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Three essays in derivatives, trading and liquidity

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Three essays in derivatives, trading and liquidity

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Content Three Essays in Derivatives, Trading and Liquidity Tong Wang A Dissertation Presented to the Faculty of the University of Southern California in Candidacy for the Degree of Doctor of Philosophy Adviser: Professor Christopher Jones August 2013 c Copyright by Tong Wang, 2013. All rights reserved. iii Abstract The work in Chapter 1 shows that hedging by option writers has a large and significant destabilizing effect on the stock market. We demonstrate that weekly return reversals are significantly stronger surrounding option expiration days. Our evidence suggests that the hedging pressure that drives weekly reversals mainly comes from index options rather than individual stock options. We find in addition that index option hedging appears to have an impact on the aggregate market, and that the strength this aggregate impact is highly related to the degree of cross-sectional reversal. We also find that index option prices tend to be high before option expiration, suggesting that option hedgers are attempting to unwind written positions that might be difficult to hedge due to price impact on the underlying stocks. Collectively, the evidence we present strongly supports the conclusion that option trading causes significant price displacements in stocks and in the market as a whole. Chapter 2 investigates the relationship between the slope of the implied volatility (IV) term structure and future option returns. A strategy that buys straddles with high IV slopes and short sells straddles with low IV slopes returns seven percent per month, with an annualized Sharpe ratio just less than two. Surprisingly, we find no relation between IV slopes and the returns on longer-term straddles, even though the correlation between the returns on portfolios of short-term and long-term straddles generally exceeds 0.9. Our evidence suggests that the return predictability we document is unrelated to systematic risk premia. We believe that our results point to two possible explanations. One is that temporary hedging pressure pushes option prices away from efficient levels. The other is that short-term options are more likely to be mispriced by noise traders than long-term options. Chapter 3 shows that the positive correlation between stock-level trading activity and market betas remains strong even using the Dimson (1979) method to correct for non-synchronous trading. This finding suggests that controlling for non-synchronous trading alone does not provide unbiased inferences regarding the effects of events on market betas. Instead, it is necessary to control for changes in trading activity explicitly. We show that controlling for trading activity significantly changes the estimated impact of seasoned equity offerings and share repurchases on market betas. iv Acknowledgements I would like to thank my thesis advisor Chris Jones for his guidance throughout my education in finance and economics. I am very grateful to the support and encouragement from my thesis committee members Maria Ogneva, David Solomon, Andreas Stathopoulos and Fernando Zapatero. I also appreciate many conversations I have had with other faculty members and Ph.D. students at USC. I benefit from the financial support from USC Marshall School of Business Department of Finance and Business Economics. v Contents Abstract iii Acknowledgements iv List of Tables vii List of Figures viii Chapter1 The Destabilizing Effects of Option Hedging and the Weekly Reversal Anomaly 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The Price and Liquidity Effect of Option Hedging . . . . . . . . . . . . . . . . . . 8 Price Pressure Following Option Expiration . . . . . . . . . . . . . . . . . . 9 Stock Price Reversals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Weekly Price Reversals in the Cross-Section . . . . . . . . . . . . . . . . . . 10 Price Reversals and Market Liquidity . . . . . . . . . . . . . . . . . . . . . . 12 Price Reversals and Stock Option Open Interests . . . . . . . . . . . . . . . 14 Market Return Reversals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Linking Cross-Sectional and Market Reversals . . . . . . . . . . . . . . . . . 18 Option Prices Before Expiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Market Liquidity Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Endogeneity Issues in Measuring Market Liquidity . . . . . . . . . . . . . . . 24 Comparing Market Liquidity in Option Expiration Weeks and Other Weeks . 25 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Chapter2 The term structure of equity option implied volatility 41 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 vi The relation between IV slope and straddle returns . . . . . . . . . . . . . . . . . 45 The determinants of the IV slope . . . . . . . . . . . . . . . . . . . . . . . . 45 The IV slope and the cross-section of straddle returns . . . . . . . . . . . . . 48 Trading strategies and time-series predictability . . . . . . . . . . . . . . . . . . . 49 Risk adjustments and profitability of trading strategies . . . . . . . . . . . . 49 Time series predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Controlling for expected realized volatility . . . . . . . . . . . . . . . . . . . . . . 52 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Chapter3 On the Relation between Trading Activity and Market Beta and Its Implications on Event Studies 63 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Data and Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Summary Statistics of SEO Firms . . . . . . . . . . . . . . . . . . . . . . . . 68 Market Betas and Trading Activity . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Theoretical Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Beta Dynamics of SEO Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Beta Dynamics and Trading Activity Surrounding SEO . . . . . . . . . . . . 75 Delayed Response to Market-Level Information . . . . . . . . . . . . . . . . 80 Cross-Sectional Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . 82 Share Repurchases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Conclusion and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 References 100 vii List of Tables 1.1 Stock Price Reversals In the Cross-Section . . . . . . . . . . . . . . . . . . . 29 1.2 The Effect of the Market Liquidity on Cross-Sectional Weekly Reversals . . . 31 1.3 Stock Option Open Interests and Cross-Sectional Weekly Reversals . . . . . 32 1.4 Market Level Reversals Surrounding Option Expiration Days . . . . . . . . . 33 1.5 Predicting Cross-Sectional Reversals with Market Reversals . . . . . . . . . . 35 1.6 The Differential Impact of the Aggregate Hedging Pressure on the Cross Sec- tion of Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.7 The Decreases in the VIX Index Over Option Expiration Weeks . . . . . . . 38 1.8 Comparing the Liquidity Measures in Option Expiration Weeks and in Other Weeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.1 Summary Statistics for Implied Volatility Slopes . . . . . . . . . . . . . . . . 54 2.2 Determinants of Implied Volatility Slopes . . . . . . . . . . . . . . . . . . . . 55 2.3 Fama-Macbeth Regressions for One-Month Straddle Returns . . . . . . . . . 57 2.4 Fama-Macbeth Regressions for Long-Term Straddle Returns . . . . . . . . . 58 2.5 Fama-Macbeth Regressions for Stock Returns . . . . . . . . . . . . . . . . . 59 2.6 One-Month Straddle Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.7 Long-Term Straddle Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.8 Time Series Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.1 Summary Statistics of the SEO Sample . . . . . . . . . . . . . . . . . . . . . 92 3.2 Event Returns of SEO Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.3 Summary Statistics of Repurchasing Firms . . . . . . . . . . . . . . . . . . . 94 3.4 Changes in Turnover and Changes in Market Beta . . . . . . . . . . . . . . . 95 3.5 Determinants of Changes in Market Betas (Pooled Regressions) . . . . . . . 96 3.6 Determinants of Changes in Market Betas (Fama-Macbeth Regressions) . . . 98 viii List of Figures 1.1 The Differential Impact of the Market Level Hedging Pressure on the Cross- Section of Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2 Cross-Sectional Weekly Reversals and the Market Liquidity . . . . . . . . . . 28 3.1 Changes in Market Beta v.s. Changes in Turnover . . . . . . . . . . . . . . . 73 3.2 Market Beta Dynamics Surrounding SEOs . . . . . . . . . . . . . . . . . . . 78 3.3 Turnover Rate Surrounding SEOs . . . . . . . . . . . . . . . . . . . . . . . . 78 3.4 Amihud Illiquidity Measure Surrounding SEOs . . . . . . . . . . . . . . . . . 79 3.5 Corporate Investment Surrounding SEOs . . . . . . . . . . . . . . . . . . . . 79 3.6 Dimson (1979) Beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.7 The Time-line For Cross-Sectional Regressions . . . . . . . . . . . . . . . . . 83 3.8 TheDifferenceinBetaandTurnoverBetweenRepurchasingFirmsandMatched Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.9 Beta and Turnover Dynamics of Repurchasing Firms . . . . . . . . . . . . . 91 THE DESTABILIZING EFFECTS OF OPTION HEDGING 1 Chapter 1 The Destabilizing Effects of Option Hedging and the Weekly Reversal Anomaly Introduction Understanding the impact of options on the pricing and liquidity of their underlying assets has been a major topic of research for several decades. Yet, the question of whether options have positive or negative impact on stock markets is still very open to debate. Theoretical arguments for positive impact emphasize the role of options in completing financial markets that are incomplete otherwise (Ross (1976)) and in creating new channels through which private information can be conveyed to relatively uninformed traders. Both Grossman (1988) and Cao (1999) provide information-based models in which the introduction of an option decreases the volatility of the underlying asset. Empirical analysis of the effects of option listing seems to support this prediction. However, treating option listings as quasi-experiments suffers from the critique that option listings are unlikely to be exogenous events (Mayhew & Mihov (2004)). Moreover, certain effects of options may not be observable immediately following listing if option volume develops slowly. These shortcomings make the empirical support for the positive effects of options unreliable. On the other side of the debate, theory also suggests that option related trading can be destabilizing, causing excessive movements in underlying stock prices that cannot be justified by fundamentals. Authors such as Gennotte & Leland (1990), Grossman & Zhou (1996), and Platen & Schweizer (1998) have argued that feedback effects that arise from option replication or hedging can amplify swings in stock prices. Policy makers also frequently express concerns about the option market’s role in abnormal stock market THE DESTABILIZING EFFECTS OF OPTION HEDGING 2 movements. Thus far, however, convincing empirical evidence that directly supports the existence of destabilizing effects is missing, partly due to the lack of identification strategies that distinguish options from other explanations of excessive price movements. This paper fills in the missing evidence by examining the link between asset price anomalies, such as the weekly reversals anomaly (Lehmann (1990)), and the option expiration calender. Monthly option contracts always expire on the Saturday following the third Friday of the month, providing an excellent opportunity for quasi-experiments. We show that several anomalous price behaviors concentrate in the trading days surrounding option expiration and can be attributed to the destabilizing trades of option hedgers. For instance, the profit to the value-weighted weekly reversal strategy, which buys past loser stocks and sells past winner stocks, is particularly large (16 bps per day) following option expiration weeks and cannot be eliminated by transaction costs. In contrast, the reversals following all other weeks is significantly weaker (7 bps per day). We also find that reversal profits are significantly correlated with market level liquidity only for post-expiration holding periods. Our further analysis of these findings shows that they are manifestations of the destabilizing effects of option hedging. How might option hedging destabilize stock markets? Under the Black-Scholes (1973) model, the option delta represents the number of shares in the underlying stock one needs to hold to replicate the payoff of an option contract. The deltas of call options are always positive, implying long hedging positions in stocks, and the deltas of put options are always negative, implying short hedging positions. In addition, option deltas increase in the underlying stock price for both call and put options. When stock prices increase (decrease), call option deltas become more (less) positive and put option deltas become less (more) negative. To maintain zero exposure to stock price movements, when the stock price rises, a call option writer needs to increase his long position in the underlying stock, and a put option writer needs to decrease his short position. This implies that option writers buy the underlying stock when the stock price rises, regardless of whether they have written call options or put options. On the other hand, option writers sell the underlying stock when the stock price decreases. Hence, delta hedging by option writers destabilizes the underlying stock market in the following sense: when an initial liquidity demand in the stock causes a price displacement, option hedgers add to the liquidity demand and amplify the initial displacement. Because options either expire in the money or out of the money, option deltas always converge to one of two values at expiration (0 or 1 for calls, 0 or -1 for puts). Deltas may become unstable in the last few days before expiration, leading to an increase in the amount of hedging trades and stock price displacements. Following option expiration, THE DESTABILIZING EFFECTS OF OPTION HEDGING 3 hedgers liquidate hedging positions, causing reversals in the direction of price pressure. 1 We predict that the surge in hedging activity prior to expiration and the reversal of those trades following expiration causes stock price reversals surrounding option expiration days. In theory, price pressure can arise from the hedging pressure of individual stock options or equity index options. To the extent that hedgers of individual stock options hold different positions on different stocks, their hedging trades could generate dispersion in pricing errors in the cross-section, which results in cross-sectional reversals as the pricing errors are corrected. On the other hand, although equity index option hedging requires trading stocks at the aggregate level, these trades could nevertheless have differential price impacts on different stocks. We first show that stock option open interest plays little role in explaining the magnitude of price reversals. In particular, stocks with no options traded on them still show price reversals, implying that hedging pressure from individual stock options cannot be the sole explanation of cross-sectional reversals. We show that index option hedging explains at least part of the return on cross-sectional reversal in two steps. First, we establish that index option hedging appears to cause significant price displacements at the market level. Specifically, when market maker inventories are high (low) prior to expiration week, then a large amount of selling orders from option writers pushes prices down (up). This effect is reversed immediately following index option expiration. We then show that index option hedging trades are the primary cause of cross-sectional reversal as well. We find that most of the cross-sectional reversal profit comes from the stocks that experienced the largest price displacements that were in the same direction as the market. For example, when the market index is more likely under hedging-related selling pressure during expiration week, then loser stocks reverse significantly in the following week while winners do not. Because the market impact that option hedgers generate through their trading is detrimental to their own performance, they may take steps to avoid it by unwinding their option portfolios prior to expiration. In order to buy back the options that they have written, however, market makers may be required to pay a higher price. We therefore study option prices around expiration weeks. Changes in the VIX index show that option prices tend to be high at the beginning of expiration weeks, and the prices decrease significantly at the end of the expiration weeks. The effect is most pronounced when market liquidity is low, which is when market impact is likely the greatest concern for option hedgers. Finally, we show that market liquidity is low during option expiration weeks using several different liquidity measures. We argue that liquidity measures constructed from large 1 See Section 1 for more discussion of the reversal in price pressure following expiration. THE DESTABILIZING EFFECTS OF OPTION HEDGING 4 stocks are biased due to an increase in index-related trading and therefore focus on liquidity measures for small stocks. We find some evidence that in markets with low liquidity, liquidity in option expiration weeks is even lower. This suggests that when the supply of liquidity is scarce, it is reallocated away from small caps to better accommodate index hedging demands. The unique findings in this paper rely on several key features of our empirical approach. Among them is our use of a relatively recent sample period. In contrast to other stock price anomalies, which often become weaker in more recent sample period, option-related anomalies may in fact be getting stronger over time due to the dramatic increase in option trading activities. According to the Options Clearing Corporation, the annual volume of equity options increased by 600% between 2000 and 2011, dwarfing the 100% increase in stock trading volume over that period. The statistic implies that the impact of options may be much larger since the first studies were published and, in addition, highlights the growing importance of the topic studied in this paper. Our work has broad implications for a number of fundamental questions that have intrigued financial economists and practitioners for decades. First, this work demonstrates several novel aspects of the short term price reversal anomaly. In previous research on weekly reversals, starting with Lehmann (1990) and demonstrated more recently in Avramov et al. (2006), small and illiquid stocks have always provided the largest component of weekly reversal profits. In contrast, we find that large stocks are more likely to experience price displacements in option expiration weeks and reversals in the following week. This finding contrasts sharply with previous work and suggests that a more nuanced view of cross section of stock level liquidity may be necessary. In particular, although large stocks appear to be more liquid than small stocks on average, they are also more likely to be targeted by traders who demand liquidity when market liquidity is low and trades must be executed quickly (e.g., in option expiration weeks). Therefore, when market liquidity is low it may in fact be possible that large stocks are more prone to pricing errors than are small stocks. We make two other contributions to the literature on short term reversals. One is to demonstrate that return reversal is stronger in periods when market liquidity is low, suggesting that liquidity in the time-series is as important as it is in the cross-section. More fundamentally, our results demonstrate that short term reversals may arise from differential responses to aggregate order imbalances. Option-related hedging may just be one source of aggregate market pressure that is driving cross-sectional reversal, and other sources may be worth exploring. Our findings also confirm those of Chordia et al. (2002), who show that aggregate order imbalance has little predictive ability for future market returns in general. We THE DESTABILIZING EFFECTS OF OPTION HEDGING 5 demonstrate, however, that there is an effect that is much stronger around option expiration and when market liquidity is low. Finally, the presence of predictable liquidity shocks in expiration weeks challenges the intuition articulated by Grossman (1988). Grossman argues that option open interest allows liquidity providers to better predict hedging demand, which leads to an improvement in market liquidity. Option hedging around expiration weeks generates predictable variation in the demand for liquidity yet nevertheless leads to large displacements in underlying asset values. Our results suggest that the information embedded in open interest may be more difficult to extract that Grossman assumes. The rest of the paper is organized as follows. Section 1 reviews the literature. Section 1 describes the data. Section 1 explains the channels through which option hedging causes price displacements and reversals. Sections 1, 1, and 1 present our empirical results. Section 1 shows cross-sectional and market level stock price reversals. Section 1 demonstrates patterns in option prices surrounding option expiration weeks. Section 1 compares liquidity measures in option expiration weeks and other weeks. Section 1 concludes the paper. Related Literature The current work is related to several strands of research in finance literature. First, it extends the literature that studies the impact of option listings on liquidity, volatility, and price levels of the underlying stocks. Despite abundant theoretical works that model the impact of option trading on the volatility and level of the underlying prices (for example, see Ross (1976), Back (1993), and Zapatero (1998)), there is little robust empirical evidence that option trading has a significant impact on stock prices. 2 One exception to this is Ni et al. (2005) who show that hedging pressure of stock options pushes stock prices toward strike prices as option expiration days are approached. Pearson et al. (2007) also provides interesting evidence on the impact of option hedging on stocks. To explain the impact of option tradings on liquidity, Grossman (1988) argues that option trading provides information about future liquidity demand to the potential liquidity providers and thus helps to improve market liquidity. The option market may lure traders away from the stock market and so reduce information asymmetry in the stock market. The model in Easley et al. (1998) provides conditions under which traders choose options over stocks. In an empirical analysis of the effect of option listings on stock market liquidity, Kumar et al. (1998) measure the change of several liquidity measures around option listings and conclude that the market quality of underlying stocks improves after option listings. 2 See the reference in Ni et al. (2005) for more examples of the past empirical work. THE DESTABILIZING EFFECTS OF OPTION HEDGING 6 Our work is also related to the literature on market crashes. Understanding how market price and liquidity can decrease over a very short period of time has been an important topic of finance research. Gennotte & Leland (1990) propose a model to explain the 1987 crash. In their model, a small amount of hedging pressure can drive down prices significantly because uninformed investors cannot perfectly distinguish hedging from informed trading. Their results hinge on the assumption that uninformed investors do not know the hedging plan, while our work shows that even well predicted, recurrent hedging can still cause stock prices to deviate from fundamentals. 3 The underlying mechanism of destabilizing trading in Gennotte & Leland (1990) may also apply in our setting. When options approach expiration, hedging pressure pushes the underlying stock price away from its fundamental level. An uninformed investor could interpret this as informed trading and hence also trade in the same direction and cause more price movement, which, in turn, causes more hedging. Here, although the timing of option expiration is well understood, the magnitude of hedging may not be well understood by less sophisticated investors. Another channel for destabilizing the trading and liquidity spiral is provided by Brunnermeier & Pedersen (2009). In their model, lower asset prices may lead to a margin call that triggers more selling when it is difficult to obtain funding. This channel could also apply to the current setting: when hedging pressure depresses stock prices, hedgers sell and thereby add to that pressure. Deeply depressed stock prices could trigger margin calls that further lower stock prices and cause more selling from hedgers. The phenomenon that we study here can be viewed as an example of a liquidity and price spiral that could arise through various channels. Although we conduct our investigation in the setting of option hedging, our work shares many links to the broad topic of market liquidity. It is a challenging task to measure the market liquidity level because of endogenous trading decisions made by traders. Obizhaeva (2011) argues that traders often employ price dependent strategies and cancel expensive orders, and the conventional estimates tend to overestimate available liquidity. Using portfolio transitions data, she finds that liquidity is lower than it is usually believed, especially in high volume markets. Collin-Dufresne & Fos (2012) find that informed traders time the market and trade when market liquidity is high. We show that although market liquidity is low during option expiration weeks, due to the fact that a significant portion of trading volume on large stocks comes from option level hedging, the measured price impact does not reflect the true liquidity condition of the market. Finally, the empirical vehicle we rely on is weekly price reversal. Short term price reversal 3 Note that our setting is different from Gennotte & Leland (1990) also in that their model does not imply that hedging pressure increases as time approaches a certain point. THE DESTABILIZING EFFECTS OF OPTION HEDGING 7 has been extensively studied in previous research. Lehmann (1990) is among the first to show price reversals at weekly frequency, arguing that it is an indication of market inefficiency. In previous research, small and illiquid stocks are thought to be the main contributors to the weekly reversal profits. For example, Avramov et al. (2006) show that the magnitude of weekly price reversals is positively correlated with stock level illiquidity measures in the cross section. The current work, on the other hand, shows that large stocks are more likely to experience price displacements in an option expiration and reversals in the following week. Data This section briefly describes the data sources used in our work. We obtain stock returns and other daily/monthly trading data from CRSP and firm information from COMPUTSTAT. The goal is to study the impact of option trading on the underlying stocks, so we choose the sample period that starts January of 1973, when the first options were listed on the CBOE, and ends in December 2011. The sample period is restricted to start in January 1993 and January 1996 when we need to combine CRSP data with TAQ data and OptionMetrics data, respectively, due to the availability of the two datasets. We apply a filter to eliminate firms whose stock prices are lower than $5 or greater than $1000. To compute cross-sectional price reversals, stocks that are smaller than the smallest NYSE size decile are excluded. When stock returns and trading volume are used to compute the weekly Amihud illiquidity measure, we require the daily data to be available for every trading day of the week for a stock to be included. To compute the Pastor-Stambaugh liquidity measure (Pastor & Stambaugh (2003)) for a week, we use the 15 consecutive trading days preceding the Saturday of the week. Stocks with missing daily returns or volume in the 15 trading days are excluded from the computation of that week. Option data on daily pricing, open interest, and trading volume is obtained from OptionMetrics, which covers the sample period from 1996 through 2010. Intra-day trading data from 1993 to 2010 is obtained from the TAQ dataset. The Sadka liquidity measure (Sadka (2006)) is obtained from Ronnie Sadka’s website. The historical values of the VIX index from 1986 to 2011 are obtained from the CBOE indexes data on WRDS. THE DESTABILIZING EFFECTS OF OPTION HEDGING 8 The Price and Liquidity Effect of Option Hedging In the Black-Scholes model (Black & Scholes (1973)), European call option deltas are given by Φ(d 1 ), where Φ(·) is the standard normal distribution function and d 1 is given by: d 1 = ln(S/K) + (rf +σ 2 /2)(T−t) σ √ T−t , (1.1) where S is the stock price, K is the strike price, rf is the risk-free rate, σ is the volatility, T is the expiration date, and t is the valuation date. The delta for a put option with the same parameters is Φ(d 1 )− 1. Clearly, both call and put option deltas increase in stock price S, i.e., gammas are positive. When option hedgers hold net written positions, decreases/increases in the underlying price cause selling/buying from hedgers. Given the assumption that the supply curve of the underlying stock is upward sloping, option hedgers who hold written positions behave like liquidity consumers in the underlying market. I.e., when an initial liquidity demand causes price displacement in the underlying stock, option hedgers add to the liquidity demand and amplify the initial price displacements. As a result, when option hedgers hold net written positions, their trades are destabilizing. Avellaneda & Lipkin (2003) study the time derivative of option deltas: ∂Φ(d 1 ) ∂t = Φ 0 (d 1 ) 2σ(T−t) 3/2 [ln(S/K)− (rf +σ 2 /2)(T−t)]. (1.2) When T−t is small, the time derivative can be approximated as: ∂Φ(d 1 ) ∂t ≈ Φ 0 (d 1 ) 2σ(T−t) 3/2 ln(S/K)      > 0 if S >K < 0 if S <K. (1.3) Equation 1.3 implies that the absolute value of time derivatives increases as T−t decreases, i.e., the closer the option is to expiration, the faster the delta changes and the more intensive the hedging-related trades. If an option hedger holds written positions in an option, then when the stock price S is higher (lower) than the strike price K, the hedger has to keep buying (selling) more stock until the option expires. With upward sloping supply curves, the hedging pressure could keep driving the stock price away from strike until expiration. In option expiration weeks, the increasing hedging pressure over time could keep stock price displacements from being corrected, or even amplify the initial displacements. After options expire, the stock price converges to fundamentals, which motivates our focus on weekly stock price reversals surrounding option expiration days in THE DESTABILIZING EFFECTS OF OPTION HEDGING 9 the empirical analysis. An initial stock price decrease not only triggers instantaneous selling by hedgers who hold written positions in options, it also makes more options satisfy the condition S <K. As time lapses, Equation 1.3 implies that deltas decrease for more options and increase for less, which causes more selling pressure from hedgers who hold written positions before options expire. As a result, the initial price displacement could be amplified before the options expire. This positive feedback effect is similar to the liquidity and price spirals described in Chowdhry & Nanda (1998) and Brunnermeier & Pedersen (2009) and the market crash modeled by Gennotte & Leland (1990). 4 Of course, the final liquidity and price effects are likely the joint work of more than one mechanism. For example, an initial price decrease caused by hedging pressure could trigger future liquidations due to margin calls (Chowdhry & Nanda (1998)), tightening of funding (Brunnermeier & Pedersen (2009)), or a misunderstanding of hedging pressure (Gennotte & Leland (1990)). These liquidations may drive the price even lower and cause more hedging pressure. Price Pressure Following Option Expiration The change in price pressure following option expiration deserves more discussion. Before options expire, option hedgers build up hedging positions and cause price impact along the way. If these hedging positions are liquidated after expiration, the order imbalance and price pressure would go in the opposite direction of the hedging pressure before expiration. In this case, reversals in order imbalance occur around expiration days. Does the liquidation of hedging positions happen after expiration? How fast does it happen? To answer these questions, notice that stock options and index options are settled differently: stock options are physically settled by delivering underlying stocks and index options are cash settled. After stock options expire, writers of in-the-money options transfer their hedging positions to option buyers, simply delivering the shares they have already accumulated. As a result, stock option hedgers effectively relinquish control of the portfolios used to hedge expired options. Whether the hedging positions are liquidated and how fast it is done is up to the owners of the options. If option buyers are not interested in holding the underlying stocks for the long run, they may liquidate the delivered positions following expiration. The situation is quite different for cash settled index options. Following expiration, delta hedgers immediately liquidate all their hedging positions. In this case, hedgers themselves generate reversals in order imbalance. Of course, if delta equivalent positions are 4 Note that our setting is different from Gennotte & Leland (1990) also in that their model does not imply that hedging pressure increases as time approaches a certain point. THE DESTABILIZING EFFECTS OF OPTION HEDGING 10 immediately established after old options expire, option hedgers do not need to liquidate their hedging positions. Our preliminary analysis suggests that option buyers are typically sluggish in rolling over their positions, which implies that it is very likely that index option hedgers generate sharp reversals in order imbalance following option expiration days. On the other hand, because stock option owners do not have the urgency to liquidate the delivered positions they take, the reversals in order imbalance they cause is unlikely to be sharp. Stock Price Reversals This section studies price reversals surrounding option expiration dates. We show the following empirical results. First, stock prices exhibit cross-sectional reversals, i.e., stocks that performed well in the past tend to deliver low returns in the future and vice versa. Second, market returns also reverse after expirations. Index option hedging pressure drives the market index away from fundamentals before settlement prices are determined and the market index reverses afterward. The magnitudes of price reversals in the cross section and at the market level are both positively correlated with market illiquidity. Third, cross-sectional price reversals and market level reversals are positively correlated in the time series. Fourth, market level price pressure exhibits differential impacts on individual stocks, which causes cross-sectional price reversals. Weekly Price Reversals in the Cross-Section In this section, we measure cross-sectional price reversals following option expiration weeks and compare them to reversals following non-expiration weeks. Previous work on weekly price reversals often defines the week as the 7 days from one Thursday to the following Wednesday. In the current setting, because we are interested in the weekly reversal after option expiration, it is proper to use calendar weeks as formation periods. The holding period is the calendar week after a formation week. Following the convention in the literature, we skip the last trading day in the formation week to avoid reversal due solely to bid-ask bounce. At the end of each week, stocks are sorted into quintiles by their returns from Monday to Thursday. A zero-investment portfolio is then constructed by buying the stocks in the past winner quintile and selling the stocks in the past loser quintile and is held for the week after formation. Holding week returns to the zero-investment portfolio are fitted to the Carhart 4-factor model to produce abnormal returns. To avoid the impacts from stocks under extreme liquidity conditions, we exclude stocks that have unusually high (larger than $1000) or low (smaller than $5) stock prices at the beginning of the formation week. We also exclude stocks that fall below the 1st NYSE size decile. THE DESTABILIZING EFFECTS OF OPTION HEDGING 11 Table 1.1 presents results on cross-sectional weekly reversals. The rows labeled as “expiration week” refer to the cases in which the formation weeks are option expiration weeks, i.e., the abnormal returns are computed in the weeks after expiration weeks. The analysis only considers option contracts that expire monthly. Hence, the expiration weeks are always those that contain the third Friday of the month. Entries in Panels A and B are average abnormal returns (percentage per day) to the zero investment portfolios that buy past winners and sell past losers. Column 1 presents the results for the full sample period from January 1973 to December 2011, and Columns 2 and 3 show the results for the sample periods before and after (including) 1993, respectively. The table shows that prices exhibit significant reversals following both expiration weeks and non-expiration weeks for both equally and value weighted portfolios. The reversals following expiration weeks are stronger than non-expiration weeks. In the second half of the sample, the difference is 9 basis points per day for the value-weighted portfolio and 5 basis points per day for the equal-weighted portfolio. The price reversals following other weeks have declined by half in the second half of the sample period, which reflects an improvement in the market liquidity condition over time. However, the magnitude of price reversals following option expiration weeks has been virtually constant over time, which suggests that the increasing liquidity demand in option expiration weeks is offsetting the improvement liquidity provision in the market. Following expiration weeks, value-weighted portfolios deliver stronger reversals than equal weighted portfolios. Considering that large stocks are usually associated with more intensive option-related trading and that index option trading also tends to have more impact on large stocks, it is not surprising that the value-weighted portfolio performs better. Panels C and D present the return spreads in the formation weeks. In each formation week, weekly returns (excluding the last trading day of the week) of stocks in the same quintile are aggregated by either equally or value weighting to produce the equal-weight or value-weight formation period returns for each quintile. Entries in the table are the differences between the formation period returns of the past winners and past losers. Comparing the rows for expiration week and non-expiration week, we see that the formation period spreads are consistently lower for expiration weeks, which seems at odds with the fact that price reversals tend to be stronger after expiration weeks. Panels E and F present the comparison between daily abnormal returns and round-trip transaction costs on daily basis. In a previous study on the weekly reversal anomaly, Avramov et al. (2006) argue that the profit to the weekly reversal strategy can be completely eliminated after accounting for transaction costs, given by effective bid-ask spreads. In contrast, we show that the profit of the value-weighted weekly reversal portfolio THE DESTABILIZING EFFECTS OF OPTION HEDGING 12 following option expiration weeks cannot be eliminated even after accounting for transaction costs. Panel E presents the results for value-weighted portfolios. Limited by the availability of the data to compute effective spreads, we consider the sample period from 1993 to 2011 and present the results for this sample period in column 1. Column 2 presents the sub-sample period starting in 1998, which represents the period following the reduction of the minimum tick size from one eighth to one sixteenth. Column 3 presents the abnormal returns and costs of a cost efficient weekly reversal strategy. Specifically, when the portfolios are formed at the end of each week, we exclude the top 20% of the stocks in the cross-section with the highest one-way effective spreads. We present the results for the reversals following expiration weeks and other weeks in separate rows as indicated by row names. Also, in the row names, α wml,t+1 indicates the daily abnormal returns during the holding period and “cost” indicates the round-trip transaction costs on daily basis. Both numbers are presented in percentage term per day and are comparable. The first two rows of Panel E show that the transaction costs (13 bps per day) are almost equal to the abnormal returns following option expiration weeks for the full sample period considered here. The transaction costs are reduced to 9 bps per day following the reduction in tick size, and they are further reduced to 7bps per day by the exclusion of the stocks with high transaction costs. In comparison, the abnormal returns stay constant at around 16bps per day in all three cases. Hence, in the recent sample period, the weekly reversal anomaly is not dominated by illiquid stocks and portfolios can be constructed such that abnormal returns exceed transaction costs. The third and fourth rows in the Panel E show that the abnormal returns following other weeks are smaller or equal to the transaction costs. Panel F shows that the transaction costs are always higher than abnormal returns for equal-weighted portfolios. Price Reversals and Market Liquidity This section demonstrates the relation between market liquidity and price reversals following option expiration weeks. Although we expect liquidity to be a significant determinant of the magnitude of price reversals, it is important to realize that price reversals can arise for other reasons. For instance, irrational traders might overpay for glamour stocks, with stock prices converging to their efficient levels afterward. There could be more overpricing and hence greater reversals when the market liquidity is high. These potentially offsetting forces make the sign of time series correlation between liquidity and reversal unclear a priori. The main market level liquidity measure we consider is the Pastor-Stambaugh liquidity measure (Pastor & Stambaugh (2003)). We do so because their measure is likely to capture THE DESTABILIZING EFFECTS OF OPTION HEDGING 13 variations in the inventory risks of market makers, which we believe should play an important role in driving price displacements in option expiration weeks. For formation week t, we compute the Pastor-Stambaugh measure PS t using data from the consecutive 15 trading days ending with the last trading day before the Saturday of week t. Formation weeks that are expiration weeks are sorted into 10 groups by Pastor-Stambaugh measures. Group 1 has the lowest market liquidity, and group 10 has the highest market liquidity. The same sorting is done for all other weeks separately. Figure 1.2 plots the average abnormal returns to the weekly cross-sectional reversal strategy for each liquidity group. All the abnormal returns are presented in the unit of percentage points per day. The results from expiration formation weeks and non-expiration formation weeks are represented by bars with different colors. Clearly, there are strong reversals following option expiration weeks when the market liquidity is low. The value-weighted long-short portfolio following expiration week delivers an average abnormal return of 38 basis points per day in liquidity group 1. In contrast, there is no clear relationship between market liquidity and reversals following other weeks. The relationship between post-expiration week reversals and liquidity is clearly non-linear, which is consistent with past findings that inventory risks have a non-linear impact on market liquidity (see Comerton-Forde et al. (2010) for an example). Next, we use regressions to confirm the correlation between liquidity and price reversals. Table 1.2 presents the coefficient estimates from the model: α wml,t+1 =a +b·PS t +c·X t , where α wml,t+1 is the abnormal return to the weekly reversal portfolio following formation week t. PS t is the Pastor-Stambaugh liquidity measure of week t, where a high value in PS t means the market is liquid. X t is the vector of control variables, which consists of the permanent-variable component of the Sadka liquidity measure (SPV) as described in Sadka (2006) and mutual fund flows in percentage term (Fund Flow). We run the regression for option expiration formation weeks and other formation weeks separately. The full sample period (1973-2011) is used for regressions without controls, and a sub-period (1983-2010) is used for for regressions with controls due to the availability of control variables. Panels A and B show that coefficient b is only significant for expiration formation weeks. Also, the R-squared for non-expiration weeks is virtually zero. Overall, the evidence shows that there is significant correlation between post-expiration week cross-sectional reversal and market liquidity level given by the Pastor-Stambaugh measure, which suggests that the reversal is likely to be related to the inventory risks of market makers. THE DESTABILIZING EFFECTS OF OPTION HEDGING 14 Price Reversals and Stock Option Open Interests Table 1.1 shows that the expiration of futures contracts cannot explain the weekly reversal pattern. This section investigates the role of stock option hedging by showing the relation between cross-sectional reversals and stock option open interest. The stock option open interest data come from OptionMetrics, which covers the sample period from January 1996 to December 2010. Hence, our analysis in the section is restricted to this shorter sample period. For each formation week, we divide the cross-section of stocks into 9 subsamples with a 3× 3 two-way sorting in size and open interest in stock options. In formation week t, the open interest of a stock is calculated as the sum of the open interests of all the expiring options on the stock at the end of the last trading day before the expiration week. The open interests are then normalized by the total number of common stock shares outstanding and reported in Table 1.3. We sort the expiration weeks into two groups — the liquid subsample and the illiquid subsample — by the ranking of their Pastor-Stambaugh measures and present the results in Panels A and B. Each entry contains the average abnormal returns to the weekly cross-sectional reversal strategy. The table does not show a clear relationship between the strength of reversal and open interest. The first column of Table 1.3 presents the reversal profits of stocks with zero or little option open interest. Panel A shows that even stocks with zero open interest have significant reversals. Our results seem to be consistent with Ni et al. (2005), who find that stock option hedgers tend to hold net long positions, so their hedging pressure is unlikely to be destabilizing. Market Return Reversals This section shows that index option hedging pressure has significant impact on the market index. We find evidence of significant price displacements at the index level during option expiration weeks, which disappear after option expiration, causing reversals in market returns. We also find that the direction of the displacements and reversals can be predicted by a proxy of market maker inventories. These findings demonstrate that index option hedging pressure generates price displacements by amplifying the existing inventory risk of market maker. The results are stronger when the market liquidity is low. Our work also makes interesting contributions to the longstanding debate on the Efficient Market Hypothesis by showing the predictability of market returns using public information. Index options and stock options expire in the same weeks, and they both imply increasing hedging demands near expiration days. Hence, separating the effects of the two can be challenging. Fortunately, a unique feature of index options allows us to distinguish them from stock options. Unlike stock options, which are settled at Friday close, the settlement THE DESTABILIZING EFFECTS OF OPTION HEDGING 15 price for an index option is determined at Friday open. 5 As a result, index option hedgers should stop increasing hedging positions at Thursday close, and price reversals should start at Friday open instead of the following Monday. In fact, because index options are cash-settled contracts, we expect option hedgers to liquidate their hedging positions on Friday if the holders of the expired options fail to roll over into delta-equivalent positions. 6 In other words, the effect of index option hedging on the inventories of stock market makers is temporary and it is likely to disappear after settlement on Friday morning. We conjecture that, during the trading days before Friday in expiration weeks, increasing index option hedging demands could generate pricing errors at the index level . Furthermore, the relative importance of over-pricing and under-pricing in an option expiration week should be predictable using market maker inventory level I 0 at the beginning of that week. 7 When I 0 is high, which suggests selling pressure in the market, it may be difficult for market makers to absorb more sell orders due to funding constraints. Hence, a surge in sell orders from hedgers is likely to push stock prices significantly lower. When I 0 is low, which suggests buying pressure in the market, it may be difficult for market makers to locate shares for buyers, so buy orders from hedgers might push stock prices significantly higher. On the other hand, buy (sell) orders are unlikely to have significant price effects when I 0 is high (low). To summarize, index option hedging generates price displacements by amplifying the existing inventory risk of market makers; when I 0 is high (low), the average effect of option hedging is to cause under-pricing (over-pricing). This further implies that I 0 negatively predicts the market returns from Monday to Thursday (before settlement) in option expiration weeks. After Friday opening, hedgers liquidate their hedging positions and the market price reverses, which leads to negative predictability of I 0 on market returns on expiration Fridays. Our data do not permit a direct measure of the inventory level I 0 . The following empirical analysis uses lagged aggregate order imbalance (buy volume minus sell volume) as a proxy for the aggregate inventory level of market makers — high order imbalance implies low inventory , low order imbalance implies high inventory. The intuition is that, if the entire market has been under heavy selling (buying) pressure, market makers have most likely accumulated high (low) inventories. Since that our goal is to study the predictability of I 0 on market returns in expiration weeks, it is crucial that the proxy we use does not cause predictability through other channels. Chordia et al. (2002) find little predictability of lagged order imbalance on market returns. Our findings in non-expiration weeks confirm 5 The am-settlement practice started in June 1987 6 See Section 1 for more discussion on this. 7 For simplicity, we assume that the inventory level in the current week contains two parts: I 0 and the additional inventories resulting from the current hedging demands. THE DESTABILIZING EFFECTS OF OPTION HEDGING 16 their results. Hence, we believe it is safe to assume that our proxy does not cause predictability that is unrelated to option hedging pressure. The following hypothesis formalizes the idea: Hypothesis 1 In option expiration weeks, the lagged aggregate order imbalance positively predicts market returns before settlement prices are determined and negatively predicts market returns after settlement. The predictability is stronger when market liquidity is low. Regression models are as follows: r m after,t =a 1 +b 1 ·lagimb t (1), r m before,t =a 1 +b 2 ·lagimb t (2). The above two models imply: r m rev,t =r m after,t −r m before,t =a 1 −a 2 + (b 1 −b 2 )lagimb t (3). Null: b 1 = 0, b 2 = 0. In the models above, r m after,t is the CRSP value weighted market return after settlement prices are set for week t. r m before,t is the CRSP value weighted market return before settlement. lagimb t is the lagged aggregate order imbalance, which captures the market maker inventory risks. For a formation week t, lagimb t is defined as the average daily aggregated S&P 500 stock order imbalance over the 15 trading days preceding the first trading day of week t− 1. Specifically, for each S&P 500 stock, we compute daily order imbalances as the difference between share volume initiated by buyers and sellers normalized by total shares outstanding. Daily order imbalances of all S&P 500 stocks are then averaged to obtain the aggregate daily order imbalance. lagimb t is given by the mean of aggregate daily order imbalances over the 15 trading days preceding week t− 1. Notice that the horizon over which the lagged imbalance is computed is chosen to ensure that it does not overlap with the return horizons we consider. Here, we only include S&P 500 stocks because they represent a large portion of the total market capitalization, and the SPX index option is the most heavily traded index option. Table 1.4 presents the results from testing Hypothesis 1. First, we specify the ranges over whichr m before,t andr m after,t are computed. Although the hypothesis specifies that week t is an option expiration week, we also run regressions in non-expiration weeks separately as a placebo test. The horizon over which r m before,t is computed ends at Thursday close in week t. We consider various starting points ranging from the Monday in week t back to the Tuesday in week t− 1. The horizon for r m after,t starts from the opening of the Friday in week t. The ending points range from the Friday closing in week t to the Friday closing in week t + 1. The column and row names in the table denotes the starting and ending points THE DESTABILIZING EFFECTS OF OPTION HEDGING 17 with standard abbreviations of weekdays. In the column and row names in Panels C and D, the commas separate the starting dates and the ending dates. Option expiration weeks in the sample period are further divided into a liquid half and an illiquid half on the basis of the Pastor-Stambaugh liquidity measure. Regressions are performed within each half to study the effect of the market liquidity on the aggregate price effect of option hedging. The same liquidity sorting process is followed for non-expiration weeks. Because there is no significant result when market liquidity is high, we omit the results from the high liquidity half from this table. Panels A and B reports values of b 1 and b 2 , respectively. The differences b 1 −b 2 are reported in Panels C and D. In these regressions, the unit of order imbalance is 0.01% of the market capitalization, and the unit of market returns is percentage points; returns here are not on a daily basis. Panel B confirms that the lagged aggregate order imbalance positively predicts market returns before settlement prices are determined. Results from different horizons suggest that most of the price displacements occur during option expiration weeks, and only the Friday of the week before an expiration week contributes positively to the pricing errors. Panel A confirms the market reversals after settlement. The first two numbers (0.091 and 0.184) from row 1 in Panel B suggest that half of the price displacements occur on the Thursday before expiration 8 , which is consistent with the theoretical prediction that hedging pressure becomes particularly strong when the time to expiration is close to 0. The first two numbers (-0.085 and -0.179) from row 1 in Panel A suggest that almost half of the reversals happen on the expiration Friday, and the other half occur on the following Monday. Interestingly, it seems that the release of the hedging pressure tends to over-correct the market prices to some extent, and the reversal peaks on the Wednesday following expiration (-0.253), which is consistent with the our conjecture that the index option hedging pressure is not only released after expiration but also reversed. Panels C and D show the values and statistical significance of b 1 −b 2 , which is a direct measure of the degree of the market reversals. Panel C is for option expiration weeks and Panel D is for other weeks. The two panels contrast sharply in terms of the magnitude of the coefficients and the statistical significance. All the coefficients in Panel D are much smaller than the coefficients in Panel C, which confirms that, in general, there is no predictability of lagged order imbalances on market returns. Hence, we draw the conclusion that the results in Panel C are indeed driven by option hedging pressure. The evidence in this section is consistent with our conjecture that, in illiquid markets, 8 Although these numbers are not regression coefficients, one can think of them as market returns to gain intuition. The only difference to notice is that the sign of coefficients is determined by both actual returns and order imbalance. This makes sure that the sign of price displacements is always positive, and reversals are always negative. THE DESTABILIZING EFFECTS OF OPTION HEDGING 18 index option hedging causes significant price displacements at the market level when the hedging pressure is in the direction that increases the inventory risks of market makers. The finding is intriguing in light of Grossman (1988), who argues that option open interest allows liquidity providers to predict hedging demand and hence improves market liquidity. Our evidence suggests that it is potentially difficult for investor to learn about future hedging pressure from option open interest. Our findings also speak to the assumptions made in Gennotte & Leland (1990)). The key assumptions there is that hedging demands from hedgers need to be unknown to uninformed traders to trigger a market crash. We show that the market price can exhibit significant displacements even when the hedging schedule appears to be well known to the market. A deeper understanding of the difficulties faced by stock market makers in option expiration weeks is an interesting research topic which we leave for the future. Linking Cross-Sectional and Market Reversals This section shows that it is the differential impact of market level hedging pressure on different stocks that drives the dispersion in stock price displacements before expiration and the cross-sectional reversals after expiration. First, we link the findings from sections 1 and 1 by measuring the time series correlation between cross-sectional reversals and market return reversals. Predicting Post-Expiration Cross-Sectional Reversals with Market Reversals in Expiration Weeks. The market-level reversals that we focus on arise from index option hedging pressure. The analysis in the previous section suggests that these reversals are likely to happen in only one direction conditioning on the lagged aggregate order imbalance, i.e., when the market has been under selling pressure, sell orders from option hedgers in expiration weeks tend to cause low returns before expiration and high returns after expiration, causing down-up reversals, while an up-down reversal is unlikely to be related to hedging pressure; when the market has been under buying pressure, buy orders from option hedgers in expiration weeks tend to cause high returns before expiration and low returns after expiration, causing up-down reversals, while a down-up reversal is unlikely to be related to hedging pressure. In other words, we argue that option hedging pressure causes price displacements by amplifying existing order imbalance and inventory risk in the market and it is highly implausible that the trading volume generated by option hedging is large enough to dominate all other sources of trading volume and overturn the direction of order imbalance in the market. To summarize, hedging pressure results in market return reversals in the following way: when the lagged aggregate order imbalance lagimb t is positive (negative), market returns following expiration should be low (high) relative to the THE DESTABILIZING EFFECTS OF OPTION HEDGING 19 returns before expiration, i.e., r m rev,t =r m after,t −r m before,t < 0 (r m rev,t > 0). These two cases can be unified and simplified to the statement that r m rev,t ∗sign(lagimb t )< 0 regardless of the sign of the lagged aggregate order imbalance, that is, low r m rev,t ∗sign(lagimb t ) indicates strong market level reversals caused by index option hedging. In the following analysis, we capture the link between cross-sectional reversals and market level reversals by the correlation between α wml,t+1 and r m rev,t ∗sign(lagimb t ). To compute r m rev,t , we specify the horizon of r m before,t and r m after,t to be Monday to Thursday and Friday respectively, i.e., r m before,t =r m M−T,t andr m after,t =r m F,t . With this choice of horizon, α wml,t+1 is measured in week t+1, and r m rev,t is measured in week t. Hence, significant correlations imply predictive ability of market reversals for cross-sectional reversals. Table 1.5 Panel A presents the value of correlations. To understand how market liquidity affects the correlation, we sort expiration weeks into two groups — high liquidity and low liquidity — based on the rankings by the Pastor-Stambaugh measure. We also introduce a set of notations: ˜ r m rev,t =r m rev,t ∗sign(lagimb t ),˜ r m F,t =r m F,t ∗sign(lagimb t ) and ˜ r m M−T,t =r m M−T,t ∗sign(lagimb t ). The first row shows that market level reversals ˜ r m rev,t are significantly positively correlated with the cross-sectional reversals. When the market liquidity is low, this correlation is 32%. Notice that we obtain positive sign because our signing convention implies that α wml,t+1 < 0 and ˜ r m rev,t < 0 indicate cross-sectional and market-level reversals respectively. Two legs of the market reversal, i.e., Monday to Thursday build-up and Friday reversal, both contribute to the correlation. The sign of corr(˜ r m M−T,t ,α wml,t+1 ) is negative because ˜ r m M−T,t represents the build-up of the price displacements and should be negatively correlated with reversals. The magnitude of price reversals could contain conflicting information. A strong Friday reversal implies that price displacements are likely to be high before Friday, but it also implies that a significant portion of pricing errors is already corrected on Friday. As a result, it is not clear what the implication is on the residual pricing errors that need to be corrected in the following week. Therefore, we also consider a trading strategy that only depends on the signs of market returns. This trading strategy is interesting in its own right since it is easily implementable. For expiration weeks and other weeks, respectively, within each liquidity subsample, we further divide the sample into 4 groups by the sign of ˜ r m F,t and ˜ r m M−T,t and study cross-sectional weekly reversal in each group. The average daily abnormal returns to the cross-sectional weekly reversal strategy within each group are reported in Panels B and C of Table 5. Interesting patterns stand out in Panel B columns 3 and 4: reversals are clearly stronger in low liquidity weeks; the strongest results (-40 basis points per day) in low liquidity weeks come from the combination ˜ r m F,t < 0 and ˜ r m M−T,t > 0, which is the most consistent with hedging pressure THE DESTABILIZING EFFECTS OF OPTION HEDGING 20 driven market level reversals in expiration weeks, i.e. both legs add to the negative value of ˜ r m rev,t ; in contrast, there is no cross-sectional reversal for the opposite combination ˜ r m F,t > 0 and ˜ r m M−T,t < 0, which is the least consistent with hedging pressure driven reversals. For the other two combinations that are partially consistent with hedging pressure effect, there are also significant cross-sectional reversals(-28 and -16 basis points per day). It appears that the Friday reversal leg plays a more important role in determining the profit of cross-sectional reversal. The possible explanation is that the Monday to Thursday pressure build-up spans four trading days, so the sign of market returns is more likely to be affected by fundamentals. The Friday involves a relatively concentrated pressure release which only need to compete with the fundamentals of one day; as a result, the price pressure is more likely to play an important role in driving the sign of market returns on Friday. Comparing columns under high liquidity and low liquidity, we find that abnormal returns following expiration weeks that satisfy ˜ r m F,t < 0 and ˜ r m M−T,t > 0 exhibit the strongest sensitivity to market liquidity, which suggests that the link between market reversals and cross-sectional reversals only exists in illiquid market. The finding that the signs of market returns alone can indicate hedging pressure in the market is striking. Because it is normally believed that news about fundamentals and market sentiment should drive most of the market returns, and liquidity effects only account for a tiny part. Our evidence, on the other hand, shows that market liquidity is of first order importance in determining asset prices in some cases. An Explanation of the Relation Between Market Reversals and Cross-Sectional Reversals. In the following analysis, we argue that the reason why cross-sectional reversals and market level reversals tend to occur around the same expiration weeks is because they are both driven by index option hedging pressure. Index level hedging pressure generates different levels of price impact on different stocks, and the subsequent correction of these pricing errors results in the cross-sectional reversal. Figure 1.1 visualizes the idea. In this figure, we use three lines to indicate the past winner portfolio (denoted as W), the market portfolio (denoted as M), and the past loser portfolio (denoted as L), respectively. If the market maker inventories are high (low) before an expiration week, selling (buying) pressure from option hedgers in the expiration week could amplify the existing inventory risk and push the index level lower (higher). The loser (winner) portfolio formed in the expiration week is likely to contain the stocks that experience the largest price impact from option hedging, and these stocks should also contribute the most to the price reversals in the following week. Again, we use the lagged aggregate order imbalance as a proxy for market maker inventory level at the beginning of option expiration weeks and the conjecture above leads to the following hypothesis: THE DESTABILIZING EFFECTS OF OPTION HEDGING 21 Hypothesis 2 When the lagged order imbalance is high, most of the cross-sectional price reversals following option expiration weeks come from past winners; when the lagged order imbalance is low, most of the cross-sectional price reversals come from past losers. This effect is particularly strong when market liquidity is low. We use raw returns instead of abnormal returns to test this hypothesis. Studying abnormal returns can be misleading in this case. For instance, when the market is under buying pressure, supposed portfolio L is not affected by the pressure at all and so does not contribute to the price reversal, but the market index is overpriced and the winners are among the most affected as depicted in the top half of Figure 1.1. Also suppose the market beta of portfolio L is not very different from 1; then a market model would produce a positive abnormal return for portfolio L, which is wrong because we assume that there is no reversal in portfolio L. To focus on the most relevant periods, we also exclude the weeks that satisfy ˜ r m F,t > 0 and ˜ r m M−T,t < 0 because these weeks show neither market level nor stock level reversals according to Panel B in Table 1.5. Including them can only obscure the relevant pattern. Lagged return sorted portfolios are formed in the same way as in Section 1. The first four columns in Table 1.6 present the average holding period returns of the lagged return-sorted portfolios from 1993 to 2011. The sample is divided into halves by the Pastor-Stambaugh liquidity measure, and we only present the results from the more illiquid half. We present the average raw returns of the weeks with positive and negative lagged aggregate order imbalance in the columns titled “lagimb t > 0” and “lagimb t < 0”, respectively. Panel A presents the average returns following option expiration weeks , and Panel B presents the average returns following non-expiration weeks. The first two columns in Panel A show that, when the market is under buying pressure for reasons unrelated to option hedging, about 80 percent (16 basis points out of 20 basis points) of the profit of the post expiration week value-weighted cross-sectional reversal portfolio comes from the past winner portfolio; when the market is under selling pressure, about 80 percent (42 basis points out of 53 basis points) of the profit of the value-weighted cross-sectional reversal portfolio comes from the past loser portfolio. For equally weighted portfolios, columns 3 and 4 show that virtually all the profits come from the portfolios that are most affected by the market level trading pressure. This is largely consistent with the pattern shown in Figure 1.1. In comparison, Panel B shows that the pattern for returns following non-expiration weeks is less clear. It is not consistent with Figure 1.1. Index option hedgers most likely use index futures to hedge the their option position for liquidity reasons, and index arbitrageurs would connect the prices of index futures and index constituents. Although the exact trading strategies used by index arbitrageurs is THE DESTABILIZING EFFECTS OF OPTION HEDGING 22 Figure 1.1. The Differential Impact of the Market Level Hedging Pressure on the Cross-Section of Stocks unknown to us, it is reasonable to assume that they trade heavily in large stocks. First, the most commonly used index option is the option on the S&P 500 index, and its constituents are all large stocks. Second, delta hedgers need to trade fast enough to follow a dynamic hedging strategy, so they are likely to prefer large stocks for better liquidity. We investigate whether large stocks are more likely to be affected by the index option hedging pressure. The last two columns in Table 1.6 present the average size of stocks in each past return quintiles. Panel A shows that, when the market is already under buying pressure for reasons unrelated to option hedging, large stocks tend to be pushed into the winner portfolio in expiration weeks; when the market is under selling pressure for reasons unrelated to option hedging, large stocks tend to be pushed into the loser portfolio in expiration weeks. Panel B shows that this pattern is unique to option expiration weeks. The evidence in this section brings a refreshing perspective to the cross-sectional weekly reversal anomaly. Previous research on the anomaly shows that small and illiquid stocks contribute dominantly to the weekly reversal profits. For instance, Avramov et al. (2006) argue that the magnitudes of the weekly price reversals are positively correlated with the stock level illiquidity measures in the cross-section. In contrast, our evidence shows that the weekly reversals following option expiration weeks are stronger for value-weighed portfolio, so large stocks play an important role in our setting. The current work suggests that although large stocks appear to be more liquid than small stocks on average, they are also more likely to be targeted by liquidity traders when the market liquidity is low and when orders need to be executed fast (e.g., in option expiration weeks). Therefore, large stocks could be more prone to pricing errors as market liquidity condition worsens. THE DESTABILIZING EFFECTS OF OPTION HEDGING 23 Option Prices Before Expiration This section shows that the prices of SPX index options tend to be high before option expiration, which suggests that option market markers raise prices to induce the holders of long positions to close their positions. This pricing behavior is consistent with our argument that written positions held by option hedgers could generate destabilizing hedging pressure when options approach expiration and option market makers adjust prices to reduce the amount of written positions they hold. First, we measure changes in the VIX index surrounding Friday openings. For each Friday F t (t is the week number), we compute the changes in VIX over various horizons that enclose F t . Denoting the starting and ending dates of the horizons as d 1,t and d 2,t respectively, changes in VIX are given by ΔVIX t =VIX(d 2,t )−VIX(d 1,t ). Notice that starting and ending dates do not have to be in week t; for example, d 1,t =M t−1 means that the horizon starts on the Monday in the week before t. Entries in Table 1.7 are the average changes: P t ΔVIX t /N, where N is the number of weeks in the (sub)sample period. We study changes in VIX over expiration Fridays and other Fridays separately. Expiration weeks and other weeks are further divided into two subsamples by a lagged Pastor-Stambaugh liquidity measure to study the role of market liquidity. Results from the four subsamples are presented in Panels A to D. Panel A shows that the VIX index decreases significantly over expiration weeks when the market liquidity is low. The panel suggests that the VIX index remains high in the week before expiration weeks and that it is also high at the beginning of expiration weeks. But the level starts to drop after the Monday in an expiration week, reaches its low at Friday close, and remains at the low level afterward. In comparison, the pattern is much weaker when the market liquidity is high as shown in Panel B. Panels C and D do not show strong patterns. The signs in these two panels are more likely to be positive, which has to be the case given that VIX decrease in expiration weeks and the fact that there is no long term trend in the VIX level. The VIX index can be viewed as a measure of risk neutral volatility that is determined by physical volatility and risk premiums. Changes in VIX could simply result from changes in physical volatility. Panel E of Table 1.7 shows the standard deviations of the S&P 500 index in each subsample. The standard deviations are calculated using all the daily returns of the S&P 500 index in each subsample. Panel E shows that the estimated physical volatility of the S&P index is in fact lower in expiration weeks and cannot explain the high VIX levels. Hence, our evidence suggests that SPX options are priced with high premiums prior to expiration weeks. This is consistent with the notion that option market makers try to reduce the written positions they hold by raising prices to attract option writers into the market. The timing of changes in VIX suggests that option market makers tend to prepare THE DESTABILIZING EFFECTS OF OPTION HEDGING 24 for the expiration weeks ahead of time by raising prices the week before. Market Liquidity Measures Endogeneity Issues in Measuring Market Liquidity In this section, we provide evidence from liquidity measures that suggests the liquidity condition in option expiration weeks is likely to be worse due to the destabilizing option hedging. The evidence in previous sections shows that market level hedging pressure in expiration weeks put significant pressure on liquidity providers. If the pools of liquidity for market level trades and stock level trades are connected, consumption of liquidity by market level trades should lower the available liquidity for stock level trades. As a result, we expect the market liquidity during option expiration weeks to be low in general. Measuring underlying liquidity is a generally challenging task due to various endogeneity problems in liquidity measures. In particular, traders tend to switch to more liquid assets when the market illiquidity is low, and they can also cancel trades when prices are unfavorable. These endogeneity problems cause bias and make the market appear to be more liquid than it actually is. In the current setting, during option expiration weeks, the entire market is under significant price pressure due to index hedging. As a result, a significant portion of the trading on large stocks is likely to be related to index level trades. Obviously, when we study the time series of market liquidity, it is necessary to compare the liquidity level of the same asset over time. However, due to the existence of indexes, this seemingly trivial requirement becomes tricky. Stocks play a “dual role” as both a individual stock and a part of the market index and, conceptually, their total trading volume can be decomposed into an index component and an idiosyncratic component. It is reasonable to assume that the two components cause a different price impact if the market makers were able to identify them to some extent, and typically the price impact of the index component would be lower due to less information asymmetry and low volatility at the index level. When the market liquidity decreases, traders tend to switch to more liquid assets and possibly from individual stocks to indexes, which causes the weight of the index component relative to the idiosyncratic component to rise. If we use λ m t to denote the market component of the price impact and λ i k,t for the idiosyncratic component of stock k, it is reasonable to assume that λ m t <λ i k,t . The total price impact per dollar traded could be written as the weighted average of the two components: λ k,t =w m k,t λ m t +w i k,t λ i k,t , where w m k,t is the weight of the trading volume that is due to index trading and w i k,t is the weight of the trading volume that is due to stock level trading, we have w m k,t +w i k,t = 1. If we compare the price impact of the stock k at time t 1 and t 2 , and assume that the market is less liquid at t 2 , λ m t 2 >λ m t 1 and λ i k,t 2 >λ i k,t 1 . As we THE DESTABILIZING EFFECTS OF OPTION HEDGING 25 have argued above, it is very likely that w m k,t 2 >w m k,t 1 due to endogenous trading decisions. As a result, it is difficult to sign the change in total price impact λ k,t 2 −λ k,t 1 . Comparing Market Liquidity in Option Expiration Weeks and Other Weeks Back to our setting, during option expiration weeks, w m k,t could increase for two reasons. First, it would increase because of intensified index option hedging. Second, strong hedging demands consume liquidity and reduce the liquidity available for stock level trades; facing low liquidity, traders might switch from trading individual stocks to trading indexes. The endogeneity problem implies that even if both λ m t and λ i k,t rise during option expiration weeks, the measurable average impact λ k,t may not rise. Given the above analysis, we propose that the liquidity change in option expiration weeks are most likely to be detected in stocks that are the least likely to be affected by index tradings. Our conjecture is that small stocks are less likely to be traded as the part of a index because, first, the most popular index option is the option on the S&P 500 index, whose constituents are large stocks; second, even if index level traders (e.g., option hedgers, index arbitrageurs) do not restrict themselves to index constituents, they are likely to prefer large stocks for liquidity considerations. To test our conjecture, we compare the liquidity measures in the expiration weeks and non-expiration weeks for stocks with different sizes. Specifically, at the end of each week, stocks are sorted into terciles based on market capitalization, and a weekly liquidity measure is computed for each size tercile by averaging the weekly liquidity measures of the stocks in the tercile. We also consider the role of the market liquidity. Section 1 indicates that the index option hedging only causes significant price impact when the market liquidity is low, so it is reasonable to expect that the liquidity effect of index option hedging is stronger when the market liquidity is already low for reasons unrelated to hedging. We divide the expiration weeks into four groups by the Pastor-Stambaugh measure, with group 1 being the most illiquid and group 4 being the most liquid. Other weeks are sorted into four liquidity groups in the same way. Within each liquidity group, we compare the sample means of the weekly liquidity measures in the option expiration weeks and other weeks for each size tercile. Table 1.8 contains the differences of the sample means for each size-liquidity combinations. The liquidity measures used here are the Amihud illiquidity measure and the effective spread. The weekly Amihud illiquidity measure is defined as 1 n P n d=1 |r d | v d · 10 6 where r d is the stock return (in percentage terms) on day d and v d is the dollar volume on day d and n is the number of days in the week. The effective spread (ESPR) of the nth trade is defined as ESPR n = 2·|log(p n )−log(m n )|, where p n is trade price and m n is the middle point of THE DESTABILIZING EFFECTS OF OPTION HEDGING 26 the prevailing quotes when the nth trade occurs, daily measure ESPR d is the dollar volume weighted average of all trades in day d, and the weekly measure is the mean of all the daily values in the week. Table 1.8 shows that when the market liquidity is low, stocks in the lowest size tercile have lower liquidity measures in option expiration weeks than in other weeks, but there is no significant difference for the stocks in the other two size terciles. This finding is consistent with the conjecture above. When the market liquidity is low, on average, the price impact of a $ 1 million trade in a small stock in an option expiration week is higher than its price impact in a non-expiration week by 1.37% of the stock price. The difference in effective spreads is 5 basis point. These differences are both statistically and economically significant. The table also seems to suggest that the liquidity condition in option expiration weeks is better than in other weeks when the market liquidity is high. But the evidence is not consistent across different liquidity measures. Conclusion This paper provides a comprehensive study on the trading environment in option expiration weeks. Based on the idea that trading pressure from delta hedgers who hold net written positions in options can be destabilizing in expiration weeks, we investigate price displacements and subsequent reversals around option expiration days. We find strong evidence that index option hedging generates significant stock price displacements in option expiration weeks. The pricing errors disappear gradually after options expire, resulting in stock price reversals both in the cross-section and at the market level. Cross-sectional and market level reversals are stronger when market liquidity is low. The direction of price displacements caused by option hedging can be explained by the inventory risk of liquidity providers in the stock market. When market maker inventories are high, selling orders cause significant underpricing, while buying orders cause little overpricing, and vice versa. Our evidence suggests that the price displacements and reversals are indeed related to market liquidity, especially the inventory risk component of the liquidity. Specifically, hedging demands from option hedgers increase inventory risks of stock market makers who demand significant price adjustments to compensate for the risks they assume. We study the link between cross-sectional reversals and market level reversals and find that a significant portion of cross-sectional reversals can be explained by market level reversals. First, the magnitudes of two types of reversals exhibit a strong positive correlation in the time series. Second, the relative contributions of winners and losers to cross-sectional reversal profits depend on market makers’ inventory risks. When the market is under selling pressure, most of the profits come from losers, and vice versa. The evidence THE DESTABILIZING EFFECTS OF OPTION HEDGING 27 supports our conjecture that index option hedging pressure has significantly different impacts across stocks. Our evidence also indicates that large stocks are more likely to be under pressure during option expiration weeks. With respect to option pricing patterns, we find that VIX index levels appear to be high in the week before and at the beginning of expiration weeks, which can not be explained by the volatility of S&P 500 index returns during expiration weeks. This pattern suggests that option market hedgers are averse to holding written positions in expiration weeks and raise option prices to attract sellers into the market. Finally, we show that market liquidity is lower in option expiration weeks, where we gauge liquidity using both the Amihud measure and the effective spread. We argue that liquidity measures of large stocks are biased upward due to an increase in index trading, so it is necessary to focus on small stocks to evaluate the condition of market liquidity. Index options are popular risk management tools. Their open interests tend to be high when market volatility is high. High market volatility also implies high inventory risks born by market makers and frequently coincides with low market liquidity. Hedging pressure from index option hedgers makes market liquidity conditions even worse in volatile markets. We believe this is an issue that deserves consideration both by regulators and by practitioners. While the specific remedies are beyond the scope of this paper, it is possible that a better awareness of the option expiration schedule by stock market traders would help to mitigate certain issues. Encouraging option exchanges and traders to expand the use of contracts with non-traditional expiration dates could result in more evenly distributed demands for liquidity from option hedgers, which might be preferred by liquidity providers as well. THE DESTABILIZING EFFECTS OF OPTION HEDGING 28 Figure 1.2. Cross-Sectional Weekly Reversals and the Market Liquidity (a) Value-Weighted (b) Equal-Weighted This figure demonstrates the relation between the market liquidity and cross-sectional weekly stock price reversals. Option expiration weeks are sorted into 10 groups by the Pastor-Stambaugh liquidity measure. Group 1 represents the low liquidity period and group 10 represents the high liquidity period. The same sorting is performed for non-expiration weeks. The height of the bars in the chart represents the average abnormal returns of the cross-sectional weekly reversal portfolios within each liquidity sorted sub-period. THE DESTABILIZING EFFECTS OF OPTION HEDGING 29 Table 1.1 Stock Price Reversals In the Cross-Section This table presents the average abnormal returns of the weekly reversal portfolios from 1973 to 2011. Stocks are sorted into quintiles at the end of each week by their cumulative returns in the week. The last trading day in each week is excluded in computing formation week returns. The value-weighted weekly reversal portfolio is formed by buying the value-weighted portfolio of the stocks in the highest return quintile (winner) and selling the value-weighted portfolio of the stocks in the lowest return quintile (loser). The equally weighted reversal portfolio is formed similarly but with equal weights. Portfolios are held for the full calendar week after formation. The raw returns are fitted to the Carhart four-factor model to produce abnormal returns. Entries in Panels A and B are the average abnormal returns of the reversal portfolios. The rows labeled as “Expiration Weeks” and “’Non-Expiration Weeks” refer to the cases where the formation weeks are option expiration weeks and other weeks, respectively. In the rows named “Expiration Weeks (No Quarter Ends)”, the averages are taken after excluding the weeks in March, June, September and December. The rows named “Difference” show the differences in the holding period abnormal returns between expiration weeks and non-expiration weeks. Panels C and D present the difference in formation week returns between the winner and loser portfolios. Stocks with unduly high (larger than $1000) or low (smaller than $5) prices are excluded. Also excluded are stocks that are smaller than the 1st NYSE size decile. T-statistics based on the Newey-West standard errors are reported in parentheses. * indicates significance at 10% level, ** indicates significance at 5% level, *** indicates significance at 1% level. All returns are in percentage points per day. Panel A: Returns to value-weighted weekly reversal portfolios Full Sample Pre 1993 Post 1993 Expiration Weeks -0.16(-6.9) -0.16(-9.2) -0.16(-3.9) Non-Expiration Weeks -0.10(-8.9) -0.13(-13) -0.07(-3.7) Difference -0.06(-2.2) -0.03(-1.25) -0.09(-1.84) Expiration Weeks(No Quarter Ends) -0.16(-5.82) -0.15(-6.33) -0.16(-3.46) Panel B: Returns to equal-weighted weekly reversal portfolios Expiration Weeks -0.12(-7.4) -0.13(-9.5) -0.11(-3.8) Non-Expiration Weeks -0.10(-12) -0.14(-17) -0.06(-4.0) Difference -0.02(-0.91) -0.01(-0.2) -0.05(-1.5) Expiration Week(No Quarter Ends) -0.12(-6.43) -0.13(-6.42) -0.12(-3.84) Panel C: The differences between the pre-formation returns of the value-weighted winner and loser portfolios Expiration Weeks 2.81 2.55 3.07 Non-Expiration Weeks 2.85 2.60 3.11 Continued on Next Page... THE DESTABILIZING EFFECTS OF OPTION HEDGING 30 Table 1.1 – Continued Panel D: The differences between the pre-formation returns of the equal-weighted winner and loser portfolios Expiration Weeks 3.26 2.97 3.59 Non-Expiration Weeks 3.35 3.03 3.67 Panel E: Transaction costs of the value-weighted portfolios 1993-2011 1998-2011 1998-2011(low cost) Expiration Weeks α wml,t+1 -0.16 -0.16 -0.17 Expiration Weeks Cost 0.13 0.09 0.07 Non-Expiration Weeks α wml,t+1 -0.07 -0.07 -0.07 Non-Expiration Weeks Cost 0.13 0.09 0.07 Panel F: Transaction costs of the equal-weighted portfolios 1993-2011 1998-2011 1998-2011(low cost) Expiration Weeks α wml,t+1 -0.11 -0.1 -0.1 Expiration Weeks Cost 0.33 0.22 0.15 Non-Expiration Weeks α wml,t+1 -0.06 -0.04 -0.04 Non-Expiration Weeks Cost 0.33 0.22 0.15 THE DESTABILIZING EFFECTS OF OPTION HEDGING 31 Table 1.2 The Effect of the Market Liquidity on Cross-Sectional Weekly Reversals This table presents the regression coefficients from the model α wml,t+1 = a +b·PS t +c·X t , whereα wml,t+1 is the abnormal return of the weekly reversal portfolio held over the week following formation week t. PS t is the Pastor-Stambaugh liquidity measure of week t computed using the daily stock returns and volumes from the 15 trading days preceding the Saturday of week t. A high value ofPS t indicates high market liquidity. X t is the vector of control variables, which include the permanent-variable component of the Sadka liquidity measure(SPV) and aggregate mutual fund flows in the month that contains the Wednesday of week t. Monthly fund flows are in percentage terms, and they are defined as (aggregate dollar flow in month t)/(aggregate TNA at the end of month t-1). The full sample period over 1973-2011 is used for regressions without controls , and the sub-period over 1983-2010 is used for for regressions with controls due to the availability of control variables. T-statistics based on the Newey-West standard errors are reported in parentheses. All returns are converted to the unit of percentage points per day. * indicates significance at 10% level, ** indicates significance at 5% level, *** indicates significance at 1% level. Panel A: Value-weighted portfolio returns as independent variables Expiration Weeks Non-Expiration Weeks Intercept -0.06(-2.02**) -0.03(-0.9) -0.09(-5.73***) -0.08(-4.07***) PS 5.74(3.13***) 7.08(3.23***) 1.14(1.3) 0.93(0.94) Fund Flow -1.31(-0.8) -0.95(-0.78) SPV 0.31(0.08) -1.27(-0.48) Adj. R-Squared 0.051 0.068 0.002 -0.001 Panel B: Equal-weighted portfolio returns as independent variables Expiration Weeks Non-Expiration Weeks Intercept -0.06(-3.58***) -0.04(-1.83) -0.08(-6.55***) -0.06(-3.66***) PS 3.18(2.84***) 3.72(2.78***) 1.03(1.48) 1.19(1.4) Fund Flow -0.44(-0.38) -1.54(-1.33) SPV 1.53(0.55) 1.45(0.68) Adj. R-Squared 0.032 0.039 0.006 0.004 THE DESTABILIZING EFFECTS OF OPTION HEDGING 32 Table 1.3 Stock Option Open Interests and Cross-Sectional Weekly Reversals This table shows the average post-expiration week abnormal returns of the cross-sectional weekly reversal portfolios based on a three-way sorting. The sample period is from 1996 to 2010. At the end of each option expiration weeks, stocks are sorted into 27 groups (3× 3× 3) by their weekly returns, option open interests and market capitalization. The open interest measure of a stock is the sum of the open interests in all front contracts (both puts and calls) on the stock normalized by the number of common stock shares outstanding. Open interests ,shares outstanding and market capitialization are measured at the end of the last trading day before each expiration week. At the intersection of each size and open interest tercile, value-weighted reversal portfolios are formed by buying the value-weighted portfolio of the stocks in the highest return tercile and selling the value-weighted portfolio of the stocks in the lowest return tercile. These portfolios are held for one calendar week after formation. The full sample period is divided into two sub-periods with equal length based on the Pastor-Stambaugh liquidity measure. The table reports the average Carhart four-factor model adjusted returns for each combination of size and open interest terciles in each sub-period. T-statistics are based on Newey-West standard errors. * indicates significance at 10% level, ** indicates significantce at 5% level, *** indicates significance at 1% level. All returns are in percentage points per day. Panel A: The low liquidity period Low Open Int. Medium Open Int. High Open Int. Small -0.16** 0.030 -0.16** Medium -0.20*** -0.09 -0.18** Large -0.26** -0.24*** -0.23** Panel B: The high liquidity period Low Open Int. Medium Open Int. High Open Int. Small -0.03 -0.1** -0.01 Medium -0.08 -0.06 -0.05 Large -0.04 -0.12** -0.1 THE DESTABILIZING EFFECTS OF OPTION HEDGING 33 Table 1.4 Market Level Reversals Surrounding Option Expiration Days This table demonstrates regression coefficients from r m after,t = a 1 +b 1 ·lagimb t (1), r m before,t = a 1 +b 2 ·lagimb t (2), and r m rev,t =r m after,t −r m before,t =a 1 −a 2 + (b 1 −b 2 )lagimb t (3), where lagimb t is the lagged aggregate order imbalanceforweekt. lagimb t iscomputedbyaveragingthedailyaggregateS&P500stockorderimbalanceoverthe 15 trading days preceding the Monday of week t-1. Daily order imbalance of a stock is the difference between the share volume initiated by buyers and the share volume initiated by sellers normalized by total shares outstanding; the daily aggregate S&P 500 stock order imbalances are the equal weighted averages of the daily order imbalances of the S&P 500 stocks. r m before,t denotes CRSP value weighted market returns over a period ending at the Thursday closing in week t. Results for various starting points are presented in the table, ranging from the Monday in week t (Mon t orM t ) back to the Tuesday in week t-1 (Tue t−1 ). r m after,t denotes the market returns over a period starting from the opening of the Friday in week t. Results for various ending points are presented in the table, ranging from the Friday closing in week t (Fri t or F t ) to the Friday closing in week t+1 (Fri t+1 or F t+1 ). In Panels C and D, the symbols before commas denote starting points and the symbols following commas denote ending points. We run regressions for the cases where week t is an expiration week and the cases where week t is a non-expiration week separately, and present results in the rows and panels labeled correspondingly. Option expiration weeks in the sample period are further divided into a liquid half and an illiquid half on the basis of the Pastor-Stambaugh measure. Regressions are performed within each half. The same process is followed for non-expiration weeks. Because there is no significant result when market liquidity is high, this table only presents the results from the low liquidity halves to save space. Panels A and B reports values of b 1 andb 2 , respectively. The differences b 1 −b 2 are reported in Panels C and D. The unit of order imbalance is 0.01% of the market capitalization. Returns are in percentage points. T-statistics are based on the Newey-West standard errors. * indicates significance at 10% level, **indicates significantce at 5% level, *** indicates significance at 1% level. Panel A: Coefficient b 1 On Fri t From Fri t From Fri t From Fri t From Fri t From Fri t To M t+1 To Tue t+1 To Wed t+1 To Thur t+1 To Fri t+1 Expiration weeks -0.085 -0.179* -0.234** -0.253** -0.223* -0.198* Other weeks 0.016 0.08 -0.004 0.055 0.082 0.062 Panel B: Coefficient b 2 On Thur t From Mon t From Fri t−1 From Thur t−1 From Wed t−1 From Tue t−1 To Thur t To Thur t To Thur t To Thur t To Thur t Expiration weeks 0.091 0.184* 0.218** 0.181* 0.199* 0.192* Other weeks -0.004 -0.021 -0.051 -0.008 -0.005 -0.091 Panel C: Coefficient b 1 −b 2 in expiration weeks P P P P P P P P P After Before M t ,T t F t−1 ,T t T t−1 ,T t W t−1 ,T t Tue t−1 ,T t F t -0.27** -0.304** -0.266 -0.284 -0.277 F t ,M t+1 -0.364*** -0.397*** -0.36** -0.378** -0.371* F t ,Tue t+1 -0.419** -0.452*** -0.415** -0.433** -0.426* Continued on Next Page... THE DESTABILIZING EFFECTS OF OPTION HEDGING 34 Table 1.4 – Continued F t ,W t+1 -0.438*** -0.472*** -0.434** -0.452** -0.445* F t ,T t+1 -0.407** -0.441*** -0.403** -0.422** -0.414* F t ,F t+1 -0.382** -0.416** -0.378* -0.397* -0.389* Panel D: Coefficient b 1 −b 2 in non-expiration weeks P P P P P P P P P After Before M t ,T t F t−1 ,T t T t−1 ,T t W t−1 ,T t Tue t−1 ,T t F t 0.038 0.068 0.024 0.021 0.107 F t ,M t+1 0.101 0.131 0.088 0.085 0.171 F t ,Tue t+1 0.017 0.047 0.004 0 0.086 F t ,W t+1 0.076 0.106 0.063 0.059 0.145 F t ,T t+1 0.103 0.133 0.09 0.087 0.173 F t ,F t+1 0.084 0.114 0.07 0.067 0.153 THE DESTABILIZING EFFECTS OF OPTION HEDGING 35 Table 1.5 Predicting Cross-Sectional Reversals with Market Reversals This table shows that market level reversals in expiration weeks predict cross-sectional reversals after expiration weeks. The sample period covers from January 1993 to December 2010. The first row in Panel A presents the correlations between the order imbalance-signed market reversals ˜ r m rev,t in week t and the abnormal returns of the cross-sectional weekly reversal portfolios α wml,t+1 in week t+1. ˜ r m rev,t is defined as (r m F,t −r m M−T,t )·sign(lagimb t ), where r m F,t and r m M−T,t are the market returns on Friday and from Monday to Thursday in the week t, respectively. lagimb t is the lagged aggregate order imbalance. To compute the holding period abnormal returns α wml,t+1 , cross-sectional weekly reversal portfolios are formed at the end of week t and held in week t+1. Results for value and equally weighted portfolios are presented under the corresponding column names. Formation weeks (week t) are sorted into two groups (high liquidity and low liquidity) based on the Pastor-Stambaugh liquidity measure, and the correlations are computed within each group and presented in different columns. Panel A presents the time series correlations between ˜ r m rev,t andα wml,t+1 for cases where week t is an option expiration week. The second and third row in Panel A present the decomposition of cor(˜ r m rev,t ,α wml,t+1 ) into cor(˜ r m F,t ,α wml,t+1 ) and cor(˜ r m M−T,t ,α wml,t+1 ). Panels B and C present the averageα wml,t+1 in the eight sub-periods formed by dividing formation weeks within each liquidity group into four sub-periods based on the signs of ˜ r m M−T,t and ˜ r m F,t . The Newey-West standard errors are used to evaluate statistical significance. * indicates significance at 10% level, **indicates significantce at 5% level, *** indicates significance at 1% level. Value-weighted cross-sectional Equal-weighted cross-sectional weekly reversal portfolio weekly reversal portfolio High Liquidity Low Liquidity High Liquidity Low Liquidity Panel A: Correlations in expiration weeks corr(˜ r m rev,t ,α wml,t+1 ) 0.07 0.32*** 0.05 0.24*** corr(˜ r m M−T,t ,α wml,t+1 ) 0.02 -0.23** -0.03 -0.17** corr(˜ r m F,t ,α wml,t+1 ) 0.03 0.21** 0.04 0.18** Value-weighted cross-sectional Equal-weighted cross-sectional weekly reversal portfolio weekly reversal portfolio High Liquidity Low Liquidity High Liquidity Low Liquidity ˜ r m F,t < 0 ˜ r m F,t > 0 ˜ r m F,t < 0 ˜ r m F,t > 0 ˜ r m F,t < 0 ˜ r m F,t > 0 ˜ r m F,t < 0 ˜ r m F,t > 0 Panel B: The predictability of the signs of the market returns in expiration weeks ˜ r m M−T,t > 0 -0.01 -0.13** -0.40** -0.16** -0.09* -0.1** -0.31* -0.12 ˜ r m M−T,t < 0 -0.08 0.02 -0.28** 0.01 -0.01 0 -0.20* -0.03 Continued on Next Page... THE DESTABILIZING EFFECTS OF OPTION HEDGING 36 Table 1.5 – Continued Panel C: The predictability of the signs of the market returns in expiration weeks ˜ r m M−T,t > 0 -0.08 -0.14** -0.11* -0.11* -0.05 -0.09* -0.09* -0.08 ˜ r m M−T,t < 0 -0.04 -0.02 -0.12** -0.01 -0.06 -0.01 -0.11 -0.05 THE DESTABILIZING EFFECTS OF OPTION HEDGING 37 Table 1.6 The Differential Impact of the Aggregate Hedging Pressure on the Cross Section of Stocks The table presents the average holding period raw returns and the average market capitalization of the lagged return sorted portfolios from 1993 to 2011. The sample is divided into halves by the Pastor-Stambaugh liquidity measure, and this table presents the results from the more illiquidity half only. Formation weeks that satisfy ˜ r m F,t > 0 and ˜ r m M−T,t < 0 are also excluded from the sample. The first four columns in each panel present the average post-formation raw returns of the lagged weekly return sorted portfolios. Columns labeled “lagimb t > 0” represent the average over the weeks where the lagged aggregate order imbalance is positive; “lagimb t < 0” represents the cases where the lagged aggregate imbalance is positive. The last two columns present the average market capitalization of the stocks in each portfolio. The Newey-West standard errors are used to computed T-statistics. * indicates significance at 10% level, **indicates significantce at 5% level, *** indicates significance at 1% level. All returns are in percentage points per day. Panel A: Cross-sectional reversals following expiration weeks Value-Weighted Returns Equal-Weighted Returns Market Capitalization (in million dollars) Lagged Return lagimb t > 0 lagimb t < 0 lagimb t > 0 lagimb t < 0 lagimb t > 0 lagimb t < 0 1(Low) 0.04 0.42** 0 0.38** 2729 4760 2 -0.03 0.22* -0.03 0.25* 4234 6466 3 -0.05 0.14 -0.02 0.17 4528 6893 4 -0.1* 0 -0.05 0.09 4988 5519 5(High) -0.16* -0.11 -0.12** -0.02 3430 3713 5-1 -0.2** -0.53** -0.12** -0.4** 701** -1047* Panel B: Cross-sectional reversals following non-expiration weeks 1(Low) 0.11** 0.1* 0.11** 0.08* 2841 3933 2 0.04 0.09 0.07 0.03 4259 5738 3 0 0.05 0.06 0.01 4499 6855 4 -0.03 0.03 0.06 0 4495 6671 5(High) -0.8 0.02 0.03 -0.03 2865 4762 5-1 -0.19** -0.08 -0.09* -0.11* 25 829** THE DESTABILIZING EFFECTS OF OPTION HEDGING 38 Table 1.7 The Decreases in the VIX Index Over Option Expiration Weeks This table shows the changes in the VIX index surrounding option expiration days 1990 – 2011. Changes in VIX over non-expiration weeks are also presented for comparison. Row/column names indicate the starting/ending points of the horizons over which changes in VIX are computed. Standard abbreviations are used for weekdays. Panels A and B present the cases where week t (as in the subscripts of the row and column names) is an option expiration week. The sample period is divided into halves based on the Pastor-Stambaugh liquidity measure. Results from each liquidity sorted period are presented in different panels. Panel E presents the volatility of the S&P 500 index returns computed within each subsample. The Newey-West standard errors are used to evaluate statistical significance. * indicates significance at 10% level, **indicates significantce at 5% level, *** indicates significance at 1% level. Panel A: Week t is an expiration week in the low liquidity period H H H H H H From To F t M t+1 W t+1 F t+1 W t -0.58** -0.47 -0.51 -0.31 M t -1.34*** -1.31*** -1.38*** -1.16** F t−1 -1.11*** -1.01*** -1.17*** -0.96** W t−1 -1.22*** -1.05*** -1.16*** -1.02** M t−1 -1.25*** -1.08*** -1.2*** -1.16** Panel B: Week t is an expiration week in the high liquidity period H H H H H H From To F t M t+1 W t+1 F t+1 W t 0.18 0.21 -0.32 -0.39 M t 0.14 0.28 -0.23 -0.34 F t−1 -0.02 0.35 -0.14 -0.29 W t−1 -0.38 -0.06 -0.5** -0.65*** M t−1 -0.31 -0.07 -0.5** -0.63** Panel C: Week t is a non-expiration week in low liquidity period H H H H H H From To F t M t+1 W t+1 F t+1 W t -0.09 0.36** 0.04 -0.2 M t -0.41** 0.11 -0.28 -0.49 F t−1 0.15 0.65*** 0.26 0.03 W t−1 0.14 0.6*** 0.21 -0.04 M t−1 -0.21 0.16 -0.13 -0.33 Panel D: Week t is a non-expiration week in the high liquidity period H H H H H H From To F t M t+1 W t+1 F t+1 W t -0.1 0.3*** 0.19** 0.06 M t -0.16 0.27** 0.1 -0.06 Continued on Next Page... THE DESTABILIZING EFFECTS OF OPTION HEDGING 39 Table 1.7 – Continued F t−1 0.24** 0.67*** 0.52*** 0.36** W t−1 -0.01 0.37*** 0.23 0.12 M t−1 -0.33** 0.13 -0.05 -0.16 Panel E: S&P 500 index return volatility Expiration Weeks Non-Expiration Weeks High Liquidity 0.0083 0.0086 Low Liquidity 0.0127 0.0131 THE DESTABILIZING EFFECTS OF OPTION HEDGING 40 Table 1.8 Comparing the Liquidity Measures in Option Expiration Weeks and in Other Weeks This table demonstrates the differences in illiquidity measures between option expiration weeks and other weeks. The sample period is from 1993 to 2010. The illiquidity measures include the Amihud liquidity measure and the effective spread. Weekly Amihud illiquidity measure of a stock is defined as 1 n P n d=1 |r d | v d · 10 6 , where r d is the stock return (in percentage terms) on day d and v d is the dollar volume on day d and n is the number of days in the week. Effective spreads (ESPR) of the nth trade is defined asESPR n = 2·|log(p n )−log(m n )|, wherep n is the trade price andm n is the middle point of the prevailing quotes when the nth trade occurs, daily measure ESPR d is the dollar volume weighted average of all trades in day d, and the weekly measure is the mean of daily values in the week. At the end of each week, stocks are sorted into size terciles by market capitalization and the results are presented in columns titled “Small”, “Medium” and “Large”, respectively. We computed the Pastor-Stambaugh liquidity measure for week t using the lagged 15 trading days preceding the Saturday of week t. Weeks are divided into 4 groups based on their Pastor-Stambaugh measure and the results from different groups are presented in different rows. The entries are the differences in the sample means of the weekly liquidity measures in option expiration weeks and in other weeks for each size tercile, within each liquidity group. The Newey-West standard errors are used to computed T-statistics. **indicates significance at 5% level. Small Medium Large Liquidity Amihud ESPRD Amihud ESPRD Amihud ESPRD 1(low) 1.37** 0.051** -0.1 0.0037 0.001 0.0004 2 -0.1 -0.004 -0.02 -0.003 -0.01 -0.003 3 -0.1 -0.007 -0.1** -0.001 0.002 -0.001 4(high) -0.6** -0.001 -0.1 -0.001 -0.003 -0.004** THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 41 Chapter 2 The term structure of equity option implied volatility Introduction Option-implied volatilities reflect risk-neutral expectations of the future future volatility of the underlying asset. Because these expectations are computed under an equivalent martingale measure, they embody expectations under the true probability measure as well as premia on risk factors such as jumps or stochastic volatility. One might therefore suppose that implied volatilities have some predictive ability to forecast the future returns on options and their underlying assets. This prediction has been confirmed directly in several recent papers and indirectly in a number of others. Bollerslev et al. (2009) show that the spread between implied and realized volatility forecasts stock market returns. Specifically, when implied volatility is high relative to realized volatility, then future stock returns tend to be higher. Goyal & Saretto (2009) find that a similar variable has strong predictive power in the cross section of at-the-money straddle returns, with high volatilities forecasting negative straddle returns. Other papers, such as Jones (2003), that estimate stochastic volatility and/or THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 42 jump models using both stock and implied volatility series generally find that nonzero premia on volatility and/or jump risk are necessary to fit both series, which implies that option expected returns will vary with the level of actual or implied volatility. In the fixed income literature, it is well known that the slope of the term structure has significant predictive power for future bond returns, with positive slopes associated with high excess returns on intermediate and long-term bonds. Work in this area goes back to seminal papers such as Fama & Bliss (1987) and Campbell & Shiller (1991). The importance of the term structure slope can be attributed to the fact that risk premia have relatively minor effects on the discounting of short-maturity cash flows and much larger effects for long maturities. Subtracting the short maturity yield from the long maturity yield seems to isolate the component of yields that is driven by risk premia rather than expected future interest rates. In this paper we ask whether the slope of the implied volatility term structure is a predictor of future stock and option returns. When predicting option returns, we focus on the returns on at-the-money (ATM) straddle, which is a portfolio of one ATM call and one ATM put, both with the same maturity. The straddle portfolio is naturally close to zero-delta, which means that it primarily represents a bet on volatility (or time varying jump risk) rather than being a directional bet on the underlying asset. We examine predictability both in the time series and in the cross section, and we examine both indexes and individual equities. Our primary finding is that the slope of the implied volatility term structure has strong and highly significant predictive power for future short-term straddle returns. For straddles with one month remaining until expiration, a one percentage point increase in the difference between long and short IV results in a 0.5 to one percent decrease in expected straddle returns. in expected return per year. This effect remains significant after controlling for the volatility difference measure of Goyal & Saretto (2009), which is the difference between implied and historical volatility, as well as numerous other controls. A long-short strategy that buys straddles with high IV slope and sells straddles with low IV THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 43 slope results in an average monthly return of 7.0%, which is accompanied by a t-statistic of 7.7. Risk adjustment with a multifactor model increases both numbers. The strength of these results makes it surprising, therefore, that we find almost no predictability in longer-term straddle returns. When we examine the one-month returns on straddles with as few as two months until expiration, the predictive ability of the IV slope vanishes almost completely. Furthermore, the return on a long-short strategy that buys high IV slope straddles and shorts low IV slope straddles generates essentially zero returns when implemented with longer maturity straddles. This is even more puzzling given that the large correlations between portfolios of short-term and long-term straddles, which range from 0.91 to 0.94. We find no predictability in the pure time-series component of straddle returns. The average IV slope across all equity options does not predict the average return on equity straddles. In addition, the implied volatility slope constructed from S&P 500 index options has no predictive power for future index straddle returns. For two reasons, we believe that it is unlikely that the predictability of short-term straddle returns is the result of cross-sectional variation in risk. Primarily, this is because of the extremely high correlations between the returns on portfolios of long and short straddles and the fact that long-term straddles show little predictability. A risk-based reconciliation of these findings would require an extremely large Sharpe ratio on the component of short straddle returns that is orthogonal to long straddle returns. This Sharpe ratio is too large to be consistent either with common sense or the “good deal” bounds of Cochrane & SaaâĂŘRequejo (2000). The second reason is the lack of time series predictability. The finance literature has been relatively conclusive in showing that factor risk premia in equity, fixed income, and option markets all vary significantly through time. If the return on short-term straddles represents some systematic factor, then its lack of predictability would make it somewhat unique among risk factors. We proceed in Section 2 by describing our data sources. Section 3 reports the results of THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 44 cross-sectional regressions of straddle returns on the IV slope and various controls. We analyze trading strategies and time-series predictability in Section 4. In Section 5 we attempt to refine the slope measure by decomposing slope into expectation and risk premium components. Section 6 concludes. Data Our study’s primary focus is on option prices, which we obtain from the IvyDB database of Optionmetrics. This database contains end-of-day quotes on equity, index, and ETF options in addition to option-level contractual characteristics (strike price, expiration date), trading data (volume and open interest), implied volatilities, and “Greeks” (delta, vega, etc.). Our primary sample is restricted to options on common equity and not ETFs or indexes, though we analyze index options separately. Our sample period starts in January of 1996, which is the start of the Optionmetrics dataset, and ends in December of 2010. Although we use daily data to construct many explanatory variables, the straddle returns we analyze are monthly. Our short-term straddle is an investment that is initiated on the Monday following expiration week, which is either the third or fourth Monday of the month. On that day (or the first day after it if that Monday is a holiday), one call and one put on each underlying are purchased. The contracts chosen are those with one month remaining until expiration, with a strike price that is closest to the current spot price. The cost of each option is assumed to be the midpoint of each option’s bid-ask spread. The short-term straddle is held until expiration, so the terminal value is determined based solely on the difference between stock and strike prices. We do not allow for early exercise. Long-term straddle returns are computed similarly, except that they are not held until expiration. The long-term straddle is defined as the straddle with the longest maturity available subject to being no longer than four months and considering only those options that are part of the traditional monthly expiration calenday. (I.e., we do not consider “weeklys.”) Regardless of the maturity, the straddle is held for one month and sold at a THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 45 price determined by option bid-ask midpoints. The schedule on which new options are introduced ensures that in any given month the cross-section of long-term straddles will be approximately equally divided between two-month, three-month, and four-month maturities. Thus, while there is significant heterogeneity in the cross-section of long-term straddles, the degree of that heterogeneity is essentially constant over time. Our primary predictive variable is the slope of the implied volatility term structure, computed as the long-maturity IV minus the short-maturity IV. The long-maturity IV is from the longest maturity options that are available subject to being no more than four months until expiration. The short-maturity IV is from the pair of options with just above one month until expiration. In both cases, IVs are taken from at-the-money options, defined as those with the smallest distance between strike and spot prices. To reduce the effects of noise in the IV slope, we compute the slope daily over the two weeks prior to expiration Friday and then take the average of these values. We follow the same approach with other explanatory variables that are based on option prices. Results are slightly weaker when we compute slopes from just one day of data. We augment the Optionmetrics data with stock prices, volume, and market capitalization data from CRSP and book values from Compustat. We obtain data on the Fama-French (1993) factors as well as the “up-minus-down” momentum factor suggested by Carhart (1997) from the website of Ken French. We augment these factors with an SPX (S&P 500 index) straddle return factor used by Goyal & Saretto (2009). We construct this factor identically to the short-term straddles described above except that it is based on S&P 500 index options. The relation between IV slope and straddle returns The determinants of the IV slope We begin our empirical analysis with an examination of the variables associated with the level of the implied volatility slope. Although it might be argued that some of the variables THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 46 we examine can be viewed as exogenous, this is not generally the case. The results we present should be regarded as establishing associations, not causality. The explanatory variables we consider fall into three broad categories. The first consists of stock market variables. This includes the prior stock return and stock return volatility computed over two horizons, the previous month and the 11 months prior to that one. This group also includes the market capitalization of the stock, the stock’s book-to-market ratio, and the Amihud (2002) measure of liquidity in the underlying stock. The second category consists of contemporaneous variables that are also based on the prices of options on the same stock. This includes the at-the-money implied volatility of the option, the risk-neutral implied skewness and kurtosis from options with just more than one month until expiration, and the slopes of the same skewness and kurtosis measures constructed from the same short and long options used to compute the IV slope measure. The third category consists of option trading variables. These include option trading volume and open interest, both computed by summing across all contracts for a given underlying stock, and then dividing the total by the trading volume of the stock. In addition to reporting totals for volume and open interest, we also compute corresponding “slope” measures by taking the difference between volume and open interest for long and short dated options. Finally, this category also includes the average bid-ask spread and the slope of the bid-ask spread for options on a given underlying stock. Table 2.2 reports the results of Fama-Macbeth regressions of the IV slope variable on subsets of these explanatory variables. T-statistics, which appear in parentheses, are computed using the method of Newey & West (1987). Here and in all results to follow, we apply the automatic lag truncation method derived by Andrews (1991). Average R-squares are time-series averages of cross-sectional regression r-squares. The first regression considers the effects of different stock market variables on the IV slope. By far the most significant predictors are the two volatility measures. The short-term measure is computed from daily returns over the previous month, while the long-term THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 47 measure is based on returns during the 11 months prior to that. High recent volatility makes the IV slope more negative, while long-term volatility makes it more positive. One explanation is that volatility appears to have short-run and long-run components (e.g., Brandt & Jones (2006)), and recent volatility mainly reflects a short-run component with a relatively short half-life. Short-run volatility has a strong impact on short-term implied volatility, but its fast mean reversion means that the impact on long-run implied volatility is minimal. The persistence of the long run component of volatility, on the other hand, means that it has a stronger effect on long-term implied volatility. The coefficients on lagged returns have a similar interpretation given that negative returns are predictors of higher future volatility. When lagged one-month returns are negative, this points to a more transient increase in volatility, leading mainly to an increase in short-term implied volatility. Low long-run past returns suggest a more persistent increase in volatility, implying higher long-term implied volatilities. Finally, the coefficients on lagged book-to-market and the Amihud liquidity measure are significant, suggesting that growth stocks and illiquid stocks tend to have steeper IV slopes. These effects appear minor, however. The second regression shows the relationship between the IV slope and other measures extracted from various options on the same underlying stock. The most important explanatory variable in this group appears to be the implied volatility level. This result merely follows from the fact that the one-month IV is the same IV that is used to represent the short end of the IV term structure when the slope is computed. The other pair of variables that appears to be important is the level and slope of risk neutral implied skewness. When skewness is high, the IV slope is greater. When the term structure of skewness is more upward sloping, the IV slope becomes more negative. The third regression examines the relations between IV slope and variables related to option trading. Higher option trading volume is associated with lower IV slopes, particularly when the volume is heavier in longer-dated options. Higher spreads, especially THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 48 in longer-term contracts, are also associated with lower IV slopes. Finally, more open interest in short-term options seems to make the slope higher, while open interest in long-term options appears to reduce the IV slope. The IV slope and the cross-section of straddle returns The cross-sectional relationship between the IV slope and future returns is established in Table 2.3. We report results from Fama-Macbeth regressions, where Newey-West t-statistics are in parentheses. The main result is easy to summarize. With or without controls, a high IV slope predicts high future straddle returns. The coefficient of 0.908 from the first regression means that if the IV slope increases by one percentage point, then the expected 1-month straddle return will increase by 0.908 percent. The second regression in the table adds two closely related control variables. The first is a pure IV measure, specifically the lagged implied volatility on the same options being held in the short-term straddle. The second is the IV deviation measure of Goyal & Saretto (2009), defined as the same lagged implied volatility minus the standard deviation of daily stock returns over the previous year. While its coefficient declines in magnitude somewhat when Goya and Saretto’s variable is included, the IV slope remains highly significant. Including IV by itself seems to have no effect on the other variables. The third regression adds some of the stock-level variables from Table 2.2 as controls. While both size and the Amihud measure appear to offer some improvements in fit, including them along with the other controls only increases the magnitude and significance of the coefficient on IV slope. The fourth regression instead includes controls related to option trading activity, which has very minor effects on our main result. Table 2.4 repeats these regressions, except now the short-term straddle is replaced with a long-term straddle, which is held over the same one month period as the short term straddle. This relatively minor change leads to completely different results. The IV slope is no longer statistically significant in any regression except the second, and in that regression THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 49 its sign is negative, opposite the result from the previous table. Interestingly, the IV difference variable remains significant, and the IV level becomes significant as well. Coefficients on other control variables are broadly similar to before. The stark contrast between the results of Tables 2.3 and 2.4 present something of a puzzle. The returns on short-term and long-term straddles are highly correlated. On average, the cross-sectional correlation between the two is 0.84, suggesting that loadings on systematic risk factors must be similar for short-term and long-term contracts. But if loadings are similar, and if the IV slope is a proxy for some systematic risk loading, then we should see similar predictability in both short-term and long-term contracts. Since we do not, the risk premia-based explanation appears tenuous at best, Our last cross-sectional regression asks whether the IV slope can predict stock returns. The mechanism that would induce predictability at the stock level is the generally negative correlation that is found between volatility and returns. If the IV slope forecasts straddle returns, and if this is the result of a stochastic volatility risk premium, then it is possible that the equity return would inherit part of this risk premium through its correlation with the volatility process. The result, as shown in Table 2.5, is negative – there is no evidence at all that the IV slope or other IV-related variables forecasts future stock returns. Trading strategies and time-series predictability Risk adjustments and profitability of trading strategies In this section we analyzed portfolios of straddles from different underlying stocks. This allows a clearer measure of the profitability of buying straddles with low IV slopes and also allows this profitability to be risk-adjusted using standard methods. Table 2.6 analyzes the performance of five portfolios of short-term straddles formed on the basis of the IV slope. These portfolios are rebalanced monthly following expiration Friday, when the previous month’s straddles expire and a new set of straddles is purchased. The top panel of the table reports the means, standard deviations, and Sharpe ratios of the five THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 50 portfolios and of the long-short portfolio that buys high-slope straddles and shorts low-slope straddles. All of these numbers are reported on a monthly basis. This panel also includes the average IV slope of each portfolio. The second panel shows alphas and betas from a regression with the Fama-French and Carhart factors in addition to the SPX index straddle factor suggested by Goyal & Saretto (2009). The third panel reports corresponding t-statistics. In short, the table shows large gains from buying high-slope straddles relative to low-slope straddles. Most of the raw return comes from shorting the straddles with low IV slope, which on average lose about six percent of their value every month. When returns are risk-adjusted with the factor model, then the long leg turns out to be the source of more abnormal return. This is due primarily to the fact that all straddle returns load positively on the SPX straddle factor, which itself has a negative mean. The alpha of the long-short portfolio turns out to exceed eight percent per month, with a t-statistic of 7.84. Consistent with the cross-sectional regressions, long-term straddle returns are unrelated to the IV slope variable. This result is shown in Table 2.7, which replicates the previous analysis with longer-term contracts. While all the portfolios now have positive alphas, the spread between high and low IV slope portfolios is insignificant, as is the spread in raw returns. The positivity of the alphas again follows from the positive loadings on the SPX straddle factor and that factor’s negative mean. The non-result here strongly undermines the risk premia explanation. The correlation between the portfolios of short-term and long-term straddles are generally above 0.9, implying that roughly 90% of the standard deviation of short-term straddle returns can be eliminated by hedging with long-term straddles. Since the expected return of the long-term straddle is close to zero, the hedge does not reduce the mean of the hedged portfolio, which implies that the Sharpe ratios of a hedged strategy may be larger by a factor of about 10. The Sharpe ratio of the low-IV slope portfolio, after this type of hedging, is well above 2. On an annualized basis, it is around 10, a number that is many times the Sharpe ratio of THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 51 the stock market. As Cochrane & SaaâĂŘRequejo (2000) argue, “good deals” such as these should be ruled out by any reasonable asset pricing model. Time series predictability If high IV slopes predict high returns in the cross section, then do they predict returns in the time series as well? Suppose that IV slopes predict returns because they proxy for the beta with respect to some priced risk factor, i.e. the slope is generated by β i λ t . Then the cross-sectional average of the IV slope will provide a measure of the average beta multiplied by the ex ante price of risk λ t . Variation in the ex post risk premium should therefore be forecastable using the average IV slope. An increase in the ex ante price of risk should also increase the dispersion in IV slopes and in subsequent realized returns. This should make the return on the long-short strategy of buying high-IV slope straddles and selling low-IV slope straddles predictable based on the spread in IV slopes. We test both of these hypotheses in Table 2.8. The dependent variable in the first two regressions is the cross sectional average of the returns on all short-term straddles. The independent variable that was motivated by the above discussion is the average IV slope, but the second regression also includes a measure of IV slope dispersion and the IV slope from S&P 500 Index options. The dispersion measure is the average IV slope of all stocks in the high-slope quintile minus the average IV slope of the low-slope quintile. The next two regressions are identical except that they replace the dependent variable with the average return on long-term straddles. The results of all four regressions are negative – there is no evidence of time series predictability in average straddle returns. Regressions 5-8 examine predictability in the long-short spread portfolios. For this variable, the key predictor was hypothesized to be the spread in IV slopes. We run regressions with only this predictor, both for short-term and long-term straddles, and with all three predictors. Again, we find no evidence of time series predictability. Finally, THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 52 regressions 9-12 examine predictability in short-term SPX straddle returns. Here, the key predictor is assumed to be the slope of the SPX IV term structure. No predictability is found here as well. Failing to find evidence of predictability does not rule out the risk premia explanation by itself, as it is possible that the risk premia reflected in IV slopes is not time varying. This would appear to be the exception rather than the rule given the extensive research documenting predictability in most risk premia, but it is possible. Given that we believe we have already made a strong case against risk premia being the driver of IV slope-based predictability, we do not find the lack of time-series predictability to be surprising. Controlling for expected realized volatility To be completed. Conclusion This paper demonstrates a remarkable level of predictability in the returns on short-term equity options. Specifically, short-term options on stocks with an upward-sloping term structure of implied volatility have returns that far exceed those of stocks with downward-sloping term structures. The relationship between IV slope and straddle returns is highly significant in the cross section, even when many other variables, including the IV difference measure of Goyal & Saretto (2009), are included. A trading strategy that buys short-term straddles on high-IV slope stocks and shorts straddles on low-IV slope stocks results in monthly returns that average seven percent, with a Sharpe ratio that exceeds that of the stock market by several times. We believe that our results rule out the possibility that the IV slope predicts returns because it proxies for the loading on some systematic risk factor. In part this is because we find no evidence of any time series predictability in average straddle returns or in the returns on the high-slope minus low-slope strategy. If variation in slope is driven by the risk THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 53 loading of some factor, then average slope should be related to that factor’s risk premium. While there is no requirement that factor risk premia be time-varying, most risk premia appear to be at least somewhat predictable. Our finding of no time series predictability therefore makes the cross-sectional effect we document look more like mispricing. More importantly, we rule out any risk-based explanation because straddles formed from options with expirations that are just a few more months longer display no predictability, their returns being essentially unrelated to the IV slope. This is problematic because these longer-term straddle returns are highly correlated with the short-term straddle returns. The average cross-sectional correlation between long and short straddle returns is above 0.8, while the time series correlations between the returns on portfolios of long and short straddles are all above 0.9. Such high correlations imply very similar risk loadings, implying that risk loadings cannot explain the predictability we document. THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 54 Table 2.1 Summary Statistics for Implied Volatility Slopes This table presents the summary statistics of the term structure slope of individual stock option implied volatility. For each stock, daily measures of implied volatility (IV) at each maturity are given by the averages of the IVs of at the money (ATM) put and call options with that maturity. Slopes are defined as the differences between the IV of the one-month contracts, which expire in one calendar month, and the IV of the longer term contracts, which are chosen to be the pair of ATM options with maturity between 2 to 4 months, and if there are multiple maturities available within this range, the contacts with the longest maturity are used. The maturity k (month) of long-term options is specified in the first column. We first obtain the time series average of the slope for each stock, then use these stock level measures to compute the statisticsreportedinthetable. Tobeconsistentwiththepredictive regressions in this paper, we only include the daily measures from the two weeks preceding the last trading day before each monthly option expiration day. The sample period is from January 1996 to December 2010. k Mean Median Std. Dev. Minimum Maximum 2.0000 0.0095 0.0073 0.0286 -0.3191 0.2578 3.0000 0.0200 0.0150 0.0361 -0.2467 0.5588 4.0000 0.0251 0.0183 0.0422 -0.2754 0.9034 THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 55 Table 2.2 Determinants of Implied Volatility Slopes In this table, we regress slopes of stock option implied volatility (IV) term structure on contemporaneous or lagged stock characteristics. For each stock, daily measures of implied volatility (IV) at each maturity are given by the averages of the IVs of at the money (ATM) put and call options with that maturity. Slopes are defined as the differences between the IV of the one-month contracts, which expire in one calendar month, and the IV of the longer term contracts, which are chosen to be the pair of ATM options with maturity between 2 to 4 months, and if there are multiple maturities available within this range, the contacts with the longest maturity are used. Monthly IV slopes are given by the averages of daily measures over the two weeks preceding the last trading day before the option expiration day of month t. We regress monthly measures of IV slopes on monthly measures of firm characteristics obtained using data preceding the expiration day of month t. "Lagged volatility in month t− 1(t− 12 to t− 2)" refers to stock return volatility calculated using daily returns in the month (the year excluding the most recent month) preceding option expiration day of month t. Similarly, we include the one-month and one year lagged cumulative returns for each stock. Both market capitalization and book-to-market ratio are computed at the month end preceding month t expiration day. The Amihud liquidity measure is computed using data from the month preceding month t expiration. The rest of the independent variables are computed by first obtaining daily measures, then averaging daily measures over the two weeks preceding the last trading day before month t expiration day to obtain monthly measures. The intermediate maturity IV refers to the IV of one-month ATM contracts. Intermediate term skewness(kurtosis) is the risk-neutral skewness(kurtosis) estimated from the one-month contracts based the method in Bakshi, Kapadia, and Madan(2003). Total option volume (open interest) is the sum of the volume (open interest) of all options on a stock. The option spread is the average quoted spreads of all options on the stock. The "slopes" of the above variables are the differences between the variables measured from long- term contracts and one-month contracts. The sample period is from January 1996 to December 2010. All variables are winsorized at 1% and 99%. T-statistics are computed using Newey-West standard errors based on the data-dependent automatic bandwidth/lag truncation parameters derived in Andrews 1991. THE TERM STRUCTURE OF EQUITY OPTION IMPLIED VOLATILITY 56 Table 2.2, cont. Determinants of Implied Volatility Slopes Lagged volatility in month t− 1 -0.6014 (-27.8900) Lagged volatility in months t− 12 to t− 2 0.2600 (8.0200) Lagged stock return in month t− 1 0.0163 (7.5300) Lagged stock return in months t− 12 to t− 2 -0.0008 (-2.4900) Log market capitalization 0.0002 (1.3200) Book-to-market ratio -0.0010 (-2.1500) Amihud illiquidity measure 0.1157 (2.2500) Intermediate maturity IV -0.0629 (-21.3200) Intermediate term skewness 0.0037 (4.6100) Intermediate term kurtosis -0.0001 (-0.4400) Skewness slope -0.0063 (-9.0600) Kurtosis slope 0.0009 (3.5600) Total option volume -2.9097 (-11.3100) Option volume slope -0.7238 (-5.4600) Total open interest 0.1746 (7.0100) Open interest slope -0.2305 (-10.8500) Option spread -0.3105 (-14.0200) Option spread slope -0.4172 (-11.3700) Adj. R-Squared (%) 11.1796 17.0727 7.4127 TRADING ACTIVITY AND BETA 57 Table 2.3 Fama-Macbeth Regressions for One-Month Straddle Returns This table presents the results of Fama-Macbeth regressions of one-month straddle returns on IV term structure slopes. Straddle returns are computed using at the money options that expire in one calendar month. These options are purchased at the end of the first trading day following the expiration day of month t and held to expiration in month t+1. Independent variables are computed using information before month t expiration. "Lagged stock return in month t− 1(t− 12 to t− 2)" refers to cumulative stock returns in the month (the year excluding the most recent month) preceding month t expiration. Both market capitalization and book-to- market ratio are computed at the month end preceding month t expiration day. The Amihud liquidity measure is computed using data from the month preceding month t expiration. The rest of the independent variables are computed by first obtaining daily measures, then averaging daily measures over the two weeks preceding the last trading day before month t expiration day to obtain monthly measures. IV slope is the difference between long-term IV and one-month IV. IV refers to the IV of one-month ATM contracts. IV deviation is the difference between one-month IV and stock return volatility measured over the 250 trading days preceding month t expiration. Total option volume (open interest) is the sum of the volume (open interest) of all options on a stock. The option spread is the average quoted spreads of all options on the stock. The "slopes" of the above variables are the differences between the variables measured from long-term contracts and one-month contracts. We perform one cross-sectional regression for each month. The sample period is from January 1996 to December 2010. All dependent variables are winsorized at 1% and 99%. T-statistics are computed using Newey-West standard errors based on the data-dependent automatic bandwidth/lag truncation parameters derived in Andrews 1991. IV slope 0.91 0.54 0.93 0.84 (8.27) (5.28) (9.20) (7.94) IV deviation -0.29 -0.35 (-7.43) (-9.54) IV -0.01 (-0.26) Lagged stock return in month t− 1 -0.06 (-1.77) Lagged stock return in months t− 12 to t− 2 0.01 (0.92) Log market capitalization -0.01 (-3.62) Book-to-market ratio 0.00 (0.01) Amihud illiquidity measure -4.09 (-5.04) Total option volume 0.03 (0.01) Option volume slope -1.35 (-0.48) Total open interest -0.35 (-1.80) Open interest slope -0.40 (-1.25) Option spread -0.77 (-2.16) Option spread slope -1.45 (-2.25) Adj. R-Squared (%) 0.24 1.41 0.46 1.90 0.98 TRADING ACTIVITY AND BETA 58 Table 2.4 Fama-Macbeth Regressions for Long-Term Straddle Returns This table presents the results of Fama-Macbeth regressions of long-term straddle returns on IV term structure slopes. Straddle returns are computed using at the money options that expire in 2 to 4 months. When multiple maturitiesareavailableinthisrange, theoptionswiththelongestmaturitiesareused. Thestraddlesarepurchased at the end of the first trading day following the expiration day of month t and sold at the end of the expiration day in month t+1. Independent variables are computed using information before month t expiration. "Lagged stock return in montht−1(t−12 tot−2)" refers to cumulative stock returns in the month (the year excluding the most recent month) preceding month t expiration. Both market capitalization and book-to-market ratio are computed at the month end preceding month t expiration day. The Amihud liquidity measure is computed using data from the month preceding month t expiration. The rest of the independent variables are computed by first obtaining daily measures, then averaging daily measures over the two weeks preceding the last trading day before month t expiration day to obtain monthly measures. IV slope is the difference between long-term IV and one-month IV. IV refers to the IV of one-month ATM contracts. IV deviation is the difference between one-month IV and stock return volatility measured over the 250 trading days preceding month t expiration. Total option volume (open interest) is the sum of the volume (open interest) of all options on a stock. The option spread is the average quoted spreads of all options on the stock. The "slopes" of the above variables are the differences between the variables measured from long-term contracts and one-month contracts. We perform one cross-sectional regression for each month. The sample period is from January 1996 to December 2010. All dependent variables are winsorized at 1% and 99%. T-statistics are computed using Newey-West standard errors based on the data-dependent automatic bandwidth/lag truncation parameters derived in Andrews 1991. IV slope -0.01 -0.10 0.01 -0.03 (-0.28) (-2.82) (0.29) (-0.78) IV deviation -0.16 -0.17 (-9.36) (-10.57) IV -0.04 (-2.61) Lagged stock return in month t− 1 0.01 (1.14) Lagged stock return in months t− 12 to t− 2 0.01 (1.71) Log market capitalization 0.00 (-1.96) Book-to-market ratio 0.00 (0.82) Amihud illiquidity measure -1.43 (-4.57) Total option volume -0.70 (-0.84) Option volume slope -0.74 (-0.67) Total open interest -0.15 (-1.72) Open interest slope -0.42 (-3.03) Option spread -0.51 (-2.98) Option spread slope -0.04 (-0.12) Adj. R-Squared (%) 0.18 1.95 0.69 2.26 1.30 TRADING ACTIVITY AND BETA 59 Table 2.5 Fama-Macbeth Regressions for Stock Returns This table presents the results of Fama-Macbeth regressions of monthly stock returns on IV term structure slopes. The stock returns are cumulative returns from the first trading day following the expiration day of month t to the last trading day before option expiration in month t+1. Independent variables are computed using information before month t expiration. "Lagged stock return in month t− 1(t− 12 to t− 2)" refers to cumulative stock returns in the month (the year excluding the most recent month) preceding month t expiration. Both market capitalization and book-to-market ratio are computed at the month end preceding month t expiration day. The Amihud liquidity measure is computed using data from the month preceding month t expiration. The rest of the independent variables are computed by first obtaining daily measures, then averaging daily measures over the two weeks preceding the last trading day before month t expiration day to obtain monthly measures. IV slope is the difference between long-term IV and one-month IV. IV refers to the IV of one-month ATM contracts. IV deviation is the difference between one-month IV and stock return volatility measured over the 250 trading days preceding month t expiration. Total option volume (open interest) is the sum of the volume (open interest) of all options on a stock. The option spread is the average quoted spreads of all options on the stock. The "slopes" of the above variables are the differences between the variables measured from long-term contracts and one-month contracts. We perform one cross-sectional regression for each month. The sample period is from January 1996 to December 2010. All dependent variables are winsorized at 1% and 99%. T-statistics are computed using Newey- West standard errors based on the data-dependent automatic bandwidth/lag truncation parameters derived in Andrews 1991. IV slope 0.00 -0.01 0.04 0.01 (0.09) (-0.60) (1.82) (0.30) IV deviation -0.01 0.00 (-0.70) (-0.20) IV -0.02 (-1.21) Lagged stock return in month t− 1 0.00 (-0.34) Lagged stock return in months t− 12 to t− 2 0.00 (-0.19) Log market capitalization 0.00 (0.15) Book-to-market ratio 0.00 (0.39) Amihud illiquidity measure -0.17 (-1.00) Total option volume -0.94 (-1.37) Option volume slope 0.16 (0.28) Total open interest 0.00 (-0.04) Open interest slope 0.00 (0.04) Option spread -0.17 (-1.45) Option spread slope 0.05 (0.33) Adj. R-Squared (%) 0.98 7.34 1.33 7.77 3.95 TRADING ACTIVITY AND BETA 60 Table 2.6 One-Month Straddle Portfolios This table presents the returns of one-month straddle portfolios sorted by IV slopes. On the last trading day before month t option expiration day, stocks are sorted into quintiles based on their IV slopes. IV slopes used for sorting are the averages of daily IV slopes over the two weeks preceding the last trading day before the option expiration day of month t. Holding period straddle returns are computed using at the money options that expire in month t+1. These straddles are purchased at the end of the first trading day following the expiration day of month t and held to expiration in month t+1. The abnormal returns are given by monthly regressions on five risk factors: Carhart four factors and the returns of straddles formed by ATM SPX options. The returns of factors are computed over the same horizon as the stock straddle returns. The sample period is from January 1996 to December 2010. IV slopes are winsorized at 1% and 99%. T-statistics are computed using Newey-West standard errors based on the data-dependent automatic bandwidth/lag truncation parameters derived in Andrews 1991. Mean Return Sharpe Average IV Portfolio Return SD Ratio IV Slope 1 (low IV slope) -0.064 0.186 -0.342 -0.056 (-4.410) (-20.330) 2 -0.033 0.223 -0.146 -0.021 (-1.830) (-13.050) 3 -0.029 0.231 -0.126 -0.008 (-1.610) (-6.710) 4 -0.014 0.245 -0.058 0.002 (-0.720) (2.390) 5 (high IV slope) 0.006 0.244 0.026 0.026 (0.340) (26.610) 5− 1 0.070 0.122 0.575 0.082 (7.720) (35.930) ˆ α ˆ β M ˆ β SMB ˆ β HML ˆ β UMD ˆ β SPX Adj. R 2 1 (low IV slope) -0.035 -0.082 0.068 -0.014 0.038 0.202 0.510 (-2.590) (-1.500) (1.040) (-0.290) (1.390) (8.570) 2 0.002 -0.060 0.017 -0.073 0.068 0.245 0.535 (0.120) (-0.880) (0.200) (-1.140) (1.740) (8.610) 3 0.009 -0.104 0.020 -0.015 0.052 0.257 0.545 (0.500) (-1.610) (0.270) (-0.220) (1.220) (8.230) 4 0.030 -0.124 0.042 -0.045 0.054 0.286 0.629 (1.710) (-1.810) (0.570) (-0.730) (1.160) (11.010) 5 (high IV slope) 0.048 -0.110 0.054 -0.004 0.034 0.280 0.571 (2.750) (-1.620) (0.640) (-0.060) (0.750) (9.440) 5− 1 0.083 -0.027 -0.014 0.010 -0.004 0.078 0.201 (7.840) (-0.790) (-0.310) (0.210) (-0.100) (5.150) TRADING ACTIVITY AND BETA 61 Table 2.7 Long-Term Straddle Portfolios This table presents the returns of long-term straddle portfolios sorted by IV slopes. On the last trading day before month t option expiration day, stocks are sorted into quintiles based on their IV slopes. IV slopes used for sorting are the averages of daily IV slopes over the two weeks preceding the last trading day before the option expiration day of month t. Holding period straddle returns are computed using at the money options that expire in month t+2,t+3 or t+4. When multiple maturities are available in this range, the straddle with the longest maturities is used. These straddles are purchased at the end of the first trading day following the expiration day of month t and sold at the end of the expiration day in month t+1. The abnormal returns are given by monthly regressions on five risk factors: Carhart four factors and the returns of straddles formed by ATM SPX index options. The returns of factors are computed over the same horizon as the stock straddle returns. The sample period is from January 1996 to December 2010. IV slopes are winsorized at 1% and 99%. T-statistics are computed using Newey-West standard errors based on the data-dependent automatic bandwidth/lag truncation parameters derived in Andrews 1991. Mean Return Sharpe Portfolio Return SD Ratio 1 (low IV slope) 0.014 0.094 0.147 (1.830) 2 0.024 0.107 0.220 (2.670) 3 0.017 0.109 0.156 (1.900) 4 0.017 0.111 0.153 (1.850) 5 (high IV slope) 0.011 0.102 0.108 (1.320) 5− 1 -0.003 0.045 -0.065 (-0.840) ˆ α ˆ β M ˆ β SMB ˆ β HML ˆ β UMD ˆ β SPX Adj. R 2 1 (low IV slope) 0.029 -0.027 0.038 -0.053 0.002 0.093 0.442 (4.040) (-0.820) (0.880) (-1.800) (0.120) (7.780) 2 0.039 -0.025 -0.002 -0.056 0.034 0.109 0.521 (4.970) (-0.750) (-0.030) (-1.560) (1.650) (8.610) 3 0.036 -0.055 -0.022 -0.041 0.017 0.111 0.533 (4.210) (-1.860) (-0.540) (-1.150) (0.930) (7.960) 4 0.037 -0.057 -0.002 -0.052 0.021 0.118 0.569 (4.460) (-1.830) (-0.040) (-1.640) (1.040) (9.300) 5 (high IV slope) 0.026 -0.030 0.018 -0.030 0.021 0.106 0.513 (3.410) (-1.100) (0.380) (-0.960) (1.160) (8.410) 5− 1 -0.003 -0.003 -0.019 0.023 0.019 0.014 0.070 (-0.590) (-0.200) (-1.100) (1.430) (1.980) (2.240) TRADING ACTIVITY AND BETA 62 Table 2.8 Time Series Predictability This table presents the predictive regressions of straddle returns on term structure slopes. Straddle returns in column "All one-month (long-term)" are the equal-weighted cross-sectional averages of one-month (long-term) ATM straddle returns of all stocks. Straddle returns in column "5-1 one-month (long-term)" are returns of the zero-investment portfolios that buy one-month (long-term) ATM straddles on stocks with high past IV slopes and sell one-month (long-term) ATM straddles on stocks with low past IV slopes. SPX one-month straddles are formed using ATM SPX index options with one month to maturity. All straddle returns are computed over the horizon that starts on the first trading day following month t option expiration and ends on the last trading day before month t+1 expiration. We consider three independent variables: "average IV slope" is the cross-sectional average of the IV slopes of all stocks; "high slope minus low slope" is the difference between the average IV slope of stocks in the high IV slope quintile and the stocks in the low IV slope quintile; "SPX IV slope" is the difference between the IV of the two month ATM SPX index options and the IV of the one month ATM SPX index options. Monthly observations of independent variables are computed by averaging over the daily observations over the two weeks preceding the last trading day before month t expiration. T-statistics are computed using Newey-West standard errors based on the data-dependent automatic bandwidth/lag truncation parameters derived in Andrews 1991. Type of straddle return: All one-month All long-term 5-1 one-month 5-1 long-term SPX one-month Regression #: 1 2 3 4 5 6 7 8 9 10 Intercept -0.034 0.009 0.006 0.034 0.086 0.086 -0.006 0.003 -0.122 0.218 (-1.780) (0.100) (0.720) (0.890) (1.760) (1.760) (-0.430) (0.150) (-2.110) (0.860) Average IV slope -0.725 -1.873 -0.873 -1.808 0.725 -0.220 -3.419 (-0.570) (-0.800) (-1.520) (-1.890) (0.670) (-0.620) (-0.500) High (5) slope minus low (1) slope -0.699 -0.485 0.725 -0.048 0.032 -0.101 -4.559 (-0.500) (-0.830) (0.670) (-0.070) (0.200) (-0.400) (-1.220) SPX IV slope 0.769 0.866 -0.360 -0.049 -4.743 -6.841 (0.300) (0.710) (-0.250) (-0.080) (-0.780) (-0.940) Adjusted R-square (%) -0.250 -1.120 1.530 1.280 -0.030 -0.910 -0.540 -1.410 -0.070 -0.180 TRADING ACTIVITY AND BETA 63 Chapter 3 On the Relation between Trading Activity and Market Beta and Its Implications on Event Studies Introduction Risk exposure, or the amount of risk, is one of the most important measures characterizing a financial asset and the underlying business. In the finance literature, much research has been devoted to understanding the determinants of the risk exposure of equities. The current work extends this literature by studying positive correlation between stock-level trading activity and systemic risk, and the implications of this correlation on studies of corporate events. While the positive correlation between trading activity and measured market risk has been the subject of a number of studies, most of the previous studies have focused on delayed responses of individual stock prices to information flow. If a less traded stock with a positive market beta responds to market-level information slowly, its market beta estimated from contemporaneous returns tends to be understated. As the trading activity of the stock increases, the stock price becomes more responsive to market information and the estimated beta increases as the downward bias diminishes. Applying this logic, Denis & TRADING ACTIVITY AND BETA 64 Kadlec (1994) argue that biases in measured market betas change surrounding SEOs and share repurchases because trading activity changes around these events, and therefore, the Dimson (1979) method should be used in order to obtain unbiased inferences regarding the impact of events on market betas. 1 The main contribution of our work is to show that, even after implementing the Dimson correction, the market beta is still positively correlated with trading activity. This finding has two important implications: first, there are mechanisms other than delayed price adjustments or non-synchronous trading that drive the positive correlation between trading activity and betas; second, explicit control of trading activity is necessary to obtain unbiased conclusions in event studies even when the Dimson adjustment is applied. Aside from delayed price adjustments to market-level information, the positive correlation between trading activity and market betas can arise from the following two channels: style-based trading and learning. First, the finance literature has recognized that stock returns may covary simply because stocks are “similar” to each other in certain ways. This phenomenon has been attributed to category- or style-based investing and trading. For example, Green & Hwang (2009) show that stock returns with similar nominal prices tend to covary with each other. In addition, BOYER (2011) shows that stock prices in the same economically meaningless index tend to move together. We argue that the same logic can be applied to trading activity. Consider a small to medium sized stock, A. When its trading activity and liquidity condition improves, it is likely to attract more traders who prefer highly traded stocks. Their preference suggests that they are also likely to have large stocks in their portfolios. As a result, the trading activity of these traders could introduce a common component in returns of Stock A and large stocks and, therefore, the market. Second, Patton & Verardo (2012) propose a learning channel that results in the positive correlation between firm level information release and market betas. Using intra-day 1 The Dimson (1979) method involves regressing stock returns on contemporaneous, leading and lagged market returns, and aggregating over all the coefficients to obtain market betas that are free of non- synchronous trading biases. TRADING ACTIVITY AND BETA 65 returns, they show that the market beta of a stock rises on its earning announcement days. Their explanation is that a firm’s earning announcements also reveal market-level information, which allows the entire market to learn from the announcing firm. This learning channel generates strong co-movement between announcing firm returns and market returns. The same logic could apply to stocks traded by informed traders who possess and trade on both market- and firm-level information. Although it might be optimal for traders to trade market level information using instruments such as index ETFs, a subset of traders might face constraints that prevent them from doing so. For example, when the manager of an under-diversified mutual fund receives negative news about the market, it may not be possible for her to short sell ETFs due to regulatory constraints. As a result, unloading some of her major holdings might be necessary. This argument suggests that the extent of informed trading in a stock is positively correlated with its market beta. Pinning down the exact causes of the positive correlation between trading activity and beta is an interesting research topic which we leave to the future. The rest of the paper is devoted to study the implications of this positive correlation on event studies. Several event studies examine the changes in equity risk exposure surrounding event dates (see Carlson et al. (2010) and Grullon & Michaely (2004) for recent examples). In these studies, the risk dynamics of event firms are often compared to those of event-free but otherwise similar firms to extract the component that is solely due to the events. Ideally, all the firms’ characteristics that are unrelated to the event under study should match. In most event studies, however, only a small number of firm characteristics, such as size, book-to-market ratios (B/M) and returns are used. The caveat of this practice is that changes in risk exposure driven by omitted characteristics may be mistakenly attributed to the events. 2 In this paper, we focus on two types of corporate events: seasoned equity offerings (SEO) and share repurchases. The trading activity and liquidity of a stock often increase or in the 2 Bessembinder & Zhang (2013) show that, after properly controlling for firm characteristics, the stock return anomalies following several types of corporate events disappear. In the same vein, understanding the relation between risk exposure and firm characteristics is important to event studies involving risk dynamics. TRADING ACTIVITY AND BETA 66 lead up to SEOs and decrease afterward. If changes in trading activity cause the systemic risk to move in the same direction, we would expect run-ups in beta before SEOs and decrease afterward, which is same pattern as those documented by Carlson et al. (2010). The authors argue that this beta pattern is driven by the build-up in real option value before SEOs and the exercising of real options afterward. In their study, SEO firms are compared to non-issuing firms matched by size and B/M only. We compare non-issuing firms that have similar time series patterns in trading activity and liquidity to SEO firms and find that they also exhibit increasing betas before matched issue dates and decreasing betas afterward. In cross-sectional regressions that include both SEO and non-SEO firms, trading volume plays a highly significant role in explaining changes in market betas. After controlling for trading activity, the estimated impact of SEOs on market betas decrease significantly. Our evidence suggests that the time series patterns in trading activity and the positive correlation between trading activity and market betas are partly responsible for the observed beta dynamics surrounding SEO firms. Furthermore, we argue that the delayed price adjustment is not the most important reason that trading activity and market betas are positively correlated, because even after we aggregate the regression coefficients of contemporaneous and 30 lagged market returns to compute the market betas, the beta dynamics are virtually unaltered for both SEO firms and matched firms. In contrast to SEO firms, firms that repurchase shares tend to experience decline in trading volume, which could potentially give rise to decline in their betas. Grullon & Michaely (2004) show that the betas of repurchasing firms decline significantly following the events. The authors argue that companies repurchase their shares because their investment opportunities, which are modeled as real options, are declining, which causes their betas to decline. We examine the possibility that the post-repurchase decline in beta is driven by changes in trading activity. A closer examination of beta dynamics surrounding repurchases also reveals that betas, in fact, tend to reach their lowest levels immediately before and TRADING ACTIVITY AND BETA 67 after repurchase dates and then begin to increase after repurchases. This pattern is puzzling in light of the investment opportunity based explanation, because it is unclear why firms tend to pay out cash when they are about to regain investment opportunities. We argue that accounting for patterns in trading activity might help to resolve this issue. An alternative explanation of our findings is that changes in real option value drive both beta and trading activity. The matched firms we consider happen to experience increases and decreases in real option value, leading to increases and decreases in betas and trading activity. We argue that this explanation is difficult to reconcile with standard real option theory, which typically predicts spikes in corporate investment when real option value and systemic risk start to decrease (see McDonald & Siegel (1986), Carlson et al. (2010) for examples). We find that there are no significant investment spikes for matched firms surrounding the peaks of their systemic risk. Additionally, the level of corporate investment of the matched firms is much lower than that of the SEO firms. An investigation the feasibility of a version of real option theory that could explain the patterns we document would be an interesting direction for future research. Data and Summary Statistics Data We obtain data on stock returns, trading volume and shares outstanding from CRSP. Information on firm fundamentals, such as book value and investment, comes from COMPUSTAT. Stock option data are provided by OptionMetrics, which covers 1996 to 2010. The sample of seasoned equity offerings (SEOs) is obtained from SDC Platinum, whose SEO database covers issuances from 1980 to 2012. Our sample only includes U.S. firms that issued common stocks traded on the NYSE, AMEX, or NASDAQ. Following the standard practice in the literature, we exclude utility and financial firms. SEOs with missing filing dates or proceeds are also excluded. TRADING ACTIVITY AND BETA 68 The sample of open market share repurchases comes from SDC’s U.S. Mergers and Acquisitions database. This dataset provides comprehensive coverage of share repurchase programs from 1984 to 2012. Repurchases with missing announcement dates or values of repurchases are excluded. Firms in our SEO and repurchase samples are linked to the CRSP database through historic CUSIP numbers and ticker symbols. We delete all firms that can not be matched to a PERMNO. Firms are excluded if they do not have valid information from COMPUSTAT annual fundamental dataset in the fiscal years before and after announcements. We also require firms to have at least 100 non-missing returns in the 12 calendar months before announcements. Summary Statistics of SEO Firms Table 3.1 provides summary statistics of the SEO sample after applying our selection criteria. The full sample covers SEOs from 1980 to 2012. The row labeled “PS” refers to the subsample in which all the proceeds come from secondary offers. We also consider two subsamples of SEOs based on option trading. Row “Option” includes SEOs where the event firms have exchange traded options. We consider a stock optionable on a trading day if it has at least one option closing bid price greater than $0.25. For a SEO firm to be included, we require its stock to be optionable on at least 10 trading days during the announcement month. Finally, “Active Option” refers to a smaller subsample of SEO firms with actively traded options. We consider a stock to have active option trading on a day if there are valid bid prices on at least four different contacts: a pair of at-the-money(ATM) calls and puts, an out-of-the-money call and an out-of-the-money put, where ATM contracts satisfy 0.95≤ strike price/underlying price≤ 1.05. The columns named “B/M and Size” in Table 3.1 present average book to market (B/M) deciles and size deciles, respectively. The column named “Primary” contains ratios of primary to total proceeds. The last column presents the ratio of total SEO proceeds to the market capitalization of TRADING ACTIVITY AND BETA 69 issuing firms prior to issuance. These statistics are largely in line with those of previous studies. As expected, firms with traded options are larger than other firms, and their proceeds-to-size ratios are smaller than those of other firms, and they also tend to have larger portions of proceeds coming from primary issues. In Table 3.2, we present stock returns surrounding SEO events. To determine the SEO firms’ abnormal buy-and-hold returns, we construct matched samples based on size and B/M using the same method as Carlson et al. (2010). Specifically, for each SEO event, we identify the firm that belongs to the same B/M decile as the issuer, has not issued equity in the previous five years, and is the closest size match satisfying these criteria. One-year run-ups are cumulative returns over the 252 trading days before announcement days. Announcement returns are three-day cumulative returns surrounding announcements days. Post-SEO returns are cumulative returns over the specified periods of time following issue dates. Similar to previous findings in the literature, SEO firms exhibit large positive returns relative to their matched firms (100% v.s. 27.4%) before announcements and significant under-performance (27.4% v.s. 43.6%) over the three years following issuance. These patterns are particularly strong for SEO firms with active option trading, which have an average pre-SEO run-up of 197% and post-issuance return of -2%. To the extent that option trading indicates high trading interest in the underlying stock, our results in the last subsample suggest that the return patterns surrounding SEOs are related to trading activity. Table 3.3 presents summary statistics of firms that had open-market share repurchases from 1984 to 2012. Compared to SEO firms, book-to-market ratios are clearly higher for repurchasing firms, which suggests that they have fewer investment opportunities. On average, repurchasing firms are also bigger than SEO firms. TRADING ACTIVITY AND BETA 70 Market Betas and Trading Activity Theoretical Motivations Previous work on the relation between market betas and trading activity tends to focus on the effects of price adjustment delays on estimated betas. In a study related to ours, Denis & Kadlec (1994) argue that if stock prices adjust slowly to market-level information, market betas estimated from contemporaneous stock and market returns tend to be downward biased. When the trading activity of a stock increases, the stock price adjusts faster to market-level information and the estimated market beta increases because the downward bias diminishes. Recent research suggests that the positive correlation between trading activity and market betas could be motivated by other factors. First, the finance literature has recognized that stock returns may covary simply because stocks are “similar” to each other in certain ways. This phenomenon has been attributed to category- or style-based investing and trading. For example, Green & Hwang (2009) show that returns of stocks with similar nominal prices tend to covary with each other. BOYER (2011) shows that prices of stocks in the same economically meaningless index tend to move together. We argue that the same logic can be applied to trading activity. Consider a small to medium sized stock A. When its trading activity and liquidity condition improves, it is likely to attract more traders who prefer highly traded stocks. Their preference suggests that they are also likely to have large stocks in their portfolios. As a result, the trading activity of these traders could introduce a common component in returns of stock A and large stocks, therefore the market. Second, Patton & Verardo (2012) propose a learning channel that results in a positive correlation between firm level information release and market betas. Using intra-day returns, they show that the market beta of a stock rises on its earning announcement days. They argue that this is because earning announcements of a firm also reveal market level information; therefore, the market can learn from announcing firms. This learning channel generates stronger co-movement between announcing firms and the market. The same logic TRADING ACTIVITY AND BETA 71 could apply to stocks traded by informed traders who possess and trade on both market and firm level information. Although it might be optimal for traders to trade market level information using instruments such as index ETFs, a subset of traders might face constraints that prevent them from doing so. For example, when the manager of an under-diversified mutual fund receives negative news about the market, it may not be possible for her to short sell ETFs due to regulatory constraints. As a result, unloading some of her major holdings might be necessary. This argument suggests that the extent of informed trading in a stock can be positively correlated with its market beta. All the above mechanisms imply a positive correlation between market betas and trading activity at the stock level. Although distinguishing between the latter two channels is not the main goal of this paper, we do provide evidence to show that the delay price adjustment channel is not the main driver behind the positive beta-trading activity relation we document. It is important to point out that mechanisms do exist to generate negative correlations between betas and trading activity. For instance, Edelen & Kadlec (2012) argues that when traders seek liquidity, they tend to sell following positive returns and buy following negative returns. This contrarian trading strategy could reduce the measured market betas of stocks that are being traded heavily by liquidity seeking traders. Empirical Evidence In this section, we present evidence to support the existence of the correlation between trading activity and market betas. At the end of each calendar year t, we measure the difference between the annual turnover of each stock in each year t and its turnover in year t-1. Then we sort stocks into 20 portfolios according to their changes in turnover rates from year t-1 to year t. For a stock to be included in the portfolios in year t, we require it to have 12 monthly volume measures both in year t and year t-1. We only include common stocks traded on the NYSE, AMEX TRADING ACTIVITY AND BETA 72 or NASDAQ in both years t and t-1. After forming the portfolios, we compute portfolio daily log returns in year t and year t-1. We regress the daily log returns of each portfolio on market returns to obtain the market betas of each portfolio in years t and t-1. The difference in the betas between years t and t-1 is the change-in-beta measure for each portfolio. To control for the effect of delayed price adjustments to market-level information, we compute market betas by including lagged market returns in our regressions and aggregate over coefficients on contemporaneous and lagged market returns to obtain market betas. This method is proposed in Dimson (1979) and we call the market betas estimated this way Dimson betas. Specifically, we run the following regressions for each firm i in each year t: r j,τ =α j + N X L=0 b L,j r M,τ−L + j,τ , where r M,τ and r j,τ are the daily log returns of market and portfolio j on day τ; N is the number of lagged days. Dimson betas are obtained by summing over coefficients b L,j for all Ls. Figure 3.1 plots the median changes in beta for each change-in-turnover sorted portfolio. Market betas in Panel A are computed using the standard OLS method involving only contemporaneous market returns, while Panel B presents results from the Dimson (1979) method. As the two panels clearly show, changes in market betas are positively correlated to contemporaneous changes in turnover regardless of whether the Dimson correction is used. These results suggest that style-based trading or learning could play a more important role than delay price adjustments to market-level information in explaining the positive correlation between betas and trading activity. In Table 3.4, we present the results from cross-sectional regressions using the Fama & MacBeth (1973) method. Specifically, in each year t, we regress the change in market beta of each portfolio j on the contemporaneous change in turnover and the contemporaneous portfolio return: Δβ j,t =a +b· ΔTO j,t +c·r j,t + j,t , where ΔTO j,t is the average of the TRADING ACTIVITY AND BETA 73 Figure 3.1. Changes in Market Beta v.s. Changes in Turnover Change in Dimson Betas -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Change in Turnover Rates 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Change in OLS Betas -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 changes in turnover of all firms in portfolio j from year t-1 to year t; r j,t is the return of portfolio j from the end of year t-1 to the end of year t. In our regression analysis, we use Dimson betas as independent variables. Results in Table 3.4 Panel A confirm Figure 3.1 and show significant positive correlation between changes in turnover and changes in beta even after controlling for portfolio returns. Turnover is typically positively correlated with stock returns in the cross-section. As a result, stocks with high turnover might appear to have high betas in years when the market return is positive, simply because high turnover stocks have high returns. To rule out this spurious correlation, we present the regression coefficients from years with negative market returns in Panel B. The coefficients on turnover remain positive and significant, suggesting that our results are not spurious. Our evidence suggests that, in event studies, adding the Dimson correction can not fully TRADING ACTIVITY AND BETA 74 address the omitted variable problem due to the correlation between trading activity and market betas. Instead, explicit control of contemporaneous changes in trading activity must be imposed. Beta Dynamics of SEO Firms In this section, we study how stock level trading activity and liquidity evolve surrounding SEOs and argue that they drive the dynamics of market betas, while SEO events themselves do not seem to have a significant impact on beta dynamics. In addition, we show that trading activity affects market betas not only because of price adjustment delays but because the positive correlation between market betas and trading activity still exists even after correcting for price delays. Using a real option model, Carlson et al. (2010) show that a SEO firm’s market beta tends to rise before issuance and continue to do so within about six months after issuance. Betas start to decrease six months following issuance. In the model, the market betas increase because the betas of growth options are higher than the betas of asset-in-place, and as growth options become a larger portion of firm value, the market betas of firms increase. In Carlson et al. (2010), the betas of the SEO firms are compared to the betas of the non-issuing firms matched by size and B/M to extract the impact of SEO events on the betas. In a recent study, Bessembinder & Zhang (2013) point out that firm characteristics such as liquidity tend to experience significant changes over the years surrounding corporate events and it is important to control for these changes when we evaluate abnormal returns following events. They show that anomalous returns following several types of corporate events disappear after properly controlling for values of firm characteristics at different points in time surrounding the events. We argue that the logic in Bessembinder & Zhang (2013) also applies to studies of risk dynamics surrounding corporate events. In particular, we observe that stock trading activity changes significantly surrounding SEOs. To the extent that trading activity affects TRADING ACTIVITY AND BETA 75 market betas through channels unrelated to SEO events, it is necessary to control for changes in trading activity to obtain an unbiased view of the role played by SEOs in driving risk dynamics. Beta Dynamics and Trading Activity Surrounding SEO To demonstrate the risk dynamics of SEO firms, we plot the market betas surrounding SEOs in Figure 3.2. We consider the time windows that start five years before announcements and end five years following issuance. For each SEO, monthly measures of market betas are computed using stock log daily returns and contemporaneous market returns from each calendar month. A monthly beta measure is then assigned an event time. If the month in which the beta is measured is prior to an announcement (after issuance), the event time is equal the number of months between the month and the announcement date (issue date) and has a negative (positive) sign. Figure 3.2 shows cross-sectional medians of market betas at each event time. The period between the announcement and issuance is treated as one period no matter how long it is, and the medians of the betas measured during this period are labeled as 0 on the event time axis. If the time lapse between the announcement and issuance is less than 15 trading days, this observation will be omitted from the calculation. Panel A confirms the findings of Carlson et al. (2010). SEO firms exhibit significant run-ups in market risk over the five-year period before announcements. The increase in market risk continues for about 6 months following issuance and then stars to decline. This pattern is not observed in the matched sample. The rest of the panels in Figure 3.2 compare the market beta dynamics of SEO firms and non-issuing matched firms chosen to have relatively high levels of trading activity surrounding issuance dates. In Panel B, in addition to the size and B/M matching procedure we describe in Section 3, we require that matched firms’ monthly turnover rates (1000·# of shares traded per month/# of shares outstanding) be higher than 90% of the TRADING ACTIVITY AND BETA 76 turnover rate of the SEO firm during the month of issuance. In Panel C, we require the Amihud illiquidity measures of matched firms be no higher than 110% of the Amihud measure of the SEO firm during the month of issuance. In Panel D, we choose matched firms from the ones with active option trading during issuance months, defined in the same way as in Section 3. In Panels B, C and D, matched firms show clear run-ups in betas before announcement dates, and their betas also tend to decline following matched issue dates. Comparing corresponding panels in Figure 3.3 and Figure 3.2, we can see a clear resemblance between the dynamics of market betas and turnover rates. One notable feature of SEO firms is that the increase in their turnover tends to be more persistent than that of the matched samples. As we can see in Panel B of Figure 3.3, when we pick matched firms to have high turnover around issue dates, the matched firms exhibit a sharp peak in turnover. This peak is, however, highly transitory. It has completely diminished within 15 months following matched issue dates. Also, the sharp peak in turnover does not result in a sharp peak in the market betas. In contrast, the pre-SEO run-up in turnover tend to be more persistent. Even after 5 years following issuance, at least 2/3 of the pre-SEO turnover increase is preserved. Also, the peak in turnover right after issue dates is associated with a peak in market betas. These detailed features suggest that not all trading activities are equal when it comes to affecting market risk. To understand this, consider a stock’s turnover spikes because some large institutional holders are liquidating their holdings. This type of increase in turnover tends to be transitory, and it may in fact reduce the market beta of the stock if liquidity traders tend to sell into market rallies 3 . In Figure 3.4, we present the dynamics of trading costs which are captured by the Amihud illiquidity measure. Compared with matched firms, SEO firms experience a much larger decrease in trading costs. In all cases, trading costs mostly flatten following issue dates and do not exhibit clear re-bounces. This is most likely a result of the long-term downward 3 See Edelen & Kadlec (2012) for an example. TRADING ACTIVITY AND BETA 77 trend in trading costs, which can be observed in Panel A. The trading costs of matched firms based on size and B/M decrease almost linearly over time. Figures 3.2, 3.3 and 3.4 suggest that changes in trading activity alone could be responsible for the market beta dynamics surrounding SEOs, while the SEO events themselves does not play any special role. Even if SEO events do not fully explain the risk dynamics, it does not follow that the real option theory proposed by Carlson et al. (2010) does not explain risk dynamics. One possibility is that the matched firms we picked happen to be investing to exercise their real options surrounding the matched issue dates, just like the SEO firms. The difference is that matched firms do not need to raise cash from equity markets. To examine this hypothesis, Figure 3.5 plots the dynamics of the corporate investment surrounding the SEOs. Corporate investment is defined as the sum of the changes in PP&E and changes in inventory, normalized by the total assets at the beginning of the fiscal year. Figure 3.5 shows a clear contrast between the SEO firms and all matched samples. The SEO firms tend to make more investment surrounding the issue dates. In comparison, the levels of investment of the matched firms are much lower and do not necessarily show peaks at the issue dates. For example, Panel C shows that investment declines over time. The evidence from alternative matched schemes poses serious challenges to the real option based explanation. It appears that neither SEO events nor corporate investments are necessary to generate the run-up and decline in market betas. TRADING ACTIVITY AND BETA 78 Figure 3.2. Market Beta Dynamics Surrounding SEOs Market Beta 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel C: Matched by Size,B/M and Illiquidity Market Beta 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel D: Matched Using Stocks with Options Market Beta 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel A: Matched by Size and B/M Market Beta 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel B: Matched by Size,B/M and Volume SEO Firms Matching Firms Figure 3.3. Turnover Rate Surrounding SEOs Turnover 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel C: Matched by Size,B/M and Illiquidity Turnover 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel D: Matched Using Stocks with Options Turnover 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel A: Matched by Size and B/M Turnover 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel B: Matched by Size,B/M and Volume SEO Firms Matching Firms TRADING ACTIVITY AND BETA 79 Figure 3.4. Amihud Illiquidity Measure Surrounding SEOs Market Beta 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel C: Matched by Size,B/M and Illiquidity Market Beta 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel D: Matched Using Stocks with Options Market Beta 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel A: Matched by Size and B/M Market Beta 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel B: Matched by Size,B/M and Volume SEO Firms Matching Firms Figure 3.5. Corporate Investment Surrounding SEOs Investment 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel C: Matched by Size,B/M and Illiquidity Investment 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel D: Matched Using Stocks with Options Investment 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel A: Matched by Size and B/M Investment 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel B: Matched by Size,B/M and Volume SEO Firms Matching Firms TRADING ACTIVITY AND BETA 80 Delayed Response to Market-Level Information Previous research has established that small and less traded firms may fail to fully respond to contemporaneous market-level information. Denis & Kadlec (1994) show that, after accounting for non-synchronous trading using the method in Dimson (1979), there are no significant changes in market betas surrounding SEOs. As argued in Denis & Kadlec (1994), when stock prices exhibit delayed responses to market-level information, regressing stock returns on contemporaneous market returns might cause downward biases in estimated market betas if the true betas are positive and if the horizon used for computing returns is short. One potential explanation of the positive correlation between trading activity and market betas is that, when stocks become more actively traded, their prices tend to adjust to market-level information more quickly, therefore, the downward biases in estimated betas start to diminish. To investigate this hypothesis, we compute Dimson betas by including lagged market returns in our regressions. Specifically, we run the following regressions for each firm i in each year n: r i,t =α i + N X L=0 b L,i r M,t−L + i,t , where r M,t and r i,t are daily log returns of market and firm i, N is the total number of lagged days. Dimson betas are obtained by summing coefficients b L,i over all lags. Panel A of Figure 3.6 shows adjusted market betas for SEO firms. Panel B shows the adjusted betas for matched firms with trading activity comparable to the SEO firms. This is the same matched sample used to produce Panel B in Figures 3.2, 3.3, 3.4 and 3.5. We choose N to be 1,15 or 30. The beta dynamics in Figure 3.6 are very similar to those in Figure 3.2. The run-up before year 0 and decline afterwords remain clear. In an untabulated analysis, we confirm that the betas in year 0 are significantly higher than the betas in years 5 and -5 for both the SEO firms and matched firms even when the market returns from 30 lagged days are included in TRADING ACTIVITY AND BETA 81 Figure 3.6. Dimson (1979) Beta Market Beta 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Event Time in Year -5 -4 -3 -2 -1 0 1 2 3 4 5 Panel B: Firms Matched By Size, B/M and Turnover Market Beta 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Event Time in Year -5 -4 -3 -2 -1 0 1 2 3 4 5 Panel A: SEO Firms N=1 N=15 N=30 PanelAplotsthemedianmarketbetaofrepurchasingfirmsindifferentmonthsintheeventwindow. Monthly beta measures are computed using daily log returns from each calendar month. Panel B plots the median turnover rate of repurchasing firms. Monthly turnover rates are defined the as ratios of numbers of shares traded in the month to numbers of shares outstanding. the regressions. These results suggest that the positive correlation between betas and trading activity is not totally driven by delayed price responses to market level information. The other two channels we proposed in Section 3 might play a more important role. TRADING ACTIVITY AND BETA 82 Cross-Sectional Regression Analysis This section presents cross-sectional regressions that analyze the determinants of the market betas surrounding SEOs. Our sample includes all the SEO firms and firms that do not issue any equities in the six-year windows centered on the matched issue dates and are in the same size decile and B/M decile as the SEO firms. This matching process creates many redundant data points, because a non-issuing firm could be used multiple times as a matched firm. To handle this issue, we collapse all the data points from the same non-issuing firm in the same calendar year into one data point. For instance, if Firm A is selected as the matched firm for an SEO with an issue date on 3/1/2000 and another one on 7/1/2000, any variable of A, such as the pre-SEO return or change in beta, in 2000 is defined as the average of the two measures associated with the two matched SEOs. 4 This concatenation process leaves us with one data point per matched firm per year. For each firm-event (or matched event), we measure the market beta, turnover and Amihud measure using data from the one-year window centered on the issue date. These measures are labeled with the superscript issue in Figure 3.7. In all the regressions in this section, we use the Dimson method to compute the market betas. Each market beta is obtained by regressing daily log stock returns on contemporaneous and 30 lagged daily log market returns and aggregating the coefficients on the market returns. Given the relatively large number of independent variables in these regressions, we require at least 220 valid returns for each stock within each one-year window. An Amihud measure is the average ratio of the absolute values of daily stock returns and daily dollar volume. Turnover rates are the summations of daily turnover rates over the time window. Basically, these variables capture the levels of beta, trading activity and liquidity at the SEO events. 4 Anon-issuingmatchedfirmisassignedwithissuanceinformation, suchasissuedate, ofthematchedSEO firm. Its issuance related variables, such as pre-SEO return, are defined accordingly using these information. TRADING ACTIVITY AND BETA 83 X post X pre X issue Issue Date Figure 3.7. The Time-line For Cross-Sectional Regressions TRADING ACTIVITY AND BETA 84 We also measure changes in the three variables during the three-year periods before and after issue dates. The changes in variable X before issue dates (ΔX pre ) are given by the differences between X issue and X pre , as shown in Figure 3.7. X pre is measured using the data from the one-year window centered at the 36th month before the issue dates in ways similar to computing X issue . Besides the three variables, we measure the cumulative stock returns over the three-year periods before the issue dates. The changes in variable X after the issue dates (ΔX post ) are given by the differences between X post and X issue , where X post is measured from the one-year window centered at the 36th month after the issue date. For each firm-event, we also provide a measure of corporate investment following issuance by summing the annual investments over the fiscal year including the issue date and the two fiscal years following it. Annual corporate investment in a fiscal year is defined as the change in PP&E plus the change in inventory, divided by the total asset at the beginning of the fiscal year. All the variables are first constructed for each firm-event. Then, for each firm, we collapse the measures of each variable associated with different events in the same year into one data point. Given these definitions, we run the following regressions: Δβ pre i,t =a +b·I seo i,t +c· ΔTODEC pre i,t +d·TODEC issue i,t +e· ΔILLIQDEC pre i,t +f·ILLIQDEC issue i,t +g·r pre i,t +h·r pre i,t ·I seo i,t + i,t (3.1) and Δβ post i,t =a +b·I seo i,t +c· ΔTODEC post i,t +d·TODEC issue i,t +e· ΔILLIQDEC post i,t +f·ILLIQDEC issue i,t +g·r pre i,t +h·r pre i,t ·I seo i,t + +i·INV post i,t + i,t , (3.2) TRADING ACTIVITY AND BETA 85 where Δβ post i,t and Δβ pre i,t represent post-SEO and pre-SEO changes in market beta of firm i that issues in year t; indicator variable I seo i,t is equal to 1 if firm i issues an SEO in year t and 0 otherwise; r pre i,t is the three-year cumulative return before the issue date; INV post i,t is the three-year corporate investment following the issue date. Instead of working with raw measures of changes and levels of turnover and illiquidity, we use their deciles as independent variables. In year t, we sort firms into deciles based on their changes in turnover before issue dates and denote their decile number as ΔTODEC pre i,t . ΔTODEC post i,t and TODEC issue i,t are defined similarly. ΔILLIQDEC pre i,t , ΔILLIQDEC post i,t and ILLIQDEC issue i,t are deciles for the Amihud illiquidity measure. To examine the robustness of our results, we run both pooled regressions and Fama-Macbeth regressions as in Fama & MacBeth (1973). The estimated coefficients are reported in Tables 3.5 and 3.6, respectively. Table 3.5 Panel A presents the coefficients in Equation 3.1. The first column shows that the market betas of the SEO firms increases by 0.188 over the 3-year period before issuance. Column 2, however, suggests that all the pre-SEO increases in beta can be explained by the contemporaneous changes in turnover and the level of turnover surrounding the issue dates. The coefficient of the contemporaneous change in turnover ΔTODEC pre is positive and highly significant. In Column 3, the coefficient of the contemporaneous change in illiquidity ΔILLIQDEC pre is negative and significant. Results from Column 2 and 3 show that market betas increase with trading activity and stock level liquidity, which is consistent with our conjecture. The coefficient of r pre in Column 4 is positive and significant, suggesting that the market beta increases with contemporaneous returns. The coefficient I seo in Column 4 is 0.107 and still statistically significant. This suggests that contemporaneous returns do not explain the changes in market betas as well as the contemporaneous changes in turnover. Table 3.5 Panel B reports the coefficients from Equation 3.2. The coefficient on I seo in the TRADING ACTIVITY AND BETA 86 first column shows that the market betas of the SEO firms decreases by 0.26 over the three-year period following SEO. Column 2 shows that controlling for changes in turnover following issue dates reduces the SEO effect on betas to 0.17. The coefficient of ΔTODEC post is positive, which confirms the positive correlation between the changes in trading activity and market betas. The negative coefficient on TODEC issue suggests that high past trading activity is associated with a decrease in the betas. Column 3 indicates that illiquidity plays little role in explaining changes in the market betas following SEOs. The coefficient of the lagged returns in Column 4 of Table 3.5 Panel B is significantly negative, and including lagged returns reduces the magnitude of the coefficient on I seo to 0.215. This finding suggests that high past returns are associated with a decrease in the betas. The coefficient on r pre ·I seo is positive and insignificant, suggesting that changes in the betas of SEO firms are less sensitive to lagged returns. This seems to be at odd with the results given by Carlson et al. (2010), who run similar regressions for a sample only including SEO firms and find negative coefficients on lagged returns. The authors argue that firms with high pre-SEO returns experience large increase in real option value before SEOs, therefore these firms also tend to have large decrease in real option value as these options are exercised following SEOs. Because real option value and beta are positively correlated, firms with higher pre-SEO returns tend to experience larger decrease in beta following SEOs. However, the positive coefficient on r pre ·I seo shows that, in fact, betas of SEO firms are less likely to show negative responses to pre-SEO returns. The coefficient on corporate investment in column 5 shows that changes in market beta are negatively correlated with contemporaneous corporate investment, which is consistent with the result in Carlson et al. (2010). The real option theory argues that firms invest to exercises their real options following SEOs, which reduces market betas. Although the investment variable has statistically significant coefficient, it has very little power in explaining changes in market beta following SEOs. The coefficient on I seo is -0.246 in column 5, compared to -0.26 in column 1. Finally, column 6 shows that ,after including all variables, the coefficient TRADING ACTIVITY AND BETA 87 on I seo is still negative and significant, suggesting that SEO events indeed play a role in driving market beta. Overall, results from our regressions suggest that, although real option theory seems to explain changes in market betas in the correct direction, trading activity may play an even more important role in driving market beta. Our results on Fama-Macbeth regressions in Table 3.6 largely confirm the qualitative results in Table 3.5. Share Repurchases Grullon & Michaely (2004) study the motivations behind share repurchases. They argue that companies repurchase their shares because their investment opportunities, which are modeled as real options, are declining. As one piece of the evidence, the authors show that, following share repurchases, market betas of repurchasing firms decline more than matched non-repurchasing firms. The authors argue that this is consistent with declining real option value of repurchasing firms. In Figure 3.8 Panel A, we confirm the finding in Grullon & Michaely (2004). For each repurchasing firm, we find a matched firm based only on B/M and size in the same way we find matches for SEO firms. Panel A plots the cross-sectional medians of the differences in beta between repurchasing firms and matched firms. We can see clear decline of repurchasing firms’ betas relative to matched firms. Figure 3.8 Panel B plots the median differences in turnover between repurchasing firms and matched firms. As we can see, the median turnover rate of repurchasing firms exhibits clear decline following announcement (i.e. following time 0). This similarity between the patterns in Panel A and B suggests the possibility that the post-repurchase decline in beta is driven by changes in trading activity instead of real option value. Figure 3.9 Panel A shows the beta dynamics of repurchasing firms. It appears that betas tend to reach their lowest levels in about 8 months following repurchases and start to increase after that. This pattern is puzzling in light of the investment opportunity based TRADING ACTIVITY AND BETA 88 explanation. If betas are positively correlated with investment option value, it is unclear why firms tend to pay out cash when they are about to regain investment opportunities. The pattern in Panel B suggests that the increase in market beta following repurchases might be driven by the increase in trading activity. Conclusion and Discussions This paper studies the implications of the positive correlation between market betas and trading activity on event studies. We find that even after correcting for non-synchronous trading, market betas are still positively correlated with trading activity, which suggests that trading activity affects betas in ways beyond correcting for price adjustment delays. We also find that controlling for dynamics in trading activity surrounding corporate events can change our view on how events affect risk dynamics of firms. Market betas appear to increase before SEOs and decrease afterwords. But after controlling for turnover rates, SEO events do not play a significant role in explaining risk dynamics. Matched non-issuing firms that happen to have high turnover surrounding issue dates also exhibit the beta run-up and decline pattern that is attributed to SEOs by previous studies. Real option theory is unlike to be able to explain the risk dynamics of matched firms. We find no significant peaks in corporate investment surrounding matched issue dates, which contradicts the prediction of standard real option theory. Aside from SEOs, we find that the risk dynamics surrounding share repurchases is likely to be driven by trading activity as well. The market beta dynamics surrounding SEOs is not affected by Dimson correction, which suggests that delayed responses to market information is not the main reason behind the positive correlation between market risk and trading activity. We proposed two alternative channels to explain this relation: style-based investing and learning. It is left to future research to determine which channel is more important. The lack of impact of SEOs and repurchases on risk dynamics pose challenges to real option theory. It is an interest research question to find out whether there exist TRADING ACTIVITY AND BETA 89 modifications to real option theory that accommodate the new findings in this paper. TRADING ACTIVITY AND BETA 90 Figure 3.8. The Difference in Beta and Turnover Between Repurchasing Firms and Matched Firms -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel B: Difference in Turnover -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel A: Difference in Beta PanelAplotsthedifferenceinmarketbetabetweenrepurchasingfirmsandmatchedfirms. Eachrepurchasing firm is matched to a non-repurchasing firm that is in the same B/M decile and have the most similar market capitalization prior to announcement dates. Monthly beta measures are computed and the differences are obtained by subtracting the betas of matched firms from that of repurchasing firms. The plotted values are cross-sectional medians of the differences at each event time. Panel B uses the similar procedure to compute differences in turnover rates. TRADING ACTIVITY AND BETA 91 Figure 3.9. Beta and Turnover Dynamics of Repurchasing Firms 2.5 3.0 3.5 4.0 4.5 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel B: Turnover 0.4 0.5 0.6 0.7 0.8 0.9 Event Time in Month -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Panel A: Beta PanelAplotsthemedianmarketbetaofrepurchasingfirmsindifferentmonthsintheeventwindow. Monthly beta measures are computed using daily log returns from each calendar month. Panel B plots the median turnover rate of repurchasing firms. Monthly turnover rates are defined the as ratios of numbers of shares traded in the month to numbers of shares outstanding. TRADING ACTIVITY AND BETA 92 Table 3.1 Summary Statistics of the SEO Sample The table presents summary statistics of the SEO sample we study in this paper. The sample includes seasonedissuanceofcommonstocksbyU.S.companiesfrom1980to2012. Weexcludeissuancesbyfinancials, utilities and firms that are not traded on the NYSE, AMEX or NASDAQ during the month of issuance. SEOs with missing filing dates or proceeds are also excluded. Firms are excluded if they do not have valid information from COMPUSTAT annual fundamental data set in the fiscal years before and after announcements. We also require firms to have at least 100 non-missing returns in the 12 months before announcements. The columns named “B/M” and “Size” present average book to market (B/M) and size deciles, respectively. The column named “Primary” contains ratios between the proceeds from primary issues to total SEO proceeds. The last column presents the ratio between total SEO proceeds and the market capitalization of the issuing firm prior to issuance. Standard deviations reported in parentheses. N B/M Size Primary(%) Proceeds/Size All SEOs 6474 3.56 3.70 0.73 0.24 (2.86) (2.45) (0.38) (0.29) PS 1057 3.59 5.13 0.00 0.17 (2.85) (2.52) (0.00) (0.21) Option 1790 3.66 4.91 0.75 0.16 (2.88) (2.48) (0.41) (0.13) Active Option 613 3.10 6.06 0.81 0.15 (2.63) (2.42) (0.38) (0.17) TRADING ACTIVITY AND BETA 93 Table 3.2 Event Returns of SEO Firms The table presents average returns over different horizons surrounding SEOs. For each SEO firm, we construct matched samples based on size and B/M. Specifically, for each SEO event, we identify the firm that belongs to the same B/M decile as the issuer, has not issued equity in the previous five years, and is the closest size match satisfying these criteria. One-year run-ups are cumulative returns over the 252 trading days before announcement days. Announcement returns are three-day cumulative returns surrounding announcements. Post-SEO returns are cumulative returns over the specified periods of time following issue dates. Standard errors are presented in parentheses. 1-yr Run-up Announcement 1-yr Post 3-yr Post SEO match SEO match SEO match SEO match All SEOs 1.0034 0.2742 -0.0159 0.0030 0.0917 0.0996 0.2737 0.4361 (0.0244) (0.0131) (0.0009) (0.0008) (0.0103) (0.0090) (0.0200) (0.0257) PS 0.6299 0.2798 -0.0175 0.0043 0.2086 0.1443 0.5587 0.5213 (0.0300) (0.0239) (0.0019) (0.0015) (0.0207) (0.0214) (0.0451) (0.0795) Option 1.1623 0.3065 -0.0163 0.0022 0.0716 0.0969 0.2394 0.4392 (0.0686) (0.0394) (0.0021) (0.0017) (0.0257) (0.0186) (0.0454) (0.0672) Active Option 1.9678 0.3743 -0.0102 0.0031 0.0263 0.0501 -0.0220 0.3308 (0.1857) (0.1069) (0.0045) (0.0037) (0.0616) (0.0300) (0.0920) (0.1205) TRADING ACTIVITY AND BETA 94 Table 3.3 Summary Statistics of Repurchasing Firms ThistablepresentssummarystatisticsofU.S.companiesthatlaunchedopenmarketcommonstockrepurchases during the period 1984 to 2011. Firms are excludes if they do not have valid information from COMPUSTAT annual fundamental data set in the fiscal years before and after announcements. We also require firms to have at least 100 non-missing returns in the 12 months before announcements. The first three columns present sizes of (sub)samples, average B/M deciles and average market capitalization deciles. The column named “$Value/ME” is the ratio of the dollar value of repurchases to firms’ market capitalization. The last column is the dollar value of repurchases. N B/M ME $Value/ME $Value(Mil.) Full Sample 5658 5.19 4.29 0.22 545.68 (2.91) (3.22) (5.97) (2896.73) Option 2450 4.02 6.48 0.14 969.22 (2.65) (2.72) (0.69) (3038.78) Active Option 1208 3.54 7.43 0.12 1490.76 (2.50) (2.40) (0.25) (3817.96) TRADING ACTIVITY AND BETA 95 Table 3.4 Changes in Turnover and Changes in Market Beta This table presents the results from cross-sectional regressions using Fama & MacBeth (1973) method. Specifically, in each year t, we regress the change in market beta of each portfolio j on the contemporaneous change in turnover and the contemporaneous portfolio return: Δβ j,t = a +b· ΔTO j,t +c·r j,t + j,t . To compute independent variables ,we use the method in Dimson (1979). Panel A presents the results from the full sample period from 1970 to 2012. Panel B presents the results from the years with negative market returns.T-statistics with White corrections are reported in parentheses. Panel A: Full Sample 1970-2012 Intercept -0.036(-0.53) -0.141(-2.09) -0.052(-0.75) ΔTO 0.052(7.52) 0.032(4.02) r 1.204(11.26) 0.498(3.45) Adj.R 2 0.53(16.82) 0.509(13.27) 0.606(18.08) Panel B: Years with Negative Market Return Intercept -0.2(-1.81) -0.082(-0.52) -0.117(-0.81) ΔTO 0.061(3.99) 0.031(2.49) r 1.472(7.22) 0.806(2.79) Adj.R 2 0.566(12.83) 0.517(7.87) 0.649(15.97) TRADING ACTIVITY AND BETA 96 Table 3.5 Determinants of Changes in Market Betas (Pooled Regressions) Panel A presents the coefficients of the pooled regressions based on the following model, Δβ pre i,t =a +b·I seo i,t +c· ΔTODEC pre i,t +d·TODEC issue i,t +e· ΔILLIQDEC pre i,t +f·ILLIQDEC issue i,t +g·r pre i,t +h·r pre i,t ·I seo i,t + i,t , and Panel B presents the coefficients of the pooled regressions based on the following model, Δβ post i,t =a +b·I seo i,t +c· ΔTODEC post i,t +d·TODEC issue i,t +e· ΔILLIQDEC post i,t +f·ILLIQDEC issue i,t +g·r pre i,t +h·r pre i,t ·I seo i,t + +i·INV post i,t + i,t , whereβ post i,t andβ pre i,t represent post-SEO and pre-SEO changes in market beta of firm i in year t; indicator variableI seo i,t is equal to 1 if firm i issues a SEO in year t and 0 otherwise; r pre i,t is the three-year cumulative return before the issue date; INV post i,t is the three-year corporate investment following the issue date. Also included as independent variables are deciles of the change and level of turnover and illiquidity. Pre-SEO (post-SEO) changes in turnover and liquidity are the differences between the levels measured from the one-year windows surrounding issue dates and the one-year windows three years prior to (following) issuance. In each year t, we sort firms into deciles based on their changes in turnover before issue dates and denote their decile number as ΔTODEC pre i,t . ΔTODEC post i,t and TODEC issue i,t are defined similarly. ΔILLIQDEC pre i,t , ΔILLIQDEC post i,t and ILLIQDEC issue i,t are deciles for the Amihud illiquidity measure. The sample includes all SEOs from 1980 to 2009 that survive our select criteria and firms that are in the same size deciles and B/M deciles as the SEO firms and do not issue any equities in the six-year windows centered at the matched issue dates. If a firm is selected as the matching firm for multiple SEOs issued in the same calendar year and therefore has multiple highly correlated copies of each variable, we take average over the copies of each variable on the firm in this year to obtain one data point per firm per year.T-statistics with White corrections are reported in parentheses. Panel A: Δβpre Continued on Next Page... TRADING ACTIVITY AND BETA 97 Table 3.5 – Continued Intercept -0.049(-2.88) -0.477(-13.32) 0.058(1.37) -0.077(-4.18) -0.758(-10.24) Iseo 0.188(3.6) -0.018(-0.33) 0.158(2.92) 0.107(1.98) -0.050(-0.89) ΔTODECpre 0.045(14.61) 0.048(14.64) TODEC 0.003(0.9) 0.005(1.44) ΔIllIQDECpre -0.008(-2.65) 0.014(4.33) ILLIQDEC -0.004(-1.23) 0.008(2.39) rpre 0.041(3.89) 0.019(1.6) rpre·Iseo 0.030(1.79) 0.043(2.47) Adj. R 2 0.001 0.015 0.001 0.003 0.017 Panel B: Δβpost Intercept -0.025(-1.44) -0.308(-6.69) 0.020(0.43) 0.013(0.68) -0.025(-1.33) -0.393(-4.12) Iseo -0.260(-4.92) -0.171(-3.12) -0.269(-4.94) -0.215(-3.93) -0.246(-4.42) -0.151(-2.54) ΔTODECpost 0.039(13.64) 0.044(12.72) TODEC -0.010(-3.46) -0.012(-3.05) ΔIllIQDECpost -0.002(-0.83) 0.016(4.65) ILLIQDEC -0.002(-0.76) -0.006(-1.7) rpre -0.056(-5.15) -0.039(-3.34) rpre·Iseo 0.022(1.25) 0.014(0.78) Investmentpost -0.022(-2.69) -0.019(-2.33) Adj. R 2 0.001 0.014 0.001 0.003 0.002 0.016 TRADING ACTIVITY AND BETA 98 Table 3.6 Determinants of Changes in Market Betas (Fama-Macbeth Regressions) Panel A presents the coefficients of the Fama-Macbeth regressions based on the following model, Δβ pre i,t =a +b·I seo i,t +c· ΔTODEC pre i,t +d·TODEC issue i,t +e· ΔILLIQDEC pre i,t +f·ILLIQDEC issue i,t +g·r pre i,t +h·r pre i,t ·I seo i,t + i,t , and Panel B presents the coefficients of the Fama-Macbeth regressions based on the following model, Δβ post i,t =a +b·I seo i,t +c· ΔTODEC post i,t +d·TODEC issue i,t +e· ΔILLIQDEC post i,t +f·ILLIQDEC issue i,t +g·r pre i,t +h·r pre i,t ·I seo i,t + +i·INV post i,t + i,t , whereβ post i,t andβ pre i,t represent post-SEO and pre-SEO changes in market beta of firm i in year t; indicator variableI seo i,t is equal to 1 if firm i issues a SEO in year t and 0 otherwise; r pre i,t is the three-year cumulative return before the issue date; INV post i,t is the three-year corporate investment following the issue date. Also included as independent variables are deciles of the change and level of turnover and illiquidity. Pre-SEO (post-SEO) changes in turnover and liquidity are the differences between the levels measured from the one-year windows surrounding issue dates and the one-year windows three years prior to (following) issuance. In each year t, we sort firms into deciles based on their changes in turnover before issue dates and denote their decile number as ΔTODEC pre i,t . ΔTODEC post i,t and TODEC issue i,t are defined similarly. ΔILLIQDEC pre i,t , ΔILLIQDEC post i,t and ILLIQDEC issue i,t are deciles for the Amihud illiquidity measure. The sample includes all SEOs from 1980 to 2009 that survive our select criteria and firms that are in the same size deciles and B/M deciles as the SEO firms and do not issue any equities in the six-year windows centered at the matched issue dates. If a firm is selected as the matching firm for multiple SEOs issued in the same calendar year and therefore has multiple highly correlated copies of each variable, we take average over the copies of each variable on the firm in this year to obtain one data point per firm per year. T-statistics with White corrections are reported in parentheses. Panel A: Δβpre Continued on Next Page... TRADING ACTIVITY AND BETA 99 Table 3.6 – Continued Intercept -0.075(-0.98) -0.472(-5.58) 0.105(1.46) -0.12(-1.49) -0.665(-5.03) Iseo 0.155(1.9) -0.044(-0.66) 0.094(1.1) 0.098(1.23) -0.055(-0.76) ΔTOpre 0.043(8.45) 0.044(8.67) TO 0.001(0.19) 0.001(0.08) ΔIllIQpre -0.01(-2.24) 0.015(4.38) ILLIQ 0.002(0.26) -0.009(-1.47) rpre 0.079(3.57) 0.047(2.75) rpre·Iseo -0.015(-0.39) 0.008(0.24) Adj. R 2 0.006(4.33) 0.036(6.05) 0.023(5.39) 0.021(4.46) 0.063(6.1) Panel B: Δβpost Intercept 0.012(0.15) -0.206(-2.03) 0.124(1.77) 0.045(0.55) 0.013(0.16) -0.083(-0.53) Iseo -0.23(-2.38) -0.151(-1.75) -0.224(-2.54) -0.189(-1.81) -0.2(-2.21) -0.125(-1.31) ΔTOpost 0.034(6.22) 0.032(5.37) TO -0.012(-1.88) -0.017(-1.74) ΔIllIQpost -0.005(-1.01) 0.011(2.08) ILLIQ -0.007(-1.3) -0.015(-1.67) rpre -0.055(-2.72) -0.03(-2.02) rpre·Iseo 0.061(1.03) 0.061(1.1) Investmentpost -0.064(-1.13) -0.031(-0.57) Adj. R 2 0.005(4.27) 0.031(7.58) 0.019(6.04) 0.017(5.5) 0.009(6.52) 0.057(9.83) THREE ESSAYS IN DERIVATIVES, TRADING AND LIQUIDITY 100 References Amihud, Y. (2002, January). Illiquidity and stock returns: cross-section and time-series effects. Journal of Financial Markets, 5(1), 31-56. Retrieved from http://ideas.repec.org/a/eee/finmar/v5y2002i1p31-56.html Andrews, D. W. K. (1991, May). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3), 817-58. 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Abstract (if available)

Abstract The work in Chapter 1 shows that hedging by option writers has a large and significant destabilizing effect on the stock market. We demonstrate that weekly return reversals are significantly stronger surrounding option expiration days. Our evidence suggests that the hedging pressure that drives weekly reversals mainly comes from index options rather than individual stock options. We find in addition that index option hedging appears to have an impact on the aggregate market, and that the strength this aggregate impact is highly related to the degree of cross-sectional reversal. We also find that index option prices tend to be high before option expiration, suggesting that option hedgers are attempting to unwind written positions that might be difficult to hedge due to price impact on the underlying stocks. Collectively, the evidence we present strongly supports the conclusion that option trading causes significant price displacements in stocks and in the market as a whole. ❧ Chapter 2 investigates the relationship between the slope of the implied volatility (IV) term structure and future option returns. A strategy that buys straddles with high IV slopes and short sells straddles with low IV slopes returns seven percent per month, with an annualized Sharpe ratio just less than two. Surprisingly, we find no relation between IV slopes and the returns on longer-term straddles, even though the correlation between the returns on portfolios of short-term and long-term straddles generally exceeds 0.9. Our evidence suggests that the return predictability we document is unrelated to systematic risk premia. We believe that our results point to two possible explanations. One is that temporary hedging pressure pushes option prices away from efficient levels. The other is that short-term options are more likely to be mispriced by noise traders than long-term options. ❧ Chapter 3 shows that the positive correlation between stock-level trading activity and market betas remains strong even using the Dimson (1979} method to correct for non-synchronous trading. This finding suggests that controlling for non-synchronous trading alone does not provide unbiased inferences regarding the effects of events on market betas. Instead, it is necessary to control for changes in trading activity explicitly. We show that controlling for trading activity significantly changes the estimated impact of seasoned equity offerings and share repurchases on market betas.

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Creator Wang, Tong (author)

Core Title Three essays in derivatives, trading and liquidity

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Publication Date 08/07/2013

Defense Date 05/11/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag liquidity,OAI-PMH Harvest,options,real options,seasoned equity offerings,term structure,trading

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Jones, Christopher S. (committee chair), Ogneva, Maria (committee member), Solomon, David (committee member), Stathopoulos, Andreas (committee member), Zapatero, Fernando (committee member)

Creator Email wangt657@gmail.com,wangtong@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-320144

Unique identifier UC11294657

Identifier etd-WangTong-1991.pdf (filename),usctheses-c3-320144 (legacy record id)

Legacy Identifier etd-WangTong-1991.pdf

Dmrecord 320144

Document Type Dissertation

Format application/pdf (imt)

Rights Wang, Tong

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA