Why Do Option Returns Change Sign from Day to Night? *

Transcription

1 Why Do Option Returns Change Sign from Day to Night? * Dmitriy Muravyev and Xuechuan (Charles) Ni a Abstract Returns for S&P 500 index options are negative and large: -0.7% per day. Strikingly, when we decompose these delta-hedged option returns into intraday (open-toclose) and overnight (close-to-open) components, we find that average overnight returns are -1% while intraday returns are actually positive, 0.3% per day. A similar return pattern holds for all maturity and moneyness categories, equity and ETF options, and VIX futures. Rational theories struggle to explain positive intraday returns. We show that returns change sign and become positive because option prices fail to account for the well-known fact that stock volatility is substantially higher intraday than overnight. Thus, option market-makers, some of the most sophisticated investors, appear to completely ignore one of the strongest volatility seasonalities, which can be easily added to option pricing models. Finally, other option investors also appear unaware of this anomaly, which may explain its persistence. JEL Classification: G12, G13, G14 Keywords: Behavioral finance, option returns, volatility seasonality, market microstructure * We greatly appreciate comments from Pierluigi Balduzzi, Hui Chen, Peter Christoffersen, Pasquale Della Corte, Mathieu Fournier, Ben Golez, Kris Jacobs, Francis Longstaff, Loriano Mancini, Alan Marcus, Neil Pearson, Jeff Pontiff, and Grigory Vilkov. We also thank seminar participants at Boston College, the Sixth Risk Management Conference in Mont Tremblant, the 2015 IFSID Conference on Financial Derivatives and the 2016 EFA Conference for their helpful comments and suggestions. We thank Nanex and Eric Hunsader for providing the trade and quote data for options and their underlying stocks. We also thank Barry Schaudt and Boston College Research Services for computing support. a Both authors are from Carroll School of Management, Boston College, 140 Commonwealth Avenue, Chestnut Hill, MA Tel.: +1 (617) muravyev@bc.edu (Muravyev, corresponding author), xuechuan.ni@bc.edu (Ni)

2 1. Introduction Derivatives are an essential ingredient of a developed financial system, they help to complete the market and allow investors to hedge and speculate in a capital-efficient way. Yet policymakers are concerned that because derivatives are so complex, even professional investors may not fully understand their risks and may even occasionally misprice them. 1 We find evidence consistent with this concern in one of the world s most widely-studied, actively-traded and transparent derivatives markets the options market. 2 Option market-makers are among the most sophisticated investors. Therefore, one would expect option prices to be efficient and fair. Yet, strikingly, we find that option prices are systematically biased, and positive intraday option returns are hard to reconcile with rational models. The prices fail to account for a well-known volatility seasonality volatility is much higher during trading hours than overnight. This conclusion is particularly striking because volatility is obviously one of the main inputs to option pricing models, and these models can be easily adjusted to account for the volatility seasonality. This result may suggest that prices in other derivatives markets may similarly deviate from their fair values preventing efficient capital allocation. To understand our main result, let us first explain the intuition behind option returns. In the Black-Scholes-Merton model, an option buyer can perfectly replicate it by hedging continuously in the underlying; thus, a delta-hedged option portfolio earns a return that equals to the risk-free rate. That is, the risk premium embedded in options is zero in this baseline case. However, the delta-hedged option returns are on average negative in practice, implying that option sellers collect a risk premium from option buyers. Option delta-hedged returns are also directly related to, but should not be confused with, the variance risk premium, i.e., that the return variance implied from option prices on average exceeds the realized variance. Although option returns have 1 Derivatives received a lot of attention in the Dodd-Frank Act in the US and the European Market Infrastructure Regulation (Emir). Merton Miller nicely explains some of the concerns: But if derivatives have really made the financial system safer, not riskier, as I have claimed, why are we hearing so many calls these days for more regulation? Part of the answer, I suspect, comes from misunderstanding by the public and the financial press about how serious the risks really are. 2 Indeed, U.S. equity options had a notional volume of 372 trillion shares in 2015, which is about one-fifth of trading volume in U.S. equities. This number is from Option Clearing Corporation website s Annual Volume Statistics 1

3 been studied extensively, 3 there is an active debate about whether these large negative returns reflect compensation for taking risk or are due to mispricing. 4 Option investors are highly sophisticated, which makes the mispricing less likely: institutional investors account for most of option trading volume while market-makers use sophisticated models to set option prices. 5 Indeed, numerous studies show that option prices and volume are informative about future unscheduled events (e.g., mergers), stock returns, and volatility. We contribute to this debate by documenting a striking pattern in delta-hedged option returns. In particular, we show that option returns are only negative during the overnight period (i.e., outside of trading hours, from close to open) and are actually mildly positive intraday (from open to close). Specifically, overnight delta-hedged returns are -1.0% per day for index options, and -0.4% for equity options. These negative overnight returns are notably persistent over our sample period (2004 to 2013). In striking contrast, during the trading day, option returns flip sign and become positive: 0.3% per day for index options and 0.1% for equity options. This day-night effect is stronger for options with high embedded leverage such as short-term and out-of-the-money options. VIX futures returns show similar albeit weaker pattern. This day-night effect is not only puzzling in itself but also makes it harder to rationalize why option returns are so negative. Indeed, the literature struggles to explain why a trading strategy that sells a delta-hedged call (or put) on S&P 500 index earns large positive returns, which are 0.7% per day in our sample. 6 This strategy however is akin to picking up nickels in front of a steamroller and lost 80% of its capital during the financial crisis. As option returns are only negative overnight, this baseline strategy can be substantially improved by only selling option volatility overnight and having no position during the day. The proposed overnight strategy not only increases average return to 1.0% per day but also more than doubles its Sharpe ratio. Moreover, the overnight strategy is profitable in every three-month period including the financial crisis! Thus, this strategy poses new challenges for rational theories of the negative option 3 E.g., Bakshi and Kapadia (2003), Carr and Wu (2009), and Bakshi, Madan, and Panayotov (2010). 4 For example, Han (2008) and Bondarenko (2014) advocate the mispricing and sentiment explanations. 5 Muravyev and Pearson (2015) show that most option trades are executed using sophisticated algorithms not available to retail inventors. 6 This estimate is consistent with the prior literature that uses older data Coval and Shumway (2001), Bakshi and Kapadia (2003), Santa-Clara and Saretto (2007), and Broadie, Chernov, and Johannes (2009). 2

4 returns. Admittedly, large trading costs in index options make this trading strategy hard to implement in practice. However, we show that in some special cases, such as options on SPY ETF, it can potentially be profitable after transaction costs. Also, although high option trading costs limit the ability of arbitrageurs to eliminate the day-night effect, the costs cannot explain why this effect exists in the first place. We consider a number of potential risk- and friction-based explanations for the striking asymmetry between overnight and intraday option returns including stochastic volatility and price jumps, inability to adjust delta-hedge overnight, funding and other carry costs. None of these explanations can reproduce its main features. These theories particularly struggle to rationalize positive intraday returns. For example, if our sample period missed an intraday rare disaster (a peso problem ), then intraday returns should be even more positive. Even zero intraday returns would be highly puzzling. Simply put, why would put options, which provide insurance against market crashes, offer positive returns intraday? Also, negative overnight returns do not depend on volatility (such as VIX) and measures of tail risk; the returns are stable and negative in every three-month period including the crisis. In contrast, most theories predict that expected option returns should depend on market conditions. Finally, the market (index returns) and volatility risks (volatility futures returns) explain little of the day-night effect. We further discuss rational explanations in Section 5. As rational theories are unable to explain the day-night effect, this implies that option market-makers (OMMs) are not fully rational in a sense that these highly sophisticated investors post systematically biased prices. 7 Instead of settling on this residual conclusion, we test and confirm a simple specific behavioral explanation. In particular, option returns change sign from negative overnight to positive intraday because OMMs, and thus option prices, fail to account for the fact that stock volatility is much higher intraday than overnight. This well-known fact is perhaps the strongest volatility seasonality. 8 Strikingly, option prices are set as if total intraday and overnight volatilities were about the same totally ignoring the seasonality; then in fact intraday 7 Option market-making is highly concentrated. According to Citadel, as of late 2008, Citadel (30% of option volume, specialist in options on 1,655 stock names), Susquehanna (1,152 stock names), Timber Hill (1,124), Citi (554), Goldman Sachs (390), Morgan Stanley (286), UBS (218) dominated this market. 8 French and Roll (1986), Lockwood and Linn (1990), Stoll and Whaley (1990), and Chan, Chan, and Karolyi (1991) among numerous other papers 3

5 volatility is about 50% higher. This failure to account for the volatility seasonality translates into option returns. Indeed, option delta-hedged returns are proportional to the difference between realized and implied volatilities (Bakshi and Kapadia, 2003). Implied volatility is usually set slightly above the expected realized volatility resulting in negative average option returns which compensate investors for taking volatility/tail risk. Therefore, positive intraday returns imply that option prices systematically understate intraday volatility, and similarly large negative overnight returns imply that overnight volatility is overstated. Option close prices are too high, and open prices are too low. We test this behavioral explanation by exploiting the cross-sectional variation in the day-night volatility seasonality. First, we study the day-night effect for options on major exchange-traded-funds (ETFs). Both the day-night effect in option returns and the volatility seasonality are pronounced for major U.S. market and industry indices. However, international equity ETFs have about the same total volatility intraday and overnight (U.S. time). Strikingly, for these ETFs, overnight and intraday option returns are both negative and have similar magnitude. For some countries, such as China, the sign flips (overnight returns are positive) as most news arrive while the U.S. market is closed. Finally, our main test applies the same idea to the cross-section of equity options. Strikingly, the ratio of intraday-to-overnight stock volatility computed from historical data explains the day-night option return puzzle. That is, stocks with stronger day-night volatility seasonality (high day-night volatility ratio) have more pronounced option return asymmetry (more positive intraday and more negative overnight returns). 9 Also, both intraday and overnight option returns become negative and of similar magnitude after controlling for the volatility seasonality ratio. To better understand these volatility ratio results, we simulate a simple Black- Scholes economy with the volatility seasonality bias and the variance risk premium. Model parameters are chosen to match average daily option returns and volatility of S&P 500 index. As we increase the parameter responsible for how much option prices 9 Specifically, the intercept (alpha) in the Fama-MacBeth regression for intraday option returns flips its sign from positive to negative after controlling for this volatility ratio; i.e., option returns are negative in both periods after accounting for the volatility bias. Finally, the coefficients for the volatility ratio in the intraday and overnight return regressions not only have expected signs but also match in absolute value but with opposite signs (positive for intraday). This last result allow us later to conclude that option prices completely ignore the day-and-night volatility seasonality. 4

6 underreact to the volatility seasonality, overnight option returns become more negative, while intraday returns become less negative and ultimately turn positive. We simulate overnight and intraday returns from this model and then estimate the same Fama- MacBeth regressions as for the actual data. We compare simulated results for different degree of underreaction with the results for actual data and conclude that option investors not simply underreact but completely ignore the day-night volatility seasonality. Importantly, the model is able to match the magnitudes for both intraday and overnight returns under realistic assumptions about the day-night volatility seasonality, thus providing further support for the seasonality bias explanation. Finally, the model suggests additional tests to validate the volatility bias that we confirm in the data. Apparently, other option investors are also unaware of the day-night anomaly as implied by the timing of their option trades. Specifically, positive intraday returns encourage option investors to move their sell trades from morning to the afternoon. Contrary to this prediction, option order imbalance is actually more positive in the afternoon; that is, investors buy more (and thus sell less) options towards close. Admittedly, few option investors have access to expensive data and computing power required to uncover the day-night effect in option returns. The fact that all option investors seem unaware of this anomaly may explain its persistence. OMMs do not lose money due to the volatility bias and thus have little incentive to correct their models. Perhaps, OMMs use a relatively simple model to forecast future volatility that accounts for first-order effects such as volatility clustering, mean-reversion, the leverage effect, but ignores lesser known stylized facts such as seasonal patterns in volatility. We conduct numerous robustness tests. Our main results are robust to alternative definitions of open and close prices, different subsamples, and delta-hedging strategies. The correlation between intraday and overnight option returns is close to zero suggesting that the day-night effect is not due to measurement error in open/close option prices. This correlation is also zero in simulated data further supporting the proposed explanation. Finally, index option returns are non-negative in all of the intraday subperiods. Empirical work in option pricing typically relies on the estimation of fully specified parametric models. Option returns are more straightforward to interpret economically than the pricing errors of such models because returns represent the actual 5

7 gains or losses to an investor on purchased securities. Several others have also noted the advantages of analyzing average option returns. 10 Overall, we document a striking asymmetry between intraday and overnight option returns. This return asymmetry is hard to reconcile with rational option pricing models. We show that OMMs set prices that completely ignore the day-and-night volatility seasonality, and, thus cause option returns to be positive intraday and negative overnight. Other option investors also appear unaware of this return anomaly. Figlewski (2016) calls for more academic research on how behavioral factors affect option pricing, which is what we do in this paper by using option prices to infer biases in OMM s believes about volatility. 11 The remainder of the paper is organized as follows. In Section 2, we provide a brief review of the related literature and outline our contribution. In Section 3, we describe the data and the methodology. Section 4 documents the striking asymmetry between overnight and intraday option returns, while in Sections 5, we explain what causes the day-night effect and rule out alternative explanations. Section 6 concludes. 2. Literature and Contribution This paper contributes to several distinct literatures. First, we contribute to the literature that studies behavioral finance and investor irrationality. To our knowledge, this paper is the first to unambiguously show how some of the world s most sophisticated investors, option market-makers (and other option investors), make systematic mistakes in setting option prices for the most important option contract, S&P 500 index options. OMMs ignore one of the largest volatility seasonalities, that volatility is higher overnight, and as a result intraday option returns are positive, which is hard to reconcile with rational models. The fact that OMMs completely ignore the day-night volatility is extremely puzzling because volatility is obviously one of the main inputs to option pricing models, and these models can be easily adjusted to account for the volatility 10 See for example Coval and Shumway (2001), Bondarenko (2003), Driessen, Maenhout, and Vilkov (2009), Duarte and Jones (2007), Broadie, Chernov, and Johannes (2009), Goyal and Saretto (2009), Bakshi, Madan, and Panayotov (2010), and Muravyev (2016). 11 One direction in which it will certainly be appropriate to extend our theoretical models is to incorporate behavioral factors beyond the von Neumann-Morgenstern axioms. 6

8 seasonality. Besides, the idea that volatility seasonality should affect option prices goes back to at least Merton (1973) and French (1984). Although options provide leverage (Black, 1975) and lottery-like payoffs (Shefrin and Statman, 2000, 1993) that can attract speculators some of whom may act irrationally, surprisingly few papers study behavioral factors in derivatives markets. 12 Both Stein (1989) and Poteshman (2001) show that options implied volatility underreacts to individual daily changes in instantaneous variance and overreacts to periods of mostly increasing or mostly decreasing daily changes in variance. Han (2008) shows that changes in investor sentiment help explain time variation in the slope of index option smile and risk-neutral skewness beyond factors suggested by the current models. Jones and Shemesh (2016) show that returns for stock options are more negative over weekends than weekdays. Overall, these studies argue that the option market reacts in the right direction but the magnitudes are too large, while in our case, the sign of intraday return is wrong, and we are able to identify the mechanism behind the puzzle. Relatedly, the literature on the optimal exercise of equity options concludes that professional investors such as market-makers almost always exercise their options optimally while retail investors occasionally make mistakes, which is hardly surprising because optimal exercise boundaries are hard to compute. 13 We focus on systematic pricing mistakes by market-makers rather than occasional mistakes by retail investors. We know of only one other paper, Sheikh and Ronn (1994), that investigates intraday patterns in option returns. Using the data on short-term at-the-money options on 30 stocks for just 21 months ending pre-1987 crisis (pre volatility skew period), they find, among other results, that the adjusted option returns are more negative overnight than intraday but the difference is not statistically significant, perhaps because their sample is too small. Sheikh and Ronn focus on returns towards the end of trading day, and do not discuss overnight versus intraday returns, nor do they study index options. They argue that differences between option and equity market returns provide evidence of 12 Barberis and Thaler (2003), Subrahmanyam (2007), Hirshleifer (2008), and Shefrin (2009) provide extensive excellent surveys of behavioral finance. 13 Barraclough and Whaley (2012) show that retail investors sometimes fail to optimally exercise deep-inthe-money put options. Poteshman and Serbin (2003) present evidence that call options are sometimes exercised when they should be instead sold them. However, Battalio, Figlewski, and Neal (2014) and Jensen and Pedersen (2015) argue that such call exercises are optimal because of large trading costs in options and short-sale constraints. 7

9 information-based trading in options. Obviously, the options market changed substantially since mid-1980s. Also, Chan, Chung, and Johnson (1995) show that option volume exhibit a U-shaped pattern similar to stock volume; however, we are the first to examine intraday patterns in option order imbalance. Growing literature examines the day-night effect in equity returns. For example, Lockwood and Linn (1990) and more recently Cooper, Cliff, and Gulen (2008) show that all of the equity risk premium in their sample comes from overnight return. Lou, Polk, and Skouras (2015) examine how stock anomalies behave intraday/overnight. Despite apparent similarity, the equity market effect is driven by a different mechanism. First, unlike option prices stock prices are not directly linked to volatility. Thus, it s hard to see how the volatility bias that explains the options day-night effect would affect the equity risk premium. Second, options are delta-hedged so that their beta is close to zero, and thus option returns are uncorrelated with the directional changes in the underlying. We confirm that controlling for stock/index returns do not affect the option day-night effect. Third, unlike in the equity market, the autocorrelation between day and night returns is essentially zero in options. Finally, the options day-night effect is order of magnitude larger than its equity market counterpart, the latter amounts to less than one basis point per day in our sample. The fact that OMMs may ignore short-term swings in the underlying volatility, such as the day-night volatility seasonality, suggests that the results of intraday event studies using option data should be interpreted with caution. For example, positive option returns immediately after an intraday announcement, such as a macro news announcement, may indicate a risk premium associated with this event or alternatively OMMs simply ignored the event. In the latter case, option returns will also be positive as they are proportional to the difference between realized volatility, which is high after the announcement, and implied volatility, which does not change if the event is ignored. Relatedly, an analysis that is limited to just open and close VIX levels or other implied volatility measures (including model-free versions) tells us little about whether option prices are cheap or expensive. VIX is mechanically higher at open and lower at close (and also higher on Monday and lower on Friday) because it is computed based on calendar instead of business time, which is essentially another manifestation of the 8

10 volatility seasonality bias. 14 The option risk premium is determined by the difference between implied and realized variances; thus these two components should be studied jointly, and option returns provide a convenient way of doing this. Finally, as discussed in the introduction, our results are important for the option returns literature. 3. Data and Methodology We obtained stock and options data from Nanex, a firm specializing in highquality data feeds, which in turn obtains its data from standard data aggregators: OPRA for options and SIP for equities (e.g., TAQ data also use SIP). Our data include intraday quoted bid and ask prices at one-minute frequency for both options and the underlying equities. Our sample period is from January 2004 to April For options, we also observe best bid and offer (BBO) for all option exchanges. Timestamps are synchronized across markets. To reduce data size, only option contracts with at least one trade on a given day are included. Even given these constraints intended to reduce data size, the compressed data are still huge and require more than twelve terabytes of storage. The data also contain all option trades as well as BBO quoted prices preceding a trade. This information lets us compute option order imbalances. We use the quote rule applied to option trade and NBBO prices to determine whether an option trade is buyer or seller initiated; if a trade is at the NBBO quote midpoint, we apply the quote rule to the quoted bid and ask prices from the reporting exchange. Open price is computed as the quote midpoint at 9:40 a.m. We skip the first ten minutes of trading (both the equity and options markets open at 9:30 a.m. EST) because, as Chan, Chung, and Johnson (1995) first show and we confirm, option quotes are sporadic and bid-ask spreads are often wide right after market open. Close prices are based on quoted prices immediately before close, which is 4:00 p.m. for equity and 4:15 p.m. for S&P index options. These closing times for options match those in the underlying market. Our main results are robust to a number of alternative specifications for open and close prices, including taking average quote midpoint during the first and last ten minutes, using only bid or only ask prices, 10 am price as open or 4pm price as 14 For example, Kaplanski and Levi (2015) document this fact and attribute it to excessive perceived risk. 9

11 close. Table A.4 reports that the magnitude of the day-night effect changes very little with these alternative open and close prices. We apply standard data filters to option quoted prices. In order to compute option return over a given time period, we exclude option contracts for which at the beginning of this period (1) the bid price is greater or equal to the ask price, (2) the bid price is not available or is below 50 cents, (3) the quoted bid-ask spread is greater than 70% of the quote midpoint or is greater than three dollars, or (4) if option delta cannot be computed. For some of robustness tests, we merge our intraday data with daily stock and option prices from CRSP and OptionMetrics by ticker and date. Delta-hedged option returns are computed using deltas from the Black-Scholes- Merton (BSM) model; and the hedge is revised five times during a trading day (approximately every 80 minutes). In untabulated results, we confirm that our main results are robust to alternative hedging frequencies, including hedging only once per day. Following the literature, we define delta-hedged option dollar profit (P&L) for option contract with price between times 1 and as & = Δ, where Δ is option delta and is the underlying price at time. Option deltahedged return is then computed as = & We study regular option returns instead of excess returns because risk-free rate is negligible compared to option returns, and thus, as we show in untabulated results, subtracting it makes little difference. Following this definition, intraday (open-to-close) returns are computed as the intraday (open-to-close) dollar P&L for a long option position divided by option price from the open, where the dollar P&L is delta hedged at the beginning of each intraday sub-period. This definition is reasonable particularly for index options given the low funding costs for index futures. 15 In untabulated results, we show that alternative ways to normalize P&L (instead of simply dividing by option price) will expectedly affect the 15 e.g., Santa-Clara and Saretto (2009) discuss margin costs in the context of the options market 10

12 magnitudes of intraday and overnight option returns but not their signs, which is the main puzzle we are trying to solve. We first compute intraday and overnight returns for each option contract and then aggregate them into their average for each underlying, and finally take an equallyweighted average across underlying stocks (this step is not needed for S&P index options). We end up with two numbers on each day, one for overnight and the other for intraday return. With slightly less than ten years of data, we end up with more than 2200 daily observations. Our analysis is conducted separately for S&P 500 index options and equity options. Then required, we compute returns for option subsamples such as out-ofthe-money (OTM) index puts in a similar way. For robustness, we also examine leverage-adjusted option returns. Options of different moneyness and maturities differ in embedded leverage. Therefore, the deltahedged returns of these option subcategories are not directly comparable. To address this issue, we implement a very intuitive version of deleveraged option returns that is common in the literature. Specifically, the deleveraged option return for is defined as: =, h = Δ, is the delta-hedged option return for time period [ 1, ] defined above. is the deleveraged factor. In general, the value of the deleveraged factor,, is well above 5. This leverage is also higher for out-of-money options. The empirical results for deleveraged option returns are reported in Table A.1. We rely on average delta-hedge option returns to access the size of the risk premium embedded in options because this method is widely used and has a number of advantages over alternatives. The alternative is to estimate the risk premium within a fully-specified option pricing model. Such estimates can be sensitive to a model choice, with no consensus on which option model is the best. In contrast, option returns offer an (almost) model-free way of estimating the aggregate risk-premium embedded in options. A delta-hedged option portfolio has zero delta and thus zero beta, thus, in theory, its value is immune to small directional changes in the underlying price. Thus, all of the excess returns on this delta-hedged portfolio corresponds to the option-specific risk 11

13 premium. By analogy, the equity premium is typically estimated as an average excess returns for a stock portfolio rather than in a context of an equity valuation model (such as DCF). The option returns approach is easier to implement and interpret. On the other hand, option returns are highly volatile and a large sample size is needed to get precise estimates of average returns. Bakshi and Kapadia (2003) and Bakshi et al. (2010) show theoretically that delta-hedged returns provide a good way to estimate the variance riskpremium. Broadie, et al. (2009) argue that delta-hedged returns estimated at daily frequency are more robust to statistical biases than unhedged or longer-term option returns. 4. The Day-and-Night Effect in Option Returns 4.1. Statistical Properties of Overnight and Intraday Option Returns In this section, we explore the main statistical properties of overnight and intraday option returns. Figure 1 presents the main result of the paper. We decompose daily deltahedged option returns into intraday (open-to-close) and overnight (close-to-open) components. It is well known that delta-hedge returns for index options (and to a lesser degree for equity options) are negative on average. We show that these negative returns are entirely due to the returns from the overnight period, which are -1.0% per day, while intraday returns are mildly positive (0.3%). As discussed in the introduction, our magnitudes for total daily option returns are consistent with the literature (e.g., Coval and Shumway, 2001). Table 2 confirms that overnight and intraday returns are both statistically significant (t-statistics of -12 and 2.6 respectively). This day-night effect is also observed in equity options but magnitudes are expectedly lower: returns are -0.4% per day overnight and 0.1% intraday. Statistical significance is higher for equity options though (t-statistics of -19 and 3) as averaging across stocks reduces estimation error. Figure 2 shows that despite high variance, overnight returns are very stable over time; unlike intraday returns, average overnight returns remain at about the same level through the sample period. Specifically, the figure compares cumulative option returns over a three-month rolling window for two trading strategies. The conventional strategy of collecting the risk premium in options is to sell a delta-hedged option portfolio and keep the position open for the entire day (thus collecting both overnight and intraday 12

14 returns) while the other ( overnight ) strategy only keeps the short position open overnight and thus has no position intraday. The conventional strategy is highly profitable but these profits are highly volatile, and it loses more than 80% of capital in late In contrast, the overnight strategy has remarkably stable and positive returns even during the financial crisis. The overnight strategy is profitable in every three-month window. As a result, it offers more than twice the Sharpe ratio of the conventional strategy. Admittedly, the overnight strategy is harder to implement in practice because it requires frequent trading. Its average daily profits are smaller than a 6% effective bid-ask spread in S&P500 index options assuming an investor takes rather than provides liquidity. In Section 5.5, we discuss how options on SPY ETF, which have similar return properties but much smaller transactions costs, can be used to make the overnight strategy potentially profitable after costs. Importantly, high trading costs can explain why the anomaly does not disappear, but not why it exists in the first place. Table A.2 complements this analysis by reporting option returns by calendar year. Intraday returns are positive in most of the years. To better understand the nature of intraday returns, we compute option returns over five equal subperiods within the trading day in Table 2. The intraday sub-period returns show a striking seasonality: returns in the morning and noon are close to zero but in the afternoon and especially before close option returns become positive (0.16% and 0.19% in the last two sub-periods). Interestingly, average option returns have a different intraday seasonality than the underlying volatility, which has a pronounced U-shape pattern: volatility is highest at the open and close. These positive intraday returns are hard to explain with conventional option pricing models. The fact that returns are non-negative for all subperiods indicates that our results are not driven by some strange price behavior at open or close and are robust to alternative definitions of open and close prices. For example, defining open and close prices as an average quoted price during the first and last 15 minutes of trading has little effect on our results. In Table A.4, we further show that average overnight and intraday option returns change very little with alternative definitions of open and close prices. In particular, we compute option returns using only ask (or only bid) prices, to alleviate the concern that bid prices are unrepresentative and may bias the quote midpoint. 13

15 In another robustness check, we also confirm in Table 2 that the overnight returns are not driven by weekends overnight returns become slightly less negative, increasing from -1.0% to -0.8% if weekends are excluded. We thus confirm the finding of Jones and Shemesh (2016) that option returns are more negative over weekends (Friday to Monday). 16 In untabulated results, we test whether the volatility seasonality bias proposed in this paper can explain the weekend effect, and it does not. Unfortunately, the weekend effect remains a puzzle. The difference in average overnight and intraday returns cannot be explained by differences in higher moments of option return distribution. Table 1 shows that day and night option returns have about the same volatility of 4.5% and similar tail quantiles (1% and 99%). Thus, at least in terms of these naïve risk measures overnight and intraday returns are similarly risky. 17 After studying index and equity options, we investigate the day-night effect for major exchange-traded funds in Table 3 including major U.S. index ETFs as well as industry, commodity, fixed-income, and international equity ETFs. Besides confirming the robustness of our results, the variation of the day-night effect across ETFs hinted us about the mechanism behind the day-night effect. The delta-hedged option returns of U.S. index and industry ETFs have similar patterns to the S&P index option returns. However, option returns of international ETFs (e.g., tickers EEM and EFA) are negative both intraday and overnight. Even more interestingly, the day-night effect flips sign for the China Large-Cap ETF: overnight returns are positive and intraday returns are negative. These interesting exceptions encouraged us to examine the intraday/overnight volatility of these ETFs. For the Chiniese ETF, intraday volatility is less than overnight volatility; for international equity ETFs, the intraday volatility and overnight volatilities are roughly the same. Finally, for most of the major U.S. index ETFs and industry ETFs, intraday volatility is significantly higher than overnight volatility. Thus, the pattern in volatility matches the pattern in option returns! This anecdotal evidence suggests that the day-night 16 Also, Panel B of Table 2 reports a similar analysis for equity option returns: results are qualitatively similar but the magnitudes are expectedly smaller particularly for intraday returns. 17 Expectedly, the median return is lower than the mean because an option payoff is non-linear. Overnight return median is -1.2%, and thus our main result is not driven by outliers. Median intraday return is slightly negative (-0.38%) reflecting the fact that a buyer of an option straddle (put plus call) will lose money on a median day because stock price remains unchanged in this median scenario, and thus an option buyer loses time value. 14

16 effect is related to the ratio of intraday-to-overnight volatility for the underlying, and is strongly consistent with our proposed behavioral explanation the volatility seasonality bias. Note that this variations of option returns across ETFs is hard to explain for conventional option pricing theories. We will further discuss rational explanations of the day-night effect in Section 5. We also find evidence of the day-night effect in the VIX futures market. We first focus on VIX futures with shortest maturity as they have highest liquidity similarly to other futures markets. In Table 9, we show that intraday returns for VIX futures are close to zero (0.01%, statistically insignificant) while overnight returns are negative and significant (-0.15%). VIX futures are traded around the clock but are highly illiquid outside of normal trading hours. Thus, we use the same open and close time as for index options to compute VIX futures returns. All futures with maturities up to six months have negative overnight returns and slightly positive (or zero) intraday returns. After launching in 2004, the market for VIX futures really took off only recently, which made futures prices volatile in the beginning of our sample. Finally, we compare S&P 500 index returns for intraday and overnight periods in Table 4. Average returns are close to zero during our sample period: 0.008% for overnight period and % for intraday (see Table 4 Panel B). That is, the difference is only one basis point and is not statistically significant. We also compare higher moments of return distribution. Intraday period is only 6.5 hours long (regular trading hours) but its total volatility is 1.5 times higher than for the longer overnight period. Similarly, return percentiles are more extreme for intraday returns. Excess kurtosis is similar for two return distributions while skewness is significantly more negative for intraday returns. Interestingly, return skewness flips sign from negative to positive during the trading day Conditional Properties of Option Returns We next document how intraday and overnight option returns depend on option parameters. Table 1 shows that the return asymmetry becomes more pronounced as option moneyness decreases. For example, out-of-the-money options have highest leverage and thus more extreme returns: 0.27% intraday and -1.74% overnight; while inthe-money (ITM) options have little leverage/optionality and have day and night returns of only 0.07% and -0.22%. There is little difference between delta-hedged call and put 15

17 returns because both produce a similar straddle position after delta-hedging. Finally, Panel B of Table 1 reports that equity options have the same stylized facts, but the magnitudes are smaller (less extreme) perhaps because volatility risk for individual stocks can be partially diversified and thus carries less systematic risk. The return asymmetry declines with time-to-expiration, and thus short-term options have more extreme returns. Table 5 shows that options with less than three weeks to expiration have overnight and intraday returns of -2.6% and 0.7%, while returns for long-term options are close to zero. Returns for equity options show similar dependence. Table 6 double-sorts options based on maturity and moneyness and shows that the daynight effect is more pronounced for short-term and more out-of-the-money options. Inthe-money long-term options have both returns close to zero, while short-term OTM options have overnight returns of -5.3% and intraday returns 0.75%. The return magnitudes are gradually decreasing in time-to-maturity and option moneyness. Importantly, overnight returns are negative and intraday returns are positive for all moneyness and time-to-expiration categories. Thus, any portfolio that combines options with positive weights will have the day-night effect as observed for individual options. For example, call and puts can be combined to form a synthetic variance swap, which is used extensively to study the variance risk premium. Thus, according to this argument, the day-night effect will be also observed for variance swaps. Leverage can explain most of option return variation across maturity and moneyness. In Table A.1, we report the deleveraged version of S&P index option returns sorted by moneyness and time-to-expiration. As expected, return magnitudes decrease sharply after deleveraging, and more importantly, returns of different moneyness and time-to-expiration become comparable. After deleveraging, the option return dependency along the time-to-expiration disappears, at least no obvious pattern is observed. There still exists a weak decreasing pattern along the moneyness dimension. The deleveraged overnight option returns remain negative and range from -0.04% to -0.10% for out-ofthe-money and from % to % for in-the-money options. The out-of-money and short-term options bear the most negative overnight returns. The intraday returns are generally equivalent across moneyness and time-to-maturity. Most t-statistics remain significant for both overnight and intraday deleveraged returns. Thus, sign and statistical 16

18 significance of delta-hedged option returns, negative for overnight and positive for intraday, are not driven by leverage. We further explore the relation between delta-hedged index option returns and option Greeks. In particular, Table A.3 double-sorts options based on Theta and Vega (i.e., the sensitivity of option price to time-to-expiration and volatility respectively). Theta and Vega are computed using the BSM model. Note that unlike bounded option delta, Theta and Vega are unbounded. Therefore, double-sorting equal-weighted option portfolios are formed by sorting options into 4 groups with same number of options along both dimensions. The magnitude of delta-hedged option returns is decreasing along Vega dimension, as well as Theta, which is consistent with the findings for moneyness and time-to-expiration sorts. Short-term options are generally possess lower Theta and lower Vega compared with long-term options. To complement the results in Figure 2, we report the day and night effect for each of the ten years in our sample in Table A.2. The overnight returns for index options are negative and large in each year. They range from -0.77% in 2008 to -1.66% per day in By construction, delta-hedge option returns have the same properties as straddle returns, i.e., being long volatility, and thus are less negative (more positive) in 2007 and 2008 when volatility was extremely high and are more negative then volatility is low (such as in 2013 and 2005). Intraday returns vary significantly from year to year but are mostly positive; they range from -0.21% in 2012 to 1.59% in The returns in the last two intraday sub-periods are consistently positive. Interestingly, morning returns are negative in the first half of the sample period but then flip sign in the second half. We next explore how option returns depend on market conditions. We sort trading days into portfolios based on market volatility, tail risk, liquidity, interest rate, and investor sentiment. As reported in Panel A of Table 10 and consistent with visual evidence in Figure 2, market conditions explain only a small portion of overnight return variation. Overnight returns are slightly more negative when VIX is high, and interest rates and investor sentiment are low. Intraday returns on the other hand are extremely positive when volatility is high (0.97% per day) or option liquidity is low (0.57% per day). Interestingly, intraday returns depend differently on the two measures of investor sentiment that we use. The returns are increasing in AAII investor sentiment, which is 17

19 based on a survey of how bullish investors are on the stock market, but are decreasing in the Baker and Wurgler (2006) sentiment, which consists of six components including closed-end-fund discounts, market turnover, equity issuance, number of IPOs, and their first day return. 18 Interestingly, the BW sentiment is the only variable that produces significant spread between high and low portfolios for both overnight and intraday returns (-0.62% and -0.54%). To check whether rare disasters or systematic tail risk can explain the day-night effect, we study how index option returns depend on tail risk measures. We use two popular tail risk measures proposed by Kelly and Jiang (2014) (hence, KJ), and Du and Kapadia (2012) (hence, DK). In Panel B of Table 10 shows that systematic tail risk only explains a small fraction of variations in overnight option returns. Overnight returns are slightly more negative when tail risk is high, as measured by KJ or DK. However, the difference is statistically insignificant. Overall, tail risks struggle to explain overnight returns let along positive intraday returns. Both index options and VIX futures are claims to market volatility and thus their prices should be tightly linked, but the day-night effect is much stronger for index options. We show that the day-night effect in S&P index options cannot be explained by market and volatility factors measured as S&P index returns and VIX futures returns. Specifically, we estimate a regression of index option returns on VIX futures returns and index returns separately for intraday and overnight periods in Panel B of Table 11. First, delta-hedging works reasonably well as the coefficient for index returns is zero for intraday case and is relatively small for overnight period. Second, intraday returns for options and VIX futures are highly correlated with t-statistics reaching 17. However, overnight returns are much less correlated as the coefficient is lower than for the intraday regression (0.66 vs. 0.92), and t-statistics is only 5.6. Thus, the options and volatility futures markets are less integrated during overnight period. Most importantly, volatility and market risk factors explain only a tiny portion of the day-night effect. Indeed, the intercept, which corresponds to alpha/abnormal returns, is 0.24% for the intraday case that is only slightly smaller than average intraday return of 0.28%. Overnight return 18 Baker and Wurgler (2006) provide sentiment index data only until

20 decreases from -1.08% to -0.89% after controlling for market and volatility factors. Perhaps, the seasonality bias is stronger in the options market than in VIX futures Are Other Investors Aware of the Day-Night Effect? Option market-makers ignore the day-night volatility seasonality and thus post biased option prices. Are other option investors aware of this anomaly? How do they respond? To answer these questions we study how option investors trade over the trading day as measured by order imbalance. The day-night effect encourages option investors who sell volatility (by selling options and then delta-hedging them) to execute their trades in the afternoon rather than in the morning. Investors who sell options in the morning suffer from positive intraday returns and are not compensated for incurring additional volatility. Surprisingly, contrary to this prediction, option order imbalance is more positive in the afternoon, i.e., investors buy more instead of selling. Perhaps, option investors are not aware of the day-night effect. Table 7 reports how option order imbalance depends on the time of day. Following the literature, order imbalance is computed as the difference between the number of buyer- and seller-initiated trades divided by total number of trades; thus, order imbalance is contained between -100% and 100% (if all trades are buys). We first confirm a well-known fact that investors are generally buying puts in index options and writing covered calls 19 in equity options. While equity options imbalance does not vary much across intraday sub-periods, imbalance for index options is more interesting. During the first half of the trading day investors tend to buy index puts (about a 2% imbalance) with no order imbalance in index calls, but in the afternoon investors start purchasing more options both calls and puts. Imbalance for call options becomes positive (2%), and put imbalance becomes even more positive (5%) in the afternoon. Obviously, a 3% increase in order imbalance during the trading day is too small to explain positive intraday returns (besides, expected order imbalances should be reflected in prices in advance), but it allows us to reject the hypothesis that most investors are aware of the overnight effect, because under this hypothesis we should observe a large negative order imbalance toward the close. 19 Covered call is selling an OTM call combined with a long position in the stock 19

21 Overall, OMMs post systematically biased prices, but they do not incur losses from doing this because other investors are similarly unware of this anomaly. Apparently, all option investors missed the day-night effect. Perhaps, many of them simply do not have intraday options data and computational resources needed to uncover this anomaly. 5. Potential Explanations In this section, we explain the puzzling empirical facts described above. We consider a wide range of potential explanations including risk-based option pricing theories, market frictions and financial constraints, and finally behavioral explanations. Almost all of these theories struggle to replicate the asymmetry between intraday and overnight option returns. However, one behavioral explanation, the volatility seasonality bias, explains the empirical facts particularly well. We confirm the volatility bias explanation in several additional empirical tests. In one of these tests, we simulate a Black-Scholes economy with the volatility seasonality for different degrees of volatility underreaction and then compare simulated and actual data to conclude that option prices completely ignore the day-night volatility seasonality Challenges for Rational Explanations So far, we ve documented a number of puzzling empirical facts about option returns. In this section, we explore a wide range of potential explanations and settle on a simple behavioral theory that explains the facts. Two of our empirical results are particularly hard to explain. First, intraday returns for index options are positive. Second, overnight returns are large and negative, which creates the day-night return asymmetry. Risk-based and friction-based theories struggle to explain these facts, particularly the positive intraday returns. First, rational theories (e.g., stochastic volatility and jump models) under mild assumptions imply that expected returns for a delta-hedged portfolio of index options should be negative. The intuition is simple. A long delta-hedged position in a call or a put option has a V-shaped payoff: it makes money if the underlying price moves up or down significantly from its current level. Thus, this option portfolio provides valuable insurance against market crashes and thus should earn negative excess returns. Numerous theoretical papers formally show this point. Second, it is even harder to construct a model where option returns changes sign from day to night. After all, the 20

22 stochastic process for the underlying is not that different across the two periods and if anything the overnight period has lower total return variance. Many conventional optionpricing models even assume that the price of volatility risk is negative and constant, which is clearly inconsistent with the return asymmetry. Third, if overnight returns are risky, then a strategy of selling volatility overnight should occasionally lose money, but it is profitable in every year in our sample including the financial crisis. Finally, rare disasters ( peso problem ) could bias our estimates of the intraday/overnight average returns, but then overnight and intraday returns should depend on the disaster likelihood as captured by tail risk measures but they do not in the data. Also, rare disasters imply that positive intraday returns should be even more positive and thus cannot explain the day-night effect. Overall, risk-based theories clearly struggle to explain the empirical results - they fail to match the sign of average option returns let along their magnitudes and cross-sectional implications (such as the results for ETF options). We next discuss several financial frictions. Perhaps, the overnight period differs from intraday in some fundamental ways. First, the underlying market is very liquid intraday but is illiquid overnight. Thus, while investors can delta-hedge frequently and seamlessly during the trading day, they cannot adjust their hedges during the night. 20 As a result, although return variance is larger intraday, volatility of an option portfolio can be substantially reduced intraday by hedging in the underlying (index futures). Thus, the overnight period has more residual volatility and thus is riskier in this sense. Therefore, option investors may require a larger premium to carry positions overnight. This theory seems very natural, and it can potentially explain why overnight returns are more negative than intraday returns but it cannot explain the remaining facts. First, this theory predicts negative intraday returns (just less negative than overnight returns), but the returns are positive. Second, overnight returns should be more negative when volatility is high (high VIX) as market-makers require higher compensation for taking the overnight risk the risk is proportional to MM s risk-aversion, position size, and volatility. But overnight returns show little dependence on volatility. Finally, as the overnight trading in index futures became more liquid in recent year, adjusting the overnight delta-hedges 20 For example, Figlewski (1989) shows that even with frequent delta-hedging the residual standard deviation that remained unhedged is large. Thus, even small transaction costs make the market incomplete. 21

23 became easier and thus overnight returns are expected to be less negative in recent years, but average overnight returns do not chance over the sample period. Even more surprisingly, we show in Section 5.3 that after controlling for the volatility bias the overnight and intraday periods carry similar option risk-premium. Overall, it is certainly harder for market-makers to hedge option positions overnight, which can potentially expose them to different jump risks. However, we find little evidence that this financial friction is priced, it certainly cannot explain the positive intraday returns. Second, although S&P 500 index options are some of the most important and actively traded options in the world, their trading costs are large. The effective bid-ask spread is more than 6%, and it decreased little over time. 21 High trading costs may explain why arbitragers do not eliminate the day-night effect, but not why this anomaly exists in the first place. In Section 5.5, we tentatively show that arbitragers can potentially make positive after-costs profits if they take liquidity in a smart way. A related concern is that the day-night effect is somehow mechanical because, following the literature, we compute option returns from the quote midpoints. To address this concern, we show that the size of the day-night effect does not depend on the option bid-ask spread and alternative price specifications (such as computing returns from only bid or only ask prices). Also, the correlation between day and night returns is close to zero unlike a pronounced negative correlation which is typical for a mechanical case (e.g., if open or close prices are special compared to other intraday prices). Borrowing and margin costs are usually computed using positions at close and are thus incurred during the overnight period. If these costs drive the overnight effect, the effect should be more pronounced during periods of high interest rates. However, the overnight returns are actually slightly more negative when the interest rates are low. Next, order imbalance is slightly positive for index option, but its price pressure is too small to explain positive intraday returns as discussed in Section 4.1. Finally, the underlying index returns are positive overnight and zero intraday. However, as discussed in Section 2, the magnitude of this effect is only one basis point per day and is small compared to the day-night effect in options. We also confirm that controlling for index 21 We compute effective spreads as twice the difference between trade price and the quote midpoint immediately before a trade, adjusted for trade sign and normalized by option price. 22

24 returns does not affect the magnitude for the abnormal day and night option returns. Overall, all these frictions are not successful at explaining the day-night effect Behavioral Explanations Given that rational explanations fail to explain the day-night effect, we now turn to behavioral explanations. First, option investors may underreact to the day-night seasonality in the underlying volatility, which as we show below, is the main cause of this puzzle. Second, they may fail to continuously adjust time-to-expiration during the trading day and thus overstate option maturity by almost one day at the end of the trading day. According to the volatility seasonality bias that we propose, option prices reflect the underlying volatility correctly in general; however, they ignore the day-night volatility seasonality. The underlying volatility is much higher (by 50%) intraday than during overnight period. This is perhaps one of the strongest known volatility seasonalities. Thus, the underlying volatility during the life of an option can be viewed as a sequence of high (intraday) and low (overnight) volatility periods stacked together. Option prices are proportional to the total volatility expected before expiration and thus should reflect this seasonality. Compared to the no-seasonality (equal volatility) case, option prices should be slightly higher at open because there are more high volatility than low volatility periods left before option maturity (one more as options expire at close). However, this intuition about option prices can be tricky to translate into expected option returns because if the underlying price does not change, an option loses time value, and thus its returns are negative. But, the underlying price rarely remains unchanged. A more intuitive way to think about returns is that option returns are proportional to the difference between implied and realized volatilities during a given period. Implied volatility is usually set slightly above the expected realized volatility, which results in the negative average option returns. The failure to account for the volatility seasonality translates into option returns. Positive intraday returns indicate that option prices systematically underestimate intraday volatility, and similarly large negative returns overnight suggest overnight volatility is overstated. That is, option prices underreact or even completely ignore the day-night volatility seasonality - option close prices are too 23

25 high and the open prices are too low. Figure 3 illustrates how the volatility bias effects the relationship between implied and realized volatilities. We also consider another plausible behavioral explanation perhaps option market-makers only adjust time-to-maturity at open instead of continuously changing it during the trading day: that is, a 30-day option remains exactly 30-day during the entire trading day, and becomes 29-day only at the next-day open. As option prices are increasing in time-to-expiration, this bias causes closing prices to be too high and thus option returns are negative overnight and positive intraday. Although this theory produces the option return asymmetry it fails on other dimensions. In particular, in untabulated results, we simulate a Heston economy with the time-to-maturity bias. The simulations produce negative overnight returns and positive intraday returns. However, if we match the overnight magnitude, the intraday magnitude is simply too large to be consistent with the data. Furthermore, the time-to-maturity bias implies that the deltahedged overnight option returns should be always negative on average and comparable across different ETFs and stocks: this prediction is inconsistent with the results in Table 2 Panel C that clearly shows a lot of variation in the overnight/intraday option returns across different ETFs. Overall, this exercise illustrates that even finding a behavioral explanation for the day-night effect is not easy, which makes even more striking how the volatility seasonality bias matches the stylized facts in the data so well Testing for the Volatility Seasonality Bias As discussed above, the volatility seasonality bias can certainly make returns for S&P index options positive intraday and negative overnight. But we don t stop there and conduct two sets of empirical tests to validate this explanation of the day-night effect. In the next section, we test whether the bias is able to produce the return magnitudes observed in the data. In this section, we explore the cross-section of stocks. The volatility bias implies that stocks with more pronounced day-night volatility seasonality should have higher asymmetry between overnight and intraday option returns. Indeed according to the bias, OMMs price options at the average volatility ignoring that actual volatility deviates significantly from the average during intraday and overnight periods. Thus, holding everything else constant, the more intraday volatility deviates from the overall volatility average (day + night), the more positive intraday option returns will be. This 24

26 deviation (i.e., the volatility seasonality) can be measured as simply the ratio between intraday and overnight volatilities, =,,. Intraday volatility,, (, ), is measured as the standard deviation of intraday (overnight) underlying returns over preceding 60 days. This volatility ratio is roughly 1.5 for an average stock in our sample. As discussed in Section 4.1 and Table 3, we first apply this test to major ETFs and confirm the predictions of the volatility bias. ETFs with high day-to-night volatility ratio (such as U.S. major and industry indices) have positive intraday and negative overnight returns (i.e., high option return asymmetry), while ETF with low volatility ratio (such as international equity indices) have negative returns for both overnight and intraday (i.e., little return asymmetry). Encouraged by this anecdotal evidence from ETFs, we conduct a formal test generalizing this idea to the cross-section of stocks. In particular, we show that the daynight volatility ratio can explain the day-night effect for equity options. In Table 8, we estimate separate Fama-MacBeth regressions for intraday and overnight option returns (as a dependent variable). In the intercept-only regressions, intercepts are 0.1 and -0.4 (%) matching average intraday and overnight returns (respectively) from Figure 1 Panel B. Next, we add the day-night volatility ratio to the regressions. As predicted by the volatility bias, the coefficient for the ratio is positive for intraday case (stronger seasonality causes more positive intraday returns) and negative for overnight (stronger seasonality more negative overnight returns). The result is highly statistically significant as t-statistics are larger than ten. Remarkably, the coefficients for the volatility ratio for day and night periods have the same absolute size but differ in sign. In the next section, we show that this pattern is not a coincidence, it is a direct implication of the volatility bias if option prices completely ignore the day-night volatility seasonality. Interestingly, after we add the volatility ratio, the intercepts change from (day = 0.1, night = -0.4) to (-0.15, -0.26). Thus, after controlling for the volatility bias, option returns are negative and of similar magnitude (though statistically different) for both intraday and overnight periods. Finally, all these results generally hold if other controls are included in the regression as reported in the last two columns of Table 8. 25

27 Overall, the results of the cross-sectional test strongly support the volatility seasonality bias explanation for the day-night effect. Specifically, the volatility ratio negatively (positively) predicts subsequent overnight (intraday) option returns in the cross-section, exactly as implied by the volatility bias. In the next section, we introduce a simple model that help us interpret some of the coefficient relationships found above A Black-Scholes-Merton Economy with Volatility Seasonality In this section, we introduce the day-night volatility seasonality in the simplest option pricing model, the Black-Scholes-Merton model. The model allows us to control for how much option prices underreact to the volatility seasonality and thus study the volatility bias in controlled settings. Strikingly, this simple model can replicate the puzzling empirical facts. This calibration exercise also indicates that option prices not simply underreact but completely ignore the day-night volatility seasonality. Perhaps, OMMs are simply unaware of it. First we explain the basic setup and then simulate it. The underlying price,, follows a geometric Brownian motion but with deterministic time-varying volatility to incorporate the day-night volatility seasonality. Specifically, = +, where is a Brownian motion with no drift, and is the annualized instantaneous volatility of the process for the underlying. In particular, to introduce the volatility seasonality, we set = for intraday periods, and = for overnight periods, i.e., a different constant for the two periods. Obviously, this is a minor adjustment to the classic BSM model, and option prices can be easily solved for. The European call and put option prices for the no dividend case are: where =, =, = [ + ], +, = +, 26

28 and is the cumulative function of standard Gaussian distribution. is sum of the intraday periods over, represented in years. Similarly, is the sum of the overnight periods over. These formulas are much simpler than they may appear, if = =, they simplify to the standard BSM prices. We choose model parameters to match key return moments for S&P index and its options. In particular, we assume expected return of =8.7%, volatility =20.6% (= 1.3% 252), risk-free rate =2.4%, the day-to-night volatility ratio (or ) =1.8, and implied volatility =28%. Panel A of Table 12 summarizes the parameter choice. Relatedly, the implied volatility is set higher than the actual volatility to produce the -0.7% daily delta-hedged option return observed in the data. Havinghigher relative to is a common way to introduce the variance risk premium in the BSM model. Obviously, we can repeat our calibration exercise with a more general model with endogenous variance premium, such as the Heston model, but we purposefully chose the simplest option model possible to make the intuition clear. As the underlying returns are not autocorrelated in the data, given and, the intraday volatility and overnight volatility can be decomposed as: = 1+, 1 = 1+, where =. 22 Option market-makers have biased believes about the day-night volatility, i.e., they price options assuming the day-night volatility ratio that in general differs from the true ratio. If =, we are back to the fully rational case with no volatility bias. If = =1, option prices completely ignore the volatility seasonality, which we refer to as full bias. We will infer from the data. After choosing the parameters, we simulate the model. 23 Figure 4 reports intraday and overnight option returns for different levels of undereaction to volatility 22 Note that ( ) is not scaled to the daily level. Therefore, when computing option prices, and are scaled to daily level. However, when we talk about the volatility ratio, such as or, we usually mean the ratio before scaling. 23 Specifically, we simulate the economy with 50-year long and 365-day in each year. The first 10% sample is treated as burn-in period and therefore, is discarded. In the benchmark case, the volatility seasonality bias parameter λ is set to Full Bias: the annualized σ =σ =σ. To be noticed that the λ is not 1 in 27

29 seasonality,λ. If there is no undereaction (λ = =1.8), i.e., market-makers have rational beliefs, then intraday return is even more negative than overnight return because total volatility is higher intraday. Similarly, for other volatility undereaction levels λ = 1.5 and 1.2, option intraday returns remain negative. However, intraday returns are increasingly less negative as we increase the degree of volatility underreaction. Finally, the Full Bias case λ =1 generates returns of -0.94% for overnight and 0.21% for intraday. These simulated returns are strikingly close to the option returns observed in the data (-1.04% and 0.28% in Table 2 Panel A)! The model is able to match the magnitudes for both intraday and overnight option returns under realistic assumptions about the daynight volatility seasonality, thus providing further support for the seasonality bias explanation. This result allows us to conclude that OMMs not simply underreact but completely ignore the day-night volatility seasonality, i.e., λ =1. The full bias case is actually more natural than partial undereaction. If market-makers are aware of the volatility seasonality, i.e. their option pricing models are adjusted to account for it, why would they misestimate the day-night volatility ratio? However, if they are completely unaware of the day-night seasonality, then this and perhaps other similar seasonalities are simply not part of their models. Panel B of Table 12 complements the graphical evidence in Figure 4. Also, note that the volatility seasonality bias does not affect the daily variance risk premium, but only the split between intraday and overnight periods. The model generates an additional prediction that we test in the data. If marketmakers underreact to the volatility seasonality, the correlation between overnight and intraday option returns must be zero. This correlation is indeed close zero in the data (0.02). This result provides an independent validation for the volatility bias hypothesis and alleviates concerns that this anomaly is simply mechanical (for instance, in the underlying stock market, the correlation between overnight and intraday returns is negative). So far, we established that the striking asymmetry between intraday and overnight index returns can be explained by option prices completely ignoring the day-night this case. Before scaling, λ = 390/ Full Bias indicates that option market makers price options using σ only and completely ignore the volatility seasonality. 28

30 volatility seasonality. We now use the same model to simulate the cross-sectional implications of the volatility bias. These simulations help interpret our cross-sectional results in Section 5.3 and Table 8, where we find that stocks with more pronounced daynight volatility seasonality have more pronounced option return asymmetry, we also find striking regression coefficient patterns for the volatility ratio. We now show these patterns are not coincidental and are implied by the model with the volatility bias. We extend the baseline model to a cross-section of assets with different volatility ratios, λ. Specifically, we assume the full volatility bias case and uniformly draw λ from the range [0.5, 2]. This range obviously affects unconditional mean of overnight and intraday option returns, but not the regression coefficient relationships of interest. For example, the coefficients of volatility ratio for intraday and overnight regressions are of similar magnitude but with opposite sign for all of the alternative specifications for λ that we tried. In Table 12 Panel C, we report results for the Fama-MacBeth cross-sectional regression for simulated data, i.e., how much intraday/overnight option returns depend on the day-night volatility ratio. We find exactly the same patterns as in the regressions for actual data. First, the intercept in the intraday return regression flips sign from positive (unconditional return) to negative after controlling for the day-night volatility ratio. The negative conditional intercept reflects the true option risk premium embodied in the model, which is negative. Note that t-statistics are huge as we can simulate as much data as we want. Second, the volatility ratio positively predicts intraday returns and negatively overnight returns. That is, the day-night return asymmetry increases with the volatility ratio. Most importantly, the coefficients for the volatility ratio λ in the intraday and overnight regressions have the same magnitude and opposite sign, β 29 and β, equal to 0.33 and -0.33, respectively. A formal statistical test confirms this relationship. Strikingly, we observe almost exactly the same pattern in the actual data (Table 8) same magnitude but opposite sign! Finally, both intraday and overnight option returns become negative and of similar magnitude after controlling for the volatility seasonality ratio; this again matches what we found in the data. In summary, we model the volatility seasonality bias in a simple BSM model with variance risk premium and volatility seasonality. Strikingly, if market-makers completely

31 ignore the day-night volatility seasonality, this model not only produces the return asymmetry but also matches the magnitudes of both intraday and overnight option returns. Even more strikingly, the cross-sectional tests implied by the model are confirmed in the data. Overall, the results in this section provide an overwhelming support for the volatility bias explanation of the day-night effect Trading Strategy Practitioners may wonder whether the day-night bias can be turned into a profitable trading strategy by capturing the large overnight returns. In short, the answer is yes, but only for a small subset of investors who are very careful about their trade execution (e.g., hedge funds that specialize in both trading options and trade execution). The costs for an average investor are too high to trade this strategy profitably; however, they can still benefit from the day-night effect (i.e., reduce costs and risks) by executing their option sales in the afternoon instead of the morning. Importantly, marginal investors, who have low execution costs, rather than average investors are responsible for arbitraging away good deals like this. At first glance, the option trading costs are ridiculously large. For example, the effective bid-ask spread for S&P 500 index options is about 6% in our sample. It is hard to find an option trading strategy that is profitable after these spreads are accounted for. Do most investors actually pay such large spreads? No, Muravyev and Person (2015, MP henceforth) show that most investors time their trades and pay lower spreads. Investors who specialize in the trade timing pay as much as one firth of the effective bid-ask spread when they take liquidity. Of course, investors can also reduce costs by providing liquidity; however, they have relatively little time to execute their trades in our case. For the trading strategy, we focus on options on SPDR S&P 500 ETF (ticker SPY) that are a close substitute for S&P index options but incur much lower transaction costs, and thus are better suited for strategies with high turnover. Next, we use trading cost measures introduced by MP (2015). That is, using the option trade data, we compute the effective bid-ask spread adjusted for the fact that many investors time their trades to reduce transaction costs. Following MP (2015), each trade is assigned the likelihood of being initiated by an execution timing algorithm, which allows us to compute trading 30

32 costs for two investor types: execution algorithms ( algos, who care about trading costs) and everybody else ( non-algos ). In Table 13, we compare overnight returns and trading costs for options on SPY. We report results for two subperiods before and after the Penny Pilot reform reduced the tick size for SPY options to one penny on September 28, 2007 (SPY options were launched in January 2005 and our sample period ends in 2013). An average overnight return for SPY options is -0.64% 24 (an intraday return is 0.18%), and is identical before and after the Penny Pilot. Trading costs however decreased a lot after the tick size reduction. The costs for non-algos, which is the conventional effective bid-ask spread, decreased from 3.9% to 1.2%. Algo-trader s costs decreased from 0.66% to 0.05%. Thus, a hypothetical trading strategy that sells SPY options overnight and incurs transaction costs typical for an algo-trader was breaking even in the pre-pilot period (-0.01% = 0.65% %), and is highly profitable in the post-pilot period (0.6% per day) as the profits didn t change but the costs decreased substantially. We are using the transaction costs for algo-traders because they are the marginal investors in this high-cost market. The costs for other investors are too high to profit from the overnight strategy. In summary, option trading costs decreased after the Penny Pilot making the overnight strategy potentially profitable for algo-traders (but only for them). So far, we focused on the bid-ask spread because it is typically much larger than other costs in the options market such as hedging costs in the underlying (e.g., Figlewski, 1989), brokerage/exchange commissions, margin/funding costs, execution uncertainty, and price impact; however, obviously, such costs should be accounted for in a more detailed analysis. Our results are consistent with OMMs using a relatively simple model for volatility forecasting that probably includes first-order effects (volatility clustering, mean-reversion, leverage effect, earnings announcements) but ignores less obvious stylized facts such as volatility seasonalities. If so, then option price underreaction to these other stylized facts can be perhaps used to generate profitable trading signals. 24 The return magnitudes are slightly lower than for S&P index options because we average over all available liquid options, and there are more short term dated options on S&P index, which lowers their aggregate time-to-expiration and thus produces more extreme option returns. 31

33 Overall, these results indicate that some investors may be able to turn the daynight effect into a potentially profitable trading strategy. Whether these investors are aware of the day-night effect is not clear though. Of course, the debate about whether there is a profitable trading strategy here does not answer a more fundamental question about why this effect exists in the first place. 6. Conclusion In this paper, we document a striking pattern in option returns. The returns are negative for the overnight period but are positive during the trading day. We also document a number of empirical properties of this day-night effect. Specifically, overnight returns are persistent over time and depend little on volatility, tail risk, and other market conditions; and the market and volatility risks explain little of this effect. These findings shed new light on the determinants of option returns. Rational theories struggle to explain the day-night effect, particularly positive intraday returns. However, a behavioral explanation related to the day-night volatility seasonality fits our empirical results particularly well. Specifically, we show that option returns become positive intraday because option traders completely ignore the wellknown fact that stock volatility is much higher intraday than overnight. Thus, option prices apparently do not properly reflect one of the strongest volatility seasonalities, suggesting that option investors may be less rational than previously thought. We also show that besides market-makers, other option investors are also unaware of this anomaly, which may explain its persistence. To our knowledge, this paper is first to unambiguously show how some of the world s most sophisticated investors, option market-makers, make systematic mistakes in setting option prices for the most important option contract, S&P 500 index options. This conclusion is particularly striking because volatility is obviously one of the main inputs to option pricing models, and these models can be easily adjusted to account for the volatility seasonality. 32

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37 Figure 1 Overnight (close-to-open) and intraday (open-to-close) delta-hedged returns for S&P index options (Panel A) and equity options (Panel B). Returns are in percentage points per day; e.g., a -1.04% daily return for overnight index options. All returns are statistically significant (see Table 2). Panel A S&P 500 Index Option Returns 36

38 Figure 1 Panel B: Equity Option Returns 37

39 Figure 2 Three-month rolling returns for two trading strategies that sell S&P500 index volatility (i.e., sell delta-hedged options). The conventional strategy keeps the position for entire day (thin dashed blue line) while the proposed strategy for only the overnight period (thick solid orange line). An investor sells calls and puts that traded at least once on a given day and then deltahedges the position in the index futures market. Option returns are computed using quote midpoints. 38

40 Figure 3 Hypothetical example of how the difference in overnight and intraday volatilities affects option returns. If overnight and intraday volatilities are equal, then option returns are similarly negative in both periods ( Vol. Ratio = 1 ). However, in a more common case then total volatility is much higher intraday than overnight ( Vol. Ratio = 2 ), implied volatility greatly overestimates overnight volatility and understates intraday volatility, which leads to large negative overnight option returns and somewhat positive intraday returns. 39

41 Figure 4 Delta-hedged option returns with volatility seasonality bias This figure shows the intraday vs. overnight delta-hedged option returns in a simulated Black-Scholes-Merton economy with different degree of the volatility seasonality bias. The structural parameter configuration follows from Table 12 Panel A: =8.7%, = 20.6% (=1.3% 252), =2.4%, (or ) =1.8, and =28%. We choose > to generate sizable variance risk premium. is the day-to-night volatility ratio of the underlying asset.,=, measures the degree of the volatility seasonality bias: if <, this means the option market maker underreacts to the volatility seasonality. In particular, Full Bias case means the option market maker completely ignores the volatility seasonality and treats: =. = 40