How to Analyze Real Estate Portfolio Volatility and Returns

Transcription

1 Asset Pricing in the Stock and Options Markets Aurelio Vasquez Bene ciario Colfuturo 2007 Desautels Faculty of Management McGill University Montreal, Quebec November 2010 A thesis submitted to McGill University in partial ful lment of the requirements for the degree of Doctor of Philosophy Copyright c 2010 by Aurelio Vasquez

2 Dedication To Claudia, Alejandro, María José and my parents.

3 Abstract This thesis comprises three essays on asset pricing on the stock and options markets. The rst essay nds a positive relation between the slope of the volatility term structure and subsequent option returns. The second essay nds a negative relation between realized skewness, extracted from high-frequency data, and stock returns. The third essay nds a negative relation between price jumps of intraday data and future stock returns. i

4 Résumé Cette thèse se compose de trois essais qui analysent l évaluation d actifs dans le marché boursier et le marché d options. Le premier essai trouve une relation positive entre la pente de la surface de volatilité implicite et les rendements futurs des options. Le deuxième essai trouve une relation négative entre le coe cient de dissymétrie, calculé a partir des données intra-journalières, et les rendements des actions. Le troisième essai trouve une relation negative entre les sauts des prix intra-journaliers et les rendements futurs des actions. ii

5 Acknowledgment I would like to thank my co-supervisors, Professors Peter Christo ersen and Kris Jacobs. Their comments and feedback on the chapters of this thesis were very helpful and showed me the path to become a better researcher and academic. I also want to thank Professor Lars Stentoft, a member of my committee, for his undivided time and his helpful comments. I want to acknowledge the nancial support of the Institut de Finance Mathématique de Montréal (IFM 2 ) as well as the Colfuturo Scholarship Loan Programme. To my wife Claudia, who supported my decision to become an academic. Most important, she unconditionally supported me and our kids, Alejandro and María José, during these ve years of doctoral studies. iii

6 Contribution of Authors The second chapter of this thesis titled, "Skewness from High-Frequency Data Predicts the Cross-Section of Stock Returns", is a joint collaboration with Diego Amaya, a PhD student at HEC Montreal. Diego is currently in the last year of his PhD. We both have made equal contributions to this paper. I am solely responsible for the rst and third chapters of this thesis, which are titled, "Volatility Mean Reversion and the Cross-Section of Option Returns" and "Explaining Stock Returns with Intraday Jumps", respectively. iv

7 Table of Contents Table of Contents 1 Introduction 3 1 Chapter 1: Volatility Mean Reversion and the Cross-Section of Option Returns Introduction Data Testing Mean Reversion in Individual Implied Volatility Portfolio Formation, Trading Strategies and Returns Portfolio Formation Characteristics of Portfolios Sorted by the Slope of the Volatility Term Structure Trading Strategies and Returns Slope of VTS and the Cross-Section of Option Returns Risk Adjusted Returns: Fama-French-Carhart Model Fama-MacBeth Regressions Transaction Costs Interaction between the Volatility Term Structure Slope and other Characteristics Robustness Checks Moneyness Sub-samples and Industries Earnings Announcements Controlling for Portfolio Weightings The Slope of the Implied-Volatility Term Structure Alternative Estimation of One-Month Implied Volatility Implied Volatility and Stock Prices Relation between the Slope of VTS and Future Realized Volatility Conclusion Chapter 2: Skewness from High-Frequency Data Predicts the Cross-Section of Stock Returns Introduction Measures of Realized Moments, Monte Carlo Simulation and Data Measures of Realized Volatility, Realized Skewness and Realized Kurtosis Monte Carlo Simulation

8 2.2.3 Data Characteristics of Realized-Moment-Sorted Portfolios Realized Moments and the Cross-Section of Stock Returns Sorting Stock Returns on Realized Volatility Sorting Stock Returns on Realized Skewness Sorting Stock Returns on Realized Kurtosis Fama-Macbeth Robustness Realized Moments and other Firm Characteristics Interaction between Realized Skewness and Volatility Realized Skewness and Realized Volatility Realized Skewness and Idiosyncratic Volatility Conclusion Chapter 3: Explaining Stock Returns with Intraday Jumps Introduction Empirical Method based on Intraday Data Data Characteristics of Portfolios Sorted by Realized Jump Realized Jumps and the Cross-Section of Stock Returns Ranking Stocks by Average Realized Jump Fama-Macbeth Cross-Sectional Regressions Interaction between Realized Jump and Realized Skewness Robustness Checks Double Sorting on Firm Characteristics Analysis for Di erent Periods Analysis by Stock Exchange Long-Term Predictability of Stock Returns Conclusion Conclusion 125 References 127 2

9 Introduction This thesis is in the area of asset pricing. Asset pricing theory studies the correctness of prices for securities such as stocks or options. The price of a security is considered low when its future rate of return is high, meaning that the price increased. Equally, the price of a security is high when its future return is negative, which means that the price decreased. Asset pricing theory tries to understand not only the sources of price di erential but also why similar securities can have di erent rates of return. In general, two main explanations are given to explain why some securities earn higher rates of return than others: 1) securities pay higher returns to compensate for higher risks and 2) market is not e cient and prices do not incorporate all the information available. This thesis consists of three essays that study the pricing of two types of securities: stocks and stock options. In particular, I explore the pricing of US stocks and options on stocks. The binding idea in this thesis is to nd individual factors that are priced and, consequently, predict subsequent returns in the stock or in the options markets. For options, I extract measures from the shape of the implied volatility surface. Iinstead of looking at the slope of the volatility surface across di erent moneyness levels (volatility smirk), I look at the slope across di erent option maturities. I discover that this measure has a positive relation with future option returns. In the second and third essays, I use high-frequency data to extract measures that are priced and predict future stock returns. The second paper does a comprehensive study that relates high-frequency moments with stock returns. In the third essay, intraday jumps are related with stock returns. In all three essays, the common denominator is the nding of a measure that is related with the future return of the stock or the option. Moreover, an extensive analysis is performed to ensure that the predictive power of the variables found in this thesis is not subsumed when the market factors are included and that the variables are not proxying for any individual measure known to be related with future returns. In the rst essay, "Volatility Mean Reversion and the Cross-Section of Option Returns", I study the pricing of at-the-money stock options in the US options market. Accurate pricing of options highly depends on the proper estimation of volatility. A stylized fact of volatility is mean reversion, meaning that, over time, volatility pulls back to some long-run average level. Hence, mean reversion indicates the most likely direction volatility might take in the future. Based on this fact, I investigate if there is a positive relation between volatility direction, as suggested by mean reversion, and options returns in the cross section. To test this premise, I use a measure of volatility mean reversion: the slope of the volatility term-structure, de ned as the di erence between the implied volatility of long-dated options and the volatility of short-dated options. Then, I rank stocks based on the slope of the volatility term-structure, form ten portfolios and analyze the returns of ve option trading strategies. I nd that option portfolios with the highest slope of the 3

10 volatility term-structure outperform option portfolios with the lowest slope of the volatility termstructure by an economically and statistically signi cant amount. In particular, straddle portfolios exhibiting the steepest slope of the volatility term-structure outperform straddle portfolios with the least pronounced slope by 27:1% per month. I conclude that volatility mean-reversion information predicts subsequent option returns. In the second and third essays, high-frequency data is used to extract relevant return distribution information that can potentially be related to stock prices. The second essay, "Skewness from High- Frequency Data Predicts the Cross-Section of Stock Returns", studies the relation between stock returns and higher moments of individual securities computed from high-frequency data. To test whether realized moments predict future stock returns, we sort stocks every week according to their realized moments, form ten portfolios and analyze subsequent weekly returns. We nd a negative relation between stock returns and realized skewness and a positive relation between stock returns and realized kurtosis in the cross section. No relation is found between realized volatility and stock returns. A trading strategy that buys stocks in the lowest realized skewness decile and sells stocks in the highest realized skewness decile generates an average raw return of 43 basis points per week with a t-statistic of 8:91. A similar strategy that buys stocks with high realized kurtosis and sells stocks with low realized kurtosis produces a raw weekly return of 16 basis points with a t-statistic of 2:12. These results are robust to di erent market periods, portfolio weightings, rm characteristic proxies and are not explained by the Fama-French-Carhart factors. In the third essay, "Explaining Stock Returns with Intraday Jumps", I explore the relation between stock returns and jumps. The presence of jumps in stock prices is widely accepted. I extract the average jump size of individual stock returns from high-frequency data and examine the empirical relation with subsequent stock returns. Jump di usion models predict a negative relation between expected stock returns and the average jump size. I nd that jumps extracted from intraday data, that I call realized jumps, are negatively related with subsequent stock returns. A trading strategy that buys stocks in the decile with large positive realized jumps and sells stocks with large negative realized jumps produces an average weekly return of 73:61 basis points with a t-statistic of 10:89. I rule out that realized jump is a proxy of rm characteristics by performing double-sortings and cross-sectional regressions with proxy variables such as size, book-to-market, previous week return, realized volatility, realized skewness, realized kurtosis and illiquidity. 4

11 1 Chapter 1: Volatility Mean Reversion and the Cross- Section of Option Returns 5

12 1.1 Introduction Option returns of indexes have so far received most of the attention in nancial literature. 1 However, in this chapter, I turn my attention to the cross-section of individual stock options. The cross-section of option returns is an interesting area to explore given that options have many more sources of price uncertainty than stocks. Studies on the cross-section of options include Cao and Han (2009), who nd that delta-hedged option returns are negatively related to total and idiosyncratic volatility. Goyal and Saretto (2009) nd that the cross-section of option returns is predicted by the individual volatility risk premium, the di erence between implied and historical volatility. Ni (2007) nds empirically that out-of-the-money call option returns are not positive and do not increase in strike price, even though theoretically they are supposed to. In this chapter, I want to determine the e ect of volatility mean-reversion information on the returns of individual stock options. The nancial industry standard for option pricing is the Black- Scholes-Merton model. One input of the option pricing model, volatility, is not observable in the market. Therefore, the estimation of volatility creates option price uncertainty. A systematic mismeasurement of volatility can lead to systematic biases in option prices and, in turn, in option returns. An evident trait that option investors ought to incorporate when estimating volatility for an individual stock is volatility mean reversion. 2 The main nding of this chapter is that volatility mean-reversion information has signi cant predictive power of future option returns in the cross-section. Volatility mean reversion information is described by two distinctive characteristics of volatility: the speed of mean-reversion and the long-run volatility level. Volatility mean reversion suggests that, over time, volatility reverts back to a long-run level. This long-run volatility level is the average value of volatility over a given period of time (i.e. one year). The speed of mean reversion indicates the time volatility takes to go back to its long-run average. To study the impact of the volatility mean reversion on the cross section of option returns, the two characteristic of mean-reversion are analysed. I nd no relation between the speed of mean reversion and subsequent option returns. However, I nd that the positioning of volatility with respect to its long-run average is priced in the cross-section of option returns. To determine the positioning of implied volatility with respect to the long-run volatility level, I employ the slope of the volatility term structure (VTS). This method looks at the slope of the VTS. The VTS refers to the relationship between the implied volatility of at-the-money options 1 Coval and Shumway (2001) study index option returns and nd that zero cost at-the-money straddle positions on the S&P 500 produce average losses of about 3 percent per week. Other studies of index option returns are done by Bakshi and Kapadia (2003), Jones (2006), Bondarenko (2003), Saretto and Santa-Clara (2006), Bollen and Whaley (2004), Shleifer and Vishny (1997), Jackwerth (2000), Buraschi and Jackwerth (2001) and Liu and Longsta (2004). 2 Andersen, Bollerslev, Diebold and Ebens (2001) show that realized volatility of individual stocks obtained from high-frequency data is mean-reverting. Lamoureux and Lastrapes (1990) and Kim and Kon (1994) nd similar results using daily stock returns. 6

13 and the time to maturity of those options. According to the rational expectations hypothesis, 3 an upward sloping term-structure is equivalent to a volatility being below its long-run average and a downward term-structure indicates that volatility is above it. Therefore, this method can be used to infer whether a stock s implied volatility is above or below its mean reversion level. Another way to determine the positioning of the volatility with respect to its mean reversion level is to actually use a long-run average. A common practice is to use historical volatility as a proxy for the long-run average of implied volatility. Goyal and Saretto (2009) nd that the variance risk premia, de ned as the di erence between implied volatility and historical volatility, predicts option returns. The variance risk premium might be predicting option returns since the volatility position with respect to its long-run average might be anticipating the most likely direction volatility will take in the future. When volatility is lower than its long-run average, volatility is expected to rise, and so is the price of the option. Conversely, when volatility is above its mean reversion level, volatility is expected to decrease, and the price of the option is expected to fall. Therefore, accurate volatility estimation must incorporate volatility mean-reversion information to properly price an option. In this chapter, I use the slope of VTS to infer the position of implied volatility with respect to its mean reversion level. Then, I test if the slope of the VTS can also predict option returns over and above the variance risk premium. I explain the fact that mean reversion predicts future volatility with a simple example that is graphically illustrated on Figure 1. Imagine two identical options on a di erent underlying but same stock price, same implied volatility, and, as a consequence, same option price. Nonetheless, the signals provided by the volatility mean-reversion information are di erent for the volatilities of the two stocks. While the rst stock has an upward sloping VTS, the second one has a downward sloping one. Equally, while the rst option s volatility is below its long-run volatility, the second one is above it. Given this information and the implications from volatility mean reversion, implied volatilities of the two stocks should be di erent; implied volatilities should be "closer" to their longrun volatility average. If volatilities are still the same, the rst option will have positive returns and the second one will have negative returns. In this study, I provide evidence that individual stock options exhibit this pattern of returns in the cross-section. [ Figure 1 goes here ] In practice, I use two measures to infer the positioning of the implied volatility in relation to the long-run volatility level. Using the Optionmetrics database, I extract once a month at-themoney puts and calls with di erent maturities and same strike price. One-month implied volatility, IV 1M ; is de ned as the average of one-month ATM call and put implied volatilities. Likewise, long-term volatility, IV LT, is de ned as the average of the farthest to maturity ATM call and put 3 The rational expectation hypothesis is mainly used to analyse the interest rate term structure in the xed income literature. In the options literature, the expectation hypothesis is assessed using the volatility term structure. The following papers study the volatility term structure: Mixon (2007), Heynen, Kemna and Vorst (1994), Campa and Chang (1995), Stein (1989), Poteshman (2001) and Cao and Li (2005). 7

14 volatilities. The long-run average, HV, is de ned as one-year historical volatility of daily returns. The rst measure, the slope of VTS, is the di erence between the long-term and the one month volatilities: IV LT IV 1M. The second measure to determine the positioning of volatility is the di erence between the long-run average and the one-month volatilities: HV IV 1M. To assess whether volatility mean-reversion information predicts future option returns, I construct ten portfolios of one-month options based on the slope of the VTS. Con rming the hypothesis that investors might be underestimating the speed of volatility mean-reversion, option portfolios with the lowest (largest) slope of VTS yield the lowest (largest) average monthly returns for the ve trading strategies. For example, the straddle portfolio with the largest slope of VTS outperforms the one with the lowest by 27.1% per month with a t-statistic of -9.06, which translates into a sharpe ratio close to one. These results are comparable to those in Goyal and Saretto (2009) who sort stocks by the variance risk premium and nd an average monthly return of 22.7% and a sharpe ratio of for the long-short portfolio of straddles. Two supplementary tests con rm the pricing e ects of the slope of VTS. In the rst one, the long-short straddle returns are regressed on the four factor model composed of the Fama and French (1993) factors and the Carhart (1997) momentum factor and I nd a signi cant alpha of 29.4% per month. In the second one, I estimate Fama and MacBeth (1973) regressions for the cross-section of straddle returns and nd that the slope of VTS has explanatory power of option returns. The slope of VTS is statistically signi cant and robust to di erent types of speci cations. As seen in Table 1, the slope of VTS and the variance risk premium used by Goyal and Saretto (2009) are related: portfolios with low values of the slope of VTS have also low values of variance risk premium and vice versa. To rule out that variance risk premium, rather than the slope of VTS, is not driving the option return di erences, I perform double sortings and recompute the raw returns. The results are robust not only to the variance risk premium, but also to size, book-to-market, historical skewness and historical kurtosis. Therefore, results for the Fama-MacBeth cross-sectional regressions further support my ndings. In addition to the straddle strategy, four other option trading strategies are implemented and analysed. Speci cally, I implement naked calls, naked puts, delta-hedged calls and delta-hedged puts. The long-short portfolio return for these strategies is also highly positive and economically signi cant. For example, the long-short portfolio for the delta-hedged call strategy has a signi cant monthly return of 4.0%. The extended Fama and French (1993) and Fama and MacBeth (1973) regressions performed on these four option trading strategies con rm the results. Extended Fama and French (1993) alphas are highly signi cant for all four strategies and the e ects of the slope of the VTS on the cross-section of option returns are also statistically signi cant. To verify that investors expectations about future volatility are not correct, I use an AR1 model to forecast an alternative one-month implied volatility. The one-month implied volatility is re-estimated for each stock by using the mean reversion level and long-term volatility. Then, all stock options are repriced and option returns are recomputed. As anticipated, the long-short 8

15 straddle returns decrease to a point where potential pro ts fall within the bid-ask spread, hence are not exploitable. Consequently, a simple AR1 model that accounts for mean reversion provides a sound estimate of short-term volatility and matching options present no overpricing (underpricing). This conclusion is similar to that of Black and Scholes (1972), who nd that options of high variance stocks are overpriced and options on low variance stocks are underpriced. The implied volatility term structure has mainly been used to examine investors rationality and behavior. Stein (1989) reports evidence of overreaction in the volatility term structure. Poteshman (2001) nds that investors underreact to daily changes in instantaneous variance, but overreact to increasing daily changes in instantaneous variance. The misreaction produces deviations in option prices of about 3% for S&P options. Cao and Li (2005) explore the economic signi cance of the misreaction present in the VTS. An abnormal return of 1-3% that cannot be exploited due to transaction costs is found for the S&P 500 index options. This chapter contributes to the literature in two ways. First, it is the rst work to uncover that the slope of the VTS predicts option returns. Even though option investors incorporate as much information as possible from market prices, it is puzzling that they do not properly account for volatility mean reversion in their estimation of volatility. A second contribution is the conclusion that volatility forecasting for individual stocks must include measures of mean reversion such as the VTS and the distance to the long run average. The remainder of this chapter is organized as follows. Section 2 describes the option data. Section 3 describes the option trading strategies and portfolio characteristics. Section 4 reports the returns from those strategies using di erent setups. Section 5 analyses the interaction between di erent characteristics of the stocks with the slope of VTS. Section 6 includes robustness checks. Section 7 proposes an alternative estimation of the one-month volatilities and examines the forecasting power of the slope of mean-reversion of future volatility. Section 8 provides a summary and conclusions. 1.2 Data The options data are from the Optionmetrics Ivy database. Optionmetrics IV database is a comprehensive source of high quality historical price and volatility data for the US equity and index options markets. The optionmetrics database has data for all US equity options and their underlying prices. The database contains US equity option market prices for the period starting on January 4, 1996 to June 30, Each quote contains information on the closing bid and ask quotes on American options, open interest, daily trading volume, implied volatilities and greeks. Implied volatilities and greeks are computed using the Cox, Ross and Rubenstein (1979) binomial model. Option data lters are similar to those in Goyal and Saretto (2009). First, I eliminate prices that violate arbitrage bounds. Call option prices that fall outside the interval (S Ke r De r ; S) 9

16 and put option prices outside of the interval ( S + Ke r + De r ; S), where S is the price of the underlying stock, K is the strike of the option, r is the risk free rate, D is the dollar dividend and is the time to expiration. Second, whenever the ask is lower than the bid, or the bid (ask) is equal to zero, or that the spread is lower than the minimum tick size, the observation is eliminated. The minimum tick size is $0.05 for options trading below $3 and $0.10 for other options. Whenever the bid and the ask are both equal to previous day quotes, the observation is eliminated. Third, I lter one-month options with zero volume or zero open-interest to ensure that one-month option prices are valid. However, this rule is not applied to long-term to maturity options since the option portfolio analysis is done only for the one-month options. Fourth, since an upward (downward) slopping VTS translates into a short-term implied volatility being below (above) its long-run average, the sign of IV LT IV 1M and HV IV 1M must be the same. Each month, stocks that do not comply with this rule are excluded from the analysis (only for that particular month). Fifth, options with underlying stock price lower than $5 are removed from the sample. Finally, the moneyness of the options must be between 0.95 and 1.05 for a stock to be included. Optionmetrics also provides stock prices, dividends and risk free rates. For all securities, high, low and close prices are found in the database. A complete history of dividends and splits is also available for each security. Risk free rates are linearly interpolated in order to match the maturity of the option. In cases where the rst risk free rate maturity is greater than the option maturity, no extrapolation is done and the rst risk free rate is used Testing Mean Reversion in Individual Implied Volatility I run some experiments to con rm that volatility is mean reverting. First, I average the daily estimates of IV 1M over one week so as to get a measure of weekly implied volatilities. Then, I compute the autocorrelation for each stock in the sample. The average autocorrelation of the whole sample is The second experiment follows Bali and Demirtas (2006). To test the mean reversion of implied volatility for each stock, I compute the correlation between t 1 and t t 1 (here refers to IV 1M ). When IV 1M is higher than the long-run average, mean reversion pulls the volatility down, and when it is lower, mean reversion pulls the volatility up. The average correlation between volatility and the change in volatility for the cross-section is -0.17, 70% of the correlations are signi cant at the 10% level and 95% of the correlations are negative. The negative relation between volatility and the change in volatility con rms the existence of a mean reverting process for individual implied volatilities. Finally, I test that the slope of the term structure of implied volatilities has a negative relation with the short-term implied volatility, IV 1M. When the short-term implied volatility, IV 1M, is above the long-term implied volatility, IV LT, according to the rational expectation hypothesis, the short-term volatility is expected to decrease; and when the short-term implied volatility, IV 1M, is lower than the long-term implied volatility, the shortterm volatility is expected to increase. The average correlation between IV 1M and IV LT IV 1M is The negative correlation of the short-term volatility and the slope of VTS corroborates that 10

17 volatility reverts to its long-run average. 1.3 Portfolio Formation, Trading Strategies and Returns Portfolio Formation This chapter investigates whether option investors incorporate volatility mean reversion information when pricing one-month options. I rank stocks based on the slope of VTS and form ten option portfolios. For each stock, the short-term volatility, IV 1M ; is the average of the one-month put and the one-month call ATM implied volatilities, as in Goyal and Saretto (2009). The long-term volatility, IV LT ; is the average volatility of the ATM put and a call options with the longest time-toexpiration available. In addition, the long-term options and short-term options must have the same strike price. Thus, the only degree of freedom to select long-term options is the time to expiration that can vary between 50 and 380 days. As a consequence, the time-to-expiration of long-term options is di erent across stocks and, for any given stock, can change across months. The average time to expiration for long-term options is around 220 days (7 months) for all decile portfolios as can be seen in Table 1. Therefore, on average, there is 6-month di erence between the onemonth volatility, IV 1M ; and the long-term volatility, IV LT : Assuming that option investors ignore volatility mean reversion, if IV 1M is above (below) IV LT then one-month options are overpriced (underpriced). Hence, I can anticipate that straddle portfolios with positive (negative) di erences of IV LT IV 1M should have positive (negative) returns. Options portfolios are formed based on one-month ATM options available on the second trading day (usually a Tuesday) after expiration of the option contracts which occurs the third Saturday of the month. I extract ATM put and call options that are one-month away from maturity. The onemonth options maturity ranges from 26 to 33 days. Since not all strikes are available for trading, most options are not exactly at-the-money (ATM), but as close as possible to it. Whenever the two options closest to ATM are not available, the next moneyness closest to ATM is selected. This means that the chosen options are the closest to ATM either from above or from below. For example, assume that the price of a stock is 17 and that the two closest option strikes are 15 and 20. Since the closest ATM strike is 15, the algorithm looks for two options with that strike. Those two options are a put and a call that expire in one month. If two options with those characteristics are not found, the algorithm looks for two options with the same features but with strike equals to 20. If those two options do not exist then the stock is dropped altogether since no ATM option is available. Each month, after all stock options have expired, a new set of options with the same characteristics described above are selected. In addition, at expiry, option returns are computed and decile portfolios are formed. With the new set of options, decile portfolios are formed and option returns 11

18 computed. The analysis is done for all ten portfolios across the 135 months available in our data sample. Since portfolios are formed based on option s availability, stocks drop in and out of the sample from month to month. A total of 3,849 stocks are included and there are, on average, 26 stocks per decile Characteristics of Portfolios Sorted by the Slope of the Volatility Term Structure Table 1 presents a summary of di erent characteristics of the ten decile portfolios formed based on the slope of VTS over the period January 1996 to June Characteristics are presented for put-call spread of IV 1M, bid-to-mid percent spread for call and put prices (de ned as 1 ((c ask 2 2 c bid )=c + (p ask p bid )=p)), rm size, BE/ME, credit rating, stock price, one-year historical volatility (HV ), one-year historical skewness (HSkew), one-year historical kurtosis (HKurt), variance risk premium (V RP ), de ned as the di erence between HV and IV 1M, future volatility (F V ), de ned as the standard deviation of the underlying stock return over the life of the option (F V ), volatility predictability, de ned as the di erence between F V and IV 1M, open interest of ATM puts and calls (in $ thousands), de ned as the open interest for calls and puts multiplied by their respective price, average maturity of long-term implied volatilities, and option greeks. [ Table 1 goes here ] As Table 1 indicates, the average slope of VTS is very di erent across all 10 portfolios. P1 is the portfolio with the lowest slope of VTS (-15.0%) and P10 is the one with the highest slope at 8.5%. Additionally, the one-month volatility in P1 is above the long-run implied volatility level and that in the P10 is below its long-run average. The portfolio with the lowest slope of VTS, P1, has the highest one-month volatility at 72.1%, and P10, the one with the highest slope of VTS, has the third highest implied volatility at 49.5%. However, P10 average one-month volatility is higher than that of nearby portfolios P3 to P9. The put-call spread of the one-month ATM implied volatility is highest for P1 and P10 with values of 1.5% and 1.9%. The percentage bid to mid option prices are increase from P1 to P10. The bid to mid percentage price di erence for P1 is 6.7% while the one for P10 is 8.4%. Additionally, portfolios 1 and 10 contain the riskier stocks of all portfolios. Historical volatility, skewness and kurtosis is the highest for P1 and P10. On average, portfolios P1 and P10 are the ones with the lowest market capitalization and the lowest credit rating quality. This means that companies in those portfolios are small and have higher probability of default given their credit rating. Goyal and Saretto (2009) nd that option returns and individual volatility risk premium are strongly related. According to Table 1, volatility risk premium and the slope of VTS are related. Low (high) values of the slope of VTS imply low (high) values of the volatility risk premium. As Goyal and Saretto (2009) anticipate, historical volatility can also be considered a measure of 12

19 the long-run volatility level. Therefore, it is to be expected that those two measures are close to each other. In section 5, I do bi-variate sorting to rule out that the two measures carry the same information. Another characteristic of P1 and P10 is that their vega, the sensitivity of the option price to movements in implied volatility, is lower compared to that of other portfolios. P1 has the lowest vega while P10 hast the third lowest vega. Even though these portfolios carry low volatility risk compared to other portfolios, their implied volatilities are the ones farther away from their long-run average. Finally, the deltas for puts and calls are decreasing from P1 to P10 and the gammas are increasing. This implies that option portfolios with the lowest slope of VTS are more sensitive to price movements but are less sensitive to changes in the price sensitivity, delta. [ Table 2 goes here ] Table 2 presents decile migration from month to month. On average 20% of the stocks of decile one continue in decile one over the next month. For decile ten, 33% of the stocks stay in decile ten. Hence, about one fth of the stock from decile one remain in decile one and one third of the stock from decile ten remain in decile ten. However, one problem with this conclusion is that only 31% of the data are analysed given that there are 24,576 month-stocks missing. This simple fact shows how di cult it is to have one-month ATM options for the same rm every month and con rms that many rms enter and leave the analysis each month. Moreover, it supports the choice of using only one-month ATM options. Even though, longer dated options or out-of-the-money options seem more attractive to study, the low availability of those options makes a cross-sectional study like this one almost impossible. In the next section, I examine the impact of the slope of VTS on future option returns Trading Strategies and Returns The analysis of option market returns is not as straightforward as that of stock returns. Option investors have several degrees of freedom when buying an option. First, investors can choose a call option or a put option. Then, they must decide on the time-to-maturity and the strike price. Given the di erent alternatives for each individual stock, Saretto and Santa-Clara (2006) analyse 23 di erent trading strategies. Yet, liquidity is a big constraint when studying individual stock options. In order to propose ready-to-implement trading strategies, I work with the most liquid options: closest to at-the-money and closest to expiration. Additionally, to avoid paying high transaction costs more than once, options are held until maturity. As displayed in Table 1, average bid to mid option price spreads for P1 and P10 are 6.7% and 8.4%. Hence, by holding the options until maturity, big transaction costs that could make the trading strategy pro tless are avoided and are only paid when opening the position. In this chapter, I study ve trading strategies: naked call and put options, delta-hedged call and put options and straddles. 13

20 Option returns are computed on a monthly basis. Once the options are bought, they are held until expiration. This methodology guarantees that returns are non-overlapping and that bid-ask costs are only incurred at the beginning of the trade. A zero cost trading strategy is implemented; one dollar is borrowed at the risk-free rate to buy an option. At maturity, that dollar with accrued interests is returned and the option pro t, if any, is collected. Naked Option A naked option consists of buying either a call option or a put option with no underlying security protection. Naked options are very risky. If the underlying asset moves in the expected direction, big returns are made. However, huge losses of up to 100 percent are recorded if the underlying stock moves in opposite direction. A zero-cost naked option traded at time t has a return equal to r call t;t = max(s T K; 0) c t r f t;t (1) r put t;t = max(k S T ; 0) p t r f t;t (2) where c t and p t are the average of the bid and ask prices of a call option and a put option respectively on trading day t, r f t;t is the future value of one dollar from time t to T, and S T is the stock price at maturity T: Straddle A straddle is an investment strategy involving the purchase (or sale) of call options and put options, simultaneously. A zero-cost long straddle return is the average of the long put option return and the long call option return: r straddle t = js T Kj (p t + c t ) p t + c t (3) Since I only work with ATM put and call options, straddles are also at-the-money. An attractive feature of long ATM straddles is that pro ts are positive irrespective of whether large movements in the stock are up or down. The downside is that investors have to buy one put option and one call option simultaneously. A long straddle is a trading strategy that bene ts from larger movements than those foreseen by market participants in the stock price. Therefore, an investor must believe that volatility is likely to increase and that belief must be di erent from that of the majority of investors. Stocks that have an increasing VTS and whose short-term volatility lies below the longrun average satisfy this rule. On the other hand, a short straddle bene ts when the stock price does not change during the life of the option. In other words, an investor that expects volatility to be lower than what most investors are expecting bene ts from selling straddles. Delta Hedged Option A more sophisticated trading scheme involves the delta of the option. Delta is the rate of change of the option with respect to the stock price. A delta neutral or delta 14

21 hedged position consists of selling the option and buying delta units of the stock so that small stock movements do not a ect the pro t and loss function of the investor. The return of a delta hedged option, a combination of long delta number of shares and short the option, is r Hedged t = (S T t T ) S t t mid t e r(t t) (4) where S t is the stock price and t is the delta of the option at time t. 1.4 Slope of VTS and the Cross-Section of Option Returns [ Table 3 goes here ] Based on the slope of VTS, stocks are ranked and ten portfolios are constructed. Option returns are weighted on the absolute value of the slope of VTS; the greater the absolute di erence of the slope, the greater the weight on the option return. Table 3 reports portfolio returns for straddles, naked calls and puts, and delta-hedged calls and puts. Option returns for all trading strategies are increasing from P1 to P10 and so is the slope of VTS. This means that, when IV LT is lower than IV 1M (IV LT IV 1M < 0), straddle returns are negative. This is an indication that, on average, IV 1M has a downward trend towards IV LM : The opposite is also true since straddle returns are positive when IV LT is larger than IV 1M (IV LT IV 1M > 0). The long-short portfolio strategy (P10-P1) yields a 27:1% monthly average return with a 34:7% monthly standard deviation. The third and fourth moments of the long-short portfolio are 0:8 for the skewness and 1:5 for the excess kurtosis. Figure 2 displays the histogram of the long-short straddle portfolio in Panel A. Panel B of Figure 2 displays the qq-plot of the long-short straddle returns. Also note that straddle returns for P1 are 12:2% and for P10 are 14:8%; hence, both portfolios contribute to the long-short portfolio return. Figure 3 plots straddle monthly returns over the entire sample from 1996 to 2007 where 88% of monthly returns are positive. [ Figure 2 goes here ] [ Figure 3 goes here ] The other four option trading strategies generate a signi cant return for the long-short trading strategy. Naked call option returns for the long-short strategy have an average of 35:7% with a standard deviation of 77:6%. The long-short portfolio for the delta-hedged call strategy has a return of 4:0% with a standard deviation of 6:1%. For both trading strategies, the return of P1 is negative and the return of P10 is positive. Strategies for put options display the same trend. Panel A, Figure 4 shows the naked-delta puts monthly return of the long-short strategy; 63% of the returns are positive. Panel B of Figure 4 displays the results for the naked-delta calls where 15

22 72% of the returns are positive. Panel A of Figure 5 displays the results for the delta-hedged puts where 83% of the returns are positive. Finally, Panel B of Figure 5 shows the delta-hedged calls monthly return of the long-short strategy; 84% of the returns are positive. [ Figure 4 goes here ] [ Figure 5 goes here ] Positive returns for all option trading strategies can further be re ned by only including volatilities that are strongly mean reverting. As previously explained, volatility follows a mean reverting process when a negative relation between volatility and the change in volatility exists. When I include stocks with negative correlation between volatility and change in volatility (with p-values lower than 10%), the long-short straddle monthly portfolio return increases to 31:5% with a standard deviation of 31:8%. This nding provides additional evidence that option investors fail to incorporate volatility mean-reversion information when pricing individual stock options Risk Adjusted Returns: Fama-French-Carhart Model In addition to the results of the raw returns for di erent option trading strategies, I regress the longshort straddle portfolio monthly returns on the various speci cations of the linear pricing model (Fama and French (1993) and Carhart (1997)). Table 4 presents the alphas of the regressions of the long-short portfolio returns on a constant, the excess market return, the size factor (SMB), the book-to-market factor (HML), the momentum factor (UMD), the S&P 500 variance risk premium 4 (VRP) and the return of the zero-beta index straddle as in Coval and Shumway (2001). To compute zero-beta straddle returns for the S&P 500 Coval and Shumway (2001) de ne the return of a zerobeta index straddle as c t r v = c + S t r call p t c c t c + S t t;t + p t c p t c r put t;t c t c + S t where c t is the call option price, p t is the put option price, S t is the underlying price, rt;t call r put t;t are the returns of the call and the put, and c is de ned as " c = S ln(s=x) + r d + 2 C N =2 # p t s and where N [:] is the cumulative normal distribution, s is the beta and d is the dividend yield of the underlying asset. We also report the t-statistics are based on Newey and West (1987) standard errors using 3 lags and the R-squared. [ Table 4 goes here ] 4 The variance risk premium of the S&P 500 is computed as the di erence between realized volatility and VIX, the market volatility index. Realized volatility is computed using intraday data of the S&P

23 Table 4 contains the results of three linear regressions done on the long-short returns of each trading strategy. No factor is highly signi cant and the intercept, that can be interpreted as the alpha, is highly signi cant for all regressions. Column (1) reports the regression with respect to the excess market return. The alpha is positive and signi cant for all ve trading strategies. Straddle long-short returns have an alpha of 28:0% with a Newey-West t-statistic of 9:28. Column (2) adds SMB, HML and UMD to the regressions and the alphas remain almost unchanged. The long-short straddle alpha almost equal to the previous one at 29:4% with a Newey-West t-statistic of 9:19. Finally, column (3) adds the variance risk premium and the zero-beta straddle return of the S&P 500. When included, the alpha and Newey-West t-statistics decrease for most trading strategies. For example, the straddle alpha is 20:7% with a t-statistic of 2:09. Most of the risk factors used, namely the excess market return, SMB, HML and UMD, have shown to predict returns on stocks. However, since the asset pricing literature for options is so small, there are no accepted risk factors that explain returns on stock options. I included the market VRP and the zero-beta straddle of the S&P 500 in an attempt to have risk factors that explain the cross section of option returns. However, these factors are not priced in the cross-section of option returns. Further research is needed to undertand the main drivers of option returns and to uncover the risk factors that can explain them Fama-MacBeth Regressions To test whether the slope of VTS has predictive power on the cross section of option returns, I run the two stage Fama and MacBeth (1973) regressions. An advantage of the Fama and MacBeth (1973) regressions is that they do not impose breakpoints for portfolio formation and that they allow to evaluate the interaction among several individual variables and the variable of interest, namely the slope of VTS. In the rst stage, I regress for each month t the following regression: ret i;t = B 0;t + B 1;t SlopeV T S i;t 1 + B 2;t V RP i;t 1 + B 3;t SIZE i;t 1 + B 4;t BkT OMkt i;t 1 + B 5;t HSkew i;t 1 + B 6;t HKurt i;t 1 + " it where ret i;t is the company i option monthly return (any of the ve trading strategies) at time t. The SlopeV T S is the slope of VTS, V RP is the variance risk premium as de ned by Goyal and Saretto (2009), SIZE is the stock size, BkT OMkt is the book-to-market, HSkew is historical skewness and Hkurt is historical kurtosis. These variables are available for each company i at time t 1 (Prior to option returns). From stage one, I obtain a time series of t coe cients, B 0;t to B 5;t, that are averaged in the second stage to obtain an estimator for each coe cient. The coe cient s signi cance is evaluated with the Newey-West t-statistic. Results are presented in Table 5 that has two regressions for each option trading strategy. 17