CFER: Summary statistics

Table 1.1, Panel A, reports the summary statistics of the estimated MF-CFER at the end of each month for each one of the four ways of estimating CFER. We can see that there are about 333,000 stock-month CFER observations for the case of the AVE-CM and ATM-CM CFER, whereas this number increases to about 347,000 observations for the case of the AVE-CLS and ATM-CLS CFER. This yields on average about 1,370 (1,420) stocks in each month in the case of AVE-/ATM-CM CFER (AVE-/ATM-CLS CFER). This is a sufficient number to form well diversified decile portfolios in the subsequent analysis. The mean and the median of the estimated CFER are about

-0.1% and -0.04% per month (30-day), respectively. Results are similar across the four construction methods of CFER. The distribution of CFER is skewed to the left and it is highly leptokurtic. The estimated CFER is sizable; it takes both positive and negative values, ranging from -1.24% to 0.89 % per month (-14% to 11% per year) in a 5th to 95th percentile range of AVE-CM CFER. Note that this range is consistent with the theoretically derived CFER bounds, equation (1.34). Lesmond et al. (1999) and Hasbrouck (2009) document that the round-trip transaction costs (i.e., 2ρ in equation (1.34)) for large stocks are in the order of 1.0%. Moreover, the fact that the distribution of the estimated CFER has longer left tail is consistent with the existence of short-sale constraints (equation (1.34)).

CFER also has fairly large variations; the standard deviation is about 1% and the interquartile range (IQR, the difference between 75th and 25th percentile points) is between 47–60 bps on average across stocks over time depending on the CFER construction method. This magnitude of variation is relatively large compared to the long-run average U.S. equity risk premium, which is about 50 bps per month (or 6% per year, see e.g.,Mehra,2012). The percentage of the negative observations of CFER is about 55% in any of the four construction ways of CFER; CFER takes more often negative than positive values over the full sample period. Table 1.1, Panel B, reports that the four ways of computing CFER are almost perfectly correlated. Therefore, the subsequent analysis is expected to be robust to the choice of the method to estimate the 30-day CFER.

[Table 1.1 about here.]

Our theoretical considerations in Section 1.2 suggest that the summary statistics of the estimated CFER should be related to various proxies of market frictions as well as other option-based measures of deviations from put-call parity. Regarding proxies of market frictions, we will extensively investigate their relation with the estimated CFER in Section 1.5. Here, we discuss the relation between the estimated CFER, DOTS, and IVS. The summary statistics of the estimated CFER are in line with those of these alternative measures based on deviations from put-call parity. For example, Cremers and Weinbaum (2010) report that the median of IVS is negative (see equation (1.28)). They also report that the IVS are highly volatile and exhibit

substantial cross-sectional variation. Goncalves-Pinto et al. (2019) document that their DOTS measure has a negative median value and exhibits large variations.

Our theoretical results in Section 1.2.4 yield more specific predictions regarding the estimated values of CFER, IVS, and DOTS. We show that our model-free CFER is (approximately) proportional to IVS and DOTS and hence the range of the estimated CFER and that of IVS and DOTS should also be proportional. We start with com- paring the range of the estimated CFER and DOTS. Proposition 1.2.2 shows DOTS and MF-CFER are approximately proportional by the factor of the gross risk-free rate, which is very close to one. Therefore, we expect that the ranges of the estimated CFER and DOTS are similar.

Goncalves-Pinto et al.(2019) report the average DOTS of the DOTS-sorted decile portfolios. They report that the average DOTS of the bottom and the top DOTS- sorted decile portfolios are -0.92% and 0.79%, respectively. We repeat the same exer- cise by using AVE-CM CFER as a sorting variable. The average AVE-CM CFER of the bottom and the top decile portfolios are -1.31% and 0.93%, respectively. There- fore, the ranges of the average DOTS and AVE-CM CFER are comparable, although the range of DOTS is slightly narrower. Note that the narrower range of DOTS orig- inates from the fact that Goncalves-Pinto et al.(2019) do not scale DOTS (equation (1.30)) by time-to-maturity. Since they use option data whose maturity are no longer than one month (hence the average maturity across their dataset is shorter than one month), the unscaled DOTS generally takes a smaller value in magnitude compared to AVE-CM CFER, which we scale to denote 30-day return.

Next, we compare IVS and MF-CFER. Goncalves-Pinto et al. (2019) also docu- ment the average IVS of the DOTS-sorted decile portfolios and the that of the bottom and the top decile portfolios are -12.2% and 9.5%, respectively. This means that the range of AVE-CM CFER is approximately ten times narrower than that of IVS. This is consistent with Proposition 1.2.1, which shows that MF-CFER is proportional to IVS by the factor ofR0

t,t+1Vt(K)/St; for one-month at-the-money options, this scaling coefficient is approximately equal to 0.1. These results suggest that the distribution of the estimated CFER is aligned with the existing measures of deviations from put-call parity in a theoretically predicted manner.

Next, we look at the time-series evolution of the estimated CFER. Figure 1.1a shows the time-series evolution of the monthly median, 25th percentile point and 75th percentile point of AVE-CM CFER. The monthly median had been mostly negative until around 2007. Accordingly, the proportion of negatively estimated CFER in each month had been about 60% to 70% until around 2007, which is higher than that of the whole observations from 1996 to 2016 (about 55%). The proportion of negatively estimated CFER before 2007 is in line with Ofek et al. (2004), who use the data between 1999 and 2001, in that they also report that the underlying stock price is greater than the synthetic stock price for about two thirds of their observations.

The monthly median CFER experienced a huge drop during the height of the financial crisis (September to November 2008). This is in line with the temporary short-sale ban during the market meltdown. The short-sale ban implies that CFER may take a very negative value, implied by equation (1.34) withς =∞. This finding is in line withGrundy et al.(2012) in that their empirical results show that underlying stock prices became significantly more expensive compared to synthetic stock prices (i.e., negative model-free CFER) more frequent for banned stocks during the ban period. After the financial crisis, the monthly median of the estimated CFER clearly changed its property; the median value fluctuates around zero and it frequently takes positive value. Unfortunately, as our model does not yield predictions on the average value of CFER, investigating reasons behind this change is out of the scope of this study.

Figure1.1bshows the time-series evolution of the monthly IQR of AVE-CM CFER. We measure the dispersion of the estimated CFER by using this statistic rather than the standard deviation because the distribution of CFER is skewed and leptokurtic. As we have discussed in Section1.2.5.2, the degree of the dispersion in CFER is deter- mined by the size of transaction costs, margin constraints and short-sale constraints among other frictions. The time-series fluctuations in the IQR are in line with this prediction: most of the spikes in the IQR correspond to market turmoils, such as Rus- sian default and LTCM crisis (August to September 1998), the collapse of Lehman Brothers and ensuing market meltdown (September to November 2008), European debt crisis (November 2011, uncertainty was the highest around the general election in Greece), and the Chinese stock market turmoils (June 2015 to January 2016). It

is well documented that the margin (and liquidity) constraints become tighter during market turmoil periods (Gˆarleanu and Pedersen,2011;Nagel,2012). Hou et al.(2016) also find that their microstructural friction measure takes greater values during reces- sions and market distress periods. On the other hand, except these distressed periods, we can see a secular decline in the size of the IQR. This may reflect the decrease in transaction costs. For example,Green et al. (2017) document that the decimalization of quotes in 2001 and the introduction of autoquoting software by the NYSE in 2003 dramatically reduced transaction costs. According to Proposition1.2.3, a reduction in transaction costs implies that both the upper bound and lower bound of CFER shrink toward zero. The behavior of the 25th and 75th percentile points of AVE-CM CFER in Figure 1.1a supports this conjecture; both the monthly 25th and 75th percentile points time-series generally shrunk toward zero from the late 1990s to the early 2000s. Finally, note that the calculation of CFER requires the existence of market option prices. As a result, our universe of stocks is confined to the optionable stocks (i.e., stocks which have options written on them). However, this should not be viewed as a shortcoming of this study. In line with the results of Cremers and Weinbaum

(2010), our optionable stocks are big stocks; the average market capitalization of stocks with (without) AVE-CM CFER is about 9.1 billion (0.5 billion) U.S. dollars over our sample period from 1996 to 2016. Relatedly, albeit we can estimate AVE- CM CFER for about 27% of stocks (about 1,350 optionable stocks out of 5,000 all common stocks in each month), these stocks on average account for about 90% of the aggregate market capitalization of U.S. common stocks over our sample period. In addition, even though our cross-section of U.S. optionable stocks is subject to smaller frictions compared to the non-optionable stock universe, still the effect of market frictions on their expected returns (i.e. CFER) is sizable as reported and we will further demonstrate in Section 1.4.17

17_{For example, the averageAmihud’s (2002) illiquidity measure of the optionable (non-optionable)}

stocks is 0.01 (5.28), and the average relative bid-ask spread of optionable (non-optionable) stocks is 0.48% (2.50%) over our sample period.