Leverage effect, feedback effect and volatility-of-volatility risk premium 31

To further explore the mechanism that volatility-of-volatility risk affects asset prices, we investigate whether the volatility-of-volatility risk contributes to the feedback effect.

To identify the timely volatility-of-volatility shocks, at the end of each day, we estimate the regression of equation (1.33) using daily stock returns over the past 22 days. We then sort stocks into quintile portfolios on the estimated _, for each day and calculate the event-time daily value-weighted portfolio returns ranging from -11 to 11 in days.

If market volatility-of-volatility is priced, an anticipated increase in market volatility-of-volatility raises the required rate of return, implying an immediate stock price decline and higher future returns. As shown in Figure 1. 2, consistent with the channel of feedback effect, stocks with negative return sensitivities to market volatility-of-volatility have lower returns before the portfolio formation and earn higher post-formation returns than stocks with positive return sensitivities.

We construct a portfolio that is long the lowest quintile and short the highest quintile and we denote the portfolio as low-minus-high. The low-minus-high portfolio has, by construction, large negative exposure to innovations in market volatility-of-volatility. The theory of the leverage effect and the feedback effect predict an asymmetric

cross-32

correlation between the aggregate volatility and the pre-formation and the post-formation low-minus-high returns.

As can be seen in the top panel of Figure 1. 3, the pre-formation low-minus-high returns are negatively correlated with VIX measured at the portfolio formation date while the correlations between VIX and the post-formation low-minus-high returns are positive, supporting the leverage effect and the feedback effect associated with the aggregate volatility.

Moreover, our theory for the leverage effect and the feedback effect similarly predicts an asymmetric cross-correlation between market volatility-of-volatility and the pre-formation and the post-formation low-minus-high returns. As can be seen in the bottom panel of Figure 1. 3, the low-minus-high return is negatively correlated with VoV at the portfolio formation date while the correlation between VoV and one-day post-formation low-minus-high return is positive. The market volatility-of-volatility carries a negative contemporaneous correlation of -0.232, which is much larger in magnitude than -0.057 for the contemporaneous leverage effect associated with market volatility. The correlation between one-day post-formation low-minus-high return and market volatility-of-volatility is 0.100, which is larger than 0.060 for the correlation between the return and the market volatility. The stronger asymmetric cross-correlation, despite less persistent, supports the leverage effect and the feedback effect associated with market volatility-of-volatility. Therefore, market volatility-of-volatility seems to be the state variable that determines the time-varying risk premium.

1.5.6 Robustness to volatility spreads

In this section, we check whether our results are robust to existing well-known volatility spreads that affect cross-sectional stock returns. We construct the

implied-33

realized volatility spread (IVOL-TVOL), which is, as described in Bali and Hovakimian (2009), defined as the average of implied volatilities by at-the-money call and put minus the total volatility calculated using daily returns in the previous month; the call-put implied volatility spread (CIVOL-PIVOL), which is, as described in Bali and Hovakimian (2009) and Yan (2011), defined as the the-money call implied volatility minus the at-the-money put implied volatility; the expected individual variance risk premium (EVRP), which is, as described in the data section and in Han and Zhou (2012), defined as the difference between the model-free implied variance and the five-minute realized variance.

Since we extract the volatility data from OptionMetrics Volatility Surface file as Yan (2011) do, we choose the 30-day maturity put and call options with deltas equal to -0.5 and 0.5, respectively.

Panel A of Table 1.8 shows the performance of portfolios sorted on each of the volatility spreads as well as on _, using stocks with available equity options. The Fama and French four factor alpha of the ‘5-1’long–short portfolio is 0.63 percent with a t-statistic of 1.76 for IVOL-TVOL quintile portfolios, 1.66 percent with a t-statistic of 6.70 for CIVOL-PIVOL quintile portfolios, 0.96 percent with a t-statistic of -2.21 for EVRP quintile portfolios, and -0.85 percent with a t-statistic of -2.38 for _, quintile portfolios. Hence, our results for market volatility-of-volatility risk remain significant in the options market and consistent with the literature, all of the three volatility variables carry significant premium in the cross-section.

We construct two-way sorted portfolios formed on intersection of each of the volatility spread quintile portfolios and _, quintile portfolios. Panel B shows the results for the _, quintile portfolios controlling for volatility spread quintile portfolios. The Fama and French four factor alpha of the ‘5-1’ long–short portfolio

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remains significant controlling for these three variables, i.e. at -0.76 percent with a t-statistic of -2.38 controlling for IVOL-TVOL, at -0.74 percent with a t-t-statistic of -2.29 controlling for CIVOL-PIVOL, and at -0.66 percent with a t-statistic of -2.20 controlling for EVRP. Hence, the low returns to high , stocks are not driven by the existing well-known volatility spreads.

As shown by Yan (2011), CIVOL-PIVOL is proxy for a disaster type jump risk that affects the cross-sectional stock returns. Our empirical finding that the pricing of _, is robust to CIVOL-PIVOL provides indirect evidence that the market volatility-of-volatility risk cannot be completely explained by a peso-problem like jump risk.