Continuous Sorting Procedure

The initial double-independent sorts require the ETFs to be ascendingly arranged in two sorts, one for the put-call ratio and one for the IV spread. Each of these sorts is divided into five categories, thus resulting in quintiles. Theoretically, intersecting these quintiles every day should give 5 × 5 = 25 portfolios that contain all the possible combinations of put-call ratio quintiles and IV spread quintiles. However, this is not the case in practice since on some days there is no intersection between certain quintiles of the two variables which should result in a portfolio. This means that in time series, some portfolios will have fewer observations than intended. Originally, this problem must come from having only 109 ETF observations per day which does not seem enough to cover all the quintile intersection possibilities. For this reason, the categorical process of sorting used so far creates some missing instances for certain portfolios on certain days. I consider this as an unnecessary noise to the double-independent sorting method, as outlined above.

Table 9

Continuous Double Sorts on Put-Call Ratio and Implied Volatility Spread

Excess Return P/C IVS

Quintile portfolio

1 (High P/C and low IVS) 0.033 81.463 −2.309

2 0.048 54.775 −0.934

3 0.049 28.350 −0.857

4 0.045 8.399 −0.593

5 (Low P/C and high IVS) 0.062 3.958 1.686

5-1 0.029^**

t-stat. 2.2

Note. In this table, the next-day ETF excess returns are independently sorted into quintiles based on a continuous sorting procedure on the put-call ratio (P/C) and on the IV spread (IVS) simultaneously, generating five equal-weighted portfolios on each day. The portfolios representing the intersection of the quintiles report the daily average excess return over the sample period. Quintile 1 (5) contains next-day ETF excess returns with the highest (lowest) put-call ratio and lowest (highest) IV spread. Column

“Excess Return” presents the average daily excess ETF return for each quintile portfolio. The average put-call ratio (P/C) and IV spread (IVS), Size and realized volatility (Rvol) are reported for each quintile portfolio. The row “5-1” reports the average next-day excess return of a portfolio which takes a long

position in the quintile portfolio 5 and a short position in the quintile portfolio 1. Newey-West (1987) t-statistics are given in the “t-stat.” row. The excess returns are obtained by subtracting the risk-free rate from each ETF return for each day. Excess returns, put-call ratio and IV spread are reported in percentages. Significance levels are indicated by ^*p < .10, ^**p < .05 and ^***p < .01.

Henceforth, in Table 9, I employ another double-independent sorting method to show that this noise can be eliminated, leaving the results in an unbiased form. Essentially, I replace the categorical sorting with a continuous sorting. Namely, after standardizing both option measures and taking their average, I sort ETF returns simultaneously in five portfolios. The first portfolio includes future ETF excess returns with the highest put-call ratio and the lowest IV spread on a particular day. In contrast, the fifth portfolio includes the excess returns of ETFs with the lowest put-call ratio and the highest IV spread. The figures of the main variables across the five portfolios from Table 9 validate this sorting procedure. Intuitively, this should generate ascending returns from the first to the fifth portfolio, as dictated by the rationale of the put-call ratio and IV spread.

The portfolio excess returns from Table 9 certainly show this. Although the third quintile portfolio does not have the highest average excess returns as does for the put-call ratio sort in Tables 4 and 5, it is still comparatively higher than the 4^th portfolio. Inferring based on these two results, the ETFs which have a put option volume of around 30% of the total option volume obtain a higher excess return than otherwise hypothesized. If this is a general option volume mechanic, the robustness check where I use the options-to-shares ratio should also reflect this.

Here as well, I initiate a trading strategy that buys the fifth portfolio and sells the first portfolio and its return is reported in Table 10. In contrast to the main analysis in Table 6, this result coincides with the exact difference between the corresponding average excess returns of the portfolios from Table 9. Although the next-day raw excess return of the strategy is lower than before, at 2.9 basis points (7.6% annualized), it is still significant at the 5% statistical level, with a slightly lower Newey-West (1987) t-statistic of 2.2. This is robust to the result in Table 6.

Additionally, the factor regressions do not explain much of this result and the alphas are still significant at the 10% level as before. Moreover, as illustrated in Table 1D in Appendix D, the predictability of the two measures together fades away completely starting from the next day. Same as before, the strategy will generate more negative and more significant returns with a longer prediction horizon, instead of reversing to zero due to the impounding of information into ETF

prices. However, the return of the portfolio strategy for returns at 10 days becomes significant and negative now. When adjusting for the four factors in column “FF4”, its alpha is significant at 10%

with a negative value of −1.7 basis points for returns at 10 days in the future.

Therefore, the results of this robustness check are validating the finding from the main analysis that the prediction of the following day’s ETF returns using both the put-call ratio and IV spread works in the portfolio analysis. Unlike the main analysis, there is some evidence to conclude that the strategy becomes significantly contrarian over longer horizons. However, I consider this proof insufficient since the results of the Fama-MacBeth (1973) regressions from the main analysis showed that there is no prediction potential for longer horizon returns. This means that the controls are explaining fully this predictability, and therefore, this case should not be taken into account.

Table 10

Main Trading Strategy using Continuous Double Sorts

Return t-stat.

CAPM FF3 FF4

Alpha t-stat. Alpha t-stat. Alpha t-stat.

Main long-short

strategy 0.029^** 2.2 0.026^* 1.7 0.028^* 1.8 0.028^* 1.8

Note. The next-day ETF excess returns are independently sorted into quintiles based ona continuous sorting procedure on the put-call ratio and on the IV spread simultaneously, generating five equal-weighted portfolios on each day. Further details on this continuous sorting aspect of this procedure are given in the methodology section. This table reports the next-day excess return of a trading strategy that goes long on the portfolio with the lowest put-call ratio and the highest IV spread and goes short on the portfolio with the highest put-call ratio and the lowest IV spread. This trading strategy is named the “main long-short strategy”. Column “Return” presents the average daily excess ETF return for the strategy. The next three columns report the alphas of the CAPM, Fama and French (1993) three-factor model (FF3), and the four factors Fama and French (1993) with Carhart’s (1997) momentum (FF4) model. For each estimate, the respective Newey-West (1987) t-statistic is given in the “t-stat.” column to the right. Excess returns are calculated by subtracting the risk-free rate from the ETF returns and are reported in percentages.

Significance levels are indicated by ^*p < .10, ^**p < .05 and ^***p < .01.

In conclusion, all these findings explicitly indicate that the continuous double sorting method is a more robust method from a technical point of view than the classic categorical double sorting. For this reason, I will employ this continuous sorting process for the double sorting analyses of the remaining three robustness checks.