Stock Options as Lotteries

Transcription

1 Stock Options as Lotteries Brian H. Boyer and Keith Vorkink 1 This Version: September 14, We acknowledge financial support from the Harold F. and Madelyn Ruth Silver Fund, and Intel Corporation. Vorkink notes support from a Ford research fellowship. We thank Greg Adams for research support. Contact Information: Both authors are from the Marriott School of Management, 640 TNRB, Brigham Young University, Provo, UT Boyer: , [email protected]. Vorkink: , keith [email protected].

2 ABSTRACT Motivated by recent theories predicting a negative relationship between asset returns and skewness, we investigate the returns on individual equity options. We construct measures of ex-ante skewness on options that have less model error than the skewness estimates used in other asset markets. Our skewness estimates for individual equity options are multiple times larger than the skewness estimates found in equity markets. Consistent with the theoretical predictions, we find that individual equity options exhibit a signficant negative relation between average returns and skewness. Options with high levels of expected skewness offer surprisingly low returns. Controlling for both market and volatility risk does little to change the results. We find that the large spread in returns is not driven by early exercise, by small-sample distributions effects, or by liquidity.

3 I. Introduction In recent years, researchers have shown an increased interest in nonstandard preferences as a mechanism for understanding patterns in asset prices considered anomalous. This interest is largely motivated by evidence that investors deviate from standard utility theory when making investment decisions in the face of uncertainty (see for example Kahneman and Tversky, 1979). One prominent theme in this literature is that investors express preferences for skewness or lottery features in asset return distributions. Models including Brunnermeier and Parker (2005) and Brunnermeier, Gollier, and Parker s (2007) endogenous probabilities model, the cumulative prospect theory of Barberis and Huang (2007), and the heterogenous skewness preference model of Mitton and Vorkink (2006) all predict that skewed assets will have low returns in the presence of skewness (or lottery) preferring investors. 1 Empirical studies such as Boyer, Mitton, and Vorkink (2010) and Conrad, Dittmar and Ghysells (2010) find that the cross-section of stock returns is consistent with these skew-preferring models: stocks with high ex-ante skewness offer low subsequent average returns. While investigations into the role of skewness preference in equity markets is relatively well-developed, little attention has been given to other markets where the impact of skew-preferring investors may be more prevalent or more meaningful. One such market is the individual equity options market; a market that offers an abundance of lottery opportunities to investors. This paper attempts to fill this gap by testing for the presence of skew-preferring investors in the individual equity options market. That little attention has been given to skewness-preferences and option markets is surprising on a number of fronts. First, if investors do express a preference for lotteries/skewness, then option markets (and in general derivative markets) offer lottery opportunities at a much greater magnitude than equity markets. The implicit leverage in options combined with their nonlinear payoff structure create skewness in option returns that an investor would find impossible to replicate in the underlying equity market. Second, investors can make cleaner bets 1 Work on skewness preferences pre-dates those articles cited above. Arditti (1967) and Scott and Horvath (1980) show that well-behaved utility functions include a preference for positive skewness. Kraus and Litzenberger (1976) and Harvey and Siddique (2000) generate asset pricing implications of skewness preference under a representative agent framework. Simkowitz and Beedles (1978) and Conine and Tamarkin (1981) show that agents that have skewness preference may prefer underdiversified portfolios in equilibrium in constrast to standard representative agent equilibrium holdings. 1

4 on skewness in options markets than in equity markets. Forecasting lottery characteristics in equity markets is difficult and requires the use of a model in most instances subjecting the investor to model error. Option markets offer more transparency between lottery features and option observables thereby diminishing the risk of skewness-forecast model error when constructing measures of ex-ante skewness. Third, existing empirical investigations of options markets have discovered substantial pricing errors. Coval and Shumway (2001) observe index option straddle alphas that are surprisingly negative when controlling for market risk. Jones (2006) finds that the low returns on index option straddles cannot be resolved using very general models of risk. In this paper, we empirically investigate the pricing implications of skewness in the cross section of individual equity option returns. Using the full cross-section of both individual equity call and put options, we find that the returns on highly skewed options are both statistically and economically lower than the returns on less-skewed options. We construct ex-ante skewness measures based on the assumption of lognormal stock prices. We find that these ex-ante measures are good predictors of future skewness in option returns. Sorting the cross-section of equity options based on our ex-ante skewness measure generates differences in option ex-ante skewness that is 3 to 4 times larger than we observe in the cross-section of equities (Boyer, Mitton, and Vorkink, 2010) illustrating the attraction of these securities to investors with lottery preferences. We also show how moneyness acts as a strong instrument for expected skewness under the assumption of lognormality and how it provides a simple characteristic to sort on ex-ante skewness as opposed to other characteristics such as return volatility. Average returns across skewness-sorted portfolios (sorts occur across both lognormal exante skewness and moneyness) generate differences in returns between high and low skewed option portfolios on the order of 10% weekly and in some cases greater than 60% weekly. These results are found in both call and put option markets. We find the strong negative relationship between average returns and skewness hold across a number of maturities ranging from 1 week through 6 months. Having found such a surprising negative average return, we then investigate the possibility that the large spread in returns between high- and low-skewed options is driven by risk. 2

5 Alphas on a one-factor market model are nearly identical in magnitude to the raw average returns indicating that variations in exposure to market risk is not driving our result. We also test whether a two-factor model of risk can resolve the puzzle, where our two factors control for both market risk and volatility risk. We construct our volatility risk factor using the returns on a zero-delta straddle on S&P500 index options. We find that the spread in two-factor alphas between high- and low-skewed option portfolios continues to remain greater than 8% weekly. Our results provide strong evidence that standard models of risk are unlikely to explain the large variations in average option returns across skewness, and results that suggest skewness preference plays an important role in the pricing of individual equity puts and calls. We investigate a number of robustness checks to our results. Given that we use both call and put options in our empirical analysis, accounting for the possibility of early exercise, especially for put options, is important. When we account for an early exercise strategy in our empirical tests, the spread in returns between high and low skewed options remains. We also investigate the possibility that the finite-sample distribution of our option returns deviates substantially from normality leading to incorrect inferences (see Broadie, Chernov, and Johannes, 2009). We find that the empirical distribution of the option portfolio alphas, both one-factor and two-factor, to be reasonably well behaved and that the spread in average portfolio returns is unlikely driven by nonnormalities. We also construct p-values for the alpha based on simulated distributions using lognormality in an attempt to rule out peso-problems in our data. Our simulation-based p-values confirm that lognormality of stock prices is unable to generate patters in option returns sufficient to reconcile the patterns in average returns we observe in the actual data. As further robustness checks, we investigate the role that liquidity may play in explaining the large return differences across skewness dimensions. Even when we constrain the sample of options to only include those with high levels of liquidity the spread in returns across skewness in options remains. Liquidity issues appear unlikely to explain away the large spread observed across skewness in equity options. Empirical asset pricing investigations of options markets are relatively scarce as compared to equity markets. A majority of interest has centered on index markets including Coval 3

6 and Shumway (2001) and Jones (2006). Bashki, Kapadia, and Madan (2003) use individual equity options to construct skew measures (for the underlying) and relate the variations in skews to the returns on the underlying. Similarly, Conrad, Dittmar, and Ghysels (2009) use model free skewness estimates taken from a cross-section of individual equity options to price the cross section of equity returns. The paper closest to ours is Ni (2009) who investigates the return properties of call options across moneyness and concludes that the variations may be due to skewness. Our approach differs from Ni (2009) in some important ways. First, we investigate both call and put options across a range of maturities, where Ni (2009) studies only one-month maturity call options. Second, we construct measures of ex-ante skewness and do not rely on moneyness as our only instrument. Third, we investigate the role that risk plays in explaining the variations in returns as opposed to Ni who reports average returns. We conduct a careful study on the distributional properties of the risk-adjusted returns in the vein of Broadie, Chernov, and Johannes (2009) study of index option return distributions. Fourth, our paper takes the testing of skewness-preferring asset pricing models as the primary purpose while Ni(2009) is an asset-pricing anomaly paper. Our paper contributes to the investor preferences literature adding strong evidence that skewness or lottery preferences are an important input into understanding asset pricing. In fact, our results suggest the lottery preferences may be of first-order importance to understanding the pricing and return properties of securities whose payoffs offer substantial amounts of skewness. The rest of the paper is organized as follows. Section 1 motivates the use of moneyness and maturity as instruments for ex-ante skewness as well as introduces our construction of an ex-ante skewness measure under lognormality of asset prices. Section 2 introduces the options data set and how we construct the option portfolios for use in our empirical tests. Section 3 contains the main empirical tests of the cross-section of individual equity options. Section 4 documents our robustness checks on early exercise, finite-sample distribution tests, as well as liquidity tests. Section 5 offers concluding remarks. 4

7 II. Skewness and Option Returns Our interest is to test the relationship between lottery preferences and the returns to options. To formalize this test we make some simplifying assumptions. First, we assume that skewness is a good proxy for lottery prospects of an option, consistent with much of the behavioral literature as in Brunnermeier and Parker (2005), Barberis and Huang (2007), and Mitton, and Vorkink (2006). Ex-ante skewness is generally unobservable for most securities, and must be estimated. Zhang (2005) estimates a firm s skewness using industry cross-sectional estimates of skewness. Boyer, Mitton, and Vorkink (2010) derive estimate s of a firm s skewness using characteristics of the firm in a predictive regression framework. Conrad, Dittmar and Ghysells (2010) use the cross-section of a firm s option prices to derive a model-free estimate of skewness for the underlying. In our case, because we are interested in pricing individual equity options, these approaches are complicated. We are able to construct a measure of return skewness for each individual option using the assumption that stock prices are lognormally distributed. Following this assumption, we can construct closed-form solutions for an options return skewness and can use this measure to test our hypothesis that options with higher expected skewness will have lower expected returns. Admittedly, the hypothesis that stock prices follow a lognormal distribution is rejected in the data, so we also use moneyness as an instrument for expected skewness. The relationship between moneyness and expected skewness should hold under a large number of assumptions regarding the distribution of underlying stock prices. We find that both approaches have a strong positive relationship with the actual return skewness observed in the data. A. Ex-Ante Skewness under Lognormality Our measure of interest, denoted as sk i,t:t, is option i s return skewness from time t to time t+t, and is defined as the centered third moment of an option return divided by the scaled second moment as shown in equation (1) below sk i,t:t = E [r i,t:t µ i ] 3 [V ar (r i,t:t )] 1.5 (1) 5

8 where µ i is the expected return of the option i, and V ar i is option i s return variance. return from holding a call option to maturity, defined as r C i,t, is simply The r c i,t:t = (S i,t X i ) + C i,t, (2) while that of a put option is r p i,t:t = (X i S i,t ) + P i,t, (3) where C and P correspond to the option premiums and X corresponds to the exercise price of the option. Rewriting equation (1) in terms of its raw moments sk i,t:t = E [ ] [ ] ri,t:t 3 3E r 2 i,t:t µi,t:t + 2µ 3 i,t:t [ [ ] ] E r 2 i,t:t µ 2 1.5, (4) i,t:t illustrates that to calculate the skewness of the option return, only the first three raw moments are required. From the payoff structures of the returns in equations (2) and (3), these moments will come from a truncated distribution where the truncation is determined by the observable exercise price, X. Lein (1985) derives the moments of a truncated lognormal distribution which we can use to construct sk i,t:t for any option contract. We demonstrate how to construct our expected skewness measure, sk i,t:t, in Appendix A. B. Moneyness To help build intuition regarding the influences of option characteristics on our expected skewness measure, sk i,t:t, we construct plots demonstrating how certain characteristics influence skewness under the assumption of lognormality. Figure 1 plots sk i,t:t as a function of moneyness ( ) X S t for both call and put options and for a number of maturities. 2 Figure 1 illustrates the strong relationship between moneyness and expected skewness in returns. For both call and put options, those options that are trading out of the money offer substantial skewness. This relationship is magnified as one decreases the maturity. For both out of the money call and put options, holding period returns can offer a skewness of over 15, which is a multiple of 2 For Figures 1, 2 and 3 we assume that the simple expected return on the stock is 8% annual and that the risk-free rate is 6% annual. All of our results are robust to these two parameter values. 6

9 the skewness coefficients offered in the equity markets (see Boyer, Mitton, and Vorkink, 2010). One other observation from Figure 1 is that put options can offer skewness opportunities that are at least as large as their corresponding put options. Looking at only call options appears to exclude securities that may be attractive to lottery preferring investors. In Figures 2 and 3 we plot the relationship between sk i,t:t and return volatility (σ). Figure 2 plots the relationship for options trading at a moneyness level of.9. This level of moneyness leads to out-of-the-money put options and in-the-money call options. We see that implied volatility can have a strong impact on skewness, but that the magnitude of the relationship is influenced by both maturity and moneyness. For in-the-money call options higher return volatility leads to slightly higher skewness, for the out-of-the-money put potions, there is a strong negative relationship - lower return volatility leads to higher skewness. Figure 3 plots the relationship for a moneyness level of 1.1 and leads to in-the-money put options and out of-the-money call options. The relationship between volatility and skewness flips for outof-the-money call options; now increasing the volatility leads to substantial declines in the return skewness. We observe essentially no relationship between volatility and skewness for in-the-money put options in Figure 3. Moneyness is the sole option characteristic that exhibits a monotonic relationship with sk i,t:t. Return volatility, σ, in some settings influences sk i,t:t substantially, but this relationship is not monotonic for all other option characteristic values. Maturity (results not shown but available upon request) is similar to return volatility in that in some cases (in particular short maturities) has a strong effect of increasing sk i,t:t, but that this relationship is not constant. In some cases, for example in-the-money call options, increasing maturity slightly increases return skewness as seen in the top graph of Figure 1. We take these results as motivation for using moneyness as an instrument for sk i,t:t. This relationship (out-of-the money increases skewness) is likely to hold even under more general assumptions regarding the distribution of underlying returns. In testing the hypothesized relationship between skewness and expected returns we will use both sk i,t:t as defined in equation (4) and moneyness. Our inclusion of moneyness, in some measure, acts a robustness check against the lognormal distributional assumption embedded in sk i,t:t. 7

10 III. Results We obtain data for options written on common stock, including end-of-day closing bid and ask quotes, underlying asset values, open interest and trading volume from the Ivy Optionmetrics database and create option portfolios on the first trading date of each month and on the second Friday of each month, one week before options expire. Before creating our portfolios we first screen out records from Ivy data that may contain errors or quotes that may not be tradable. This procedure, detailed in Appendix A, eliminates options from each portfolio using information observable on or before the corresponding formation date. For example, we screen out options that do not trade on the formation date, options that have zero open interest on the trading day immediately prior to the formation date, or options that have excessive bid-ask spreads. Portfolio formation dates extend from February 1, 1996 through October 1, For our analysis we also need the underlying asset value on each option s expiration date. We observe this value in Ivy for approximately 98.3 percent of our screened data. After filling in as many of these missing values as possible using CRSP stock prices, we observe underlying asset values on expiration dates for about 99.5 percent of our observations. The other 0.5 percent are unobservable due to events such as mergers and delistings. We also eliminate these few records from our data even though this information is not observable on formation date. On each portfolio formation date, we estimate expected skewness under the assumption that the underlying asset is lognormally distributed as discussed above in Section 2. To do this we need estimates of the expected return and volatility for every underlying asset and formation date in our sample. We use six months of daily data, from CRSP, immediately prior to each formation date to estimate these moments. Other variables needed to compute the skewness of the option include the underlying stock price on the formation date, as well as the time to maturity, strike, and price of the option. All of these are readily obtained from the Ivy database. We define the price as the midpoint of the bid-ask spread. 3 The Ivy database currently begins January 4, 1996 and ends October 30, Since we cannot observe open interest on the trading date immediately prior to the first trading date of January, 1996, we exclude this formation date from our sample. 8

11 On each portfolio formation date, we then divide all calls and puts into 8 expiration bins. The first expiration bin contains options that expire in one week. We observe these options only on formation dates which are the second Friday of each month. Further, we do not create portfolios of any other expiration on these formation dates. The second expiration bin contains options that on average expire in 18 days. We observe these options on portfolio formation dates which are the first trading date of month m. These options will expire on the third Friday of month m. The third expiration bin contains options that on average expire in 48 trading days. These options, observed on the first trading date of month m, will expire on the third Friday of month m + 1. The fourth through eighth expiration bin contains options that respectively expire on average in 78, 108, 138, 168, and 198 trading days. These options, observed on the first trading date of month m, will expire on the third Friday of month m + 2, m + 3, m + 4, m + 5, and m + 6, respectively. On each portfolio formation date we then sort options within each expiration bin into 5 expected skewness quintiles. If any bin on any given portfolio formation date does not have at least 10 options, we exclude this bin/date from the analysis. Panel A of Table II provides some insight regarding the number of options within each of our 40 bins. Panel B shows how many bins were eliminated because of insufficient data. For example, there are 280 options in the lowest skewness quintile bin for options that expire in 7 days. Across time, we had to eliminate 10 of these bins from the analysis because, on certain dates, there were less than 10 options in the bin. Since we form bins once each month from February 1996 through October 2009, there is a maximum number of 165 bins/formation dates. We therefore had to eliminate 6 percent of the bins for options in the lowest skewness quintile among options that expire in 7 days. Table III reports the average, across time, of the median skewness measure sk i,t:t across all options in each portfolio at each formation date. Within each expiration group, skewness increases across the quintiles by construction. The variation in expected skewness across these quintiles is large, especially among short-term options. For example, among options that are going to expire in 7 days, expected skewness ranges from 0.39 to In comparison, the typical skewness for a stock varies from around 0 up to 3 (see, for example Boyer, Mitton, and Vorkink, 2010). 9

12 On the appropriate expiration date for each skewness/maturity bin, we calculate the return for each option, initially assuming it is held to expiration. The return, for example, on call option i purchased on formation date t and held to expiration, T, recall is given by equation (2). Although returns calculated in this manner ignore the possibility of early exercise, this simplification should have little impact on our relative results. Ignoring the possibility of early exercise biases downward the returns of options that become optimal to exercise early. The likelihood of optimal exercise increases with moneyness. But options that are in-the-money tend to be less skewed as discussed in Section II. Therefore ignoring early exercise should, if anything, tend to bias downward the returns of in-the-money, less skewed stocks. The point of our paper is to show that such options earn higher risk-adjusted returns than out-of-the money, skewed options. In any event, we later adjust our returns for the possibility of early exercise, and show that doing so does not change the results. We first verify that our expected skewness measure actually does a good job forecasting skewness beyond the results of Table III. Empirically estimating skewness from the timeseries of options returns is challenging, especially for out-of the money options, since small probability events are often not observed within a short period of time. We therefore choose to follow Zhang (2005) and empirically estimate skewness in the cross section. Since there are many more options than time periods, it is easier to capture small probability events in the cross-section. Intuitively, the higher the (idiosyncratic) skewness across time among options within a given bin, the higher will be the average cross-sectional skewness. Table IV reports the time-series average of the cross-sectional skewness estimate, where the skewness estimate uses the cross-section of options returns within each portfolio. These results provide some evidence that our expected skewness measure, sk i,t:t, does a good job as a forecast. The average crosssectional skewness increases across the skewness quintiles for each maturity group. The bottom row of each panel tests for a significant difference in average cross sectional skewness across the bottom and top skewness quintiles. Since our returns overlap, these standard errors are adjusted for autocorrelation using the approach of Newey and West (1987). We then calculate equally-weighted portfolio returns for each maturity/skewness bin. The average of these returns, across time, are reported in Table V. In each case returns are scaled to be weekly. This table provides some initial evidence on the effect of lottery-preferring 10

13 investors on option prices. The returns decrease dramatically across skewness bins for every maturity group, especially among short-term options. For example among call option that will expire in 7 days, the average weekly return is 0.34 percent for the low skewness bin, and for the high skewness bin. The t-statistic for the difference is These results are within the range of average returns reported by Ni (2009). Among puts that will expire in 7 days, the average weekly return is for the low skewness bin, and for the high skewness bin. The t-statistic for this difference is Low average returns for the high skewness bin indicate many of these are probably out-of-the-money to start off, consistent with the figures discussed above. Given the dramatic relation in average returns across skewness quintiles reported in Table V, we now turn to the task of determining whether differences in these average returns can be explained by risk. In Table VI we report the portfolio CAPM betas for each of our option portfolios, which are estimated by regressing the excess portfolio return on the market return over the same time period as r pt r f = α + β(r mt r f ) + e t In general here we see that while betas are lower in the 5th skewness quintile than the first, betas take on a hump-shape within each expiration group. For example, among options that expire in seven days, the beta for the low skewness quintile is 16.69, then increases to about 20 for the second and third skewness quintiles, and then deciles to 9.87 for the high skewness quintile. Unlike average returns which are monotonically decreasing in skewness quintiles, the betas take on a non-linear relationship across the skewness quintiles, suggesting risk cannot fully explain the patterns documented in Table V. In Table VII we report the alphas of these single-factor regressions. The results here are dramatic. CAPM alphas are monotonically decreasing across skewness quintiles similar to the average returns in Table V. For example, among call options that will expire in 7 days, the alpha for the low skewness portfolio is percent per week and for the high skewness portfolio it is percent per week. The t-statistic for the difference is Among put options that will expire in 7 days, the alpha for the low skewness portfolio is percent 11

14 per week and for the high skewness portfolio it is percent per week. The t-statistic for the difference is Note that the difference in alphas across the high and low skewness portfolios remains significant for call options that have up to 168 days to maturity, while for put options, the difference is significant for options up to 18 days to maturity. Is it econometrically appropriate to estimate alphas for options? In particular, how do the highly skewed distributions for options returns affect the small sample properties of our estimators? We are not the first to estimate alphas for option portfolios. Broadie, Chernov, and Johannes (2009) estimate alphas for index options. We have some reason to believe the distributional properties of option portfolios are more well-behaved than that of individual index options. Nonetheless, in Figure 4 we plot the histogram of the returns for the portfolio 5 (high skewness quintile) for options that expire in 7 days, 18 days, 48 days, and 78 days. These distributions confirm that even portfolio returns of these options are still quite skewed. We therefore turn to estimating the small-sample distribution of the alphas we estimate for Table VII using a bootstrap technique. To do this, we create non-overlapping samples for options that expire in 18 days or 48 days, forming portfolios every-other month. We then sample portfolio returns in the time series with replacement, creating a new sample of the same size as the original. We then estimate alphas using this new sample. We repeat this procedure 10,000 times and create histograms of our alpha estimates in Figure 5. Here we see that the small sample distribution of our alphas are not that far from normality. Further, we can use the boot-strapped estimates to test the null hypothesis that alphas are zero. For call (put) options that expire in 18 days, the average alpha is -13.5% (-16.5%) with 0 being greater than zero in either case. For call (put) options that expire in 48 days, the average alpha is -2.2% (-1.0%) with 0.1% (22%) being greater than zero. Hence, we can reject the null hypothesis that alpha is zero for call options that expire in 18 or 48 days, and for put options that expire in 18 days. These results line up exactly with those of Table VII. Another concern about estimating alphas for options is the fact that low probability events are not observed very often, and perhaps our sample excludes some of these events. For example, one may argue that the reason out-of-the-money options prices are so high (returns are low) is because investors were pricing in the chance that these actually would expire in the money, and we just don t happen to observe a sufficient number of such events, similar to 12

15 a peso problem. To address this issue, we perform simulations as in Broadie, Chernov, and Johannes (2009). Using our sample of non-overlapping option returns used to perform the bootstrap exercise above, we simulate underlying asset values under lognormality. In doing so, we match the ex-ante moments of the underlying data. In particular, on each portfolio formation date we first simulate log-index returns over the period until the first option expiration date (average is 18 days) as r mt1 = (r f +.08)τ 0 + ξv τ 0 where r f is the annual risk-free rate, ξ N(0, 1), v is annualized volatility estimated using six months of daily data prior to the portfolio formation date and τ 0 is the appropriate time to expiration (average is 18/365). We then simulate index returns over the subsequent period until the next option expiration date (average is 48 days) by adding r mt2 to r mt1 where r mt2 is defined as r mt2 = (r f +.08)(τ 1 τ 0 ) + ξv τ 1 τ 0 On each portfolio formation date, we therefore have simulated market returns over an 18-day horizon, and a 48-day horizon (average). We then simulate underlying asset values over the short horizon (average 18 day) as r it1 = β r mt1 + ξσ i τ 0 + (r f βr f )τ 0 where σ i is the annualized idiosyncratic volatility of asset i estimated using 6 months of daily data prior to the portfolio formation date. We then estimate asset values over the longer horizon (average 48 day) as r it1 + r it2 where r it2 is defined as r it2 = β r mt2 + ξσ i τ 0 + (r f βr f )τ 0. By simulating underlying asset returns in this manner, we match not only the first and second moments of each individual asset, but we also preserve the contemporaneous correlation across assets. 13

16 Using the simulated lognormal asset returns, we simulate asset values as S e r it1 0i and S e r it1+ r it2 0i where S 0i is the value for stock i on the portfolio formation date. Using the simulated asset values, we construct option returns and alphas as before, and repeat the process 1000 times. We then calculate the fraction of samples with alphas at least as negative as those given in Table VII. Results are given in Table VIII and the small p-values in this table indicate that peso problems, under the assumption of lognormally distributed returns, cannot reconcile the large spread between high- and low-skewness options. IV. Robustness Checks In this section we test the robustness of our results in three broad dimensions: risk, early exercise, and liquidity. First, we re-estimate alphas where we control for both market and volatility risk. Second, we reconstruct portfolio returns introducing an early-exercise strategy and test the average returns of these conditional portfolios. Last, we perform tests aimed at determining the role that liquidity plays in explaining the spread in returns across skewness. All three of our robustness check lead to similar spread in returns, raw and risk adjusted. A. Two-Factor Model In this test, we estimate alpha after accounting for not only market risk, but volatility risk. Several papers have documented the existence of a volatility risk premium in options, which helps explain why options earn low returns in general. To account for this volatility risk premium, we follow Ang, Hodrick, Xing, and Xang (2006) and estimate the return on at-themoney zero-delta straddle on S&P 500 index options. Straddles on indexes are very sensitive to volatility, and earn returns on the order of -3 percent per week (Coval and Shumway (2001)). We create a daily zero-delta straddle return, rebalanced daily. We then compound these returns over the appropriate time period to match the horizon of our option returns. We then regress excess option portfolio returns on excess market returns and excess straddle returns. Results are given in Table IX. Results for this table look similar to those of Table 14

17 VII. For example, among call options that expire in 7 days, the alpha of the low skewness quintile is -2.22% per week, while the alpha of the high skewness quintile is % per week. B. Early Exercise We then adjust out CAPM alphas for the possibility of early exercise. To do this, we note that it will never be optimal to exercise a call option at time t if the price of the call is greater than S t X, since an investor receives more by writing a call option with the same maturity and strike. Similarly, it will never be optimal to exercise a put option at time t if the price of the put is greater than X S t. On the other hand, we should rarely see American call option prices less than S t X, or put option prices less than X S t since these scenarios provide an opportunity to make a riskless arbitrage. Hence, if there are no arbitrage opportunities, early exercise will only be optimal for a call option if the call price is equal to S t X and for a put option if the put price equals X S t. In our world with bid and ask prices, it will only be optimal to exercise options early if and when the bid-ask prices straddle S t X for call options and X S t for put options. After each portfolio formation date, we therefore test on each day if this condition holds for each option. If it does, we immediately exercise the option, and invest the proceeds in a risk-free t-bill for the remainder of the option s life. Doing so should only increase the alphas of our portfolios if t-bills indeed earn zero alpha. Hence, our procedure is conservative in that we exercise as soon as it may be possible to do so, and perhaps sooner than it is optimal to do so. We report our results in Table IX. Again, results here are little changed from before. For example, among call options that expire in 7 days, the alpha of the low skewness quintile is -2.06% per week, while the alpha of the high skewness quintile is % per week. Alphas of longer term options are slightly higher, and alphas of short term options are insignificantly changed. C. Liquidity Last, we test to see if our results are driven by low liquidity. Perhaps prices of highly skewed options are high because buyers have to entice sellers to take a short position which is difficult to hedge because of illiquidity. We first give an idea of volume for this market by reporting 15

18 average of the cross-sectional median volume within each bin in Table XI. Here we see that volume is highest among short term options, and higher for short term options in portfolio 5 (high skewness) than among longer term options which earn lower alpha. This table therefore provides some evidence that liquidity alone isn t driving variation in alphas across skewness portfolios. In Table XII, we report results similar to Table XI, only in this table we report the average dollar volume which takes into account the price of the option in addition to the volume. Options in the high-skewness portfolios have relatively lower dollar volume than options in the low-skewness portfolios primarily due to the fact that most high-skewness options are out-of-the-money and these options have lower prices than low-skewness (in-themoney) options. To see if the negative skewness-average return relationship in options is driven by illiquid options, we reproduce the alphas of Table VII but only include the most liquid options and report these results in Table XIII. Specifically, we first sort options within each skewness/expiration bin into volume terciles, and exclude options in the bottom terciles, so that we include just the most liquid options. In the Table XIII, the alphas are less extreme than those of Table VI, but still quite dramatic. For example, among call options that expire in 7 days, the alpha of the low skewness quintile is -1.80% per week, while the alpha of the high skewness quintile is % per week. We conclude that low liquidity is not driving our results. While buyers may have to entice sellers to take illiquid short positions, this alone still doesn t explain variation in alphas across portfolios, and why buyers are especially willing to pay high prices for options that are more skewed. In sum, the spread we observe between portfolios of high-skewed and low-skewed options does not appear to be driven by variations in volatility risk, by early-exercise premiums, or by variations in liquidity. V. Conclusion Only recently have higher moments of asset returns found significant space in the asset pricing literature. The change is likely attributed to the recent theoretical advances indicating that idiosyncratic skewness, and not just co-skewness, may be priced. So while some evidence has 16

19 come in support of these theories, the literature seems unsettled on the important of lottery characteristics (or skewness) in asset returns. We find that in the individual equity options market, that skewness or lottery preferences may have as much to say (possibly more) than risk when pricing securities. We believe the evidence in the equity markets may give rise to a more serious inclusion of skewness when investigating the asset pricing of all non-normally distributed securities. 17

20 Appendix A. Expected Skewness Calculations In this appendix, we demonstrate how our expected skewness measure, sk i,t:t is constructed assuming lognormal stock prices. We make use of Lien s (1985) theorem regarding truncated lognormal distributions. We restate Lien s (1985) theorem 1 below, noting that Lien s theorem applies to bivariate distributions and our use will be univariate: Theorem 1 Let (u 1, u 2 ) be a normal random vector with mean (0, 0) and covariance matrix = σ2 1 σ 12 σ 12 σ 2 2. Then ( ) h a exp [ D/2Q] E(exp(ru 1 + su 2 ) u 1 > a) = N σ 1 N( a, σ 1 ) where h = rσ sσ 2 2, D = Q (r 2 σ rsσ 12 + s 2 σ 2 2), Q = σ 2 2σ 2 1 σ 2 12, and N(.) is the CDF of the normal. Lien s (1985) Theorem can be used to construct the first three raw moments of the truncated distribution which then can be substituted into equation (4) to construct sk i,t:t. first three moments of a call option return can be expressed as: E [r] = E [ r 2] = [ ] σ [1 S t exp 2 + µ ( )] ( ) 2 N d1 XN d2 S 2 t exp [2σ 2 + 2µ] [ N ( d3 )] 2XSt exp C The 1 (5) [ ] σ 2 + µ N ( ) d1 2 + X 2 N ( ) d2 (6) C 2 E [ r 3] = S3 t exp [ 9 2 σ2 + 3µ ] N ( d4 ) 3XS 2 t exp [2σ 2 + 2µ] N ( d3 ) [ ] +3X 2 σ S t exp 2 + µ N ( ) d1 2 + X 3 N ( ) d2, C 3 C 3 (7) 18

21 where C is the call premium, d 1 = ln( S t N(.) is the CDF of the normal. X )+σ2 +µ σ The corresponding measure for The corresponding raw moments for a put options are E [r] = E [ r 2] = E [ r 3] =, d 2 = d 1 σ, d 3 = d 1 + σ, d 4 = d 1 + 2σ, and XN ( d ) [ ] 2 σ St exp 2 + µ N ( d ) (8) P X 2 N ( d ) [ ] 2 σ 2XSt exp 2 + µ N ( d ) S 2 t exp [2σ 2 + 2µ] [ N ( d )] 3 (9) P 2 X 3 N ( d ) [ ] 2 3X 2 σ S t exp 2 + µ N ( d ) 2 1 P 3 (10) +3XS 2 t exp [2σ 2 + 2µ] N ( d 3 ) S 3 t exp [ 9 2 σ2 + 3µ ] N ( d 4 ) P 3, where P is the put premium. Equations (5) through (10) can be used to construct sk i,t:t for both call and put options for any level of moneyness and maturity. B. Option Database Screening Procedure We create portfolios on the first trading date of each month. Let t i be the formation date for portfolio i. We eliminate all options from portfolio i with any of the following characteristics observable in the Ivy database on or before date t i. 1. Underlying Asset is an Index: Optionmetrics index flag is non-zero. 2. Underlying Asset is Not Common Stock: Optionmetrics issue type for underlying is non-zero. 3. AM Settlement: The option expires at the market open of the last trading day, rather than the close. 4. Non-standard Settlement: The number of shares to be delivered may be different from 100, additional securities and/or cash may be required, and/or the strike price and premium multipliers may be different than $100 per tick. 5. Missing Bid Price: The bid price on date t i is 998 or 999. Ivy uses these as missing codes for some years. 19

22 6. Abnormal Bid-Ask Spread: The bid-ask spread on date t i is negative or greater than $5. 7. Abnormal Delta: The option delta on date t i, as calculated by Ivy, is below 1 or above Abnormal Implied Volatility: Implied volatility on date t i, as calculated by Ivy, is less than zero. 9. Extreme price: The mid-point of the bid and ask price is below 50 percent of intrinsic value or $100 above intrinsic value. 10. Duplicates: Another record exists on date t i for an option of the same type (call or put), on the same underlying asset, with the same time-to-maturity and same strike price. 11. Zero Open Interest: Open interest on the trading date immediately prior to date t i is zero. 12. No Trade: The Optionmetrics last date value is before t i. 13. Underlying Price History in CRSP is too Short: The underlying asset does not have at least 100 non-missing daily returns in CRSP over the 6-month period prior to date t i. 14. Expiration Restrictions: The expiration month is greater than m i + 6, where m i is the month in which portfolio i is formed, or the option expires after Screens 1 and 2 allow us to focus on options written on common stock. We follow Duarte and Jones (2007) in applying screens 3 through 11. Screen number 12 helps exclude stale option quotes from the analysis. We apply screen 13 because we use six months of daily data from CRSP prior to date t i to estimate moments of underlying assets, and we apply screen 14 because of data limitations. 20

23 Bibliography Ang, A., R.J. Hodrick, Y. Xing, and X. Zhang High idiosyncratic volatility and low returns: International and further U.S. evidence. Journal of Financial Economics, forthcoming. Arditti, F. D Risk and the required return on equity. Journal of Finance 22: Barberis, N. and M. Huang Stocks as lotteries: The implications of probability weighting for security prices. American Economic Review, forthcoming. Brandt, M.W., A. Brav, J. R. Graham, and A. Kumar The idiosyncratic volatility puzzle: Time trend or speculative episodes?, Review of Financial Studies, forthcoming. Broadie, M., Chernov, M., and M. Johannes Understanding expected option returns. Review of Financial Studies. 22: Brunnermeier, M., C. Gollier, and J. Parker Optimal beliefs, asset prices and the preference for skewed returns. American Economic Review Papers and Proceedings 97: Brunnermeier, M., and J. Parker Optimal expectations. American Economic Review 95: Conine, T. E. Jr., and M. J. Tamarkin On diversification given asymmetry in returns. Journal of Finance 36: Coval, J. and T. Shumway Expected Option Returns. Journal of Finance. 56: Duarte, J., and C Jones The market price of volatility risk. working paper, USC. Fama, E. F., and K. R. French Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33:3-56. Harvey, C. R., and A. Siddique Autoregressive conditional skewness. Journal of Financial and Quantitative Analysis 34: Harvey, C. R., and A. Siddique Conditional skewness in asset pricing tests. Journal of Finance 55: Jones, C A nonlinear factor analysis of S&P 500 index Option Returns. Journal of Finance. 61:

24 Kahneman, D., and A. Tversky Prospect theory: An analysis of decision under risk. Econometrica. 47(2): Kapadia, N The next Microsoft? Skewness, idiosyncratic volatility, and expected returns. Working paper, Rice University. Kraus, A., and R. H. Litzenberger Skewness preference and the valuation of risky assets. Journal of Finance. 31: Lien, D Moments of truncated bivariate log-normal distributions, Economic Letters. 19: Mitton, T., and K. Vorkink Equilibrium underdiversification and the preference for skewness. Review of Financial Studies 20: Newey, W., and K. West A simple, positive definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55: Ni, S., Stock option returns: A puzzle. working paper Hong Kong University of Science and Technology. Scott, R. C., and P. A. Horvath On the direction of preference for moments of higher order than the variance. Journal of Finance 35: Simkowitz, M., and W. Beedles Diversification in a three-moment world. Journal of Financial and Quantitative Analysis 13: Zhang, Y Individual skewness and the cross-section of average stock returns. Working paper, Yale University. 22

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28 Figure 4: Histogram of call option returns, portfolio 5 (high-skew portfolio) 28

29 Figure 5: Histogram of estimated alphas, portfolio 5 (high-skew portfolio) 29

30 Table I Number of Option Quotes Year Screened Data S T from Ivy S T from CRSP S T Observable ,237 51, , ,329 73, , ,061 94, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,580 31, ,693 Total 2,454,522 2,411,803 31,370 2,443,173 % of Total 98.3% 1.3% 99.5% This table reports summary statistics for individual equity options taken from Ivy Database. We report summary statistics for each year including options that survive our data filter as described in Appendix B, as well as where we obtain the final stock price used in the holding period returns as detailed in equations (2) and (3). 30

31 Table II Portfolio Dimensions: Breadth and Length Panel A. Calls Missing Portfolio Returns in Time Average # of Securities Series Expiration Month Expiration Month 1 (Low) (High) Panel B: Puts Missing Portfolio Returns in Time Average # of Securities Series Expiration Month Expiration Month 1 (Low) (High) This table describes portfolio characteristics for our expected skewness sorted portfolios. Based on the expiration date we sort options into one of five portfolios based on the expected skewness measure detailed in equation (4) and Appendix A. We report the average number of securities in each portfolio across the time series of the data ( ) across the five expected skewness portfolios for eight different maturities as defined in the top row of each panel. Panel A reports the results for call options while Panel B reports the results for put options. On the right side of each panel we report the number of periods where we are unable to calculate a portfolio return due to missing data. 31

32 Table III Average Expected Skewness Panel A: Calls 1 (Low) (High) Panel B: Puts 1 (Low) (High) This table reports the average expected skewness for individual equity option portfolios taken from the Ivy database over the period The portfolios are constructed by sorting on expected skewness as in equation (4) which is also used to construct the ex-ante skewness measures reported in this table. Each period we take the median of the ex-ante skewness measure for each portfolio and then report the time series average of these medians. Panel A reports the results for call options while Panel B reports the results for put options. 32

33 Table IV Average Cross-Sectional Skewness Panel A: Calls 1 (Low) (High) (25.22) -(22.41) -(17.51) -(16.01) -(11.58) -(11.92) -(9.56) -(7.55) Panel B: Puts 1 (Low) (High) (28.96) -(24.46) -(18.00) -(15.57) -(13.50) -(12.16) -(12.86) -(11.42) This table reports the average cross-sectional skewness for individual equity option portfolios taken from the Ivy database over the period The portfolios are constructed by sorting on expected skewness as in equation (4). The skewness estimates that are reported come from each period calculating the cross-sectional skewness coefficient across option returns in the portfolio and then we report the time series average of this skewness coefficient. The last two rows of each panel report the difference between the portfolio 1 (low skewness) and portfolio 5 (high skewness) portfolios along with robust t-statistics testing whether the difference is equal to zero. Panel A reports the results for call options while Panel B reports the results for put options. 33

34 Table V Average Weekly Returns Panel A: Calls 1 (Low) (High) Diff t-stat (7.66) (3.97) (4.16) (3.56) (3.38) (2.31) (1.51) Panel B: Puts 1 (Low) (High) Diff t-stat (13.87) (3.76) (1.70) (1.69) (1.35) (0.95) (1.17) (1.08) This table reports the holding period returns for individual equity option portfolios taken from the Ivy database over the period The portfolios are constructed by sorting on expected skewness as in equation (4) and the returns are holding period returns as in equations (2) for calls and equation (3) for puts. Panel A reports the results for call options while Panel B reports the results for put options. The final two rows of each panel report the long short portfolio return that is long the low expected skewness portfolio (portfolio 1) and short the high expected skewness portfolio (portfolio 5) along with robust t- statistics, that test whether the long-short portfolio returns are equal to zero. 34

35 Table VI CAPM Betas - Single Factor Model Panel A: Calls 1 (Low) (High) Diff t-stat (2.44) (2.85) (2.82) (2.48) (1.78) (1.76) (1.37) (1.33) Panel B: Puts 1 (Low) (High) Diff t-stat (2.13) (2.79) (1.96) (1.95) (1.78) (1.69) (1.58) (1.57) This table reports the estimated betas for individual equity option portfolios taken from the Ivy database over the period The portfolios are constructed by sorting on expected skewness as in equation (4) and the returns are holding period returns as in equations (2) for calls and equation (3) for puts. We estimate a one-factor model as in equation (5) on the portfolio returns and report the betas (on the market portfolio) from those estimations. Panel A reports the results for call options while Panel B reports the results for put options. The final two rows of each panel report the long-short portfolio beta that is long portfolio 1 (low expected skewness portfolio) and short portfolio 5 (high expected skewness portfolio) along with robust t-statistics, that test whether the beta on the longshort portfolio is equal to zero. 35

36 Table VII Weekly Alphas - Single Factor Model Panel A: Calls 1 (Low) (High) Diff t-stat (8.78) (4.24) (4.86) (4.26) (4.17) (3.08) (2.27) (1.70) Panel B: Puts 1 (Low) (High) Diff t-stat (13.12) (4.26) (1.43) (1.22) (0.88) (0.47) (0.68) (0.58) This table reports the estimated alphas for individual equity option portfolios taken from the Ivy database over the period The portfolios are constructed by sorting on expected skewness as in equation (4) and the returns are holding period returns as in equations (2) for calls and equation (3) for puts. We estimate a one-factor model as in equation (5) on the portfolio returns and report the alpha from those estimations. Panel A reports the results for call options while Panel B reports the results for put options. The final two rows of each panel report the long-short portfolio alpha that is long portfolio 1 ( low expected skewness portfolio) and short portfolio 5 (high expected skewness portfolio) along with robust t-statistics, that test whether the alpha long-short portfolio returns are equal to zero. 36

37 Table VIII Weekly Alphas-Single Factor Model Bootstrapped p-values and Simulation p-values Panel A. Panel B. Calls Puts (Low) (Low) (High) (High) Diff Diff t-stat -(5.89) -(3.30) t-stat (4.26) (1.43) b-strap (0.00) (0.00) b-strap (0.00) (0.17) sim (0.01) (0.00) sim (0.00) (0.00) This table reports the estimated alphas for individual equity option portfolios taken from the Ivy database over the period The portfolios are constructed by sorting on expected skewness as in equation (4) and the returns are holding period returns as in equations (2) for calls and equation (3) for puts. We estimate a one-factor model as in equation (5) on the portfolio returns and report the alpha from those estimations. Panel A reports the results for call options while Panel B reports the results for put options. The final two rows of each panel report the long-short portfolio alpha that is long portfolio 1 ( low expected skewness portfolio) and short portfolio 5 (high expected skewness portfolio) along with robust t-statistics, that test whether the alpha long-short portfolio returns are equal to zero. 37

38 Table IX Weekly Alphas Two-Factor Model Panel A: Calls 1 (Low) (High) Diff t-stat (7.98) (3.69) (5.06) (4.73) (4.20) (2.89) (1.78) (1.15) Panel B: Puts 1 (Low) (High) Diff t-stat (10.98) (3.80) (1.90) (1.69) (0.82) -(0.20) -(0.17) -(0.49) This table reports the estimated two-factor alphas for individual equity option portfolios taken from the Ivy database over the period The portfolios are constructed by sorting on expected skewness as in equation (4) and the returns are holding period returns as in equations (2) for calls and equation (3) for puts. We estimate a two-factor model as in equations (6) - (9) on the portfolio returns where factor one is the value-weighted market portfolio and factor two is the return on a delta-neutral straddle on S&P500 index options. We report the alpha from those estimations. Panel A reports the results for call options while Panel B reports the results for put options. The final two rows of each panel report the long-short portfolio alpha that is long portfolio 1 (low expected skewness portfolio) and short portfolio 5 (high expected skewness portfolio) along with robust t- statistics, that test whether the alpha long-short portfolio returns are equal to zero. 38

39 Table X Weekly Alphas - Single Factor Model Early Exercise Panel A: Calls 1 (Low) (High) Diff t-stat (9.13) (4.03) (4.66) (4.29) (3.95) (3.17) (2.52) (1.85) Panel B: Puts 1 (Low) (High) Diff t-stat (12.74) (4.31) (1.44) (1.24) (0.93) (0.29) (0.47) (0.28) This table reports the estimated alphas for individual equity option portfolios taken from the Ivy database over the period The portfolios are constructed by sorting on expected skewness as in equation (4) and the returns are holding period returns as in equations (2) for calls and equation (3) for puts. This table also accounts for early exercise by checking each day to see if exercise is optimal and, if the option satisfies early exercise conditions, we exercise immediately and then hold the proceeds for the remainder of the horizon in a risk-free t-bill. We estimate a one-factor model as in equation (5) on the early-exercise portfolio returns and report the alpha from those estimations. Panel A reports the results for call options while Panel B reports the results for put options. The final two rows of each panel report the long-short portfolio alpha that is long portfolio 1 ( low expected skewness portfolio) and short portfolio 5 (high expected skewness portfolio) along with robust t-statistics, that test whether the alpha long-short portfolio returns are equal to zero. 39

40 Table XI Average Portfolio Volume Panel A: Calls 1 (Low) (High) Panel B: Puts 1 (Low) (High) This table reports the trading volume for individual equity option portfolios taken from the Ivy database over the period Portfolios are constructed by sorting on expected skewness as in equation (4) over a range of maturities detailed in the top row of each panel. Each period we take the average number of contract traded on the day we open the portfolio and then we report the time series average of this measure. Panel A reports the results for call options while Panel B reports the results for put options. 40

41 Table XII Average Portfolio Dollar Volume Panel A: Calls 1 (Low) (High) Panel B: Puts 1 (Low) (High) This table reports the dollar volume for individual equity option portfolios taken from the Ivy database over the period Portfolios are constructed by sorting on expected skewness as in equation (4) over a range of maturities detailed in the top row of each panel. Each period we take the average number of contract traded multiplied by the price of the contract (dollar volume) on the day we open the portfolio and then we report the time series average of this measure. Panel A reports the results for call options while Panel B reports the results for put options. 41

42 Table XIII Weekly Alphas - Single Factor Model Top Volume Options Panel A: Calls 1 (Low) (High) Diff t-stat (6.18) (3.93) (3.55) (2.94) (3.54) (2.71) (1.00) -(0.46) Panel B: Puts 1 (Low) (High) Diff t-stat (10.04) (2.68) (1.56) (1.66) (1.03) (0.42) (0.07) -(0.54) This table reports the estimated alphas for individual equity option portfolios taken from the Ivy database over the period The portfolios are constructed by sorting on expected skewness as in equation (4) and the returns are holding period returns as in equations (2) for calls and equation (3) for puts and in addition we only include options that are in the top 1/3 of the volume distribution of options in each period when we open the option positions. We estimate a one-factor model as in equation (5) on the portfolio returns and report the alpha from those estimations. Panel A reports the results for call options while Panel B reports the results for put options. The final two rows of each panel report the long-short portfolio alpha that is long portfolio 1 (low expected skewness portfolio) and short portfolio 5 (high expected skewness portfolio) along with robust t-statistics, that test whether the alpha long-short portfolio returns are equal to zero. 42

43 Table XIV Average Moneyness Panel A: Calls 1 (Low) (High) Diff t-stat -(34.29) -(30.83) -(27.45) -(20.33) -(16.28) -(14.42) -(14.09) -(12.96) Panel B: Puts 1 (Low) (High) Diff t-stat (32.55) (32.92) (26.66) (20.31) (17.26) (14.33) (13.07) (13.87) This table reports the average moneyness (X/S t ) for individual equity option portfolios taken from the Ivy database over the period Portfolios are constructed by sorting on expected skewness as in equation (4) over a range of maturities detailed in the top row of each panel. Each period we take the average moneyness (X/S t ) on the day we open the portfolio and then we report the time series average of this measure. Panel A reports the results for call options while Panel B reports the results for put options. 43

44 Table XV Average Stock Volatility Panel A: Calls 1 (Low) (High) Diff t-stat -(12.31) -(11.80) -(11.35) -(10.98) -(10.46) -(10.04) -(9.54) -(9.27) Panel B: Puts 1 (Low) (High) Diff t-stat (7.15) (9.54) (8.56) (7.77) (6.90) (5.93) (5.66) (5.57) This table reports the average stock volatility for individual equity option portfolios taken from the Ivy database over the period Portfolios are constructed by sorting on expected skewness as in equation (4) over a range of maturities detailed in the top row of each panel. Each period we take the historical stock return volatility using daily returns over the previous six months on the day we open the option position. We take the average measure across options in the portfolio and then we report the time series average of this measure. Panel A reports the results for call options while Panel B reports the results for put options. 44