Cross-section of Option Returns and Stock Volatility

Transcription

1 Revised November 2010 McCombs Research Paper Series No. FIN Cross-section of Option Returns and Stock Volatility Jie Cao Chinese University of Hong Kong Bing Han McCombs School of Business University of Texas at Austin This paper can also be downloaded without charge from the Social Science Research Network Electronic Paper Collection: Electronic copy available at:

2 Cross-section of Option Returns and Stock Volatility Jie Cao and Bing Han Current Version: September 2010 We thank Henry Cao, John Griffin, Jingzhi Huang, Joshua Pollet, Harrison Hong, Alessio Saretto, Sheridan Titman, Stathis Tompaidis, Grigory Vilkov, Chun Zhang, Yi Zhou and seminar participants at Chinese University of Hong Kong, Tsinghua University, UT-Austin and for helpful discussions. We have benefited from the comments of participants at the Conference on Advances in the Analysis of Hedge Fund Strategies, Quantitative Methods in Business Conference, and Shanghai Winter Finance Conference. All remaining errors are our own. Cao is from the Chinese University of Hong Kong, and Han is from McCombs School of Business, University of Texas at Austin. Contact authors at and Electronic copy available at:

3 Abstract We document a robust new finding on the cross-section of stock option returns: delta-hedged option return decreases monotonically with total volatility of the underlying stock. This is entirely driven by stock s idiosyncratic volatility. A portfolio strategy that sells calls on high volatility stocks and buys calls on low volatility stocks (both delta-hedged) earns about 2% per month. Our results are stronger when cost of arbitrage between stocks and options is higher. They can not be explained by standard stock market risk factors, stock characteristics or volatility risk premium. They are distinct from known volatility related option mispricing. They are consistent with market makers charging higher premia for options on high idiosyncratic volatility stocks because they have higher arbitrage costs and more informed trading. Electronic copy available at:

4 1 Introduction Ang, Hodrick, Xing, and Zhang (2006) first document that stocks with high idiosyncratic volatility tend to earn abnormally low return. This is a puzzling finding, as it goes against positive risk-return tradeoff and the notion that only systematic risk is priced. Subsequent papers argue the result may not be robust to the weighting scheme, only exists in a small subset of stocks, becomes insignificant after controlling for past stock returns or skewness, or is sensitive to different measures of volatility. 1 This paper documents a new finding: there is a negative relation between the cross-section of delta-hedged option returns and the idiosyncratic volatility of the underlying stock. This result is robust to the weighting scheme. It holds for both large and small stocks. It is robust to controlling for size, book-to-market ratio, past stock returns and skewness. It is not sensitive to measurement of volatility: it holds for historical volatility measure used by Ang et al (2006), EGARCH volatility estimate used by Fu (2009) as well as at-the-money option implied volatility. Specifically, we construct a cross-section of individual stock options in each month. The options are written on different underlying stocks, but are all approximately at-the-money and have a common time-to-maturity (about one and a half month). At-the-money options are most sensitive to changes in volatility. For each optionable stock and in each month, we evaluate change in the value of a self-financing portfolio that buys one call or put option, delta-hedged with the underlying stock. The delta-hedge is rebalanced daily so that the portfolio is not sensitive to stock price movement. 2 Our results are obtained from about 210,000 delta-hedged option returns for 6,000 underlying stocks between 1996 and We find that on average, delta-hedged options have negative returns, especially when the underlying stock volatility is high. The delta-hedged option returns decrease monotonically with the total volatility of the underlying stock. This result is entirely driven by stock s idiosyncratic volatility. The same pattern holds for both call options and put options. The delta-hedged options on stocks with high idiosyncratic volatility on average earn significantly 1 See Bali and Cakici (2008), Fu (2009), Huang, Liu, Ghee, and Zhang (2009), Boyer, Mitton, and Vorkink (2010). 2 By construction of delta-hedged option portfolio, our finding is distinct from Ang et al s result for the stock returns. During our sample period, Ang et al s result is insignificant for stocks with traded options. 1 Electronic copy available at:

5 lower returns than those on low idiosyncratic volatility stocks. For example, a portfolio strategy of selling call options on high volatility stocks and buying options on low volatility stocks (both delta-hedged) earns about 2% per month. We explore a number of potential explanations for our results. The first explanation is that the average delta-hedged option returns embeds a negative volatility risk premium whose magnitude increases with the volatility level. If options can be perfectly replicated by the underlying stock (e.g., under the Black-Scholes model), delta-hedged options are riskless and should earn zero return on average. In reality, options are not redundant and deltahedged options are risky. It is well known that stock return volatility is time-varying. Deltahedged options are positively exposed to the volatility risk. In cross-sectional regressions, we find that delta-hedged option return is significantly positively related to the volatility risk premium of the underlying stocks, or the difference between a model-free measure of riskneutral expected volatility and the expected volatility under the physical measure computed from high frequency return data. 3 But after controlling for the stock volatility risk premium, the regression coefficient of delta-hedged option returns on stock volatility remains negative and significant. We also run time-series regressions of the returns to the portfolio strategy of selling options on high volatility stocks and buying options on low volatility stocks on several proxies of market volatility risk and common idiosyncratic volatility risk. Only a small portion of our portfolio s returns can be explained by the exposure to the market volatility risk or the common idiosyncratic volatility risk. Our portfolio strategy still has significant positive alpha after controlling for these volatility-related risk factors. Therefore, our result can not be explained by volatility risk premium or by exposures to either market volatility risk and common idiosyncratic volatility risk. The profitability of our option strategy can not be explained by the Fama-French three factors, the momentum factor or the jump risk. A second potential explanation of our result is volatility-related option mispricing. Stocks with high current volatility are likely to have experienced increase in volatility recently. If investors overreacted to recent changes in volatility (see, e.g., Stein (1989), Poteshman(2001)), and paid too much for options on stocks with high current volatility, then it could explain 3 Our measure is identical to that used in Bollerslev, Tauchen and Zhou (2009). The only difference is they measure market volatility risk premium while we do it for individual stocks. 2

6 our result. However, after we control for recent change in volatility, we still find a significant negative relation between delta-hedged option return and the volatility of the underlying stock. To further examine whether volatility-related option mispricing can explain our result, we control for the difference between historical realized volatility and at-the-money implied volatility. This variable is motivated by Goyal and Saretto (2009) who argue that large deviations of implied volatility from historical volatility are indicative of mis-estimation of mean reversion in volatility. Consistent with Goyal and Saretto (2009), we find that delta-hedged options on stocks with large positive difference between historical volatility and implied volatility have higher returns. But after controlling for the difference between historical and implied volatility, we find a more negative relation between delta-hedged option return and stock volatility. Thus, volatility-related mispricing of Goyal and Saretto (2009) exacerbates rather than explains our result. A third potential explanation is that our result reflects limits to arbitrage between stocks and options. Figlewski (1989) shows that market imperfections (e.g., transaction costs) limit the effectiveness of option arbitrage. Prior studies have found that idiosyncratic volatility is correlated with transaction costs and imposes a significant holding cost for arbitrageurs (e.g., Pontiff (2006)). Consistent with the impact of limits to arbitrage between stocks and options, we find that the delta-hedged option return is significantly related to option bid-ask spreads, option trading volume or open interests, Amihud illiquidity measure of the underlying stock and stock price level. But after controlling for these variables, the negative relation between delta-hedged option return and stock volatility remains significant. A voluminous literature has studied the cross-section of stock returns, but papers that examine the cross-section of option returns are sparse. We provide a comprehensive study of the cross-sectional determinants of delta-hedged option returns. Duarte and Jones (2007) regress delta-hedged individual stock option returns on their exposure to the market volatility risk. They do not examine how delta-hedged stock option return is related to the total or idiosyncratic volatility of the underlying stock, which is the focus of our study. In addition, our paper examines additional theory-motivated variables (not examined in Duarte and Jones (2007)) that are expected to be related to delta-hedged stock option returns. Goyal and Saretto (2009) focus on volatility related option mispricing and relate volatility sensi- 3

7 tive option positions such as straddles and delta-hedged options to the difference between historical realized volatility and at-the-money option implied volatility. Bali and Murray (2010) study the return of skewness asset constructed from a pair of option positions (one long and one short) and a position in the underlying stock. By design, their skewness asset is not exposed to changes in the price of the underlying stock or changes in stock volatility. Ang and Bali (2010) find that stocks with high returns over the past month tend to have call option contracts that exhibit increases in implied volatility over the next month. They do not directly study option returns. The rest of the paper is organized as follows. Section 2 describes the data and the key variables. Section 3 presents Fama-MacBeth regression results and test several potential explanations of our results. Section 4 presents an option trading strategy and portfoliosorting results. Section 5 discuss additional explanations of our results and concludes the paper. 2 Data and Delta-hedged Option Returns We use data from both the equity option and stock markets. For the January 1996 to October 2009 sample period, we obtain data on U.S. individual stock options from the Ivy DB database provided by Optionmetrics. The data fields we use include daily closing bid and ask quotes, trading volume and open interest of each option, implied volatility as well as the option s delta and vega computed by OptionMetrics based on standard market conventions. We obtain daily and monthly split-adjusted stock returns, stock prices, and trading volume from the Center for Research on Security Prices (CRSP). For each stock, we also compute the book-to-market ratio using the book value from COMPUSTAT. Further, we obtain the daily and monthly Fama-French factor returns and risk-free rates from Kenneth French s data library. 4 At the end of each month and for each optionable stock, we collect a pair of options (one call and one put) that are closest to being at-the-money and have the shortest maturity among those with more than 1 month to expiration. We apply several filters to the extracted option data. First, U.S. individual stock options are of the American type. Our main analyses 4 The data library is available at 4

8 use call options whose stocks do not have ex-dividend dates prior to option expiration (i.e., we exclude an option if the underlying stock paid a dividend during the remaining life of the option). These call options we analyze are effectively European (e.g., Merton (1973)). 5 Second, we exclude all option observations that violate obvious no-arbitrage conditions such as S C max(0, S Ke rt ) for a call option C where S is the underlying stock price, and K is the option strike price, T is time to maturity of the option, and r is the riskfree rate. Third, to avoid microstructure related bias, we only retain options that have positive trading volume, positive bid quotes and where the bid price is strictly smaller than the ask price, and the mid-point of bid and ask quotes is at least $1/8. We keep only the options whose last trade dates match the record dates and whose option price dates match the underlying security price dates. Fourth, the majority of the options we pick each month have the same maturity. We drop the options whose maturity is longer than that of the majority of options. Thus, we obtain, in each month, reliable data on a cross-section of options that are approximately at-the-money with a common short-term maturity. Our final sample in each month contains, on average, options on 1514 stocks. Table 1 shows that the average moneyness of the chosen options is 1, with a standard deviation of only The time to maturity of the chosen options range from 47 to 52 calendar days across different months, with an average of 50 days. These short-term options are the most actively traded, have the smallest bid-ask spread and provide the most reliable pricing information. We utilize this option data to study the cross-sectional determinants of expected option returns. To measure delta-hedged call option return, we first define delta-hedged option gain, which is change in the value of a self-financing portfolio consisting of a long call position, hedged by a short position in the underlying stock so that the portfolio is not sensitive to stock price movement, with the net investment earning risk-free rate. Following Bakshi and Kapadia (2003a), we define delta-hedged gain for a call option portfolio over a period [t, t+τ] as ˆΠ(t, t + τ) C t+τ C t t+τ t u ds u t+τ t r u (C u u S u )du, (1) where C t is the call option price, t = C t / S t is the delta of the call option, r is the riskfree 5 For the short-maturity options used in our study, the early exercise premium is small. We verify that our results do not change materially when we include options for which the underlying stock paid a dividend before option expire. 5

9 rate. Our empirical analysis uses a discretized version of (1). 6 Specifically, consider a portfolio of a call option that is hedged discretely N times over a period [t, t + τ], where the hedge is rebalanced at each of the dates t n, n = 0, 1,, N 1 (where we define t 0 = t, t N = t + τ). The discrete delta-hedged call option gain is Π(t, t + τ) = C t+τ C t N 1 n=0 C,tn [S(t n+1 ) S(t n )] N 1 n=0 a n r tn 365 [C(t n) C,tn S(t n )], (2) where C,tn is the delta of the call option on date t n, r tn is annualized riskfree rate on date t n, a n is the number of calendar days between t n and t n+1. Definition for the delta-hedged put option gain is the same as (2), except with put option price and delta replacing call option price and delta. 7 We define delta-hedged call option return as delta-hedged option gain scaled by the price of the underlying stock Π(t, t+τ)/s t. 8 Merton (1973) shows that option price is homogeneous of degree one in the stock price and the strike price. So for a fixed moneyness, the option price scales with the price of the underlying stock. We scale the delta-hedged option gains by the prices of the underlying stocks so that they are comparable across stocks. 3 Empirical results 3.1 Average Delta-hedged Option Returns First, we examine the time-series average of delta-hedged option returns for individual stocks. Table 1 Panel A and B show that for both call options and put options, the cross-sectional mean and median of the time-series average delta-hedged option returns are negative. For 6 Discretely rebalanced hedge may introduce a bias in the expected discrete delta-hedged gain. However, both Bakshi and Kapadia (2003a) and Duarte and Jones (2007) show that the bias is small, especially when the rebalance is done frequently (such as daily). 7 Following Carr and Wu (2009) as well as Goyal and Saretto (2009), our delta hedges rely on implied volatilities and Greeks from the Black-Scholes model. We also compute option delta based on GARCH volatility estimated using the returns of the underlying stock and obtain similar results. In fact, the deltahedged option returns from these two measures of volatilities have means that are not significantly different from each other and their average correlation is as high as We verify that our results are robust when we scale the delta-hedged option gain by the option price Π(t, t + τ)/c t (see Table 7). 6

10 example, held to maturity, delta-hedged call option position on average loses 0.48% (4.89%) of the the stock (call option) value at the beginning of the period. 9 This result is similar for one month holding period. Table 1 Panel C reports result of t-test for the time-series mean of individual deltahedged option returns. We have time series observations of call options on 6141 stocks. About 75% of them have negative average delta-hedged call option returns and 36% of them have significantly negative average delta-hedged returns. In contrast, the average deltahedged call option return is significantly positive (t >2) for about only 1.2% of the cases. The pattern for the put options is similar. 3.2 Delta-hedged Option Returns and Volatility All results in the rest of Section 3 are based on monthly Fama-MacBeth regressions. For Tables 2 to 6, the dependent variable in month t s regression is delta-hedged call option return, i.e., Π(t, t + τ)/s t, delta-hedged call option gains till maturity (about one and a half month) scaled by the underlying stock price at the beginning of the period. In Table 7, we do robustness checks using put options and different measures of delta-hedged option return (e.g., scaled delta-hedged option gains by the option price, or holding for one month). Table 2 Model 1 is the univariate regression of delta-hedged option returns on stock return volatility V OL. Following Ang, Hodrick, Xing, and Zhang (2006), we measure V OL as the standard deviation of daily stock returns over the previous month. The V OL coefficient estimate is , with a significant t-stat of Thus, delta-hedged option return decreases with the total volatility of the underlying stock. In Model (2), we add option vega as a regressor to control for cross-sectional difference in option moneyness. 10 All options in each month s cross-sectional regressions have a common maturity. They may differ in moneyness although we select the options that are closest to being at the money in each 9 Bakshi and Kapadia (2003a) report that for close to at-the-money call options on S&P 500 index with days left to maturity, the average delta-hedged option return till maturity (measured as delta-hedged option gain scaled by initial stock price) is -0.13%. Their sample period is from 1988 to In comparison, the average delta-hedged option return for these call options on S&P 500 index during our sample period of 1996 to 2009 is -0.25%. 10 In addition to option vega, we also control for theta. Options on high volatility stocks have higher theta than those on low volatility stocks. In univariate regression, option theta is negatively correlated with deltahedged option return. In the presence of other control variables, theta loses its significance but volatility is still significantly negative. 7

11 month. The point estimate and t-stat of the V OL coefficient in Model 2 barely change from Model 1. In Model 3 of Table 2, we decompose individual stock volatility into two components: idiosyncratic volatility IV OL and systematic volatility SysV ol. We measure idiosyncratic volatility as the standard deviation of the residuals of the Fama-French 3-factors model estimated using the daily stock returns over the previous month, and systematic volatility is V OL2 IV OL 2. When both idiosyncratic volatility and systematic volatility are included as regressors, the estimated coefficient of idiosyncratic volatility is with a t-stat of In contrast, the estimated coefficient of systematic volatility is with a t-stat of Thus, the significantly negative relation between delta-hedged option return and the volatility of the underlying stock is entirely driven by the idiosyncratic volatility. Table 2 Models 4, 5 and 6 show that our results are robust to alternative measures of stock volatility, such as option implied volatility, the fitted value or the one-period-ahead forecast from the EGARCH(1,1) model used in Fu (2009) Controlling for Volatility Risk Premium Under the Black-Scholes model, the call option can be replicated by trading the underlying stock and riskfree bond. In this case, the discrete delta-hedged gain in Equation (2) has a symmetric distribution centered around zero (e.g., Bertsimas, Kogan, and Lo (2000)). When volatility is stochastic and volatility risk is priced, the mean of delta-hedged option gain would be different from zero, reflecting the volatility risk premium. 13 Specifically, consider a generic stochastic volatility model, where the dynamics of the 11 Duan and Wei (2009) find a positive relation between the option implied volatility and the proportion of stock s total volatility that is systematic. 12 Each month and for each stock in our sample, we estimate the EGARCH(1,1) model using all available historical monthly stock returns since 1963, if at least 5 years of historical data are available. 13 Coval and Shumway (2001) and Bakshi and Kapadia (2003a) find strong evidence of a negative price of market volatility risk. Carr and Wu (2009) study the variance risk premiums by examining the average return to long variance swaps. Out of the 35 individual stocks they study, only seven generate variance risk premiums that are significantly negative. Driessen, Maenhout and Vilkov (2009) compare the model free implied variance and realized variance. They find no evidence for the presence of a significant negative variance risk premium in individual stock options. Buraschi, Trojani and Vedolin (2009) exploit the difference in the volatility risk premia of individual and index options. 8

12 underlying stock and its return volatility under the empirical measure are given by ds t S t = µ t [S t, σ t ]dt + σ t dwt 1, dσ t = θ t [σ t ]dt + η t [σ t ]dwt 2, with Corr(dW 1 t, dw 2 t ) = ρdt. Without imposing restrictions on the pricing kernel or the volatility process, Bakshi and Kapadia (2003a) show that E t [ˆΠ(t, t + τ)] = t+τ ( ) dm where λ t [σ t ] Cov t m t, dσ t is the volatility risk premium. t ( E t λ u [σ u ] C ) u du, (3) σ u Equation (3) says that the delta-hedged option return is positively related to the volatility risk premium. Existing option pricing models with stochastic volatility specify volatility risk premium as a function of the volatility level. For example, in Heston (1993), the volatility risk premium is linear in volatility. Thus, the negative relation between delta-hedged option return and stock volatility is consistent with a negative volatility risk premium whose magnitude increase with the volatility level. To examine whether this explains our result, we control for volatility risk premium in Table 3. We measure volatility risk premium of stock i in month t as 14 V RP i,t = RV i,t IV i,t where IV i,t is a model free estimate of the risk-neutral expected variance extracted from stock options and RV i,t is a model free estimate of the physical expected variance based on high-frequency stock trading data (see Appendix 1 for details). Our estimates of implied variance IV and realized variance RV are identical to those in Bollerslev, Tauchen and Zhou (2009) and similar to Buraschi, Trojani and Vedolin (2009). 15 Table 3 Model 1 is identical to Table 2 Model 1, except the sample size now is about 14% smaller because we require enough observed option prices for a given stock in order to 14 For this expression of volatility risk premium, see also Buraschi, Trojani and Vedolin (2009) who use the negative of volatility risk premium IV i,t RV i,t. 15 Buraschi, Trojani and Vedolin (2009) estimate realized variance based on daily stock returns while our measure is based on intra-daily stock returns. 9

13 measure the stock s volatility risk premium. The coefficient on stock volatility is significantly negative in Table 3 Model 1. Model 2 shows that our measure of individual stock volatility risk premium is significantly positively related to delta-hedged option return. This result suggests that part of the negative average delta-hedged option return is compensation for option sellers who are unable to eliminate stock volatility risk. 16 After we control for individual stock volatility risk premium, the volatility coefficient estimate is reduced in magnitude but remains statistically significant. Table 3 model 3 shows that, just like in Table 2 Model 3, the negative relation between delta-hedged option return and the total volatility of the underlying stock is entirely driven by the idiosyncratic volatility. In the presence of individual stock volatility risk premium, we still find a significant negative coefficient for the idiosyncratic volatility and a significant positive coefficient for the systematic volatility. Thus, our result can not be explained by volatility risk premium. In Table 3 Model 4, we examine whether our result can be explained by a state-dependent jump risk premium. For example, in Pan (2002), the jump-arrive intensity is linear in the volatility level, and the jump-risk premium is linear in stock volatility V OL. Following Bakshi and Kapadia (2003a), we control for the jump risk by including the option implied risk-neutral skewness and kurtosis of the underlying stock return. Appendix 2 provides details of these measures. Table 3 Model 4 shows that the coefficients of risk-neutral skewness and kurtosis are negative and statistically significant. However, after controlling for the jump risk proxies, there is still a significant negative relation between delta-hedged option return and stock return volatility. Thus, our result can not be explained by jump risk premium. 3.4 Controlling for Volatility-related Option Mispricing Another potential explanation of our result is some volatility related option mispricing. First, it could be stocks with high current volatility are those that have experienced increase in volatility recently. If investors overreacted to recent changes in volatility (see, e.g., Stein (1989), Poteshman(2001)), and paid too much for options on stocks with high current volatility, then the subsequent return of delta-hedged option return would be low. In Table 4, we control for the average monthly change in stock volatility V OL. Delta-hedged option return 16 For example, Figlewski and Green (1999) show that option sellers are exposed to volatility risk which can produce very large losses. 10

14 tends to be lower after recent increase in volatility. This is consistent with the overreaction to volatility story. More importantly, after we control for recent change in volatility, we still find a significant negative relation between delta-hedged option return and the volatility of the underlying stock. Second, Goyal and Saretto (2009) provide evidence of volatility mispricing due to investors failure to incorporate the information contained in the cross-sectional distribution of implied volatilities when forecasting individual stock s volatility. They argue that large deviations of implied volatility from historical volatility are indicative of mis-estimation of mean reversion in volatility. They find that options with high implied volatility (relative to historical volatility) earn low returns and cheap options with low implied volatility (relative to historical volatility) earn high returns. 17 Table 4 Model 3 controls for the log difference between historical and at-the-money option implied volatility, the same variable used by Goyal and Saretto (2009). 18 Consistent with their result, we find that delta-hedged option return increases with the log difference between the underlying stock s historical volatility and implied volatility. Interestingly, after controlling for the log difference between historical and implied volatility, the coefficient for stock s volatility becomes even more negative. The V OL coefficient estimate is now (Model 4), compared to (Model 1) without controlling for the Goyal and Saretto variable. The t-statistic of the V OL coefficient also doubles. Thus, volatility-related mispricing documented by Goyal and Saretto (2009) exacerbates rather than explains our result. Finally, Model 6 further controls for the change in the implied volatility of the same option over the same time period as the dependent variable, the delta-hedged option return. If the negative relation between delta-hedged option return and the stock volatility at the beginning of the period just reflects the correction of some kind of volatility-related option mispricing, then it should become insignificant once we control for the contemporaneous change in implied volatility. highly significant. However, we find that the V OL coefficient continues to be 17 The finding of Goyal and Saretto (2009) is based on returns of straddles and delta-hedged options. Unlike our study, they do not rebalance the delta-hedges. 18 Our results do not change when we use the difference (rather than log difference) between historical and at-the-money option implied volatility. 11

15 3.5 Controlling for Stock Characteristics In Table 5, we control for several stock characteristics such as past stock returns. It is well known that stock return and volatility are negatively correlated. Huang, Liu, Ghee, and Zhang (2009) report that the volatility-return relation in the cross-section of stocks becomes insignificant when past one-month return is used as a control variable. Amin, Coval, and Seyhun (2002) show that past 60 days stock market index return affects the put-call parity relation and the slope of index option implied volatility smile. We examine how the crosssection of delta-hedged option returns depends on stock s past returns, and whether past stock returns affect the relation between delta-hedged option returns and stock volatility. We find that delta-hedged call option return is significantly and positively related to the underlying stock return over past one year as well as between three years and one year ago. The coefficient of past one month return is positive and significant by itself, but is not robust to additional regressors, especially the volatility mispricing variable of Goyal and Saretto (2009). These findings can not be explained by stock return predictability by past returns. First, we examine delta-hedged options that are not sensitive to stock price movement by construction. Second, it is hard to explain why past return between three years and one year ago is positively related to delta-hedged call option return but negatively related to stock return. Third, it is hard to explain why the effects of past stock returns are the same for delta-hedged put option returns (see Table 7 Panel B). Ang and Bali (2010) find that stocks with high returns over the past month tend to have call option contracts that exhibit increases in implied volatility over the next month. This is consistent with our result that delta-hedged option return is positively correlated with past stock returns. In unreported regressions, we find that the positive relation between delta-hedged option return and past stock return remains significant after controlling for contemporaneous change in option implied volatility. Table 5 also controls for control for the size (ME) and book-to-market ratio (BE/ME) of the underlying stock as they are significant predictors of the cross-section of stock returns. Following Fama and French (1992), we measure ME as the product of monthly closing stock price and the number of outstanding common shares in previous June. BE/ME is the previous fiscal-yearend book value of common equity divided by the calendar-yearend 12

16 market value of equity. We find no reliable relation between delta-hedged option return and either size or book-to-market ratio of the underlying stock. The V OL coefficient is highly significant and negative in all regressions reported in Table 5. Controlling for stock characteristics such as size, book-to-market ratio and past stock returns does not materially affect the significant negative relation between delta-hedged option return and stock volatility. 3.6 Limits to Arbitrage Delta-hedged options can be viewed as arbitrage positions between options and the underlying stocks. Recent studies document that there are limits to arbitrage in the options market. Thus, no-arbitrage approach can only establish very wide bounds on equilibrium option prices and investors net buying pressure importantly affects option prices (e.g., Figlewski (1989), Bollen and Whaley (2004), Garleanu, Pedersen, and Poteshman (2009)). Investors demand for options could be higher when the underlying stock has higher volatility, either for hedging or speculative purpose. We expect that delta-hedged option return depends on option demand pressure. To test this idea, and to examine whether it can explain the negative relation between delta-hedged option return and stock volatility, Table 6 controls for the effect of option demand pressure using individual option s open interest as a proxy. We use (option open interest / stock volume) 10 3, where open interest is measured at the end of the month and stock volume is the monthly total trading volume. Our results are qualitatively the same if we use option trading volume instead of open interest, or if we scale by stock s total shares outstanding. Table 6 also controls for several proxies of transaction cost and price impact. The motivation is that the arbitrage between stock and option is more difficult to implement for illiquid stocks, which tend to have high volatility. We control for various liquidity measures for options and underlying stocks, such as option s bid-ask spread, stock price and the Amihud (2002) measure of the price impact for stocks. The Amihud illiquidity measure for stock 13

17 i at month t is defined as IL i,t = 1 D t D t d=1 R i,d /V OLUME i,d, where D t is the number of trading days in month t, R i,d and V OLUME i,d are, respectively, stock i s daily return and trading volume in day d of month t. Table 6 Model 1 shows that delta-hedged option returns decrease with option open interest, which has a significantly negative coefficient of (t-stat -9.44). This is consistent with the idea that option market makers charging higher premium for options with large end-users demand. Consistent with the impact of market friction and limits to arbitrage, we find delta-hedged option returns are more negative when the option bid-ask spread is higher, when the underlying stock less liquid and when the underlying stock price is low (see Table 6 Model 2 to 4). Importantly, the negative relation between delta-hedged option return and stock volatility remains significant after controlling for the limits of arbitrage proxies above, although the magnitude of the V OL coefficient is reduced. Thus, while our regression results support the idea that limit to arbitrage plays an important role in determining delta-hedged option return, it can not fully explain our main finding, the negative relation between delta-hedged option return and stock volatility. 3.7 Robustness Checks In previous regression tables, the dependent variable, delta-hedged option returns, are measured as delta-hedged option gains till maturity scaled by the initial stock price. In Table 7 Panel A, the measurement of dependent variable changes but the regressors are the same as in previous tables. We try alternative holding periods, such as one week or one month, and we scale by the initial option price rather than by the initial stock price as in the previous tables. In all variations, we still find a significant negative V OL coefficient. Previous tables report the results for call options. In Table 7 Panel B, we re-run the regression tests for put options. We find that delta-hedged put option return on average decrease with the volatility of the underlying stock. The estimated V OL coefficient in the 14

18 case of put options is virtually the same as that for call options (for the same specification). Using a panel regression on 25 individual stock options, Bakshi and Kapadia (2003b) find that delta-hedged option returns is negatively related to individual stock volatility (with a t-stat of -1.64). Our results are obtained from a much larger sample using Fama-MacBeth regressions. In an unreported robustness check, we re-estimate our models using panel regressions and find again a negative relation between delta-hedged option returns and stock volatility, consistent with the results from Fama-MacBeth regressions. To summarize, the strong negative relation between delta-hedged option returns and stock volatility is robust to alternative measures of delta-hedged option returns, different holding periods, and holds for both call options and put options. Our result is robust to regression methodology. The next section documents our result using portfolio sorting approach. 4 Volatility-based Option Trading Strategy This section analyzes an option trading strategy motivated by the Fama-MacBeth regression results in the previous section. At the end of each month and for each stock, we form a deltaneutral position that sells calls against a long position of the underlying stock. This position is held for one month and then closed out. It is essentially the opposite of delta-hedged call position studied in the previous section, except that here we do not rebalance the delta hedge. The return to selling a covered call over [t, t+1] is H t+1 /H t 1 where H t = S t C t, with C and S denoting call option price and the underlying stock price, being the Black- Scholes call option delta at initial t. Our option strategy sells (delta-hedged) call options on high volatility stocks and goes long (delta-hedged) call options on low volatility stocks. 4.1 Portfolio Sorts Table 8 reports the average return of selling delta-hedged call options for various portfolios sorted by total stock volatility (Panel A) or by idiosyncratic volatility (Panel B). We try three weighting schemes in computing the average portfolio return: equal weight, weight by the market capitalization of the underlying stock or by the market value of total option open interests on each stock (at the formation of option portfolio). Our results are consistent 15

19 across different weighting schemes. Results are very similar for total volatility sorts and for idiosyncratic volatility sorts. Table 8 shows that the average return of selling delta-hedged calls is positive. Corresponding to the significant negative relation between delta-hedged option return and stock volatility in the regressions, we find that the average return to selling delta-hedged calls on high volatility stocks is significantly higher than that on low volatility stocks. When we sort on total stock volatility, the difference ranges from 1.67% to 2.47% per month, depending on the weighting scheme. It is between 1.63% and 2.31% per month when we sort on idiosyncratic stock volatility. All of these return differences are significant both statistically and economically. Table 8 Panel C reports subsample results. The average return to the strategy of selling delta-hedged calls on high volatility stocks and buying delta-hedged calls on low volatility stocks is significantly positive in all subsamples of stocks sorted by size, although its magnitude decreases monotonically with the market capitalization of the underlying stock. Our option strategy is profitable both in January and in the rest of the year. It exists four sub-periods of our sample (1996 to 1999, 2000 to 2003, 2004 to 2006 and 2007 to 2009). The average equal-weighted return to our strategy is over 2% per month in all subsamples. Remarkably, our option strategy does not experience a single down month between 2004 and 2006, a period of relative low market volatility. It also delivers positive return more than 90% of the months between 1996 to 1999, and between 2007 to 2009 (see Figure 1). 4.2 Controlling for Common Risk Factors Next we examine whether return of our option strategy can be explained by common risk factors. We consider four risk factors from the stock market (the Fama-French three factors plus the momentum factor) and two volatility risk factors. The first volatility risk factor is the zero-beta straddle return on the S&P 500 index, which proxies for the systematic market volatility risk (e.g., Coval and Shumway (2001) and Carr and Wu (2009)). For robustness, we also measure the market volatility risk by change in the VIX index from the Chicago Board Options Exchange following Ang et al (2006). The second volatility risk factor is the common individual stock variance risk used in Driessen, Maenhout and Vilkov (2009). It 16

20 is measured as value-weighted zero-beta straddle returns on the individual stocks that are components of the S&P 500 index. All risk factors are measured monthly, just as our option portfolios. We perform time-series regression of the monthly return of our option strategy on the risk factors above. The result is reported in Table 9. We find that none of the four stock market risk factors is significantly related to the return of our option strategy. The estimated coefficients for the volatility risk factors are negative and significant in univariate regressions, suggesting that our option strategy tends to do poorly when volatility risks increase. In multivariate regressions, only the common individual stock variance risk has a significant negative coefficient. The two proxies of market volatility risk are not significantly related to the return of our option strategy. Controlling for the volatility risk factors reduce the alpha of our option strategy from 2.3% to about 2% per month, which is still economically and statistically significant. Thus, exposure to the volatility risk factors can not fully explain the positive average abnormal return of our option portfolio strategy. 4.3 Transaction Costs In all of our previous results, we use the mid-point of the closing option bid and ask quotes. Table 10 examines the impact of transaction cost on the profitability of our volatility-based option portfolio strategy. To take into account the costs associated with buying or selling options, we assume the effective option spread is equal to 50%, 75%, or 100% of the quoted spread. Corresponding to these transaction cost assumptions, the average return to the equal weighed portfolio strategy of selling delta-hedged calls on high volatility stocks and buying delta-hedged calls on low volatility stocks decreases from 2.31% per month when evaluated at the mid-point of bid and ask to 1.19%, 0.65% and 0.13% respectively. The t-stat of the average return to our option strategy also declines as transaction cost gets bigger. When the effective spread equals the quoted spread (i.e., buy option at the ask and sell at the bid), the average monthly return of our strategy is no longer statistically significant. The results are the same when we consider the Fama-French three-factor alphas rather the raw returns. Hence, our option strategy can be taken advantage of only by market participants who face relatively low transaction costs. 17

21 Table 10 Panel B documents how the profitability of our option strategy varies with stock or option liquidity. Each month, we first sort the option sample into five quintiles by the underlying stock price or its Amihud (2002) illiquidity measure, or by the option bid-ask spread. Then within each quintile, we further sort by the volatility of the underlying stock. Panel B shows that the average return of our option strategy is significantly higher for illiquid stocks, low priced stocks, as well as for options with high bid-ask spread. We also verify these results in unreported Fama-MacBeth regressions that include V OL, stock price, Amihud illiquidity measure, option bid-ask spread, as well as V OL stock price, V OL Amihud measure, and V OL option bid-ask spread as regressors. 5 Discussions and Conclusions This paper provides a comprehensive study of individual stock option returns (after deltahedging the exposure to the underlying stocks). We find that the average delta-hedged option return tends to be negative and decrease monotonically with the idiosyncratic volatility of the underlying stock. This finding holds for both call options and put options. It is roust and significant, both statistically and economically. Our tests rule out explanations based on common stock market risk factors or stock characteristics. We find that part of the negative average delta-hedged option return is compensation for option sellers who are unable to eliminate stock volatility risk. 19 But the negative relation between delta-hedged option return and stock volatility remains significant after controlling for volatility risk premium. Exposure to market volatility risk or common idiosyncratic volatility risk only explains a small portion of the profitability of an option portfolio strategy that sells delta-hedged options on high volatility stocks and buys delta-hedged options on low volatility stocks. Further, known volatility-related mispricing exacerbate rather than explain our result. Our results could not be fully explained by existing explanations for why high volatility assets have low average returns. One such explanation is investors preference for positive skewness (e.g., Boyer, Mitton, and Vorkink (2010)). By definition, options have positively 19 Investors who hold undiversified portfolios would be willing to pay a premium for delta-hedged options because they provide valuable hedges (i.e., they tend to have positive returns when stock price drops and volatility increases). 18

22 skewed payoffs. We verify that call option skewness increases with the volatility of the underlying stock. Hence, it is possible that investors with skewness preference are willing to pay a higher price and accept a lower expected return for call options on high volatility stocks because such options offer more positive skewness. However, the same argument does not apply to the put options: we find that put options on high volatility stocks offer lower, not higher, skewness. So skewness preference could not explain the negative relation between delta-hedged put options and the volatility of the underlying stocks. Another explanation for why high volatility assets have low average returns is realization utility. Barberis and Xiong (2009) show that investors with realization utility hold onto risky asset till they have a sufficient gain. Our results are based on options with about one and a half month till maturity. Investors may not have the luxury of holding onto these options until they have a gain. In addition, unlike stocks, options lose value over time. The time decay of option value is especially severe for high volatility stocks. Thus, it is unlikely that realization utility investors would find short term options on high volatility stocks attractive. We find support for the idea that limits to arbitrage proxies importantly affect the crosssection of delta-hedged option returns. Moreover, the relation between the delta-hedged option return and idiosyncratic volatility of the underlying stock is stronger when it is more costly to arbitrage between options and stocks. Thus, our results are consistent with market makers charging a higher premium for options on high idiosyncratic volatility stocks because these options have higher arbitrage costs (e.g., Pontiff (2006)). Our results are also consistent with informed trading in options. 20 It is likely that there is more private information in stocks with high idiosyncratic volatility (e.g., Durnev, Yeung and Zarowin (2003)). Back (1993) shows that asymmetric information can make it impossible to price options by arbitrage. Market makers get hurt by the informed trading in options. They charge a higher premium for options on high idiosyncratic volatility stocks because there are more informed trading. Finally, another new empirical finding of this paper is that delta-hedged option returns for past winner stocks are significantly higher than delta-hedged option returns for past loser stocks. This option momentum pattern holds for both individual stock call options 20 See e.g., Amin and Lee (1997), Cao, Chen, and Griffin (2005), Pan and Poteshman (2006) for evidence of informed trading in options. 19