A Strange Disposition? Capital Gains Overhang in the Options Market

Transcription

1 A Strange Disposition? Capital Gains Overhang in the Options Market Kelley Bergsma Andy Fodor Emily Tedford September 2017 Abstract In the individual equity options market, we document a linear disposition effect with a deviation in this pattern for extreme gains and losses. We adapt the methodology of Grinblatt and Han [2005. Prospect theory, mental accounting, and momentum. Journal of Financial Economics 78, ] to generate option capital gain overhang. As option capital gains overhang increases, delta-hedged option returns rise and option sell-buy imbalance increases, except for options with extremely high or low options capital gains overhang. These findings reconcile prior studies reporting a strictly linear or a V- shaped disposition effect by offering a constructive middle ground in a novel market setting. Kelley Bergsma is an Assistant Professor in Finance at the College of Business, Ohio University, Athens, OH. Andy Fodor is the Leona Hughes Associate Professor in Finance at the College of Business, Ohio University, Athens, OH. Emily Tedford is an Analyst at in Cincinnati, OH. For the curious reader, a strange disposition refers to Shakespeare s MacBeth (Act 3, Scene 4) and Hamlet (Act 1, Scene 5). We appreciate helpful comments from Diego Amaya (discussant), Sinan Gokkaya, Danling Jiang, Xi Liu, Thomas Maurer, Andrea Rossi, Jitendra Tayal, and Lee Wakeman and seminar participants at the 2017 Midwest Finance Association and Ohio University.

2 I. Introduction Created by Grinblatt and Han (2005), capital gains overhang is the first measure to comprehensively quantify the disposition effect (Shefrin and Statman 1985; Odean 1998) in the aggregate market using stock prices and turnover. Grinblatt and Han (2005) find a linear relationship between capital gains overhang and stock returns. An (2016) modifies the original capital gains overhang variable to isolate gains from losses and her results suggest a V-shaped disposition effect, rather than a linear disposition effect, in stock returns. Furthermore, using data from a large retail brokerage, Hartzmark (2015) documents a new rank effect whereby investors sell top and bottom ranked securities in their portfolios. Based on this evidence, we suggest a middle ground between a strictly linear and a V-shaped relationship between capital gains overhang and returns: A linear disposition effect for all securities except those with extreme gains or extreme losses. In this paper, we explore the relationship between capital gains overhang and security returns in a new setting. We document that returns monotonically increase as capital gains overhang increases, except for securities with extreme gains or losses. Our findings are in a novel setting the individual equity options market. Unlike stocks that can be held for only a few days or up to many years, options have finite lives. Therefore, since an option trader s holding period is more defined, a capital gains overhang variable generated from past option prices and turnover is perhaps a more relevant proxy for unrealized gains or losses. Using OptionsMetrics data from 1996 to 2015, we generate an options capital gains overhang variable using the same methodology as Grinblatt and Han (2005) except we use option inputs over a 20-day holding period. For both calls and puts, we show a linear relationship between option capital gains overhang (OCGO) and delta-hedged option returns (Cao and Han 2013), 1 with a deviation from this pattern for options in top and bottom OCGO deciles. 1 We examine delta-hedged returns in order to remove the effect of underlying stock price movements. 2

3 An obvious concern is that OCGO is related to moneyness or other option and even underlying stock characteristics. Therefore, we present not only portfolio sorts, but also daily Fama- MacBeth regressions that control for these characteristics. After controlling for option moneyness, implied volatility and bid-ask spread as well as underlying stock market capitalization, B/M, prior month s return, momentum, and turnover, the direct relation between OCGO and delta-hedged option returns remains strong for all but the extreme OCGO deciles. This general linear relationship holds in both January and non-january months (Grinblatt and Han 2005). The evidence suggests that capital gains overhang is a relevant measure to quantify the disposition effect in the options market, and Grinblatt and Han s (2005) established linear relationship between returns and capital gains overhang holds for all but securities with extreme gains or losses. Thus, our findings contribute to both general disposition effect literature (Shefrin and Statman 1985; Odean 1998; Ben-David and Hirshleifer 2012; An 2016) as well as recent research specifically on the disposition effect in derivatives. A few studies document a traditional disposition effect in derivatives, including executive stock option grants, futures, and bank-issued warrants (Heath et al., 1999; Frino et al., 2004; Coval and Shumway, 2005; Choe and Eom, 2009; Schmitz and Weber, 2012). Poteshman and Serbin (2003) find a disposition effect in the early exercise of exchange-traded options; however, they do not consider option trading or pricing. Our study is the first to our knowledge to adapt Grinblatt and Han s (2005) capital gains overhang methodology to individual equity options in order to explore three potential shapes of the disposition effect: Linear, V-shaped, or Modified Linear with deviation in extreme deciles. Our results indicate the third shape best fits the data, offering a compromise between Grinblatt and Han (2005) and An (2016). In addition to exploring the disposition effect s shape, we investigate the role of volatility in this effect. Prior evidence on volatility s role is mixed. Kumar (2009) finds investors exhibit a stronger disposition effect when the Chicago Board Options Exchange volatility index (VIX) is higher. Yet, 3

4 Calvet et al. (2009) find an inverse relationship between household portfolio standard deviation and households tendency to exhibit the disposition effect. Kumar (2009) and Han and Kumar (2013) report that traders are more prone to a disposition effect when idiosyncratic volatility is greater. However, Henderson s (2012) model generates a lower probability of selling winners when volatility increases. We test whether aggregate volatility and security-specific volatility lead to a stronger or weaker disposition effect in options markets using the VIX and option implied volatility, respectively. The evidence suggests that the disposition effect is stronger when the VIX is high, supporting Kumar (2009), but is weaker when implied volatility is greater, which is consistent with Henderson (2012). Last, we examine International Securities Exchange (ISE) option signed volume by trader class in order to determine whether retail option traders demonstrate a greater disposition effect as compared with institutional option traders. Shapira and Venezia (2001), Feng and Seasholes (2005), Locke and Mann (2005), Dhar and Zhu (2006), Calvet et al. (2009), and Hur et al. (2010) find an attenuated disposition effect among professional traders 2 and a more robust effect among less sophisticated traders. In the spirit of Barber and Odean (2008), we use ISE data to calculate the sellbuy imbalance as close sell volume-open buy volume and de-trend the series by subtracting the moving average over the past 20 trading days. We construct sell-buy imbalance for customer and firm traders to capture retail and institutional investors propensity to sell, respectively. Our results show a strong relationship between sell-buy imbalance and OCGO for customers, not for firms. This finding confirms prior work that retail investors are more prone to trade based on unrealized capital gains or losses. 2 In options markets, professional traders who combine options in a trading strategy, such as a straddle, would likely be less concerned about unrealized gains or losses on a specific option. 4

5 The rest of the paper is organized as follows. Section II describes the motivation and hypotheses. Section III outlines our data and methodology. Section IV discusses our empirical results. Section V concludes. II. Motivation and Hypotheses The disposition effect refers to the phenomenon whereby investors sell winners too soon and hold losers too long (Shefrin and Statman, 1985; Odean, 1998). Prior literature explains the disposition effect as arising from prospect theory and realization utility. Investors show diminishing sensitivity through risk-averse behavior in the gain domain and risk-seeking behavior in the loss domain (Kahneman and Tversky, 1979; Li and Yang, 2013). Traders also view investing as a series of good or bad episodes depending on whether they sell at a gain or a loss (Barberis and Xiong, 2009, 2012). Numerous studies have documented a disposition effect in stock trading and returns, yet recent evidence has uncovered a disposition effect in derivatives. Shapira and Venezia (2001), Coval and Shumway (2005), and Choe and Eom (2009) identify a disposition effect in futures trading in markets across the globe. In contrast, Liu et al. (2010) report a reverse disposition effect in the Taiwanese futures market. Frino et al. (2004) and Locke and Mann (2005) find that professional floor futures traders exhibit a disposition effect in their trading behavior, but these investors hold losing positions only when it is rational to do so. In other derivative markets, Schmitz and Weber (2012) report a strong disposition effect for call and put bank-issued warrants among investors at a large German discount broker. Heath et al. (1999) document that executives exercise stock options in response to a stock reaching its 52-week high. Poteshman and Serbin (2003) show that option traders irrationally exercise call options early after the stock reaches its yearly high or has exhibited high returns over the past few weeks or months. Chiang et al. (2016) find a disposition 5

6 effect in index option returns using moneyness-based propensity to sell and adjusted capital gains overhang measures. Relative to the stock market, the individual equity options market represents an interesting setting to test for disposition effect for three main reasons. First, stockholders will not lose the entire value of their investment unless the firm becomes insolvent, whereas option traders face the real possibility of options expiring worthless. Second, stocks have a theoretically infinite life, while options have finite lives. Due to options finite lives, traders are forced to realize gains or losses at expiration if positions are not closed prior, i.e. an option expiring worthless is equivalent to a sale at a price of zero. Third, individual equity options have the unique feature in which one can control for both option and underlying stock characteristics, while also removing the effect of the underlying stock returns using delta-hedged option returns (Cao and Han 2013). Given prior literature finds a disposition effect in derivatives, we expect a disposition effect in individual equity options as well. We quantify the disposition effect in call and put options using option capital gains overhang, calculated using a modification Grinblatt and Han s (2005) methodology as described in Section III. However, the shape of the disposition effect is an open research question. It could be a linear relationship (Grinblatt and Han 2005), a V-shape, or a modified linear combination where deviation from a linear relation is evident in the extremes. Our three competing hypotheses describe the three possible disposition effect shapes. Our first hypothesis is that the relationship between option capital gains overhang and deltahedged option returns will be similar to the positive linear relationship between stock capital gains overhang and stock returns documented in Grinblatt and Han (2005). This notion is formalized in our Hypothesis 1 below. Hypothesis 1: As option capital gains overhang increases, delta-hedged option returns will increase. In other words, there exists a linear relationship between option capital gains overhang and delta-hedged option returns. 6

7 However, recent research has cast doubt on the generalization that investors always hold losers too long. Ingersoll and Jin (2013) derive a model of realization utility in a dynamic setting with reinvestment that generates voluntarily realized losses. 3 In empirical studies, Calvet et al. (2009) document that Swedish households are more likely to sell extreme winners and extreme losers. Ben- David and Hirshleifer (2012) and An (2016) find a V-shaped disposition effect in which propensity to sell increases with the absolute magnitude of gains and losses. Choi et al. (2010) find a U-shaped disposition effect for stocks around earnings announcements and when abnormal volume is high. 4 These studies suggest a V-shaped disposition effect rather than a linear relationship implied by Grinblatt and Han (2005). This possibility is outlined in Hypothesis 2. Hypothesis 2: Delta-hedged option returns will increase with the magnitude of gain or loss. In other words, there exists a V-shaped relationship between option capital gains overhang and delta-hedged option returns. There exists, however, a third possibility: A linear relationship between capital gains overhang and returns except for extreme gains or losses. The model of Henderson (2012) demonstrates that investors will only liquidate large losses that are very far below breakeven, as investors give up on a losing position when the asset has a low Sharpe ratio. Using data from a large retail brokerage, Hartzmark (2015) uncovers a rank effect in which investors are more likely to sell securities with the highest gain and greatest loss in their portfolio as compared with all other securities in their portfolio. Without account level data, a proxy for the top and bottom ranked securities in a portfolio is the highest and lowest capital gains overhang deciles. Thus, we expect the disposition effect to be different in extreme deciles because selling behavior is different for securities with large losses or gains. 3 In Ingersoll and Jin s (2013) model, investors optimize their utility by realizing losses because subsequent reinvestment resets reference prices, making realizing a future gain more likely. 4 Chang et al. (2016) find that investors at a large discount broker exhibit a classic disposition effect for non-delegated assets, but a reverse disposition effect for delegated assets, except for preferred stock and options. The discount broker dataset does not distinguish between call and put options. 7

8 This prediction is distinct from a V-shaped net selling schedule (An 2016) in that we fully expect for options moderate losses through moderate gains a linear relationship between capital gains overhang and returns. However, for options with the most extreme losses or gains, the relationship is likely to change it could be direct, inverse, or flat as investors propensity to sell changes for top and bottom ranked securities in their portfolios as compared with other securities. This idea is captured in Hypothesis 3. Hypothesis 3: There exists a linear relationship between option capital gains overhang and delta-hedged option returns, except for options with extreme gains or losses. In addition to identifying the shape of the disposition effect in options markets, we have two other secondary research questions. First, we aim to investigate how volatility impacts the disposition effect. Prior literature is not in unanimous agreement on the role of volatility in the disposition effect. Using individual investor trading data, Kumar (2009) finds investors exhibit a more pronounced disposition effect when the Chicago Board Options Exchange volatility index (VIX) is higher, as behavioral biases are magnified as uncertainty increases. In addition to examining aggregate volatility, Kumar (2009) and Han and Kumar (2013) consider idiosyncratic volatility and document that individual investors are more prone to a disposition effect when idiosyncratic volatility is high. 5 An (2016) finds that high idiosyncratic volatility stocks have a stronger V-shaped disposition effect as compared with that of low idiosyncratic volatility stocks. Yet, Calvet et al. (2009) show that households with portfolios having higher standard deviations are less likely to sell winners and more likely to sell losers. Henderson s (2012) model generates a lower probability of selling winners when volatility increases. 6 Therefore, our goal to determine whether aggregate volatility and security-specific volatility 5 Stocks with greater idiosyncratic volatility are both more difficult to value (Kumar 2009) and more lottery-like (Han and Kumar 2013). 6 In addition, Chiyachantana and Yang (2013) find high idiosyncratic volatility stocks are associated with a weaker disposition effect. 8

9 lead to a stronger or weaker disposition effect. The most relevant measures of aggregate volatility and security-specific volatility for options markets are the VIX and option implied volatility, respectively. Our last objective is to determine whether retail option traders demonstrate a greater disposition effect as compared with institutional option traders. Feng and Seasholes (2005) find investors with less experience and trading sophistication are more likely to exhibit behavior consistent with the disposition effect, and Dhar and Zhu (2006) and Calvet et al. (2009) document that wealthier individuals and households show a reduced disposition effect. Shapira and Venezia (2001) report a disposition effect among both professional and individual investors, although the effect is much stronger among retail investors. Singal and Xu (2011) find that only 30% of professionally managed mutual funds exhibit the disposition effect. Moreover, Barberis and Xiong (2012) assert that institutional investors are more likely to view securities in terms of their contribution to overall portfolio returns rather than as individual investing episodes. This notion is particularly relevant to professional traders who frequently combine options in a trading strategy, such as a straddle. As such, we expect the disposition effect in option markets to be stronger for less sophisticated investors. The next section describes the data and methodology used to test these hypotheses and research questions. III. Data and Methodology Our main sample is comprised of daily individual option data from OptionsMetrics from 1996 through We include only options with between 14 to 60 days to expiration (Bakshi and Kapadia 2003) and we exclude options that have at the outset less than 90 days or more than 365 days to expiration. To obtain underlying stock characteristics, we merge the options data with the Center for Research in Security Prices (CRSP) and Compustat files. We exclude firms with stock prices less than $2, following An (2016), and with market capitalization less than $2 million. Our main dependent 9

10 variable is delta-hedged option returns, D-H Ret, calculated as delta-hedged dollar return on day t+1 scaled by option price on day t. Our proxy for option capital gains overhang is derived from Equation (11) on p. 320 of Grinblatt and Han (2005), OCGO = P t 2 R t 1 P t 2 (2) where reference price, R t-1, is equal to the weighted average option price over the past 20 trading days. Specifically, R t is calculated as in Equation (9) on p. 319 of Grinblatt and Han (2005): 20 n 1 R t = (V t n [1 V t n+τ ]) P t n n=1 τ=1 (3) where V is option turnover and P is option price. Option turnover is used for weighting to better reflect average underlying asset price when long option positions were opened, where option turnover is daily volume divided by open interest. The weights multiplied by each daily stock price are scaled to sum to one. If an option has traded for less than 20 prior days, we use all available data. We winsorize OCGO for calls and puts separately at the 1% and 99% levels in accordance with An (2016). In order to test our three competing hypotheses, we conduct portfolio sorts and daily Fama- MacBeth regressions. We regress delta-hedged option returns on OCGO and control variables to test Hypothesis 1. To explore whether a V-shaped relationship (Hypothesis 2) exists between option capital gains overhang and delta-hedged option returns, we define Gain as equal to OCGO if OCGO is greater than zero and as equal to zero otherwise. Loss is equal to OCGO if OCGO is less than zero and zero otherwise. 7 Hypothesis 3 predicts a linear relationship between OCGO and D-HRet holds except for the extreme OCGO deciles. To test this hypothesis, we construct XLoss and XGain 7 Choi et al. (2010) define CGOHigh as MAX(Normalized CGO 0.5, 0) and CGOLow as MIN(Normalized CGO 0.5, 0) to capture the U-shaped of capital gains overhang in the stock market. An (2016) uses a different method than Choi et al. (2010) to generate Loss Overhang and Gain Overhang and also creates V-shaped Net Selling Propensity (VNSP t) = Gain t 0.23Loss t. 10

11 variables. XLoss as equal to OCGO if OCGO is in the lowest decile that day and zero otherwise. XGain is equal to OCGO if OCGO is in the highest decile that day and as equal to zero otherwise. In order to examine the role of volatility in the disposition effect in the options market, we consider both aggregate volatility and individual option implied volatility. Similar to Kim et al. (2017) who split their sample into high and low Investor Sentiment months, we split our sample into high and low VIX days. A day is classified as High VIX if VIX is above the median of the entire sample that day; otherwise it is classified as Low VIX. We also investigate the impact of option implied volatility on the disposition effect. Implied volatility, ImpVol, is from day t-1 where day 0 is the date of dependent variable observation. We interact ImpVol with our main capital gains overhang variables. Our control variables are as follows. We control for option implied volatility, bid-ask spread, time to expiration, and moneyness. BidAsk is the bid-ask spread scaled by option price from day t-1. Time to expiration in days is denoted as DTE. K/S is the option strike price divided by the underlying stock price and captures option moneyness. Following An (2016), we also control for firm size, stock book-to-market, momentum, and turnover. Size is the natural logarithm of the market capitalization at the end of the prior month. Stock book-to-market (B/M) is defined as in Fama and French (1992). Momentum is the cumulative monthly stock return for months t 12 to t 2. Turnover is stock turnover of the prior month. Table 1 presents summary statistics. For comparison purpose, we report stock capital gains overhang (SCGO) calculated using Equations (1) and (2) except stock price and turnover is used in place of option price and turnover. We also break down the data by OCGO decile and present summary statistics for each decile as well as for the entire sample. Panel A summarizes calls, while Panel B summarizes puts. <<Table 1>> 11

12 For our last test involving option trading and the disposition effect, we merge our sample with the International Securities Exchange (ISE) Open/Close Trade profile dataset, available from May 2005 to December The ISE dataset provides daily open buy, close buy, open sell, and close sell volume by trader class. We solely consider open buy and close sell option volume to focus on trading in long option positions. The two main trader classes are firm and customer. Observations with missing or zero trading volume or open interest are excluded. Options with fewer than 10 non-zero close sell or open buy trades in a month are dropped. Our key measure constructed from ISE volume is sell-buy imbalance, SB. We argue that sell-buy imbalance captures traders propensity to sell. In the spirit of Barber and Odean (2008), we define SB as SB = Close Sell Open Buy Close Sell + Open Buy (4) where Close Sell refers to close sell volume and Open Buy refers to open buy volume. Volume is the sum of customer and firm volume from ISE. Sell-buy imbalance captures traders propensity to sell relative to propensity to buy. We create customer sell-buy imbalance (CSB) and firm sell-buy imbalance (FSB) using the trader class information provided by ISE. We de-trend all sell-buy imbalances by their respective 20-day moving averages. Our focus is whether customer sell-buy imbalance exhibits a more pronounced disposition effect as compared with firm sell-buy imbalance. IV. Results A. Portfolio Sorts In Table 2, we report the average delta-hedged return for decile portfolios sorted each day by option capital gains overhang (OCGO). Options are divided based on negative or positive OCGO and further into loss quintiles or gain quintiles by OCGO magnitude. For the remainder of the paper, loss quintiles will be referred to as OCGO deciles 1-5 and gain quintiles will be referred to as OCGO deciles 12

13 6-10. Delta-hedged option returns are on average negative, yet these return become less negative as OCGO increases. The direct relationship between delta-hedged returns and OCGO is monotonic with the exception of the highest decile. Specifically, there is a robust difference in call returns between Deciles 10-1 of 8.79% (t = 24.62) and the put return Deciles 10-1 difference is 4.71% (t = 12.92). This evidence is more consistent with Hypotheses 1 and 3, but is inconsistent with Hypothesis 2. << Table 2 >> Nevertheless, we test whether delta-hedged options manifest a V-shaped disposition effect (Hypothesis 2) by calculating the Decile 10-6 (Large gain Small gain) and Decile 5-1 (Small loss Large loss) differences. If the V-shape holds, we would expect the Decile 10-6 difference to be positive and the Decile 5-1 difference to be negative. We find the opposite results: The Decile 10-6 differences are negative and Decile 5-1 difference are positive for both calls and puts. Thus, delta-hedged option returns do not follow a V-shaped pattern as capital gains overhang increases. In Table A.1 in the Appendix, we repeat our portfolio sorts using stock capital gains overhang (SCGO) and find qualitatively similar results. 8 Hypotheses 1 and 3 predict both predict a linear relationship between OCGO and delta-hedged option returns for most options, but Hypothesis 3 predicts a deviation from this pattern for options with extreme gains and losses. As described in Section III, we classify options in the top and bottom OCGO deciles as having extreme gains and losses, respectively. Therefore, we examine the differences Deciles 9-2. The Deciles 9-2 differences is 4.63% (t = 26.86) for calls and 3.35% (t = 20.98) for puts. Although the Decile 10-1 differences are larger in economic magnitude than those of Decile 9-2, the 8 The interpretation for the Table A.1 results is the same for calls as in Table 2, but different for puts. Puts gain value as stocks have larger losses, so the Decile 10-1 difference is the difference between a large loss and large gain from a put trader s perspective. Thus, it is to be expected that the Decile 10-1 and Decile 9-2 differences are negative and significant for puts, whereas both differences are positive and significant for calls. 13

14 Decile 9-2 differences are greater in statistical significance as the standard errors of Decile 1 are nearly twice as large as the other deciles. Taken together, the decile portfolio results support the notion that Grinblatt and Han s (2005) results hold in the option market: Option capital gains overhang (OCGO) has a generally linear relationship with delta-hedged option returns. There is some evidence that options in the top and bottom OCGO deciles exhibit a slightly different pattern or more variation. The deviation in the extreme deciles may be related to the rank effect (Hartzmark 2015) whereby top and bottom ranked securities in a portfolio are more likely to be sold. B. Main Results In Table 3, we formally test OCGO s relationship with delta-hedged option returns using daily Fama-MacBeth regressions. First, we regress delta-hedged option return (DHRet) on OCGO and option and stock characteristic control variables as described in Section 3. This analysis tests Hypothesis 1. Next, we examine variations of OCGO to test Hypotheses 2 and 3. Table 3 presents the results. 9 In model (1), the OCGO coefficient is 2.46% (t = 5.74), indicating a strong linear relationship between delta-hedged call returns and OCGO. The OCGO coefficient is 1.21% (t = 3.27) for put returns in model (4), suggesting an analogous relationship for puts. These results provide strong evidence that the findings of Grinblatt and Han (2005) extend to individual option returns. << Table 3 >> Next, we explore whether a V-shaped relationship between capital gains overhang and deltahedged option return patterns (Hypothesis 2) is more consistent with the data than a linear pattern. To investigate Hypothesis 2, we replace OCGO with Loss and Gain in models (2) and (5). If a V-shaped 9 Regression results without control variables are presented in Table A.2 of the Appendix. 14

15 disposition effect holds, then the Loss coefficient should be negative and Gain coefficient should be positive. The evidence demonstrates instead the Loss coefficients are positive and Gain coefficient are negative for call and put delta-hedged returns. Thus, there is no evidence that capital gains overhang has a V-shaped relationship with delta-hedged option returns. 10 We next consider whether OCGO has a different impact on option returns if options are in the top and bottom OCGO deciles, indicating a modified linear OCGO relationship with returns. To explore this question, delta-hedged option returns are regressed on OCGO, XLoss, and XGain and control variables in model (3) for calls and model (6) for puts. The OCGO coefficients remain positive and are greater in statistical significance as compared with models (1) and (4). This evidence supports Grinblatt and Han s (2005) linear disposition effect but demonstrates that the linear relationship is stronger when modified to allow extreme OCGO deciles deviation. The XLoss coefficient is positive and significant for calls, but insignificant for puts, suggesting a traditional disposition effect among calls with extreme losses and no disposition effect among puts with extreme losses. In contrast, both models (3) and (6) show negative XGain coefficients. XGain represents the marginal impact of OCGO when an option is in the top OCGO decile, after controlling for the main capital gains overhang effect. 11 Thus, for options with extreme gains, the linear relationship between OCGO and delta-hedged options returns is significantly weaker, supporting Hypothesis 3. The negative XGain coefficient could mean that the selling pressure for options with the largest paper gains could temporarily depress security returns, especially over a very short time horizon (i.e. one day). Hartzmark (2015) finds a higher propensity to sell within the top and bottom 20 stocks, but the greatest selling pressure for the 10 If nothing else, there is some evidence of a reverse V-shaped disposition effect, but models (3) and (6) show that the negative coefficient on Gain is likely driven by the pattern in the top OCGO decile. 11 Untabulated results demonstrate that if OCGO is not included in the regressions, the XGain coefficient is insignificant for calls and nearly half the economic and statistical significance for puts. 15

16 top and bottom two securities in a portfolio. 12 In other words, the rank effect (Hartzmark 2015) could lead to a modified linear disposition effect with different patterns in the extremes. For completeness, the coefficients on the control variables are consistent with prior literature. K/S enters negatively for calls and positively for puts, demonstrating that call options are less valuable and put options more valuable as strike prices rise relative to a fixed stock price. The ImpVol coefficient is negative in sign and the LRet coefficients are positive for calls and negative for puts, consistent with Cao and Han (2013). 13 Stock turnover has a positive and significant relationship with delta-hedged option returns. Also, for robustness, we report in the Appendix that the OCGO coefficient remains positive and significant while controlling for stock capital gain overhang (SCGO), indicating that OCGO has additional predictability beyond that of SCGO. Prior work on the disposition effect acknowledges the impact of tax-loss motivated selling in December (Odean 1998; Grinblatt and Han 2005; An 2016). Following An (2016), we re-estimate our regression specifications in models (3) and (6) of Panel A for January only and February-December. Panel B presents the results. The disposition effect in options is most robust in non-january months, indicating our main results are not driven by a tax loss selling effect. In sum, there is a linear relationship between option capital gains overhang and delta-hedged returns, yet this effect is significantly weaker for extreme OCGO deciles. This modified linear relationship (Hypothesis 3) reconciles the linear relationship found in Grinblatt and Han (2005) with the V-shaped disposition effect found in An (2016). D. Volatility 12 This reasoning could also explain the positive XLoss coefficient for calls as selling pressure could temporarily depress securities with the most extreme losses. 13 Cao and Han (2013) report the impact of momentum on monthly delta-hedged call option returns and find past one month, one year, and three year momentum have a positive impact on call returns. Since our delta-hedged option returns are daily rather than monthly, it is no surprise that the momentum is more short-lived. 16

17 In addition to establishing the shape of the disposition effect in option markets, we also examine the role of volatility in the disposition effect. First, we consider aggregate volatility as proxied by the VIX. The sample is split into High VIX and Low VIX days to determine whether the relationship between OCGO and delta-hedged option returns is stronger when the VIX is higher (Kumar 2009). Thus, we re-estimate our regression specifications in Table 3 Panel B for High VIX and Low VIX days. Table 4 Panel A reports the results. The OCGO coefficient is greater in economic magnitude during High VIX days as compared with Low VIX days. These results suggest greater aggregate market uncertainty is accompanied by a strong disposition effect in the options markets, mirroring the results of Kumar (2009) for the stock market. << Table 4 >> Next, we consider whether security-specific volatility magnifies or reduces disposition effect by examining the role of option implied volatility. Specifically, we interact implied volatility (ImpVol) with OCGO, XLoss, and XGain and add these interaction terms to our main specifications. ImpVol is already a control variable in Table 3, but we use interaction terms in order to quantify the incremental effect of implied volatility. Also, to ensure that implied volatility is not simply capturing aggregate volatility, we report the results for High and Low VIX periods separately. Table 4 Panel B reports the results. For calls, the OCGO ImpVol coefficient is negative and significant regardless of whether the VIX is high or low. For puts, the OCGO ImpVol coefficient is insignificant for High VIX days, but is significant and negative for Low VIX days. 14 The negative coefficient implies that for a given level of OCGO, an option with higher implied volatility will be associated with a weaker disposition effect in delta-hedged option returns. Our evidence fully supports 14 Comparing the coefficients on High and Low VIX days, implied volatility has a more negative marginal impact on OCGO during Low VIX days as opposed to High VIX days. This result suggests that implied volatility further reduces the positive relationship between OCGO and delta-hedged returns when aggregate uncertainty is lower. 17

18 Calvet et al. (2009) and Henderson (2012) who find an inverse relationship between the disposition effect and volatility. Yet, we acknowledge that implied volatility plays a different role in options market than that of idiosyncratic volatility in stock markets, which could explain why our results differ from Kumar (2009) and Han and Kumar (2013). Turning to the other interaction terms, XLoss ImpVol is negative and significant in 3 of 4 specifications. Thus, conditional on the lowest decile of OCGO and for a given level of OCGO, implied volatility has a positive effect on option returns. The XLoss coefficient itself becomes negative in sign in 3 of 4 specifications. This evidence underscores that the shape of the disposition effect changes in the extreme OCGO deciles (Hypothesis 3) so that even the relationship between implied volatility and the disposition effect changes. Overall, options with greater implied volatility have an attenuated relationship between capital gains overhang and returns. However, there is some evidence that this relation is stronger as implied volatility increases if options have extreme losses. In contrast, heightened aggregate volatility as proxied by the VIX is associated with a stronger disposition effect, consistent with Kumar (2009). E. Sell-buy Imbalance The direct relationship between option capital gains overhang and delta-hedged option returns is premised on investors reluctance to realize losses and readiness to realize gains. We test this assumption by creating option sell-buy imbalances using ISE daily option trading data. In the spirit of Barber and Odean (2008), we construct a customer sell-buy imbalance (CSB) and firm sell-buy imbalance (FSB) measures using de-trended option long position trading volume as described in Section III. Without de-trending, CSB and FSB range from -100 to 100 in percent. The sample period is from May 2005 through December 2015 based on data availability. 18

19 First, we form OCGO decile portfolios each day and plot the average customer and firm sellbuy imbalances in Figure 1A for calls and Figure 1B for puts. Customer sell-buy imbalance demonstrates a clear monotonic linear relationship with OCGO with the exception of the top and bottom deciles. In contrast, firm sell-buy imbalance exhibits a less stable, weaker pattern across OCGO deciles. This graphical illustration is consistent with recent research in the equity markets documenting less sophisticated investors exhibit a strong disposition effect (Feng and Seasholes 2005; Dhar and Zhu 2006; Hur et al. 2010). << Figure 1 >> In Table 5, we formally test the relationship between sell-buy imbalances and OCGO using daily Fama-MacBeth regressions. In models (1) and (3), we regress CSB on OCGO, XLoss, and XGain and option and stock characteristic control variables as described in Section 3. For both calls and puts, the OCGO coefficients are positive and highly significant in models (1) and (3), indicating a strong linear relationship between CSB and OCGO. The XLoss coefficients are negative, while the XGain coefficients are insignificant. This evidence suggests, relative to the general linear relationship between OCGO and sell-buy imbalance, there is a greater sell-buy imbalance as OCGO falls for extreme losses. In the remaining models (2) and (4), the OCGO, XLoss, and XGain coefficients are all insignificant when FSB is the dependent variable. In untabulated results, when we aggregate customer and firm traders volume to generate an aggregate sell-buy imbalance, this measure has a significant positive relationship with OCGO and negative relationship with XLoss. << Table 5 >> These results support the notion that less sophisticated option traders more strongly exhibit trading behavior consistent with the traditional disposition effect and are also more likely to close long positions on options with extreme losses. Given ISE data is only available for a shorter time horizon 19

20 and a subset of options, these results must be interpreted with caution, yet the evidence suggests a modified linear relationship between option capital gains overhang and customer sell-buy imbalance. V. Conclusion This paper reconciles the findings of a linear (Grinblatt and Han 2005) and a V-shaped disposition effect (An 2016) by documenting a linear disposition effect with deviation in extreme gains and losses. Our setting is the individual equity options market and we generate an options capital gains overhang variable (OCGO) using a similar methodology to Grinblatt and Han (2005). Since options have finite lives, we argue that capital gains overhang is particularly well-suited to capture unrealized gains in options. We find OCGO has a positive linear relationship with delta-hedged option returns, except for options in top and bottom OCGO deciles. This modified linear disposition effect shape is robust to controls for option and underlying stock characteristics and holds in both January and non- January months. Moreover, the paper sheds light on the role of volatility in the disposition effect. The linear disposition effect in the options market is stronger when the VIX is higher (Kumar 2009), but is generally weaker when implied volatility is greater, supporting Henderson (2012). Last, using signed option trading volume, we show that option sell-buy imbalance also exhibits a modified linear disposition effect, but only for customer traders and not firm traders. This evidence is consistent with the notion that the disposition effect is more robust among individual investors rather than professional traders (Feng and Seasholes 2005; Dhar and Zhu 2006; Hur et al. 2010). Our study illustrates that adapting a respected methodology (Grinblatt and Han 2005) from the stock market to the individual equity options market yields rich insights. This paper focuses on the disposition effect, but there are many other avenues for creatively repurposing methodologies from stock to option research. Future research using innovative adaption holds great promise. 20

21 References An, L. (2016) Asset pricing when traders sell extreme winners and losers, Review of Financial Studies 29, Bakshi, G. and Kapadia, N. (2003) Delta-Hedged Gains and the Negative Market Volatility Risk Premium, Review of Financial Studies 16, Barberis, N., and Xiong, W. (2009) What drives the disposition effect? An analysis of a long-standing preference-based explanation, Journal of Finance 64, Barberis, N., and Xiong, W. (2012) Realization utility, Journal of Financial Economics 104, Barber, B. and Odean, T. (2008) All that glitters: The effect of attention and news on the buying behavior of individual and institutional investors, Review of Financial Studies 21, Ben-David, I., and Hirshleifer, D.A. (2012) Are investors really reluctant to realize their losses? Trading responses to past returns and the disposition effect, Review of Financial Studies 25, Calvet, L.E., Campbell, J.Y. and Sodini, P. (2009) Fight or flight? Portfolio rebalancing by individual investors, Quarterly Journal of Economics 124, Cao, J. and Han, B. (2013) Cross section of option returns and idiosyncratic stock volatility, Journal of Financial Economics 108, Chang, T., Solomon, D. and Westerfield, M. (2016) Looking for someone to blame: Delegation, cognitive dissonance, and the disposition effect, Journal of Finance 71, Chiang, M.-H., Chiu, H.-Y. and Chou, R.K. (2016) Measuring the disposition effect on the option market: New evidence, unpublished working paper, National Chengchi University. Chiyachantana, C.N. and Yang, Z. (2013) Reference point adaptation and disposition effect, unpublished working paper, Singapore Management University. Choe, H., and Eom, Y. (2009) The disposition effect and investment performance in the futures market, Journal of Futures Markets 29, Choi, W., Hoyem, K. and Kim. J.-W. (2010) Capital gains overhang and the earnings announcement premium, Financial Analysts Journal 66, Coval, J., and Shumway, T. (2005) Do behavioral biases affect prices? Journal of Finance 60, Dhar, R. and Zhu, N. (2006) Up close and personal: Investor sophistication and the disposition effect, Management Science 52, Feng, L., and Seasholes, M. (2005) Do investor sophistication and trading experience eliminate behavioral biases in financial markets? Review of Finance 9, Frino, A., Johnstone, D., and Zheng, H. (2004) The propensity for local traders in futures markets to ride losses: Evidence of irrational or rational behavior? Journal of Banking and Finance 28, Grinblatt, M., and Han, B. (2005) Prospect theory, mental accounting, and momentum, Journal of Financial Economics 78, Han, B., and Kumar, A. (2013) Speculative retail trading and asset prices. Journal of Financial and Quantitative Analysis 48,

22 Hartzmark, S. (2015) The worst, the best, ignoring all the rest: The rank effect and trading behavior, Review of Financial Studies 28, Heath, C., Huddart, S., and Lang, M. (1999) Psychological factors and stock option exercise, Quarterly Journal of Economics 114, Henderson, V. (2012) Prospect theory, liquidation, and the disposition effect, Management Science 58, Hur, J., Pritamani, M. and Sharma, V. (2010) Momentum and the disposition effect: The role of individual investors, Financial Management 39, Ingersoll, J.E., and Jin, L.J. (2013) Realization utility with reference-dependent preferences, Review of Financial Studies 26, Kahneman, D., and Tversky, A. (1979) Prospect theory: An analysis of decision under risk, Econometrica 47, Kim, J.S., Kim, D-H., and Seo, S.W. (2017) Investor sentiment and return predictability of the option to stock volume ratio, Financial Management 46, Kumar, A. (2009) Hard-to-value stocks, behavioral biases, and informed trading, Journal of Financial and Quantitative Analysis 44, Li, Y., and Yang, L. (2013) Prospect theory, the disposition effect and asset prices, Journal of Financial Economics 107, Liu, Y-J., Tsai, C-L., Wang, M-C., and Zhu, N. (2010) Prior consequences and subsequent risk taking: New field evidence from the Taiwan futures exchange, Management Science 56, Locke, P.R., and Mann, S.C. (2005) Professional trader discipline and trade disposition, Journal of Financial Economics 76, Odean, T. (1998) Are investors reluctant to realize losses? Journal of Finance 53, Poteshman, A.M. and Serbin, V. (2003) Clearly irrational financial market behavior: Evidence from the early exercise of exchange traded stock options, Journal of Finance 58, Schmitz, P., and Weber, M. (2012) Buying and selling behavior of retail investors in option-like securities, DBW - Die Betriebswirtschaft 72, Shapira, Z. and Venezia, I. (2001) Patterns of behavior of professionally managed and independent investors, Journal of Banking and Finance 25, Shefrin, H., and Statman, M. (1985) The disposition to sell winners too early and ride losers too long: Theory and evidence, Journal of Finance 40, Singal, V. and Xu, Z. (2011) Selling winners, holding losers: Effect on fund flows and survival of disposition-prone mutual funds, Journal of Banking and Finance 35,

23 Figure 1: Customer and Firm Sell-Buy Imbalance This figure presents average portfolio sell-buy imbalance by OCGO decile. CSB is customer sell-buy imbalance and FSB is firm sell-buy imbalance, both calculated according to equation (4). Sell-buy imbalances are de-trended by their 20-day moving averages. OCGO is the stock s capital gains overhang as defined in Equation (1). Figure 1A reports the results for calls, while Figure 1B does so for puts. The sample period is from January 1996 to December % Figure 1A: Customer and Firm Sell-Buy Imbalance for Calls 20% 15% 10% 5% 0% CSB FSB Figure 1B: Customer and Firm Sell-Buy Imbalance for Puts 25% 20% 15% 10% 5% 0% CSB FSB 23

24 Table 1: Summary Statistics This table reports the summary statistics for calls and puts sorted by option capital gains overhang (OCGO) decile. Each day, options are sorted into deciles based on OCGO is option capital gains overhang, which is defined in equation (1). SCGO is stock capital gains overhang, which is a modification of Equation (1) in that stock prices and stock turnover is used. K/S is the option strike price divided by the underlying stock price. ImplVol is implied volatility of the day t-1 (annualized). BidAsk is the option bid-ask spread. Size is market capitalization in billions. B/M (book-to-market) is defined as in Fama and French (1992). LRet is the prior month s return. Momentum is the cumulative monthly stock return from month t 12 to t 2 in percent. Turnover is the prior month s stock turnover. ImpVol, LRet, and Momentum are reported in percent. The sample period is from January 1996 to December

25 A: Calls OCGO Decile N OCGO SCGO K/S ImpVol BidAsk Size B/M LRet Momentum Turnover 1 573, % % 42.46% 0.27 [1.37] [0.19] [0.18] [25.91%] [0.17] [60.80] [0.51] [15.06%] [127.72%] [0.28] 2 576, % % 39.13% 0.26 [0.31] [0.16] [0.14] [25.47%] [0.20] [58.38] [0.52] [14.65%] [117.36%] [0.28] 3 576, % % 39.35% 0.26 [0.16] [0.15] [0.13] [25.37%] [0.22] [62.82] [0.53] [14.47%] [115.38%] [0.29] 4 576, % % 39.66% 0.26 [0.09] [0.14] [0.13] [25.55%] [0.24] [70.83] [0.51] [14.36%] [113.86%] [0.28] 5 574, % % 39.79% 0.25 [0.05] [0.14] [0.12] [25.77%] [0.26] [76.63] [0.51] [14.38%] [108.98%] [0.28] 6 415, % % 36.76% 0.25 [0.02] [0.13] [0.12] [24.42%] [0.27] [80.02] [0.55] [14.02%] [100.59%] [0.28] 7 418, % % 36.66% 0.25 [0.03] [0.12] [0.12] [23.95%] [0.28] [77.94] [0.53] [14.00%] [100.04%] [0.27] 8 418, % % 36.38% 0.25 [0.04] [0.12] [0.12] [23.36%] [0.29] [71.55] [0.52] [13.96%] [97.04%] [0.26] 9 418, % % 35.18% 0.25 [0.05] [0.13] [0.12] [23.24%] [0.32] [61.92] [0.53] [14.49%] [96.02%] [0.25] , % % 34.98% 0.25 [0.26] [0.15] [0.14] [24.07%] [0.34] [50.49] [0.54] [16.44%] [104.41%] [0.25] All 4,965, % % 38.35% 0.26 [0.90] [0.16] [0.15] [24.91%] [0.26] [67.77] [0.53] [14.78%] [109.92%] [0.27] 25