關於選擇權市場處置效果與相似度衡量期貨交易策略的兩篇論述 - 政大學術集成

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(1)國立政治大學金融研究所博士論文. 論文題目 Two Essays on the Disposition Effect of the Options Market and Similarity-based Futures Trading Strategies. 指導教授: 江彌修研究生：邱信瑜撰中華民國一○五年五月.

(2) 中文摘要第一篇論文中文摘要選擇權市場處置選擇權市場處置效果之衡量與實證. 處置效果係指投資人在處分資產時，傾向盡快賣出有未實現利得的投資部位，並且繼續持有有未實現損失的投資部位的行為偏誤現象。文獻上有關處置效果的實證多半集中在股票市場而少有於選擇權市場的實證。選擇權市場一般認為是具有私有資訊及較具備金融知識與經驗的投資人會選擇交易的市場。本文實證處置效果在指數選擇權市場上的影響。我們認為對於選擇權投資人來說，價內外程度是最重要且顯而易見的資訊，是很直觀可以衡量可能利得及損失的參考點。相較於傳統衡量根據過去交易價格所形成的未實現損益指標，價內外程度更能吸引投資人的注意力。以本文所提出的基於價內外程度衡量之賣出傾向指標 (Moneyness-based Propensity to Sell, MPS)以及根據 Grinblatt and Han (2005)所形成的調整後未實現資本利得指標(adjusted Capital Gains Overhang, ACGO)，每周將買權(賣權)排序成五等分後，我們發現持有最高等分的 MPS 或 ACGO 的買權 (賣權)並賣出最低等分的買權(賣權)所形成的投資組合能夠產生超額報酬，顯示處置效果在指數選擇權市場亦存在。利用雙重排序(double sorting)的方法，我們發現 MPS 相較於 ACGO，是較能夠在選擇權市場捕捉處置效果的指標。關鍵詞：處置效果，選擇權市場，Delta 避險選擇權報酬率，未實現資本利得關鍵詞. i.

(3) 中文摘要第二篇論文中文摘要相似度衡量期貨交易策略之獲利能力實證. 文獻上對於技術交易是否能產生顯著的報酬結果並不一致，然而實務上分析過去的價格走勢並使用技術指標所產生的訊號，是廣泛被接受的。現有測試技術交易指標獲利能力的文獻，通常假設投資人在實證測試的樣本期間一致性的參考某個交易指標產生的交易訊號並依此交易。然而實務上投資人可能同時參考不同的交易指標，每次交易可能根據不同交易指標所產生的訊號，且投資人會從歷史交易價格走勢中尋找類似於現有走勢的狀況，以這些歷史走勢接續的報酬率做為現有走勢未來報酬率的預期值。本文中我們提出一個較符合實際狀況的決策過程來描述技術交易投資人的行為，並重新檢視技術交易的獲利能力。我們提出的決策過程包含三個步驟。首先投資人建立一個特徵向量，包含投資人所認為足以預測未來報酬率並足以描述現況的指標。第二個步驟，投資人從過去某段期間中尋找相似於現有特徵向量的歷史狀況，並以這些歷史狀況接續的報酬率來作為預測的根據。最後，投資人依照過去的歷史狀況與現在有多相似，作為接續報酬率的加權權重，並以相似度權重加權平均報酬來做為未來報酬率的預測值，我們將依照相似度加權報酬所產生交易訊號所形成的策略稱為相似度衡量交易策略 (Similarity-based trading rules)。我們檢視相似度衡量交易策略在九個不同的期貨市場中的獲利能力，在考量 data-snooping 及交易成本後，每日相似度衡量交易策略仍在其中六個市場中獲得顯著的報酬率。關鍵詞：技術交易，相似度衡量交易策略，期貨市場，交易策略獲利能力關鍵詞. ii.

(4) 英文摘要英文摘要 Essay I Measuring the Disposition Effect on the Option Market: New Evidence. Abstract: The disposition effect, which refers to the tendency of investors to selling their winning investments too soon and to hold losing investments too long, has been well-documented in the extant literature. However, while empirical researches focus on examining the behavioral bias in the stock market, little attention is paid to the option market, where most informed investors and sophisticated traders gather. This essay tests for the disposition effect on the index options market. We argue that moneyness, the most salient and readily available information for option investors, is a natural reference point for potential gains and losses, which likely attracts market participants’ attention more than traditional measures that are based on past trading prices. Based on the Moneyness-based Propensity to Sell (MPS) measure that we introduce and an adjusted capital gains overhang (ACGO) measure of Grinblatt and Han (2005), we find that a strategy formed by buying calls/puts in the highest MPS or ACGO quintile and selling those in the lowest quintile would generate significant abnormal returns, suggesting the presence of the disposition effect. Using double sorting method, we find that the MPS is better as a measure in capturing the disposition effect on the options market than the ACGO. Keywords: disposition effects, options market, delta-hedged option returns, capital gains overhang. iii.

(5) 英文摘要英文摘要 Essay II Examining the Profitability of Similarity-based Futures Trading Strategies. Abstract: While the literature documents mixed results for the profitability of technical trading rules, the use of technical buy/sell signals based on analyzing past prices is widely accepted by practitioners. The existing literature on testing the predictive ability of technical trading mostly assumes that a technical investor consistently makes investment decisions based on the buy/sell signals according to one particular trading rule during the entire sample period. However this may be far from reality. Technical investors may simultaneously make predictions based on different technical indicators and follow different technical signals. Furthermore, they analyze historical price patterns that are similar to the current market condition and make assessment of future returns based on the subsequent returns of these similar patterns. The process is known as charting. We attempt to propose a more realistic decision-making process that incorporates the similarity-based predictors to account for technical investors’ decisions in the real world and reexamine the profitability of technical trading rules. The proposed process includes three steps. First, the investor attempts to predict future returns based on a vector of current characteristics that is sufficient for his assessment of the future returns and to depict the present scenario of the stock market. Second, the investor searches for the similar patterns in a specific time window prior to the current date and make an assessment of the future returns based on how similar these past patterns and the current pattern are and how rewarding the subsequent returns of the similar patterns are. Third, the investor is assumed to form a similarity-based indicator which is an assessment of the future returns depended on the similarity-weighted average of all previously observed values of the subsequent returns. The technical investor is then assumed to buy/sell according to the signals generated by the similarity-based trading rules (SBTR). We examine the profitability of the SBTR in nine futures markets and find significantly positive and robust returns after considering the data-snooping adjustments and transaction costs in six of the nine markets. Keywords: technical trading rules, similarity-based trading rules, futures markets, profitability of trading rules iv.

(6) 目錄 Introduction ............................................................................................................................... 1 Essay 1: Measuring the Disposition Effect on the Option market: New Evidence ................... 5 1. Introduction ....................................................................................................................... 5 2. Disposition effect............................................................................................................... 9 3. Empirical Methodology ................................................................................................... 11 3.1 Measures of Disposition Effect ................................................................................. 11 3.2 Data and Empirical Tests........................................................................................... 13 4. Empirical Findings .......................................................................................................... 19 4.1 Summary statistics and univariate tests ..................................................................... 19 4.2 Regression tests of the option market disposition effect ........................................... 24 4.3 Double sorts ............................................................................................................... 27 4.4 Robustness check....................................................................................................... 36 5. Conclusion ....................................................................................................................... 41 Essay 1 References .............................................................................................................. 43 Essay 2: Examining the Profitability of Similarity-based Futures Trading Strategies ............ 48 1. Introduction ..................................................................................................................... 48 2. Universe of technical indicators ...................................................................................... 53 3. Trading model ................................................................................................................. 55 3.1 Decision-making process for similarity-based technical traders ............................... 55 3.2 Data-snooping adjustment ......................................................................................... 57 4. Empirical findings ........................................................................................................... 60 4.1 Comparison of SBTRs and traditional technical rules .............................................. 61 4.2 The best SBTR return ................................................................................................ 68 4.3 Data-Snooping adjusted statistical tests and sub-sample analysis ............................. 72 4.4 Transaction cost ......................................................................................................... 78 5. Conclusion ....................................................................................................................... 80 Essay 2 References .............................................................................................................. 82. v.

(7) 表目錄 Essay I Measuring the Disposition Effect on the Option Market: New Evidence Table 1: Filters on sample options ............................................................................... 14 Table 2: Descriptive statistics for sample options ........................................................ 19 Table 3: Quintile portfolio returns formed on ACGO.................................................. 22 Table 4: Quintile portfolio returns formed on MPS ..................................................... 23 Table 5: Regression tests for the 5-1 portfolio returns formed on ACGO ................... 25 Table 6: Regression tests for the 5-1 portfolio returns formed on MPS ...................... 26 Table 7: Double-sorted portfolio returns formed on MPS and ACGO ........................ 29 Table 8: Regression tests for the double-sorted 5-1 portfolio returns formed on MPS and ACGO ........................................................................................................... 30 Table 9: Double-sorted portfolio returns formed on ACGO and MPS ........................ 33 Table 10: Regression tests for the double sorted 5-1 portfolio returns formed on ACGO and MPS .................................................................................................. 34 Table 11: Robustness checks considering options’ greeks: Regression tests for the delta-hedged portfolio returns formed on MPS ................................................... 37 Table 12: Robustness checks with different option maturities: Regression tests for the 5-1 delta-hedged option portfolios formed on MPS ............................................ 38 Table 13: Robustness checks for regression tests on the 5-1 delta-hedged option portfolios formed on MPS ................................................................................... 40. vi.

(8) 表目錄 Essay II Examining the Profitability of Similarity-based Futures Trading Strategies Table 1: Descriptive statistics for the daily returns on futures ..................................... 62 Table 2: Best traditional technical strategies................................................................ 64 Table 3: Similarity-based strategies based on the best traditional technical indicators .............................................................................................................................. 65 Table 4: Comparison of SBTR and traditional technical rules .................................... 67 Table 5: Best similarity-based technical strategies on a daily basis ............................ 70 Table 6: Best similarity-based technical strategies on a weekly basis ......................... 71 Table 7: Sub-period analysis for the best daily SBTRs ............................................... 73 Table 8: Data-snooping adjustment for the best daily SBTRs ..................................... 74 Table 9: Sub-period analysis for the best weekly SBTRs ............................................ 76 Table 10: Data-snooping adjustment for the best weekly SBTRs ............................... 77 Table 11: Transaction cost ............................................................................................ 79. vii.

(9) Introduction Traditional finance theory assumes that investors are rational in the sense that they are efficient and unbiased processors of information and that they are consistent decision makers in accordance with utility maximization. With the expected utility hypothesis and the axioms of the von Neuman-Morgenstern utility theorem, in the presence of uncertainty, investors are assumed to evaluate risky prospects by considering the probabilities of occurrence, the payoffs and investors’ preference. However, many studies provide empirical and experimental evidences on individual trading biases and anomalies in the cross-sectional stock returns or trading volumes that cannot be easily explained in the traditional framework and emphasize psychological factors of investors. For example, De Bondt and Thaler (1985), Odean (1999), Daniel and Titman (1999), Barber and Odean (2000), and Statman, Thorley and Vorkink (2006) find that overconfidence increases expected trading volume and causes markets to underreact to the relevant information of rational traders, suggesting that some investors overestimate the precision of their private information. Ang, Hodrick, Xing Zhang (2006, 2009), Baker, Bradley, and Wurgler (2011) and Frazzini and Pedersen (2014) find that stocks with high-beta or high idiosyncratic risk tend to underperform those with low-beta or low idiosyncratic risk, while Barberis and Huang (2008) and Bali, Cakici and Whitelaw (2011) show that the anomaly is related to probability weighting for security prices and investors’ preference for positive skewness. Behavioral finance assumes that investors are subjected to behavioral biases, suggesting that their financial decisions can be less than fully rational and provides a new approach to study the influence of psychology on the trading behavior of investors and the aggregate effects on financial markets. Kahneman and Tversky (1979) demonstrate several phenomena that violate the axioms of expected utility theory based on the responses of subjects to several choice problems. They show that when making decision under uncertainty, subjects can be biased by how the information is presented and subjects’ preferences systematically violate the principle that the utilities of outcomes are weighted by their probabilities. Given these empirical results, Kahneman and Tversky (1979) propose the prospect theory as an alternative model to account for individual decision making. The theory includes two stages to describe the decision-making process of investors: editing phase and evaluation phase. 1.

(10) The operations in the editing phase include coding, combination, segregation, cancellation, simplification and detection of dominance. The phase assumes that investors perceive outcomes as gains and losses defined relative to a reference point, rather than final states of wealth. Also the editing phase is designed to account for isolation effect, framing effect, certainty effect and other anomalies. Then in the evaluation phase, investors evaluate the edited prospects with a value function which is S-shaped and the value of each outcome is multiplied by a decision weight which is a monotonic transformation of probabilities. In their later work, Tversky and Kahneman’s (1992) cumulative prospect theory modify the prospect theory and assume that the transformed probabilities are obtained from objective probabilities by applying the probability weighting function. The S-shaped value function which is concave above the reference point and convex below the point and the reverse S-shaped probability weighting function which overweighs tail events have been well documented and examined intensively (Shefrin and Statman, 1985; Levy and Wiener, 1998; Baberis and Huang, 2008; Fang, 2012; Polkovnichenko and Zhao, 2013). This thesis focuses on two major implications of behavioral finance. The first essay attempts to measure the disposition effect on the options market. The disposition effect labeled by Shefrin and Statman (1985) refers to the tendency of investors to sell winners too early and hold losers too long. The effect is an implication of the S-shaped value function of the prospect theory and Thaler’s (1980) mental accounting. Since prospect theory investors are risk averse over stocks that have unrealized capital gains and risk seeking over stocks with unrealized capital losses, the difference in risk attitude induces investors to have a greater propensity to sell stocks that have risen in value. The disposition effect has been closely examined among stock investors, futures traders and mutual fund managers. For example, Odean (1998) and Grinblatt and Keloharju (2001) analyze the trading records of individuals and institutions and find evidence that investors demonstrate the reluctance to realize losses and that past returns and historical price patterns affect trading. Coval and Shumway (2005) find T-bond futures traders tend to take longer to unwind the losing position than winning position. Frazzini (2006) finds that mutual fund managers also exhibit the disposition effect. Another strand of literature on behavioral biases discusses the relationship between investors’ sophistication and the disposition effect. For example, Dhar and 2.

(11) Zhu (2006), Brown, Chappel, Rosa and Walter (2006), Hur, Pritamani and Sharma (2011) find wealthier investors, large investments traders, institutional investors tend to exhibit lower disposition effect while Locke and Mann (2000), Coval and Shumway (2005) and Frazzini (2006) document that the disposition effect still can be found among professional investors. However, while empirical researches focus on examining the disposition effect in the stock market, relatively little attention is paid to the options market where most informed investors and sophisticated traders gather. In the first essay, we introduce a moneyness-based propensity to sell measure to determine the disposition effect in option trading. We argue that moneyness is a natural reference point for potential gains and losses and likely attracts option traders’ attention better than the traditional unrealized capital gains or losses. We compare the explanatory power of the proposed measure and the traditional measure of capital gains overhang. With these measures we find that disposition effect do exist in S&P index option market. A disposition-induced momentum strategy is established on a weekly basis by buying all calls (puts) in the quintile with highest disposition measure and selling all calls (puts) in the lowest quintile. The strategy generates significant abnormal return after controlling well-known risk factors in the literature on option asset pricing. More importantly, using double sorting method, we find that the proposed measure is better as a measure in capturing the disposition effect in the options market than the traditional capital gain overhangs. The second essay attempts to propose a decision-making process for technical investors that incorporates Tversky and Kahneman’s (1973) availability heuristic where investors predict future returns by judging probabilities of possible states by the ease with which relevant and similar instances come to mind. The process employs the case-based decision theory of Gilboa and Schmeidler (1995) who document the difficulties in fitting investment decision problems to the framework of expected utility theory, since all possible outcomes and their probabilities are not easy to be analyzed and imaged in the real world. The decision-making process for technical investors includes three steps. First, the investor attempts to predict future returns based on a vector of current characteristics that is sufficient for his assessment of the future returns and to depict the present scenario of the stock market. Second, the investor searches for the similar patterns in a specific time window prior to the current date and make an assessment of 3.

(12) the future returns based on how similar are these past patterns and the current pattern and how rewarding are the subsequent returns of the similar patterns. Third, the investor is assumed to form a similarity-based indicator which is an assessment of the future returns based on the similarity-weighted average of all previously observed values of the subsequent returns. The empirical literature on technical trading documents mixed results for the profitability of technical trading rules. For example, Brock, Lakonishok and LeBaron (1992), Gencay (1998), Lo, Mamaysky and Wang (2000), Wong, Manzur and Chew (2003), Zhu and Zhou (2009), and Shynkevich (2012) find supportive evidence on the profitability of technical trading rules, while Fama and Blume (1966), Jensen and Benington (1970), Fong and Yong (2005), Savin, Weller and Zvingelis (2007), Bajgrowicz and Scaillet (2012) report no support. However, the use of technical buy/sell signals based on analyzing past prices is widely accepted by practitioners. We argue that the traditional tests on the profitability of technical trading rules may not be applicable in practice since they mostly assume that investors always follow a single trading rule during the whole sample period and that the magnitude of the technical indicators plays little role in the decision-making process. The similarity-based decision-making process in the second essay is designed to be more realistic and to closely follow the steps where the practitioners make trading decision through “charting”. With the proposed decision-making process, we test the profitability of the similarity-based trading rules in several futures markets. We find that the daily strategies based on the similarity-based indicators generate significant returns in six of the nigh futures markets after considering data-snooping adjustments. Overall, this essay contributes the literature on the technical trading by proposing a realistic decision-making process for the technical investors and confirming the profitability of the similarity-based trading rules. To sum up, this thesis contributes to the existing literature on behavioral biases by providing empirical evidences on the following two subjects. First, we examine the disposition effect on the index option markets and propose a salient measure to better capture the disposition effect. Second, we propose a similarity-based decision-making process for technical investors and examine the profitability of the similarity-based trading rules in the futures markets. 4.

(13) Essay 1: Measuring the Disposition Effect on the Option market: New Evidence 1. Introduction The disposition effect, which refers to the tendency of investors to sell their winning investments too soon and to hold losing investments too long, has been well-documented in the extant literature. While a set of studies regarding disposition effect argues that greater investor sophistication is positively correlated to lower disposition effect (Feng and Seasholes, 2005; Dhar and Zuh, 2006; Hur, Pritamani and Sharma, 2010), the other set of studies documents that the disposition effect also has influence on the trading behavior of professional investors. The debate raises a question of whether disposition effect can be found in option markets. On one hand, the high leverage achievable with options and the downside protection feature may induce informed investors and sophisticated traders to choose to trade in the option market1. On the other hand, several researches document findings are consistent with the implication of stock market evidence on behavioral biases and suggest that option investors may deviate from standard utility theory. However, while empirical researches focus on examining the behavioral bias in the stock market, little attention is paid to the option market. The existing literature notes the unlimited upside potential and limited downside risk of call options, which tend to attract investors with skewed preferences (speculative motives). Heath, Huddart and Lang (1999) find that employees are more likely to exercise their options when stock prices exceed a maximum price over the previous year. Poteshman (2001) examines options market reaction to the instantaneous variance of the underlying asset and finds that option investors tend to underreact to daily changes in volatility. Poteshman and Serbin (2003) find that option investors may engage in irrational early excises of exchange-trade options should the underlying stock price ever attain its highest level over the previous year. Bali and Murray (2013) and Boyer and Vorkink (2014) show that low returns of individual stock options are associated with high expected skewness, and option investors exhibit positively skewed preferences. In this essay, we test for whether the disposition effect exists in the options 1. Black (1975) argues that options are more attractive to informed investors than the underlying stocks because the payoff of an option is truncated at the strike price. Chakravarty, Gulen, and Mayhew (2004), Pan and Poteshman (2006) and Ni, Pan, and Poteshman (2008) provide empirical evidence that informed traders use options to trade on directional and volatility information. 5.

(14) market and more importantly, we propose a moneyness-based disposition measure for testing the disposition effect of option trading. We argue moneyness is a natural reference point for potential gains and losses, which likely attracts option traders’ attention more than the traditional unrealized capital gains or losses measures based on past trading prices, which is commonly used in the prior literature for testing the disposition effect on the stock market. Moneyness is a salient feature of options that is not applicable to stock market and thus is unique to the option market. To test this hypothesis, we devise a “Moneyness-based Propensity to Sell” (MPS) measure calculated as the trading-volume weighted average of an option’s intrinsic value over the 5-, 10- and 20-day periods prior to the trading date. We then compare the explanatory power of MPS to that of the traditional measure of the disposition effect, i.e., the “Adjusted Capital Gains Overhang” (ACGO) measure similar to that of Grinblatt and Han (2005), which uses the trading-volume weighted average of lagged option prices (i.e., cost basis) as the reference price for gain and loss calculations. While option prices are vital for calculating trading profits and losses, an option’s intrinsic value is the most salient and readily available information that are likely to attract market participants’ immediate attention – a scarce cognitive resource (Kahneman, 1973; Hirshleifer and Teoh, 2003; Peng and Xiong, 2006). Our research design mostly follows that of Grinblatt and Han (2005) and Bhootra and Hur (2014). We use daily transactions of S&P 500 futures options obtained from OptionMetrics. On a weekly basis, all options on the S&P 500 futures are first sorted into an ascending order of five quintiles by MPS and ACGO, respectively. Within each quintile, we construct two portfolios – equally weighted portfolios of options with either the unhedged or delta-hedged calls or puts belonging to that quintile. We then calculate the average weekly return for every MPS and ACGO quintile portfolio. Grinblatt and Han (2005) associate the disposition effect to prospect theory and mental accounting.2 For an asset with higher (lower) capital gains overhang, the disposition effect induces a positive (negative) spread between the asset’s. 2. Grinblatt and Han (2005) argue that Kahneman and Tversky’s (1979) prospect theory and Thaler’s (1980) mental accounting framework is the leading explanation for the disposition effect. The prospect theory employs an S-shaped value function that is risk aversion in the domain of gains and risk seeking in the domain of losses. The domain of gains and losses is measured relative to a reference point. The mental accounting depicts the way in which decision makers set reference points for the accounts that determine gains and losses. 6.

(15) fundamental value and its equilibrium price, which then generates price momentum.3 Price momentum, as a direct consequence of spread convergence of the two, ensures that if a strategy were formed by buying quintile portfolio of the highest MPS or ACGO ranking (i.e., winners) and selling quintile portfolio of the lowest MPS or ACGO ranking (i.e., losers), the long-short strategy should generate abnormal returns, if price momentum due to the disposition effect is significant. In this study, we accordingly construct such long-short portfolios and refer to them as the “5-1” portfolio, which longs options in the highest MPS or ACGO quintiles and simultaneously shorts options in the lowest MPS or ACGO quintile. To test the hypothesis that the abnormal returns of the 5-1 portfolios are driven by the disposition effect, we regress the portfolio weekly returns on a set of risk factors that are well known to be important determinants of option returns. We allow the quintile portfolios to consist of both unhedged and delta-hedged options, so that the 5-1 portfolios subsequently formed proxy for the difference in the weekly unhedged or delta-hedged returns of the winner and the loser. Due to the well known overpriced puts puzzle (Broadie, Chernov and Johannes, 2009; Bondarenko, 2014), which is particularly pronounced for the out-of-the-money (OTM) put options, as robustness checks, we remove options with a moneyness below 0.98 and above 1.02 and re-conduct all analyses.4 Our empirical findings are as follows. First, the average returns of the quintile portfolios with delta-hedged puts or calls are monotonically increasing across all MPS and ACGO quintiles, indicating that options with larger moneyness and capital gains overhang tend to outperform options with lower moneyness and capital gains overhang. For example, we find that the returns of quintile portfolios with delta-hedged calls increase from -0.67% in the lowest MPS quintile to 0.07% in the highest MPS quintile whilst the returns of quintile portfolios with delta-hedged puts increase from -0.41% to 0.21%. Second, the average returns of the 5-1 portfolios are statistically significantly 3. Jegadeesh and Titman (1993) report a predictable price pattern that firms with high returns over the past three months to one year continue to outperform firms with low past returns over the same period. Rouwenhorst (1998), Jegadeesh and Titman (2001) and Kang, Liu and Ni (2002) document that a momentum strategy utilizing the predictable price pattern by buying past winners and selling past losers generates abnormal returns. 4 The overprized put puzzle refers to the empirical phenomenon that historic prices of the S&P index puts are overly high and cannot be explained by existing asset-pricing models. 7.

(16) greater than zero, even after controlling for known risk factors, indicating the presence of the disposition effect on the option market, because the 5-1 strategy utilizing the disposition effect-induced momentum generates significantly positive abnormal returns. The results are more pronounced for the case of delta-hedged puts than for delta-hedged calls. The results on the returns of the 5-1 delta-hedged put portfolios remain robust even when we include only the options with their moneyness ranging from 0.98 to 1.02 and time-to-maturities being less than thirty days. More importantly, the 5-1 portfolio constructed by sorting delta-hedged options with the MPS measure produces a higher abnormal return than that with the ACGO measure. Using double sorting methods, we find that the returns of the 5-1 delta-hedged option portfolio formed on MPS cannot be fully explained by the ACGO measure. This supports our hypothesis that moneyness is a natural reference point for option buying and selling decisions. Option traders implicitly measure gains and losses based on option moneyness, a salient and readily available feature that is unique to the option trading, which attracts option traders’ attentions more than the unrealized capital gains and losses. Attention, being a limited cognitive resource, is likely allocated to the most salient information that can be easily recognized and readily available and, in our case, such information is more likely reflected by the option’s moneyness than by capital gains overhang. To our knowledge, this is the first paper analyzing the disposition effect on the index options at an aggregate level and utilizing the disposition measures to form options portfolios to generate abnormal returns. Related works such as Heath, Huddart and Lang (1999), Poteshman and Serbin (2003) and Choe and Eom (2010) focus on examining the disposition effect using account-level data. Our empirical findings are important for understanding whether the disposition effect exhibited by part of individuals could have impacts on aggregate index options market where most informed and sophisticated investors gather. The remainder of this essay is organized as follows. Section 2 reviews related literature on the disposition effect and behavioral biases on the options market. Section 3 describes the data and the methodology for the empirical tests. Section 4 provides an in-depth discussion of the empirical results. Section 5 concludes the eesay. 8.

(17) 2. Disposition effect Investors tend to hold on to their losing investments too long and sell their winning investments too soon. The phenomenon is first label as the disposition effect by Shefrin and Statman (1985) and widely documented in experimental and empirical studies. Shefrin and Statman (1985) employ an extension of the behavioral model based on the prospect theory of Kahneman and Tversky (1979) and Thaler’s (1985) mental accounting and argue that the observed patterns of loss and gain realization are consistent with the implications of their proposed model. Under prospect theory, investors evaluate outcomes according to their perception of gains and losses relative to a reference point rather than final wealth and are more sensitive to losses than to gains. Moreover, investors are risk-seeking for losses and risk-averse for gains. The findings of Shefrein and Statman (1985) suggest that the disposition effect can be attributed to the non-standard utility of prospect theory. The possible explanation to drive the disposition effect is still in dispute. Klye, Ou-Yang and Xiong (2006) and Henderson (2012) analyze the liquidation problem for a prospect theory agent and find that the trading pattern is consistent with the prediction of disposition effect. Li and Yang (2013) develop a general equilibrium model to further examine asset prices and trading volume and find the preference under prospect theory can explain the disposition effect, the momentum effect and the equity premium puzzle. On the other hand, Barberis and Xiong (2009) argue that the annual gains or losses do not predict the disposition effect and develop a realization utility (Barberis and Xiong, 2012) to account for it. Kaustia (2010) and Hens and Vlcek (2011) also show that prospect theory is unlikely to explain the disposition effect. Despite the debate on the question of whether prospect theory drives the disposition effect, the trading pattern is robust and widely documented. Odean (1998) analyzes the trading records of 10,000 customer accounts provided by a nationwide discount brokerage and find that these stock investors exhibit strong preference for realizing winners rather than losers. He argues that the behavior is not motivated by the desire to rebalance portfolios or to avoid the higher trading costs. Grinblatt and Keloharju (2001) employ a unique data set to monitor the buys, sells, and holds of individuals and institutions in the Finnish stock market. They find evidence that investors demonstrate the reluctance to realize losses and that past returns and 9.

(18) historical price patterns affect trading. Coval and Shumway (2005) use the history of transactions from the Chicago Board of Trade (CBOT) T-bond futures traders during 1998 and provide strong evidence for behavior biases. They find that the traders taking a losing position into the afternoon tend to take longer to unwind the position than those with a winning position. Frazzini (2006) analyzes mutual fund holdings from 1980 to 2002 and constructs a new measure of unrealized capital gains. His results show that mutual fund managers also exhibit the disposition effect. The disposition effect is also related to momentum anomaly (Jegadeesh and Titman, 1993) and price underreaction to information (Barberis, Shleifer and Vishny, 1998, Hong and Stein, 1999). Grinblatt and Han (2005) argue that the disposition effect creates a spread between a stock’s fundamental value and its equilibrium price thus the spread convergence generates predictable price momentum. Hur, Pritamani and Sharma (2010) further test the hypothesis that the disposition effect-induced momentum should be stronger in stocks with greater individual investors’ presence. Frazzini (2006) argues that the disposition effect induces underreaction to news and provides evidence that the magnitude of the post-announcement price drift may depend on the capital gains overhangs. Another strand of literature on behavioral biases examines the relationship between investors’ sophistication and the disposition effect. Dhar and Zhu (2006) analyze the trading records of a major discount brokerage house and find empirical evidence that wealthier investors or investors with professional occupations exhibit a lower disposition effect. Brown, Chappel, Rosa and Walter (2006) examine daily Australian Stock Exchange share registry data and find larger investments traders are affected less by the disposition bias. Hur, Pritamani and Sharma (2010) find evidence for the hypothesis that the disposition effect-induced momentum is stronger in stocks with greater individual investors’ presence and conclude that individual investors are more prone to the disposition effect. Although greater investor sophistication seems to be associated with lower disposition effect, the disposition effect still can be found among professional traders. For example, Locke and Mann (2000) and Coval and Shumway (2005) find disposition effect among futures traders, and Wermers (2003) and Frazzini (2006) find its presence among fund managers. The negative correlation between investor sophistication and the disposition effect raises a question of whether the disposition effect exists in option markets. 10.

(19) Option trading activities are widely considered containing information about future stock prices or future realized volatility. For example, Amin and Lee (1997) find that the buy/sell activities of option traders foreshadow subsequent earnings news. They show that the positive midquote returns of active-side option trades increase during earnings announcement. Easley, O’Hara and Srinivas (1998) investigate the informational links between options and equity markets and find that under certain conditions, the informed traders may choose to trade in the options market. The empirical results show that particular option volumes lead stock price changes, that is, option volumes may contain information about future stock prices. Pan and Poteshman (2006) also provide evidence that stocks with low put-call ratios constructed from option volume outperform stocks with high put-call ratios and suggest that option trading volume contains information about future stock prices. Chakravarty, Gulen and Mayhew (2004) apply Hasbrouck’s (1995) methodology and find significant price discovery in the option market, however the proportion of information revealed first in the option market varies across stocks. Ni, Pan and Poteshman (2008) investigate the implications of volatility information trading in the option market and provide evidence that option volume is informative about future realized volatility of underlying stocks. In general, the informational role for options can be attributed to the argument of Black (1975) and Chakravarty, Gulen and Mayhew (2004), that is, the high leverage achievable with options and the downside protection create an ideal venue for informed trading. 3. Empirical Methodology 3.1 Measures of Disposition Effect To measure the disposition effect, we first modify the capital gains overhang (CGO) measure of Grinblatt and Han (2005) by substituting option trading volumes for turnover ratios as the weights for calculating the reference prices, because turnover ratios are not well defined for options that are in zero net supply. We define the reference price, , for the option holder’s mental account on date ,as:. 1 = , (1) ∑

(20) .

(21) . where is the lagged option prices on date − , and is the lagged option. trading volumes on date − .. 11.

(22) Our empirical tests are conducted on lagged time periods of three different lengths in days, namely, = 5, 10, and 20. We find that varying lengths of lagged time period neither induces significant impacts on our empirical findings nor affects our conclusions. Having established the market’s cost basis by the reference price ( ), we can then proceed to derive an adjusted capital gains overhang (ACGO) measure as follows: =. − , (2) . where represents the capital gains overhang and is the closing mid-price of. option on date .. The “moneyness-based propensity to sell”, abbreviated hereafter as the MPS, adopts the trading-volume weighted average of an option’s intrinsic value over its sampling period. In other words, it is the trading-volume weighted average of the difference between an option’s strike price and the price of its underlying asset. The MPS measure is calculated as:. 1 = ( − ) , (3) ∑

(23) .

(24) . where denotes the underlying index price on date − and denotes the option strike price. It is appealing to adopt an option’s moneyness as the reference point set by investors’ mental accounting in deriving the MPS measure. Moneyness, which reflects the intrinsic value of an option, is the most readily available information in the options market. Should the intrinsic value of an option become in-the-money (ITM), this would suggest a potential exercisable gains for the option holder, and would likely induce a motive for the option holder to exit his/her positions. An option being out-of-the-money (OTM), on the other hand, is a clear indication of it being worthless, holders of which are therefore likely to hold on to his/her positions in a hope for possible future market reversals. Thus, we conjecture that holders of ITM options are more likely to sell than those of OTM options. In this study we argue that, while using the past option prices as the reference price is one way of inferring the gains or losses for option investors, mental 12.

(25) accounting may also involve a process of attention allocation, which induces option investors to rely on a more intuitive measure of exercisable gains and losses of moneyness. Attention, being a limited cognitive resource (Kahneman, 1973; Hirshleifer and Teoh, 2003; Peng and Xiong, 2006), is likely allocated to the most salient information that is readily available and can be easily absorbed. We argue that such information is more likely to be captured by the option’s moneyness. 3.2 Data and Empirical Tests The OptionMetrics database contains the end-of-day quotes of European call and put options on S&P 500 index futures from January 1996 to August 2015. To ensure that our portfolio consists of only options with reliable quotes, we apply several data filters suggested by Constantinides, Jackwerth and Savov (2013), Bali and Murray (2013), Bondarenko (2014) and Boyer and Vorkink (2014). Table 1 presents the total number of transactions obtained from the OptionMetrics database and the number of observations that are removed upon applying these filters. Specifically, we remove options with their bid prices being missing or less than or equal to $0; with bid-ask spreads being less than $0 or greater than $5; with deltas being less than -1 or greater than 1. We also remove transactions with missing volumes or open interest; with unrealistic trading dates; and with moneyness (S/K for calls and K/S for puts) being less than 0.94 or greater than 1.06.5 Furthermore, we remove trades that violate the arbitrage conditions, or have repetitive entries under the same strike/expiration listings on the same date. After filtering, the dataset consists of 1,562,861 observations. We adopt a research design similar to that of Grinblatt and Han (2005) and Bhootra and Hur (2014). Weekly data of call and put options, either unhedged or delta-hedged, are sorted into an ascending order of five quintiles according to the MPS and ACGO measures. Within each quintile, we construct an equally weighted portfolio of options (the quintile portfolio) and calculate the average weekly return for each MPS and ACGO quintile portfolio. Comparisons are drawn between the returns. 5. Constantinides, Jackwerth and Savov (2013) remove all quotes with moneyness below 0.8 ore above 1.2 as these options have thin trading volume and little value beyond their intrinsic value. Broadie, Chernov and Johannes (2009) and Bondarenko (2014) use a dataset that includes only options with moneyness greater than 0.94 and less than 1.06. In this article, we remove options with moneyness below 0.94 or above 1.06 to construct the weekly portfolio returns. However, our empirical results remain unchanged, when using options with moneyness below 0.8 or above 1.2. 13.

(26) generated by the unhedged and the delta-hedged option portfolios across different MPS and ACGO quintiles. This step provides us with primitive observations of the impacts of disposition effects on the cross-sections of option returns. Table 1: Filters on sample options This table reports the employed filters to ensure that only options with reliable quotes enter our portfolios. In detail, we remove transactions with zero implied volatility and missing implied volatility, with a maturity date earlier than trade date, with a missing bid or bid price less than zero, with a bid-ask spread less than zero or greater than five, with a delta less than minus one or greater than one. We also remove trades that have missing open interest, missing volume or zero volume, and trades that have an indicator of no trades (the OptionMetrics “last_date” value is before trade date. Finally, we remove options with moneyness below 0.94 or greater than 1.06 and with duplicate trades. Period: 19960101-20150831 Total trades Filter. 7,385,062 Deleted. Implied volatility=0 or missing implied volatility. 1,583,400. Maturity date earlier than trade date. 243. Missing bid or bid price<=0. 189. Unrealistic bid-ask spread (best_offer-best_bid<0 or best_offer-best_bid>5). 945,733. Unrealistic greeks (delta<-1 or delta>1). 5,787,504. Missing open interest, volume or volume=0. 5,309,755. Unrealistic trade dates (last_date<date). 5,311,312. Subtotal. 5,562,054. Moneyness<0.94 or Moneyness>1.06. Remaining. 1,808,850. 913,019. Total. 895,831. Next, long-short portfolios are constructed by buying all calls (puts) in the highest MPS and ACGO quintiles, respectively, and simultaneously selling all calls (puts) in their lowest quintiles, respectively. We examine the mean, the standard deviation, and the statistical significance of the long-short portfolio return distributions. The long-short portfolio is denoted as the “5-1” portfolio. If the average return of the 5-1 portfolio is significantly positive, then it indicates that higher MPS or ACGO predicts higher option returns and a strategy that longs options in the highest MPS (or ACGO) quintile and shorts options in the lowest quintile generates positive returns. The 5-1 portfolio returns are constructed in two ways: either with the unhedged 14.

(27) options,. ! , ,. or with the delta-hedged options, ∆#$%&$%. ! , .. Based on closing. mid-prices, the weekly returns of the unhedged options are approximated by: ,'(. =. '( − , (4) . where '( is the option price one week later and thus * equals to a week. denotes the option return over the period +, + *-.. ,'(. Following Cao and Han (2013), who define the delta-hedged returns to be dollar gains scaled by absolute values of the assets considered, ∆., − , we derive the delta-hedged option returns as:6 ∆/01201. ,'( =. 34. 5 '( − − ∑

(28) '( ∆., ( ' − ) − ∑

(29) '( 8 − ∆., 9

(30)

(31) 67!. ∆., − . (5). where ∆., is the delta of the call option on date , : is the annualized risk-free. rate on date , and ∆/01201 period +, + *-.. ,'(. denotes the delta-hedged option returns over a. To control for the effect of risk on the 5-1 portfolio returns, we regress the portfolio returns on a set of risk factors. The set of risk factors needs to be chosen with care. Broadie, Chernov and Johannes (2009) and Bondarenko (2014) suggest inclusion of the jump-risk premiums in option returns. Pan (2002), Jones (2006), and Driessen and Maenhout (2013) suggest volatility-risk and jump-risk premiums to be important determinants of option returns. The set of risk factors that we consider include the Fama and French’s (1993) three factors: market excess returns (;_:), small minus big market capitalization portfolio returns (=), and high minus low book-to-market ratio portfolio returns (>?), as well as the momentum risk factor () suggested by Jegadeesh and Titman (1993) and Carhart (1997)7. To control. 6. Bakshi and Kapadia (2003) normalize the delta-hedged gains by the index level and the option price. Our empirical findings and implications are qualitatively the same by using their methods. 7 These stock-related risk factors and linear factor models are widely documented in the literature on analyzing cross-sectional returns of index options and individual options. For example, Jackwerth (2000), Coval and Shumway (2001), Broadie, Chernov and Johannes (2009), Bakshi and Kapadia (2003), and Driessen and Maenhout (2013) examine whether cross-sectional index option returns are consistent with the single-factor models or the linear models with volatility risk and jump risk. Goyal and Saretto (2009) ,Cao and Han (2012) and Boyer and Vorkink (2014) investigate the anomalies of cross-sectional individual option returns based on the risk factors of Fama and French (1993), the momentum risk factor and other common risk factors. 15.

(32) for the volatility risk and jump risk, we follow Constantinides, Jackwerth and Savov (2013) and include changes-in-VIX (@A), changes in volatility smirk (B ;), and changes in the 30-day realized volatilities (). The regression model is as follows: ! ,. = C + D ;E4 + DF = + D6 >? + DG + D! @A . +D7 B ; + DH + I (6) and ∆/01201. ! ,. = C + D ;E4 + DF = + D6 >? + DG . +D! @A + D7 B ; + DH + I, (7) where. ! ,. is the 5-1 portfolio return constructed by buying calls (puts) in the. highest MPS or ACGO quintile and selling calls (puts) in the lowest quintile. ∆/01201. ! ,. is the 5-1 delta-hedged portfolio return constructed by buying. delta-hedged calls (puts) in the highest MPS or ACGO quintile and selling delta-hedged calls (puts) in the lowest quintile. Given the findings of Bakshi and Kapadia (2003), where returns of the delta-hedged option portfolios are found to be significantly affected by market volatility, in equation (7), we consider therefore changes in the VIX and changes in the options’ 30-day realized volatilities. To further address this concern, we adopt changes in implied volatilities of the ATM call options, denoted by @BL. NO, in place of changes in the VIX for our regression analysis. Note that both changes in the VIX and changes in the ATM option implied volatilities reflect an option investor’s expectation about the option’s unrealized volatility in the future, while changes in the 30-day realized volatilities reflect the past volatility estimates. Volatility smirk, B ;, is defined as the difference between the implied volatilities of OTM put options and the implied volatilities of ATM call options. The volatility smirk represents the negative skeweness of the underlying return distributions. For example, Bates (2000) suggests that the volatility smirk captures the crash fears of option investors. Pan (2002) also shows that the volatility smirks are primarily due to investors’ fear of large adverse price jumps. By incorporating both a jump risk premium and a volatility risk premium, Pan (2002) shows that the jump risk premium component represents 80% of total risk premium for OTM put options and 16.

(33) only 30% for OTM calls. Furthermore, Xing, Zhang and Zhao (2010) find that stocks with the steepest option-volatility smirks underperform stocks with the least pronounced option-volatility smirks. They argue that option investors tend to choose OTM puts to express their concerns about possible future negative jumps. Since our 5-1 portfolio involves long and short positions in calls or puts across different strike prices, controlling for the volatility smirk ensures that the abnormal returns of the 5-1 portfolio are not driven by the time-varying magnitudes of the volatility smirks. To further control for the overpriced puts puzzle (Pan, 2002; Jones, 2006; Driessen and Maenhout, 2013), we incorporate in our regressions a price-jump factor (PQBL) and a volatility-jump factor (NO. PQBL) that capture respectively price and volatility jump risks: ! ,. = C + D ;E4 + DF = + D6 >? + DG + D! @A + D7 B ; . +DH + DR PQBL + DS NO. PQBL + I (8) and ∆#$%&$%. ! ,. = C + D ;E4 + DF = + D6 >? + DG + D! @A . +D7 B ; + DH + DR PQBL + DS NO. PQBL + I , (9) where PQBL is the sum of all daily S&P 500 returns lower than -4% over the past. month, and NO. PQBL is the sum of all daily increases in the ATM call implied volatility that are greater than 4% over the past month. As an alternative for controlling for the presence of jump risks in both the price and volatility of options, we introduce BPQBL and BVNO. PQBL , which is defined as respectively the minimum daily index returns and the maximum daily increase in the implied volatilities of ATM call options over the past month. In particular: ! ,. = C + D ;E4 + DF = + D6 >? + DG + D! B ; + D7 . +DH @BL. O. + DR BPQBL + DS BWVNO. PQBL + I , (10) and ∆/01201. ! ,. = C + D ;E4 + DF = + D6 >? + DG + D! B ; . +D7 + DH @BL. O. + DR BPQBL 17.

(34) +DS BWVNO. PQBL + I. (11) The regression tests of equations (6) to (11) for the 5-1 portfolio returns with unhedged and delta-hedged options aim to establish empirical evidence for the presence of the disposition effects on the options market. If the intercept is significantly different from zero, we then conclude the disposition effect is present on the option market. To further test our hypothesis that MPS is a more salient reference point for inducing the disposition effect than ACGO for the investors’ mental account, we employ a double sorting technique similar to Grinblatt and Han (2005). Specifically, we first sort options into five quintiles according to the ACGO measure and within each ACGO quintile, we further sort them into five quintiles by the MPS measure. For each ACGO quintile, we examine the returns of the 5-1 portfolio constructed by buying calls (puts) in the highest MPS quintile and selling calls (puts) in the lowest MPS quintile. If the returns of the 5-1 portfolio formed by MPS within each ACGO quintile are still significantly different from zero, or the intercepts of the regression tests are significant different from zero, then we conclude that MPS has a significant explanatory power over the disposition effect, even after controlling for ACGO. Alternatively, we first sort all options into five quintiles by the MPS measure and in each ACGO quintile sort all options further into five quintiles by the ACGO measure. If the returns of the 5-1 portfolio formed by ACGO within each MPS quintile become insignificantly different from zero, or the intercepts of the regression tests are not significantly different from zero, then we can conclude that the explanatory power of the ACGO measure over the disposition effect is subsumed by that of the MPS measure. Finally, to check the robustness of our results, we remove those options with moneyness below 0.96 and above 1.04, or alternatively those with moneyness below 0.98 and above 1.02. The filters on options’ moneyness alleviate the impact of the overpriced put option puzzle. We also remove options with maturity that are longer than thirty days to prevent the possibility that the 5-1 portfolio returns are induced by the differences in the sample option maturities.. 18.

(35) 4. Empirical Findings 4.1 Summary statistics and univariate tests Table 2 reports the summary statistics for the option returns, ACGO, and MPS. At the end of each week, we include only options with available ACGO and MPS measures. Namely, we require sample options to have transactions for at least five days prior to the trading date. Table 2: Descriptive statistics for sample options This table reports a number of summary statistics for the filtered dataset. The dataset includes all daily transactions of S&P 500 futures options in the OptionMetrics database from 19960101-20150831 and filtered by the screening conditions in table 1. At the end of each week, the average of the subsequent 1-, 5-, 10- and 20-day returns are presented for the unhedged and delta-hedged option returns. Furthermore, we only include options with available ACGO and MPS in the past five days. The unhedged option returns are calculated using daily closing mid-prices as equation (4) and the delta-hedged returns are defined as equation (5). Panel A: call options Variables No. of obs Mean (%) Stdev (%) Median (%) 122,680 Unhedged option 1-day return -5.92 29.40 -2.06 122,680 Unhedged option 5-day return -15.27 60.62 -4.30 85,586 Unhedged option 10-day return -14.80 63.22 -4.89 59,502 Unhedged option 20-day return -14.22 66.67 -4.23 122,680 Delta-hedged 1-day return 0.04 1.90 -0.02 122,680 Delta-hedged 5-day return -0.20 0.89 -0.11 85,586 Delta-hedged 10-day return -0.28 5.33 0.02 59,502 Delta-hedged 20-day return -0.11 6.85 0.19 122,680 ACGO (past 5 days) -16.99 166.41 0.00 122,680 MPS (past 5 days) -16.50 47.93 -14.98 122,680 Maturity 85.86 122.23 43.00 122,680 Moneyness 0.99 0.03 0.99 Panel B: put options Variables No. of obs Mean (%) Stdev(%) Median (%) 115,509 Unhedged option 1-day return -7.60 30.41 -4.05 115,509 Unhedged option 5-day return -24.02 62.19 -12.52 79,433 Unhedged option 10-day return -28.24 58.30 -18.73 54,992 Unhedged option 20-day return -37.07 59.74 -28.22 115,509 Delta-hedged 1-day return -0.14 1.93 -0.24 115,509 Delta-hedged 5-day return -0.08 0.98 -0.15 79,433 Delta-hedged 10-day return -1.72 4.88 -1.94 54,992 Delta-hedged 20-day return -3.03 5.94 -3.27 115,509 ACGO (past 5 days) -14.99 74.89 -2.04 115,509 MPS (past 5 days) -16.76 44.47 -14.54 115,509 Maturity 92.90 134.94 45.00 115,509 Moneyness 0.99 0.03 0.99. The subsequent 1-, 5-, 10- and 20-day returns are calculated using daily closing mid-prices according to equation (4). The ACGO and MPS measures over the past 10and 20-day are also presented. Panel A of Table 2 shows the summary statistics for 19.

(36) call options while Panel B shows the results for put options. Our results in Table 2 are generally consistent with the findings in the extant literature. First, option returns are non-normal as the mean of option returns deviates from the median. For example, the average 5-day return of unhedged call options in Panel A is -15.27% and the median is -2.06%. The results are consistent with Broadie, Chernov and Johannes (2009) who document non-normality for option returns and suggest that the linear factor model may not be appropriate to examine the option returns. On the other hand, the delta-hedged option returns show relatively less positive skewness and kurtosis. Second, call and put option returns are on average negative. The average 5-day returns of unhedged calls and puts are -15.27% and -24.02%, respectively. The delta-hedged call and put returns are also on average negative. This is in line with Bakshi and Kapadia (2003), who document that the delta-hedged strategy generates negative returns and suggest a negative market volatility risk premium. The ACGO and MPS measures are generally negative. Both ACGO and MPS tend to decrease with the lengths of time, suggesting that options are on average traded below the market’s cost basis or the average exercise prices. Finally, the average maturities of the call and put options are 85.86 days and 92.90 days, respectively. The average moneyness of the options is 0.99, which is near the money. Table 3 reports the unhedged and delta-hedged options returns for each of the five ACGO quintiles ranked by their ACGOs over the past five days at the end of each week. Quintile 1 (5) portfolio consists of options in the lowest (highest) ACGO quintile. Changing the benchmark time-periods of measuring ACGO does not materially alter our results. A couple of interesting empirical findings emerge from Panel A of Table 3. First, the five-day returns of the call portfolios increase with the ACGO measure. The average five-day return in the lowest ACGO quintile is –26.52%, which increases to –12.29% in the highest ACGO quintile. Second, returns of the 5–1 call portfolio are significantly different from zero at the 5% significance level. The delta-hedged call option returns also tend to increase monotonically with the ACGO measure. For example, the delta-hedged 5-day return of call options increases from -0.41% to -0.16% with ACGO. For delta-hedged call options, the 5-1 portfolio returns are statistically significantly positive expect for the 1-day return. 20.

(37) The results for put options in Panel B of Table 3 are also strong. The patterns across ACGO quintiles for the five-day returns of the put portfolio tend to be monotonic. The average five-day return is –36.27% in the lowest ACGO quintile, which increases to –17.87% in the highest quintile. All 5–1 put options and delta-hedged put options returns are significantly positive except for the 1-day and 20-day returns of delta-hedged portfolios. This finding suggests that ACGO has predictive power for option returns. Next, we sort all options by their past five-day MPS. Panels A and B of Table 4 present the results for the unhedged and delta-hedged call and put portfolio returns, respectively. The returns of the 5–1 call portfolio in Panel A are all positive and significant at the 1% level for the unhedged return. The average five-day returns increase with MPS quintiles from –18.41% to -3.93%. The return differences by MPS quintiles for put options, shown in Panel B, are again, as in Table 3, much stronger than those of call options. The returns of the 5–1 unhedged put portfolio are all positively significant at the 1% level. The average returns of the quintile portfolios increase with the MPS measure. For example, the average five-day returns increase from –23.97% in the lowest MPS quintile to –8.74% in the highest quintile. For the delta-hedged option portfolios, the delta-hedged call and put option returns are all significantly positive at the 1% level for weekly returns. The returns of the 5-1 delta-hedged option portfolios are monotonically increasing with MPS regardless of return frequency. For example, the 5-day returns of delta-hedged put options increase from -0.41% to 0.21%. Overall, Table 3 and 4 show that the delta-hedged returns increase with both the ACGO and MPS measures and the weekly 5-1 delta-hedged option portfolios generate significantly positive returns, which indicate the existence of significant disposition effects on the option market. More importantly, it is observed that the patterns delta-hedged option returns tend to increase with the measures of disposition effects are more pronounced in terms of t-value when options are sorted by the MPS measure than by the ACGO measure. This suggests that the MPS measure is likely a more relevant estimate for the disposition effect on the option market.. 21.

(38) Table 3: Quintile portfolio returns formed on ACGO This table reports the average returns of unhedged options and delta-hedged options, and the average ACGO for each quintile portfolio sorted by the ACGO measure. The first group labeled as “1” includes all unhedged options (delta-hedged options) in the lowest ACGO quintile while the fifth group labeled as “5” includes all unhedged options (delta-hedged options) in the highest ACGO quintile. The “5-1”portfolio is formed by buying all unhedged options (delta-hedged options) in the highest ACGO quintile and selling all unhedged options (delta-hedged options) in the lowest one. Panel A: call options ACGO quintile portfolio 5-1 portfolio Variables 1 2 3 4 5 Mean Stdev t-value Unhedged option 1-day return -9.58 -3.62 -2.63 -3.15 -3.92 5.66 21.85 8.29 Unhedged option 5-day return -26.52 -11.20 -8.57 -9.88 -12.29 14.23 44.18 10.30 Unhedged option 10-day return -31.19 -15.10 -8.64 -7.15 -6.80 25.94 44.55 18.12 Unhedged option 20-day return -14.40 -8.63 -9.00 -10.88 16.31 48.46 10.03 -23.67 Delta-hedged 1-day return 0.14 0.14 0.09 0.04 -0.06 -0.20 1.52 -4.25 Delta-hedged 5-day return -0.41 -0.28 -0.22 -0.19 -0.16 0.25 0.76 10.40 Delta-hedged 10-day return -1.25 -0.29 -0.17 -0.29 -0.39 1.05 3.94 8.26 Delta-hedged 20-day return -0.55 -0.13 -0.08 -0.46 -0.78 0.44 5.88 2.25 ACGO (past 5 days) -27.45 -8.56 -1.21 5.48 18.08 Panel B: put options ACGO quintile portfolio 5-1 portfolio Variables Mean Stdev t-value 1 2 3 4 5 Unhedged option 1-day return -12.09 -5.52 -4.26 -4.56 -5.91 6.19 21.45 9.22 Unhedged option 5-day return -36.27 -18.70 -14.33 -14.87 -17.87 18.41 45.25 13.01 Unhedged option 10-day return -41.98 -28.78 -20.56 -18.26 -18.73 23.43 43.26 16.81 Unhedged option 20-day return -38.40 -38.54 -31.43 -28.09 -31.30 7.98 45.45 5.14 Delta-hedged 1-day return -0.08 -0.11 -0.15 -0.23 -0.31 -0.24 1.31 -5.77 Delta-hedged 5-day return -0.22 -0.06 -0.03 -0.02 -0.02 0.20 0.76 8.27 Delta-hedged 10-day return -2.36 -1.86 -1.64 -1.72 -1.55 0.89 3.32 8.30 Delta-hedged 20-day return -2.84 -3.30 -3.14 -3.17 -3.23 -0.12 4.36 -0.83 ACGO (past 5 days) -26.72 -10.06 -2.57 3.98 15.36. 22.

(39) Table 4: Quintile portfolio returns formed on MPS This table reports the average returns of the unhedged and delta-hedged quintile portfolios formed on ascending orders of their MPSs. Group “1” reports all portfolio returns of the lowest MPS quintile while group “5” reports all portfolio returns of the highest MPS quintile. The 5-1 portfolio is a long-short strategy of buying all unhedged/delta-hedged options of the highest MPS quintile and selling all unhedged/delta-hedged options of the lowest MPS quintile. The average returns and MPSs are in percentages. Panel A: call options MPS quintile portfolios 5-1 portfolio Variables 1 2 3 4 5 Mean Stdev t-value Unhedged option 1-day return -8.80 -8.27 -4.33 -1.32 -0.54 8.26 15.71 16.70 Unhedged option 5-day return -18.41 -25.13 -15.32 -6.46 -3.93 14.48 29.32 15.74 Unhedged option 10-day return -21.35 -19.16 -11.06 -5.01 -3.78 20.54 38.99 16.12 Unhedged option 20-day return -15.85 -17.01 -10.46 -6.31 -6.71 16.68 44.80 10.77 Delta-hedged 1-day return 0.21 0.09 0.04 0.05 -0.04 -0.25 1.72 -4.60 Delta-hedged 5-day return -0.67 -0.37 -0.21 -0.12 0.07 0.74 1.18 20.47 Delta-hedged 10-day return -0.24 -0.64 -0.48 -0.24 -0.41 0.22 4.30 1.44 Delta-hedged 20-day return 0.25 -0.41 -0.41 -0.33 -1.10 -0.19 6.41 -0.82 MPS (past 5 days) -44.41 -28.23 -10.69 -1.71 -2.24 Panel B: put options MPS quintile portfolios 5-1 portfolio Variables Mean Stdev t-value 1 2 3 4 5 Unhedged option 1-day return -10.77 -9.49 -6.11 -3.84 -2.17 8.60 15.41 17.98 Unhedged option 5-day return -23.97 -30.03 -23.29 -15.10 -8.74 15.23 25.15 19.69 Unhedged option 10-day return -26.78 -31.50 -25.63 -19.39 -13.88 15.44 31.50 14.87 Unhedged option 20-day return -26.85 -36.08 -35.46 -29.72 -22.19 11.31 38.42 8.16 Delta-hedged 1-day return -0.05 -0.21 -0.12 -0.18 -0.28 -0.23 1.48 -4.78 Delta-hedged 5-day return -0.41 -0.15 -0.04 0.05 0.21 0.62 0.92 21.92 Delta-hedged 10-day return -1.51 -2.04 -1.72 -1.63 -1.76 0.03 3.76 0.06 Delta-hedged 20-day return -2.20 -3.13 -3.22 -3.15 -3.16 -0.16 5.33 -0.92 MPS (past 5 days) -23.74 -19.41 -10.64 -3.72 -5.08. 23.

(40) 4.2 Regression tests of the option market disposition effect To control for risk factors that are likely generating part of the differences in the 5-1 portfolio returns, we run several regressions with control variables. Intuitively, if the 5-1 portfolio returns are driven by the disposition effect, then the intercept of the linear risk factor model should be significantly different from zero. Namely, if the returns of the 5-1 portfolio cannot be fully explained with the risk factors specified, then we can be more confident in drawing the conclusion that the 5-1 portfolio returns formed on the ACGO or the MPS measures are more likely to be driven by the disposition effect. Table 5 reports our regression results for the 5-1 portfolios by the ACGO measure. In Panel A, the intercepts for the 5–1 call portfolios of all models are significant at the 1% level. In Panel B, the intercepts for the 5–1 put portfolios are all significant different from zero at the 1% level. For the delta-hedged returns, the results also suggest that the delta-hedged returns of the 5-1 portfolio cannot be fully accounted for by these risk factors. For example, from Panel B of Table 5, the intercept is 0.16 and significant for the delta-hedged put portfolio in Model 3, which indicates that the abnormal 5-day return of the 5-1 portfolio is on average 0.16% during the sample period. Table 5 shows that price jumps and volatility jumps cannot explain the returns of the 5–1 option portfolios; their coefficients are not significantly different from zero. The inclusion of volatility smirks also fails to explain the portfolio returns. These findings suggest that the returns of the 5–1 portfolio constructed by sorting options according to their ACGO are not related to the expected skewness of underlying index and the crash fears (i.e., the jump-related factors) of option investors. Overall, we find that the option market trading exhibits strong disposition effects that cannot be explained by risk factors.. 24.

(41) Table 5: Regression tests for the 5-1 portfolio returns formed on ACGO This table reports the regression coefficients from regressing the 5-1 portfolio returns formed on ACGO against the following variables: Mkt_Rf, the market excess returns; SMB, small market capitalization minus big portfolio returns; HML, high book-to-market ratio minus low portfolio returns; MOM, the momentum risk factors; VIX, the changes in VIX index; Smirk, the weekly changes of the differences between the implied volatility of an OTM put option and that of an ATM call option; RV, the changes in the realized volatility computed using daily prices of last thirty days; Jump, the sum of past one month all daily S&P 500 returns lower than -4%; Vol. Jump, the sum of past one month all daily increases in the implied volatilities of ATM calls that are greater than 4%; Imp.Vol., the changes in the implied volatilities of ATM calls; minJump, the past one month minimum daily index returns; maxVol Jump, the maximum daily increases in the implied volatilities of ATM calls. The unhedged/delta-hedged 5-1 portfolio returns formed on ACGO is constructed by buying all unhedged/delta-hedged options of the highest ACGO quintile and selling those of the lowest ACGO quintile. Model 1 refers to equations (6) and (7); Model 2 refers to equations (8) and (9); Model 3 refers to equation (10) and (11). The t-statistics are reported in parentheses. Panel A: call option portfolios Regression Dependent variable model Model 1 5-1 unhedged call portfolio return. Model 2 Model 3 Model 1. 5-1 delta-hedged call portfolio return. Model 2 Model 3. Intercept. Mkt_Rf. SMB. HML. MOM. VIX. Smirk. 15.36 (10.40) 15.09 (5.62) 14.19 (5.30) 0.21 (9.69) 0.23 (5.86) 0.23 (5.69). -7.51 (-7.16) -7.66 (-7.23) -6.84 (-6.62) -0.01 (-0.59) -0.01 (-0.55) 0.00 (0.19). 0.71 (0.60) 0.84 (0.71) 0.57 (0.47) -0.01 (-0.73) -0.01 (-0.84) -0.02 (-0.91). 1.51 (1.29) 1.55 (1.32) 1.74 (1.47) -0.02 (-1.30) -0.02 (-1.36) -0.02 (-1.29). -0.39 (-0.54) -0.67 (-0.89) -0.45 (-0.61) -0.03 (-2.69) -0.03 (-2.38) -0.03 (-2.27). -0.04 (-4.81) -0.04 (-4.90). 1.52 (1.21) 1.43 (1.14) 1.10 (0.88) -0.04 (-2.27) -0.04 (-2.20) -0.04 (-2.27). Intercept. Mkt_Rf. SMB. HML. MOM. VIX. Smirk. 19.94 (12.75) 27.39 (9.72) 28.15 (10.17) 0.20 (7.98) 0.16 (3.46) 0.16 (3.59). 1.88 (1.69) 1.34 (1.20) -0.38 (-0.35) -0.07 (-4.16) -0.07 (-4.00) -0.09 (-4.82). 0.56 (0.45) 0.36 (0.29) 0.20 (0.16) 0.01 (0.70) 0.02 (0.79) 0.01 (0.70). 0.44 (0.36) 0.16 (0.13) 0.16 (0.13) -0.04 (-2.05) -0.04 (-1.95) -0.04 (-1.94). 1.34 (1.75) 0.98 (1.24) 0.98 (1.27) 0.00 (-0.25) 0.00 (-0.21) 0.00 (-0.17). -0.02 (-2.08) -0.03 (-2.84). -2.82 (-2.13) -2.80 (-2.12) -3.01 (-2.31) -0.10 (-4.44) -0.10 (-4.46) -0.10 (-4.58). 0.00 (-1.36) 0.00 (-1.28). Jump. Vol. Jump. Imp. Vol.. 1.18 (1.53) 1.00 (1.29). 0.02 (0.34) 0.04 (0.65). -3.85 (-4.06). -0.01 (-1.01) -0.01 (-1.17). 0.00 (-0.77) 0.00 (-0.61). 0.00 (-0.30). Jump. Vol. Jump. Imp. Vol.. 1.29 (1.57) 1.46 (1.81). -0.19 (-2.88) -0.19 (-3.06). -5.15 (-5.25). 0.00 (-0.02) 0.00 (0.06). 0.00 (1.10) 0.00 (1.08). -0.05 (-3.04). Panel B: put option portfolios Dependent variable. Regression model Model 1. 5-1 unhedged put portfolio return. Model 2 Model 3 Model 1. 5-1 delta-hedged put portfolio return. Model2 Model3. 0.00 (-2.23) 0.00 (-1.97). 25.