. (2007) test IV skews as a predictor of aggregate market returns, but they study one-day ahead returns to skews, and ignore any longer effects. Other studies on the conditionality of forward equity market returns to other volatility-type of measures are: Ang and Liu (2007) for realized variance, Bliss and Panigirtzoglou (2004) for risk-aversion implied by risk-neutral probability distribution function embedded in cross-sections of options, Bollerslev et al. (2009) for variance risk premium, Driessen et al. (2013) for option-implied correlations, Pollet and Wilson (2008) for historical correlations, for implied volatility indices Rubbaniy et al. (2014) and Vilkov and Xiao (2013) for the risk-neutral tail loss measure. Most of these studies document a short-term negative relation between risk measures and equity market movements.

§ We acknowledge that the implied correlation and the correlation risk premia measures of Driessen et al. (2013) and Buss et al. (2017) are also jointly extracted from the index and single stock option markets. Nevertheless, the implied correlation is calculated using the entire cross-section of strikes, whereas our measure focuses on OTM options, i.e. the tails of the implied distribution.

† We thank Barclays Capital for providing the implied volatility data. Barclays Capital disclosure: ‘Any analysis that utilizes any data of Barclays, including all opinions and/or hypotheses therein, is solely the opinion of the author and not of Barclays. Barclays has not sponsored, approved or otherwise been involved in the making or preparation of this Report, nor in any analysis or conclusions presented herein. Any use of any data of Barclays used herein is pursuant to a license.’

† We also test a percentile normalization and find results qualitatively similar to the use of Z-scores.

‡ A IC (or dispersion trading) strategy buys (sells) index options and sells (buys) single stock options, while delta hedging, to arbitrage price differences in these two volatility markets.

§ Strategy return series used are, respectively, the CBOE S&P 500 BuyWrite Index, the CBOE Investable Correlation Index, the S&P 500 index, CBOE put writing index, the CBOE 110-95 collar, the DB G10 FX carry index, the JPMorgan Equity Momentum index, and the Credit Suisse Managed Futures index.

¶ As our IV-sentiment measure requires less (cross-sectional) IV data than the Delta minus Gamma spread to be calculated, we can extend our sample, from March 19, 2013, until December 4, 2015.

∥ Results provided by tables 1–3 are all based on the three-month option maturity. The choice made for the three-month maturity is due to the higher robustness suggested by the IV-sentiment-based strategies reported in figure 1. Results for the six-month maturity are qualitatively the same as the three-month maturity ones. For the twelve-month maturity results for active strategies are poorer as reported by figure 1, though, results for table 3 for this maturity are similar to the ones for shorter maturities.

† These results are shown in figure 6 contained in the online Appendix C.1, available at https://github.com/luizfelix/IV-Sentiment.

† Welch and Goyal (2008) monthly data was updated until December 2014 and is available at http://www.hec.unil.ch/agoyal/.

‡ These variables are: the dividend price ratio, the dividend yield, the earnings-price ratio, the dividend-payout ratio, the book-to-market ratio, the net equity issuance, the Treasury bill rate, the long-term yield, the long-term return, the term spread, the default yield spread, the default return spread, the inflation rate, and the stock variance.

† Rapach et al. (2010) classify their ensemble methods in two classes: the first class uses a mean, median, and trimmed mean approach. The second class uses a discounted mean square prediction error method, which combines weights as a function of the historical forecasting performance of the individual models during the out-of-sample period. This method weights more recent forecasts heavier than older ones by the use of one additional parameter. Despite the desirable features of such a second class combination method, we prefer to stick to the first class methods because they are more transparent and do not require the choice of an additional parameter.

† The undesirable graphical pattern of $R_{OS}^2$ is caused by the normalization through $\sum _{k=q_{0+1}(r_{m+k}-{\overline{r}}_{m+l}{)}^2}^q$, which at the start of the sample tends to be very small relative to $CSSE{D}_{OS}$. Note that $R_{OS}^2=CSSE{D}_{OS}/\sum _{k=q_{0+1}(r_{m+k}-{\overline{r}}_{m+l}{)}^2}^q$.

‡ A full correlation matrix among the individual predictive factors tested by Rapach et al. (2010) and IV-sentiment factors can be provided upon request.

§ The ‘kitchen sink’ includes all 14 predictive variables used in our univariate models.

¶ We tune Ridge regression by using cross-validation with 10 folds. We tune our Random forest model using a single pass of out-of-bag errors to estimation of the optimal number of predictors sampled for spliting at each node. We use cross-validation in the estimation of our Deep Neural Networks model to come up with the number of layers and neurons (among a set of pre-defined structures) only. We do not apply any early-stopping procedure. A detailed description of these models and tuning procedures is out of scope of this paper. For specifics on these models, see Hastie et al. (2008)

† The Fama and French factors SMB, HML, RMW and CMA stand, respectively, for small minus big (size), high minus low (valuation), robust minus weak (profitability), and conservative minus aggressive (investments).

‡ The regressions that include the BAB factor have monthly frequency as this factor is not available in a daily frequency.

† Note that as $\sum _{i=1}^n{w}_i^2{\sigma }_i^2$ is always positive, the approximation provided by equation (7(7) $\overline{\rho }\approx \frac{{\sigma }_I^2}{(\sum _{i=1}^n{w}_i{\sigma }_i{)}^2},$(7) ) always overstates the true implied correlation. Given that $\sum _{i=1}^n{w}_i^2{\sigma }_i^2$ is typically small, the current analysis remains valid. See Appendix A.2 for details.

† This assumption implies that investors are somewhat rational, which is not inconsistent with the CPT-assumption that the representative agent is less than fully rational. The CPT suggests that investors are biased, not that decision makers are utterly irrational to the point that their subjective density forecast should not correspond, on average, to the realized return distribution.

† Available at http://people.stern.nyu.edu/jwurgler/.

‡ The complete set of variables provided by Welch and Goyal (2008) that is employed here is discussed in Appendix B. To avoid multicollinearity in our regression, we exclude all variables that correlate more than 40 percent with others.

§ This sample is only possible because Welch and Goyal (2008) and Baker and Wurgler (2007) have updated and made available their datasets after publication.

† Our results suggest that overweight of small probabilities is much less pronounced at the twelve-month options, at least for single stock options (see table 5), and that the Delta minus Gamma spread is disconnected to sentiment (SENT) at this same maturity. Beyond that, IV-sentiment using twelve-month options is not reliable as an active management signal (see figure 1), likely by not properly capturing sentiment but rather reflecting risk-neutral pricing. That said, we believe our twelve-month metrics might still be very useful from a risk management perspective, as the consideration of risks in the long-run is also important.

‡ While these relations are widely acknowledged, Longstaff (1995) provide a formal theorem for the link between IV skew and risk-neutral moments, whereas Bakshi et al. (2003) offer a comprehensive empirical test of this proposition for index options.

† The regression results reported here use RND kurtosis and skewness from index options (m = io). The results when RND is extracted from single stock options (m = sso) are unreported but qualitatively the same as the coefficient signs are equal to the reported ones, and regressions' explanatory power are roughly in the same range.

‡ Complete analytics comparing IV-sentiment with IVSentSingle and IVSentIndex strategies are reported in table 8 available in the online Appendix C.2, available at https://github.com/luizfelix/IV-Sentiment.

† Full results for equations 6c(6c) $\begin{array}{rl}IVSen{t}_d& ={\alpha }_d+(Mkt-RF{)}_d+SM{B}_d+HM{L}_d\\ & +WM{L}_d+RM{W}_d+CM{A}_d+{ϵ}_d,\end{array}$(6c) and 6d(6d) $\begin{array}{rl}IVSen{t}_m& ={\alpha }_m+(Mkt-RF{)}_m+SM{B}_m+HM{L}_m\\ & +WM{L}_m+RM{W}_m+CM{A}_m+BA{B}_m+{ϵ}_m,\end{array}$(6d) with the usage of IVSentSingle and IVSentIndex as additional explanatory variables are reported in table 9 of the online Appendix C.2, available at https://github.com/luizfelix/IV-Sentiment.

† Full results for this Section are provided by tables 10 and 11 reported in the online Appendix C.3, available at https://github.com/luizfelix/IV-Sentiment.

‡ Data use in Buss et al. (2017) is kindly provided by Grigory Vilkov at http://www.vilkov.net/codedata.html.

§ The estimated correlation matrix is shown in figure 7, which is reported in the online Appendix C.4 (available at https://github.com/luizfelix/IV-Sentiment).

¶ Complete results of the estimation of equation (27(27) $\begin{array}{rl}I{C}_t\left[\tau \right]-IVSen{t}_t\left[\tau \right]& ={\alpha }_t\left[\tau \right]+SKE{W}_t^m\left(\tau \right)\\ & +KUR{T}_t^m\left(\tau \right)+{ϵ}_t.\end{array}$(27) ) are reported in table 12 contained in the online Appendix C.4, available at https://github.com/luizfelix/IV-Sentiment.

† A drawback of the CPT model is that it allows for non-strictly increasing functions, which would not allow invertibility. This is the reason why the newer literature on probability distortions functions favors other strictly monotonic functions, such as Prelec's (1998) $w\left(p\right)=e^{-(-\ln (p){)}^{\delta }}$, as the weighting functions. Nevertheless, because the CPT parameters of our interest ($\gamma =0.61$; $\delta =0.69$) impose strict monotonicity, we can obtain the inverse of the probability function, $w^{-1}\left(p\right)$ numerically.

† Available at http://www.hec.unil.ch/agoyal/.