Does Option Market Volume Signal Bad News?

Transcription

1 Does Option Market Volume Signal Bad News? Travis L. Johnson Stanford University Graduate School of Business Eric C. So Stanford University Graduate School of Business June 2010 Abstract The concentration of trading volume in option markets, relative to equity markets, is a negative cross-sectional signal for future equity returns. In a model of trade in equity and option markets with asymmetric information and short sale costs, we show that option volume is a negative signal of future firm value because informed traders use options more frequently for bad news. We empirically confirm this hypothesis using the ratio of option volume to equity volume (OVR); a strategy long in the lowest decile of OVR and short in the highest earns 1.17% monthly on a risk-adjusted basis. Our model and empirical tests both indicate that OVR is a stronger signal when short sale costs are high or option leverage is low. We also find that OVR predicts both the earnings surprise and abnormal returns for quarterly earnings announcements in the following month, consistent with OVR reflecting private information about the firm. Contact s: [email protected], [email protected]. We thank Anat Admati, Darrell Duffie, Jeff Harris, Sebastian Infante, Stefan Nagel, Paul Pfleiderer and seminar partipants at the Stanford Joint Accounting/Finance Workshop for their helpful comments and suggestions. Please do not cite without permission.

2 Does option market volume signal bad news? We provide theoretical and empirical evidence that the concentration of volume in option markets, relative to equity markets, is an indication of negative private information. We model the capital allocation decision of privately informed traders who can trade in option and equity markets. Facing both capital constraints and short sale costs, the informed traders must decide how to allocate resources across the two markets. Short sale costs lead informed agents to trade options more frequently for bad news, thus predicting a negative relation between relative option volume and future equity value. We empirically document that the concentration of trading volume in options markets is a negative cross-sectional signal for future equity returns, suggesting that market frictions prevent equity prices from immediately reflecting the option volume signal. In each month of our sample, firms are sorted by their relative option volume ratio, OVR, defined as the ratio of total option market volume (aggregated across call and put contracts) to total equity market volume. After waiting one trading day, a portfolio is formed consisting of a short position in stocks with high OVR and a long position in stocks with low OVR. This portfolio provides an average riskadjusted hedge return of 1.17% in the month following the formation date (15% annualized) after controlling for the Fama-French, momentum and liquidity factors. The paper most closely related to our own is Easley, O Hara, and Srinivas (1998) [hereafter referred to as EOS], which contains a multimarket equilibrium model wherein privately informed traders are allowed to trade in both option and equity markets. The authors point out that asymmetric information violates the standard assumptions underlying complete markets and, therefore, the option trading process is not redundant. 1 Their model highlights the conditions under which informed traders prefer to transact in option markets and predicts that 1 Consistent with this idea, Bakshi, Cao, and Chen (2000) examine S&P500 options and find that option prices frequently move in the opposite direction of equity prices. Their findings indicate that options are not redundant securities and that the underlying equity s return distribution does not unilaterally dictate the price of options. 1

3 directional option volume signals private information not yet reflected in equity prices. Specifically, their model predicts that positive trades (i.e. purchases of calls and selling of puts) are positive signals of equity value and that the converse holds for negative trades. An interesting but otherwise unexplored empirical finding in EOS is that option activity carries greater predictive power for negative price changes. EOS comment on this finding in the following excerpt: An interesting feature of our results is the asymmetry between the negative- and positive-position effects... suggesting that options markets may be relatively more attractive venues for traders acting on bad news. An often-conjectured role for options markets is to provide a means of avoiding short-sales constraints in equity markets... Our results support this conjecture, suggesting a greater complexity to the mechanism through which negative information is impounded into stock prices [Page 458]. We provide a formal means of understanding their finding by introducing short sale costs into a microstructure framework with asymmetric information. Our model is similar to EOS in that we examine the interaction of informed traders and a risk-neutral market maker. The broader intent of our paper is to characterize the price-setting process in a multimarket context, and to formally analyze the capital allocation decision of informed traders. Similar to EOS, we allow informed traders to buy or sell stock, buy or sell a call, or buy or sell a put. Capital constraints and short sale costs play a central role when informed traders select a trading venue. When shorting a stock, traders must borrow shares from a third party who demands an interest payment despite retaining a claim to all dividends and capital gains. It is comparatively cheaper to capitalize on bearish private signals in option markets because traders can buy puts or sell calls, and in both cases they can create new option contracts without first borrowing them from a third party. In equilibrium, the costs associated with short-selling makes informed traders 2

4 more likely to use options for bad signals than for good ones, and therefore option volume is an indication of negative private information. Our model compliments and extends the EOS framework along a number of important dimensions. First, we model a continuum of private signals, which allows us to formally characterize the decision rule of informed traders as a function of the sign and magnitude of their signal. Second, we use continuous range of terminal values for the underlying stock. This allows us to make additional predictions about the future equity price distribution conditional on the relative concentration of trading activity across option and equity markets. Third, we incorporate short sale costs and capital constraints. Short sale costs are an important institutional friction that creates an asymmetry in the cost of trading on good and bad news in equity markets. Capital constraints are necessary for the leverage in options to be an attractive feature, and also cause the options to be non-replicable. Like EOS, we solve a static model and therefore need an additional assumption, in particular some friction preventing equity prices from immediately reflecting the information in options, in order for the model s prediction about the conditional mean equity value to translate into return predictability. Our main result, that option volume signals bad news on average, differs from EOS in that it can be tested empirically without signing the direction of trades. We predict and empirically confirm that contrasting publicly available totals of firm-specific option and equity volume portends directional prices changes. We take several steps to mitigate concerns that the OVR-return relationship reflects compensation for taking positions in illiquid and/or risky equities. First, we demonstrate that the relationship holds after controlling for exposure to the three Fama- French, momentum, and Pastor-Stambaugh liquidity factors. Second, we document that predictive power of OVR for future returns is relatively short-lived, failing to predict returns beyond the first month following portfolio formation. Third, we demonstrate that a decile strategy using 3

5 changes in OVR also generates positive risk-adjusted alphas. Having established the negative relationship between OVR and future returns, we next test our model s prediction that this relationship varies with the costs of short selling. Following Nagel (2005), we proxy for firm-specific short sale costs with the percent of the firm s outstanding shares held by institutions after controlling for firm size. Nagel (2005) argues that institutional shareholders are the primary lender of shares used for shorting and, hence, residual institutional ownership varies negatively with short sale costs. In our model, option volume is indicative of negative news as long as short sale costs are positive, but is a worse signal when short sale costs are higher. We independently sort firms into deciles of OVR and quintiles of short sale costs. Consistent with the model s prediction, we find that portfolio alphas associated with OVR are monotonically increasing in short sale costs. A surprising empirical prediction resulting from our model is that relative option volume is a stronger predictor of value when option leverage is lower. The risk neutral market maker sets bid and ask prices to break even in expectation conditional on the relative leverage available in option markets. As leverage increases, informed traders use options more frequently and therefore bid-ask spreads for option trades increases. While informed traders generally prefer to transact in option markets due to their leverage, the positive relationship between leverage and bid-ask spreads induces a negative relationship between leverage and the difference in the frequency with which informed traders use options for good versus bad news. As a result, when option leverage is low, informed traders use options for a much greater range of negative signals than positive ones, thus strengthening the correspondence between option volume and negative private information. Empirically we find that hedge returns associated with OVR are most pronounced in stocks for which the corresponding average leverage in traded options is lowest. Conditional upon low average leverage in options traded, sorting stocks based on OVR results in an average 4

6 annual hedge return of 21.3% and provides positive hedge returns in 12 out of 13 years in the sample. Under the maintained assumption that market frictions prevent option market signals from being immediately reflected in equity prices, our model predicts a negative relationship between OVR and future equity returns, but predicts no relationship between mean returns and differences between call and put volume. The model does, however, predict a positive relationship between equity return skewness and call-put volume differences. Both of these predictions are confirmed in our empirical tests using the mean and skewness of future returns. We also find that OVR predicts the sign and magnitude of earnings surprises, standardized unexplained earnings, and abnormal returns at quarterly earnings announcements in the following month. These tests show that the same OVR measure we use to predict monthly returns also contains information about future earnings announcements that occur in the subsequent month. This is consistent with OVR reflecting private information that is incorporated into equity prices following at a subsequent public disclosure of the news. The results of this paper relate to the literature on information discovery and multimarket information flow. 2 Pan and Poteshman (2006) use proprietary CBOE option market data and document strong evidence of informed trading. The authors find that sorting stocks on proprietary measures of directional volume in puts relative to calls foreshadows future returns but they conclude the predictability is due to non-public information rather than market inefficiency. Similarly, Cremers and Weinbaum (2010) and Zhang, Zhao and Xing (2010) find that asymmetries in implied volatility across calls and puts predicts future returns. While we do not consider 2 The question of whether option markets lead equity markets or vice versa remains an open question. Anthony (1988) examines the interrelation of stock and option market volume and finds that call-option market activity predicts volume in the underlying equity with a one-day lag. Similar findings are reported in Manaster and Rendleman (1982). In contrast, Stephan and Waley (1990) find no evidence that option markets lead equity markets. While our results do little to resolve this debate, they do demonstrate that combined signals from option and equity markets serve as a leading indicator of future equity market returns. 5

7 transaction costs and cannot rule out an unknown risk factor, our results rely on publicly available volume totals and therefore suggest that market inefficiencies may exist with respect to signals embedded in the concentration of volume across option and equity markets. Two recent empirical papers also focus on option volume. The first, Roll, Schwartz, and Subrahmanyam (2009), documents that firms with higher unscaled option volume have a higher Tobin s q. The authors interpret this as evidence that liquid option markets increase firm value because they help complete markets and generate informed trade. Our model and empirical tests confirm this intuition, demonstrating that option markets are an attractive venue for informed traders, particularly those with negative private information. The second, Roll, Schwartz, Subrahmanyam (2010), shows substantial intertemporal and cross-sectional variation in the relative trading volume across option and equity markets, and explain a significant part this variation in a regression framework. They find that the option volume ratio is increasing in firm size and implied volatility but decreasing in option bid-ask spreads and institutional holdings. Our model and empirical results shed additional light on the variation in relative trading volume by examining the theoretical determinants of the option volume ratio when a subset of market participants is privately informed and the empirical relationship between relative option volume and future returns. 3 In modelling the relationship between short sale costs and informed trading, our paper is also related to Diamond and Verrecchia (1987), who model the impact of short sale constraints on the speed of adjustment of security prices to private information when informed traders only have access to equity markets. In their model, short sale constraints cause some informed parties with 3 Roll, Schwartz, and Subrahmanyam (2010) also document that the pre-announcement option volume ratio predicts the magnitude of returns at earnings announcements but only after conditioning on the sign of the earnings news. Conditioning on there being positive (negative) earnings news, they find that relative option volume predicts higher (lower) announcement returns. In contrast, our analysis demonstrates a predictive relationship between OVR and announcement returns without conditioning on the sign of the earnings news. 6

8 negative information not to trade. Thus, the absence of trade in their model is a negative signal of future firm value. In our model, trading options is an alternative to abstaining from trade when the cost of shorting is high. As a result, the option volume ratio rather than the absence of trade reflects negative private information. Later in their paper, Diamond and Verrecchia suggest that the introduction of traded options would effectively lower short sale costs by providing a lower cost venue to capitalize on negative news. Our model and empirical results confirm this intuition by predicting and empirically documenting a negative relationship between relative option volume and future equity values. The rest of the paper is organized as follows. In Section I we model the multimarket price discovery process and formalize the equilibrium strategy of informed traders. The model results in five main findings regarding the information content of relative trading volumes, which we translate into empirical predictions. Section II discusses the data, methodology, and empirical results. Section III presents results pertaining to quarterly earnings announcements, and Section IV concludes. I. The Model We present a model of information trading in both equity and options markets in the presence of short sale costs and a borrowing constraint. Informed traders facing a capital constraint build a portfolio by trading sequentially with competitive, risk-neutral market makers. As argued in Black (1975), privately informed traders may prefer to trade in options markets because of the additional leverage they provide. The borrowing constraint, which creates a lower bound on the trader s position in the risky assets as well as the risk-free asset, makes it impossible for informed traders to match the leverage in options by using a combination of the risk-free asset and the stock. As a result, we find that option trades are more likely to originate from an informed 7

9 trader than stock trades regardless of parameter choices. The key feature of our model is that traders must pay a lending fee to a third party in order to short stock. Because it is costly to trade on bad news in the stock market, in our model s equilibrium the mean value of the equity conditional on an option trade is lower than the mean conditional on a stock trade. There are three tradable assets in the model, an equity, a call, and a put. The stock liquidates for Ṽ at some time t = 2 in the future. The value of Ṽ is unknown prior to t = 2, but it is common knowledge that V = µ + ε + η, (1) where µ is the exogenous mean equity value, and ε and η are independent, normally distributed, shocks with zero mean and variances σ 2 and σ 2. The call and put are both struck at µ, and both ε η expire at time t = 2. We define C and P as the value of the call and put at t = 2, and note that C = max(ṽ µ,0) = max( ε + η,0) (2) P = max(µ Ṽ,0) = max( ( ε + η),0). (3) We focus on the case of a single strike price because our aim is to model the choice between trading in options markets and trading in stock markets, as opposed to the choice amongst options of different strikes. The results given here do not change if we use a strike price other than µ. To avoid complexities arising from early exercise, we assume that all options are European. Trade occurs at time t = 1, at which point a fraction θ of the traders (henceforth informed traders ) already know the realization of ε but the remaining traders, and the market maker, do not. Informed traders only have a noisy signal of the true value Ṽ since they do not know the 8

10 realization of η. The distribution of Ṽ conditional on the information that ε = ε is: Ṽ ( ε = ε) N(µ + ε,σ 2 η ) (4) The informed traders are risk neutral, and therefore one unit of the stock, call, and put are worth: E(Ṽ ε = ε) = µ + ε (5) ε ε E( C ε = ε) = Φ ε + φ σ η (6) σ η σ η ε ε E( P ε = ε) = Φ ε + φ σ η (7) σ η respectively, where Φ is the standard normal s cumulative distribution function, and φ is its probability distribution function. The first term of the option valuations (6) and (7) represents the contribution of the conditional mean of Ṽ, while the second represents the contribution of the remaining volatility σ η. The conditional expectation of a call s value is increasing in both ε and σ η, while the put s value is decreasing in ε and increasing in σ η. Informed traders have no endowment and select the portfolio of risky assets and a risk-free asset that maximizes their expected final period wealth. We require that each trade be in exactly one type of risky asset, resulting in six possible trades: buy or sell stock, buy or sell calls, and buy or sell puts. 4 Because they have no endowment, traders need to borrow in order to take long positions. All investors have the same maximum borrowing $κ. Traders can borrow and short the same number of shares they can buy, and write ϕ options contracts for every share they can 4 At equilibrium prices, the informed traders have a strict preference among the assets except at 5 cutoff points. Allowing trades in bundles of multiple assets, for example 1 call and 2 shares, complicates the analysis without changing our results or providing additional insight. The bundles serve as "intermediate" portfolios the informed trader uses upon getting a signal that makes them close to indifferent between the two assets in the portfolio. σ η 9

11 short, where ϕ > 1. The exact value of ϕ is determined by the margin requirements imposed on the trader. A fraction 1 θ of the traders are uninformed and trade for reasons outside the model, possibly a desire for liquidity, the need to hedge other investments or human capital, or a false belief that they have information. As noted in EOS, hedging is a commonly cited reason to trade in options markets. Regardless of their motivation, uninformed traders choose among the same possible portfolios as the informed traders. Uninformed traders volume is distributed among these trades (buy and sell stock, calls, and puts) with the probabilities γ 1, γ 2,... γ 6, where 6 i=1 γ i = 1. A competitive and risk-neutral market maker posts bid and ask prices for all three assets, along with fixed order sizes, that result in zero expected profit for each trade. For notation, we use a s, b s, a c b c, a p, and b p for the ask and bid prices of the stock, calls, and puts, respectively. Similarly, q b s, q s s, q b c, q s c, q b p and q s p denote the quantities available at each type of trade, for example q b s gives the quantity of stock available for the trader to buy. As in EOS and Glosten and Milgrom (1985), trades occur sequentially and at fixed order sizes. The key difference, however, is that here these order sizes are endogenously determined. The market maker is aware of the traders budget constraint, and determines order sizes from the exogenous budget parameter κ and the endogenously determined prices. With only one chance to trade and in possession of valuable information, informed traders take the largest position allowed by the capital constraint. The traders makes their portfolio choice, the market maker decides on prices, and jointly these determine the quantities traded. This approach is different from EOS, in which the fixed order sizes are exogenous parameters used to capture the leverage available in each asset. In their model and using their notation, the stock trades are in round lots of γ shares, and the option trader are in a single contract control- 10

12 ling θ shares. When θ > γ, options are more levered in the sense that trading options provides more exposure to the underlying than trading the stock itself. When θ < γ, the opposite is true and stocks have more leverage than options. Confirming the intuition in Black (1975), they show that trades carry more information in whichever market has greater leverage. However, the EOS model provides no guidance on the relationship between γ and θ. We specify a relationship between these two by noting that leverage in options is only relevant when there is a capital constraint or some other cost of borrowing. Modelling this relationship allows us to pin down the order sizes because the total value of each trade must satisfy the budget constraint; specifically, it tells us that at equilibrium trade quantities, options have more leverage than equities (or θ > γ in the language of EOS). Another difference between our model and EOS is the use of a continuum of noisy signals rather than a discrete space of perfect signals. The benefit of this change is that informed traders follow a pure strategy mapping their signal to a specific optimal trading strategy. The form of this strategy, discussed below, indicates for what set of signals the informed traders use each market, and also provides empirical predictions not found in EOS about the skewness of stock returns. The cost is that we can no longer find a closed form solution for the equilibrium bid and ask prices, meaning that while we can prove some results hold generally, others we can only illustrate in numeric examples. Additionally, there is a cost of short selling paid by the trader to a third party who lends them the shares. The market maker posts a bid price for the stock b s, and corresponding quantity q s s. In order to sell to the market maker, traders must borrow from an unmodelled third party who charges a lending fee equal to a fraction ρ > 0 of the total amount shorted q s s b s. The lender is able to charge such a fee because they have some market power, or because there is some unmodelled counterparty risk. No such fee exists when writing options because there is no need to find a 11

13 contract to borrow the market maker can simply create a new contract. The parameter ρ can also be thought of as a reduced form of any cost to shorting stock, for example recall risk or the indirect costs described in Nagel (2005). The result is that the market maker pays q s s b s in exchange for q s s shares, but the trader only nets q s s b s (1 ρ) from the transaction. It is important for our argument that options are not perfectly replicable using a dynamic combination of the risk-free asset and underlying equity, as they are under Black-Scholes assumptions, for example. In that case, option prices would be driven by the cost of replication, meaning short sale costs in equity markets end up being reflected in option prices thus breaking the asymmetry across markets in the cost of trading on negative news 5. In our model, Black-Scholes style replication fails because there are only two periods and Ṽ is normal, and because traders face a leverage constraint. The first is a modelling convenience that simplifies reality; we feel the second is realistic, because of either a literal leverage constraint or because of a de-facto one driven by risk aversion, agency constraints, or anything preventing the arbitrageur from taking an arbitrarily large position. Without replication, option prices in our model are determined by the equilibrium between transaction costs and demand from informed traders. While informed traders demand shifts due to short sale costs, the cost of shorting is not directly incorporated into option prices, making it cheaper to trade on negative news in the options markets. A. Equilibrium An equilibrium in our model consists of an optimal trading strategy for the informed traders as a function of their signal, and bid-ask prices and quantities that yield zero expected profit for the market maker. The informed traders optimal strategy is a function f (ε) that maps the range 5 From Leland (1985) we know that in the presence of transaction costs and with discrete trading, arbitrarily accurate replication is possible when borrowing is unconstrained. The addition of transaction costs does cause replication to only yield bounds on possible option prices rather than a unique no-arbitrage price. 12

14 of possible signals ε R to the space of possible trades. 6 In equilibrium, they follow the cutoff strategy: buy puts for ε k 1 sell stock for ε (k 1, k 2 ] sell calls for ε (k 2, k 3 ] f (ε) = make no trade for ε (k 3, k 4 ] sell puts for ε (k 4, k 5 ] buy stock for ε (k 5, k 6 ] (8) buy calls for ε k 6 For extremely good or bad signals, the informed traders buy options despite the large informationbased transaction costs in these markets because of the greater leverage they provide. The large bid-ask spread makes options unattractive for weaker signals, and so informed traders use stock instead. For even weaker good or bad signals, however, informed traders value the stock near its unconditional mean and therefore cannot profitably trade stock; however, they value the options well below their unconditional mean because extreme outcomes are unlikely, and therefore sell options. If bid prices are so low they are below both the unconditional mean option value and the informed trader s conditional mean option value for a given signal, the informed traders choose not to trade. The cutoff points k i arise endogenously in equilibrium and are chosen so that informed traders strictly prefer writing puts for all ε < k 1, selling stock for all k 1 < ε < k 2, etcetera. Some regions may be empty in equilibrium, meaning k i = k i+1 for some i. The addition of short sales costs has the effect of shrinking the region of signals for which informed traders short stock, making k 2 k 1 smaller than it would be otherwise. The bid and ask prices for each asset (a s, b s, a c, b c, a p, and b p ), the associated quantities 6 Here we look at the possible risky security trades. Given the quality of their information, the informed trader always wants to borrow the maximum allowed, either directly or by shorting a security. The uninformed traders choose among the same set of possible trades for simplicity. 13

15 (q b s, q s s, q b c, q s c, q b p and q s p ), and the informed traders cutoff points k i are the 18 equilibrium parameters. Together they satisfy 18 equations, presented fully in the appendix, which assure that the market maker s expected profit is zero for each trade, that the quantities and prices match the budget constraint, and that informed traders are indifferent between the two relevant trades at each cutoff point. B. Results and Empirical Predictions Due to the non-linearity of the simultaneous equations, no closed form solution for the equilibrium parameters is available. We proceed by first deriving results and empirical predictions which hold generally from the simultaneous equations themselves, and second examining a numeric example. The focus of this paper is on the information content of option volume when there are short sale costs. To this end, we assume throughout that ρ > 0 and derive results pertaining to the role of short sale costs. Proofs are in the appendix. Result 1. When each asset is equally likely to be bought or sold by an uninformed trader, the stock is worth less conditional on an option trade than it is conditional on a stock trade. Empirical Prediction 1. Option volume, scaled by volume in the underlying equity, is negatively related to future stock returns for the underlying. The main result is that an option trade is bad news for the value of the stock and a stock trade is good news. The reason is that informed traders use stocks more frequently to trade on good news than they do for bad news because of the extra cost of short selling stock. Therefore, the expected equity value is lower conditional on an option trade than the unconditional mean, which is in turn lower than the expectation conditional on a stock trade. The result requires uninformed traders to buy and sell each asset with equal probability, but it holds regardless of how uninformed traders distribute their demand across the different assets; for example, uninformed 14

16 traders may trade stock much more frequently than options or trade calls more than puts. In order to translate Result 1 into an empirical prediction, we consider the implications of our static model in a multiperiod setting. In the model, option trades plays a larger role in the equity price discovery process than equity trades. If equity markets fully incorporate the information revealed through options trading into their valuations, stock prices will immediately reflect the new conditional expectation of Ṽ after each new option trade. However, if stock market participants do not fully and immediately react to information revealed in options markets, then we expect that stock prices do not reflect the information content of options trades for the time between when the informed option market trading occurs and when the information becomes public through another channel. In this case, abnormal returns will be higher for low option volume stocks than for high option volume stocks during the period between the option signal and the public release of the information. Empirical Prediction 1 is, therefore, a joint hypothesis that (a) short sales costs make option volume a negative signal, and (b) some of the information in option volume reaches equities through other channels like earnings announcements that occur after the option is traded. Our model makes no prediction about the overall volume in options and stocks together, only that option trades are bad news relative to stock trades. Our goal is to focus on informed traders choice between using stock and options, conditional on having a signal about the future value of a firm. Therefore, our predictive measure is the ratio of option volume to equity volume and not raw option volume. Result 2. The disparity in mean conditional equity values between option and stock trades, as a fraction of the mean stock price µ, is weakly increasing in the short sale cost ρ. 7 7 We scale the difference in conditional means by µ because we want the conditional mean difference to be relative to the unconditional mean. In the language of investment, we consider returns rather than raw profits. 15

17 Empirical Prediction 2. The predictive power of the relative option volume ratio for future stock returns is increasing in the cost of shorting equity. While Empirical Prediction 1 does not rely on cross-sectional differences in short sale costs, if such differences exist the model predicts that option volume is a worse signal for high short-sale cost equities than those with low short-sale cost, but is still a valuable signal as long as short sale costs exist. When short sale costs are higher, option volume is worse news because the informed trader deviates from shorting stock to trading options for a larger range of signals. We discuss the empirical proxy for short sale costs used to test Empirical Prediction 2 in Section 2. Result 3. The disparity in mean conditional equity values between option and stock trades, as a fraction of the mean stock price µ, is weakly decreasing in the option s leverage λ C S S = µ C 2 σ 2 ε +σ2 η Empirical Prediction 3. The predictive value of relative option volume for future stock returns is decreasing in the average Black-Scholes λ of options traded.. In order to understand Result 3, we need to take a closer look at Result 1. The driver of Result 1 is a range of signal strengths for which informed traders trade options when there are short sale costs but short equity when there were no short sale costs. Calling this set of signals W, and defining k i as the cutoff points in the no short-sale cost case ρ = 0, and k i as the cutoff points when there are short sale costs ρ > 0, we can write W as follows: W = {ε k 1 < ε < k 1 or k 2 < ε < k 2 } (9) Cutoff points k 3,... k 6 change only a very small amount due to the indirect effect of new spreads caused by the shift in informed traders demand. To illustrate Result 3 we ignore this effect and say k i = k i for i 3. Our proof in the appendix accounts for these changes. The below 16

18 illustration depicts the change in cutoff points when a short sale constraint is added. The region W is the combination of the two shaded areas. For signals in W, without short sale costs, selling stock is optimal for informed traders. However, with short sale costs, trading options is preferable despite smaller gross trading profits than those achieved by shorting because of the extra borrowing cost paid to access those profits. The switch region is therefore those signals for which the benefit from avoiding the short sale cost outweighs the cost from foregone trading profits. All else equal, increased leverage does not change the benefits of options trading in this region but does increase the costs, so the net effect is to reduce the size of the switch region and weaken predictability. Result 3 may be surprising at first because leverage is usually an attractive feature of options. Indeed, in our model leverage allows informed trader s investment to be more sensitive to their private information on a per dollar basis and therefore the overall use of options by informed traders increases with leverage. However, this very attractiveness creates large information-based transaction costs when using options. Larger bid-ask spreads make it more expensive for informed traders to divert their demand from stocks to options in order to avoid the short sale costs, meaning that when option leverage is low, informed traders use options for a much greater range of negative signals than positive ones. This strengthens the correspondence between option volume and negative private information. The end result is that the difference in option use between good and bad signals decreases with leverage even though the overall option use increases. Empirically, Result 3 suggests that volume in options markets with higher leverage provide 17

19 a weaker signal than volume in options markets with lower leverage. For a measure of leverage, we use λ C S S C because Result 3 s measure of leverage is exactly the λ that emerges from our model, as derived in the appendix. The value of λ is the elasticity of C with respect to S, reflecting the bang-for-the-buck" notion of leverage. Result 3 indicates there exists a spread between conditional means of Ṽ regardless of the leverage λ but that the spread is larger for smaller values of λ. Empirically, the prediction is therefore that OVR predicts returns for all levels of λ, but most strongly for low λ. Two results that hold regardless of whether ρ > 0 or ρ = 0 bear mentioning because they pertain to another volume signal often considered informative: the ratio of call to put volume. Result 4. Equity value has positive skewness conditional on a call trade and negative skewness conditional on a put trade. Result 5. The mean equity value conditional on a call, or put, trade may be above or below the unconditional mean depending on the parameters. Empirical Prediction 4. The ratio of call volume to put volume varies positively with the future skewness of stock returns. Empirical Prediction 5. The ratio of call volume to put volume has no predictive value for future mean stock returns. Results 4 and 5 follow from the equilibrium trading strategy described in Section 1A. Following the notation used to describe the informed trader s strategy in equation (8), skewness conditional on a put trade is negative because, if informed, it reflects either moderately good news (i.e. ε (k 4, k 5 ]) or extremely bad news (i.e. ε k 1 ). Similarly, skewness conditional on a call trade is positive because, if informed, it reflects either moderately bad (i.e. ε (k 2, k 3 ]) or 18

20 extremely good news (i.e. ε k 6 ). Empirical Prediction 4 restates Result 4 as a relationship between the current ratio of call volume to put volume and future return skewness. Because equity has non-zero skewness unconditionally in our model but not in reality, we cannot be sure of the change in sign indicated in Result 4. Instead, we expect a positive relationship between future equity skewness and the ratio of call to put volume. A common intuition is that informed traders will use calls to trade good news and puts to trade bad news, such that the ratio of call volume to put volume is positively correlated with future returns. However, in our model informed traders use both calls and puts for both good and bad signals; for example, an informed call trade can reflect a moderately bad signal or extremely good signal. Because the range of signals for which informed traders sell calls occurs more frequently, the net result for the mean of Ṽ conditional on a call trade is unclear. Result 5 indicates that this conditional mean may be above or below the unconditional mean, depending on the parameters. Translating this result into an empirical test as we did for Result 1, Empirical Prediction 5 indicates that the ratio of call volume to put volume does not have cross-sectional predictive power for future equity returns. To summarize the results up to this point, our model provides three principle predictions regarding the price discovery process across option and equity markets. First, we predict that the concentration of option volume, relative to equity volume, is a negative cross-sectional signal of returns. Second, relative option volume carries greater predictive power when short sale costs are high or leverage is low. Third, call-put volume differences do predict the skewness of future returns but not the mean. In Section 2, we explore these predictions empirically. C. A Numeric Example One of the primary goals of our analysis is to model the equilibrium decision rule of informed traders in a setting where the range of possible private signals and terminal asset values 19

21 are continuous. While these continuity assumptions allow for a richer set of predictions not available in the discrete case, they make the model sufficiently complex and non-linear that it can only be solved by a computer given numeric values for exogenous parameters. We present the solution for exogenous parameters chosen to match the characteristics of a publicly traded US stock. The stock has mean value µ = 30, reflecting the average price of a publicly traded share of equity, and standard deviation σ 2 + σ 2 = 3, so that volatility is 10% of value, approximately ε η the average monthly return volatility. We examine the case of a minority of traders θ = 10% having fairly precise information, meaning the known shock s volatility σ 2 ε = 5 is slightly larger than the unknown shock s volatility σ 2 η = 4. Equity volume is, as we document in this paper, about 20 times option volume on average. For this reason, while uninformed traders are equally likely to buy or sell each asset, they are twenty times more likely to use stocks than to trade calls or puts, meaning γ 1 = γ 2 = 40 and γ 84 3 = γ 4 = γ 5 = γ 6 = 1. The capital constraint κ is 100, which 84 in equilibrium acts as a numeraire and impacts only the quantities trades but not the prices or information properties. Regulation T requires that margin accounts for shorting equity contain 50% of the trade s value. Contrasting this with the CBOE margin requirement when writing options of 20% of the underling equity s value, we find that by posting the same margin traders can write ϕ = 2.5 options for share they can short. Finally, we use a short sale cost of ρ = 1%. We solve for the equilibrium prices and cutoff points numerically given the above parameters. The exact simultaneous equations used to find this solution are in the appendix. The process converges quickly and it does not appear that there are multiple solutions. The equilibrium bid and ask prices, presented in Panel A of Table 1, show that spreads are bigger in the options markets than the stock market (1.41 and 1.36 vs. 0.07). Panel B of Table 1 summarizes informed traders equilibrium strategy, in which they use options for 84% of signals despite the enormous 20

22 bid-ask spread in those markets. The primary reason is the leverage options afford: given their budget constraint κ = 100, traders can buy 45.7 options contracts but only 3.3 shares of stock, and due to the margin restriction ϕ = 2.5 they can write 8.3 options contracts but short only 3.3 shares of stock. Note that equity market bid-ask spreads are not symmetric around the mean because stock buys are more likely to come from informed traders than stock sells. Option spreads are also asymmetric around their mean values because informed traders buy options for extreme signals and sell them for moderate ones. Informed traders use stocks less frequently, and options more frequently, for bad signals than they do for good news signals because of the short sale cost ρ. In this example, informed traders buy stock for 10.2% of signals (or 20.4% of positive signals) but only short stock for 5.4% of signals (10.8% of negative signals). Since they use options for the remainder of signals, there is a spread in conditional expectations as discussed in Result 1, namely that conditional on a stock trade the mean of Ṽ is 30.01, while conditional on an option trade it is The asymmetry in informed traders behavior leads to a larger transaction cost for buying stock (0.04), than for selling (0.03) 8. Short sale costs result in informed traders writing calls and buying puts for part of the range of signals that they would otherwise short stock. Hence, there is a higher concentration of informed trade and higher transaction costs in put buys than call buys, and similarly a higher concentration in call sells than put sells. Figure 1 shows the equilibrium profits to an informed trader from each possible transaction as a function of their signal, as well as the equilibrium cutoff points. The signal is expressed in terms of what it implies for the conditional expectation of Ṽ. As a reminder, the standard deviation of the private signal is , so loosely speaking informed traders buy options when they expect the equity to be more than one standard deviation from the mean. The profit 8 Transaction costs are defined here as the difference between the quoted price and the mean asset value. For example, the transaction cost for buying equity is the ask prices less the mean stock value, or =

23 lines for buying options are extremely steep because of the leverage they provide; those for selling options are also steep but to a lesser extent. The cutoff points and profit lines are not symmetric around the mean asset value µ = 30 (for example, the cutoff k 2 is farther from µ than the cutoff k 5 ) because the the short sale cost ρ shifts the profit line for shorting stock downward. This shift results in a smaller region of signals for which shorting stock is optimal when compared to the region for which buying stock is optimal. Note that trading profits using the optimal strategy are always positive so the informed trader never chooses not to trade. Increasing ρ to 2% increases the difference in conditional means from 0.18% to 0.36%, as expected from Result 2. We illustrate Result 3 by varying the mean equity value µ while keeping the other exogenous parameters fixed. The leverage in options increases with µ, which reduces the difference in conditional means for stock and option trades (scaled by µ) from 0.20% for µ = 20 back to 0.18% for µ = 30. Finally, we confirm Result 4 in this setting; because informed traders write options for moderate signals and buy options for extreme signals, the skewness conditional on a call trade is higher (0.15) than the skewness conditional on a put trade (-0.10). II. Empirical Tests The option data for this study comes from the Ivy OptionMetrics database. The database provides end-of-day summary statistics on all exchanged-listed options on U.S. equities. The summary statistics include option volume, quoted closing prices, and option Greeks. The Option- Metrics database, and hence the sample for this study, spans from 1996 through The final sample for this study is dictated by the intersection of OptionMetrics, Compustat Industrial Quarterly and CRSP Monthly data. We restrict the sample to firm-months with at least 50 call and 50 put contracts traded to reduce measurement problems associated with illiquid option markets. We also eliminate close-end funds, real estate investment trusts, American Depository 22

24 receipts, and firms with a stock price below $1. The intersection of these databases and data restrictions results in 175,654 firm-months corresponding to approximately 150 months and 1,700 total unique firms. For each firm i in month m, we sum the total option and equity volumes, denoted by OPVOL i,m and EQVOL i,m, respectively. Specifically, OPVOL i,m equals the total volume in option contracts across all strikes for options expiring in the 30 trading days beginning 5 days after the trade date. 9 We report EQVOL i,m in round lots of 100 to make it more comparable to the quantity of option contracts, which each pertain to 100 shares. We define relative option volume, or OVR i,m, as: OVR i,m = OPVOL i,m EQVOL i,m (10) Panel A of Table 2 contains descriptive statistics of OVR i,m (hereafter OVR for notational simplicity) for each year in our sample. The sample size increases substantially over the window. The number of firm-months increases from 7,501 in 1996 to 18,363 in The remainder of Panel A presents descriptive statistics of OVR for each year of the sample. The mean value of OVR for the entire sample is 4.34%, which indicates that there are roughly 23 times more equity round lots traded than option contracts with times to expiration between 5 and 35 trading days. Average OVR has increased in recent years due to a relatively high amount of growth in option trading. The 25th and 75th percentiles indicate a relative tight band of OVR around its median. OVR is positively skewed throughout the sample period, which indicates a high concentration of relative option volume among a small subset of firms. Panel B of Table 2 presents firm characteristics by deciles of OVR. LMC (LBM) equals the log of market capitalization (book-to-market) corresponding to the firm s most recent quarterly 9 We exclude options expiring within 5 trading days to avoid measuring mechanical trading volume associated with option traders rolling forward to the next expiration date. The results are qualitatively unchanged if we include options with longer expirations. 23

25 earnings announcement. High OVR firms tend to be larger and have lower book-to-market ratios. Although low OVR firms tend to be smaller firms, our initial data requirement of 50 calls and 50 puts traded in a month tilts our sample toward larger and more liquid firms, which mitigates, but fails to eliminate, transaction cost explanations of the observed returns associated with OVR. The average market capitalization of firms exceeds $2 billion in each OVR decile. VLC and VLP indicate the level of call and put contracts traded in a given month. Across all deciles of OVR, the number of call contracts traded exceeds the number of put contracts, which is consistent with calls being more liquid than put contracts. High OVR firms also tend to have higher levels of both option and equity volume. Our model predicts that high OVR is a negative signal for asset values and hence our univariate trading strategy based on OVR consists of taking a short position in higher equity volume stocks (i.e. high OVR stocks) and a long position in lower equity volume stocks (i.e. low OVR stocks). This raises concerns that the predictive power of OVR may reflect compensation for taking positions in low liquidity firms. We attempt to mitigate these concerns in several ways, which are discussed in greater detail below. Panel A of Table 3 presents return characteristics for each OVR decile. MOMEN equals the cumulative market-adjusted return measured over the 6 months leading up to portfolio formation. Market-adjusted returns equal the return of firm i minus the CRSP value-weighted index inclusive of dividends. RET(m) equals the portfolio s market-adjusted monthly returns in the m-month following portfolio formation, where portfolios are formed on the last trading day of month 0. To ensure that portfolios are formed on the basis of publicly available information, we exclude the last trading day of month m = 0 when calculating OVR. For example, we measure total volume from October 1st to October 30th and match the observed volume totals with returns accumulated from the close of trading on October 31st. Excluding volume on the last trading day of each month enforces a one-day separation between observing OVR and the formation of 24

26 portfolios. 10 Table 3 demonstrates that high OVR firms tend to have high momentum suggesting that a strategy that takes a long position in low OVR firms and short position in high OVR firms results in a momentum reversal bet. The RET(1) column contains market-adjusted returns in the month following portfolio formation, where the bottom row contains one-month returns resulting from an equal-weighted short position in the highest decile of OVR and an equal-weighted long position in low OVR firms. Low OVR firms outperform high OVR firms by 114 basis points per month (14.54% annualized) prior to risk-adjustments. The corresponding t-statistic, based on the monthly time-series, is indicating that OVR is a negative cross-sectional signal of future equity returns. The results of Panel A suggests that relative option volume may be indicative of expected momentum reversals and is consistent with Lee and Swaminathan (2000), who find that equity volume provides information about investor neglect. Our findings build upon Lee and Swaminathan (2000) by documenting that a univariate strategy based on relative option volume predicts future returns without conditioning upon past momentum. Panel B of Table 3 presents the results of a pooled regression of RET(1) on deciles of OVR, MOMEN, log market capitalization, and log book-to-market. Deciles of each variable are formed at the conclusion of each month and are sorted such that higher values correspond to the 10th decile. 11 For example, in column (1) the coefficient on OVR is indicating that firms in the highest OVR decile outperform the lowest decile by an average of 95.4 (= ) basis points per month. The coefficient on OVR has a corresponding t-statistic of where standard errors are clustered at the monthly level. We demonstrate that OVR offers explanatory power for future returns, incremental to a firm s momentum alone in column (2), and incremental to a 10 This restriction amounts a to skipping day between observing OVR and forming portfolios. This restriction is important because of non-synchronous closing times across option and equity markets. Removing this restriction does not materially affect our results. 11 MOMEN, LMC, and LBM are allocated to decliles for ease of interpretation of the regression coefficients. The inferences are qualitatively identical when using non-ranked versions of these variables 25

27 firm s momentum, size, and book-to-market in column (3). Panel A of Table 4 presents portfolio alphas for each OVR decile using a five-factor riskadjustment. To calculate portfolio alphas, we regress the monthly return corresponding to each OVR decile on the three Fama-French factors, the Carhart momentum factor, and the Pastor- Stambaugh liquidity factor. Specifically, we estimate the following regression for each decile of the relative option volume ratio, OVR: r p r f = α + β mk t m m 1 (r r f ) + β m m 2 HML m + β 3 SMB m + β 4 UMD m + β 5 LIQ m + ε m (11) where r p m is the return on an equal-weighted portfolio of stocks in a given OVR decile, r f m is the risk-free rate, and r mk t m is the market return in month m. HML and SMB correspond to the monthly returns associated with high-minus-low market-to-book and small-minus-big strategies. Similarly, UMD and LIQ equal the monthly returns associated with high-minus-low Carhart momentum and Pastor-Stambaugh liquidity factors, respectively. All factor returns are obtained from Ken French s data library via WRDS. The table contains portfolio alphas corresponding to lagged decile ranks of OVR. A one-month lag indicates that the portfolio is formed at the conclusion of month m 1 and the returns accumulated in month m. The one-month lag portfolio alphas provide a risk-adjusted return comparable to the univariate RET(1) results of Table 3. Notice that risk-adjustment does not significantly change the returns associated with the OVR strategy. Low (High) OVR firms earn a factor-adjusted alpha of (-59.77) basis points in the month following portfolio formation with a t-statistic of 3.45 (-3.04). The long-short strategy results in an average risk-adjusted return of 1.17% per month or, equivalently, 15.03% annualized. Next, note that the portfolio fails to accumulate economically and statistically significant abnormal returns in the second and third months following the observation of OVR. This short-term 26

28 predictive power of OVR is consistent with relative option volume reflecting private information about near term changes in price, rather than serving as a proxy for a static risk characteristic that is priced in the cross-section. Panel B of Table 4 presents an analogous test of future returns using changes in OVR. OVR is defined as the difference between OVR in the portfolio formation month and its average over the prior six months, all scaled by this average. OVR captures changes in OVR in which the relative amount of trading volume appears to deviate from its historical average. Paralleling the construction of OVR, all of the information needed to calculate OVR is available at the portfolio formation date. Firms in the lowest decile of OVR earn a factor-adjusted alpha of 1.17% per month with a corresponding t-statistic of Similarly, firms in the highest decile of OVR earn -0.46% per month with a t-statistic of The alpha associated with a long-short strategy is 1.34% per month (t-statistic=4.56), which corresponds to 17.31% on an annualized basis. In contrast to the Panel A results, we find that OVR retains predictive power for future returns in the second month following portfolio formation. The lowest OVR decile continues to outperform the highest decile in the second month, earning an additional basis points (tstatistic=3.49). This result does not extend to the third month following the portfolio formation suggesting that OVR also provides a transitory signal of future returns. Panel A of Table 5 presents a monthly transition matrix for deciles of OVR, where firms are sorted by deciles of OVR and a one-month lag of OVR. Hence, the diagonal terms indicate the percentage of firms that remain within their respective portfolio in consecutive months. Approximately 56% of low OVR firms change portfolios from month to month. Similarly, over 40% of high OVR firms change portfolios. The average of the diagonals is 26.91% indicating that on average approximately 75% of all firms change OVR deciles in adjacent months. That approximately half of the extreme portfolios turn over each month reduces concerns that the 27

29 OVR-return relation reflects compensation for bearing a time-invariant risk characteristic. Panel B re-examines portfolio alphas associated with deciles of OVR when the sample is partitioned by whether the firm changed OVR deciles relative to the preceding month. The left half of Panel B contains the five-factor alphas among firms whose OVR decile is unchanged from the preceding month. The extreme OVR deciles continue to significantly predict one-month ahead returns in the hypothesized direction but results in lower low-high portfolio alphas and t-statistics compared to the pooled sample results. The right half of Panel B contains the results corresponding to firm-months whose OVR decile has changed from the preceding month. While the low-high portfolio alphas increases to 1.38% (t-statistic=3.45) among firms with changes in their OVR decile ranking, the portfolio alphas across firms with and without OVR decile changes is not statistically distinguishable at conventional levels (t-statistic=1.17, result untabulated). Consistent with Empirical Prediction 2, Panel A of Table 6 demonstrates that the predictive power of OVR for future returns is increasing in short sale costs. Following Nagel (2005), we measure firm-specific short sale costs as the level of residual institutional ownership RI i,t. We define RI i,t as the percentage of shares held by institutions for firm i in quarter t, adjusted for size in cross-sectional regressions. Specifically, RI i,t equals the residual ε i,t from the following regression: I N ST i,t logit(i N ST i,t ) = log( ) = α + β 1 LM C i,t + β 2 (LM C 2 1 I N ST ) + ε i,t i,t, (12) i,t where I N ST i,t equals the fraction of shares outstanding held by institutions as reflected in the Thomson Financial Institutional Holdings (13F) database. We calculate quarterly holdings as the sum of stock holdings of all reporting institutions for each firm and quarter. Values of I N ST i,t are winsorized at and Low levels of RI i,t (hereafter referred to as RI) correspond 28

30 to high short sale costs because stock loans tend to be scarce and, hence, short selling is more expensive when institutional ownership is low (Nagel, 2005). We match RI to a given firm-month by requiring a three-month lag between the Thompson Financial report date and the first trading day of a given month. Panel A of Table 6 contains OVR portfolio alphas across quintiles of RI. In each month, firms are independently sorted into deciles of OVR and quintiles of RI. Within each RI quintile, we estimate portfolio alphas associated with an equal-weighted long-short position in the extreme deciles of OVR. We repeat this process each month yielding a time series of month long-short returns. Portfolio alphas equal the intercept from regressing the monthly long-short portfolio returns on the five factors used in equation (11). Panel A demonstrates that the OVR portfolio alphas are monotonically decreasing across the RI quintiles, consistent with our model prediction that the predictive power of OVR is most pronounced when short sale costs are high. The alpha for the low RI quintile is 2.19% (t-statistic=3.20) while the alpha for the highest RI quintile is less than one-third of the alpha for the lowest RI quintile. The difference in alphas across the extreme RI quintiles is 1.58% with a t-statistic of This finding is consistent with informed traders with negative news shifting their capital allocation toward option markets when short sale costs are high. Consistent with Empirical Prediction 3, Panel B of Table 6 demonstrates that the predictive power of relative option volume for future stock returns is strongest when option leverage is lowest. For each firm-month, leverage is defined as the open-interest-weighted average of C S as provided by OptionMetrics, which we refer to as LM. 12 Panel B contains the long-short OVR portfolio alphas across quintiles of LM. The OVR alphas are monotonically decreasing across quintiles of LM, where the difference in portfolio alphas across the extreme LM quintiles is significant at the 1% level (t-statistic=4.40). The results are consistent with informed traders 12 The results are qualitatively similar when using volume-weighted average option leverage. S C, 29

31 moving a larger portion of their bets from shorting stock to trading options when available leverage is lowest. While we find that OVR alphas are decreasing in leverage, we do not find that the OVR strategy has positive alpha in each leverage quintile. Specifically, in the highest leverage quintile alpha is negative but not statistically significant. One possible explanation for this finding is that certain levels of leverage attract more volume than others for institutional reasons, a feature not described by our model. Table 7 presents annual cumulative returns to three long-short strategies assuming monthly portfolio rebalancing for each year in the sample. Column (1) contains the annualized returns corresponding to the equal-weighted long-short position in the extreme OVR deciles. The longshort strategy is implemented each month and the monthly returns are accumulated within each calendar year. The unconditional long-short strategy results in positive returns in 10 out of 13 years, with a mean return of 14.71% and a standard deviation of 20.24%. Column (2) contains cumulative long-short returns for firms in the lowest quintile of residual institutional ownership (RI). In each calendar month, the strategy implements a long-short OVR strategy among firms in the bottom quintile of RI, corresponding to firms with the highest short sale costs. The strategy produces positive returns in every year of the sample, while significantly increasing the mean and decreasing the standard deviation of the annual cumulative returns. 13 The increased performance of the long-short strategy among firms with high short sale costs is consistent with the model prediction that informed traders with negative news shift their capital allocation toward option markets in response to short sale costs. Column (3) contains analogous results for firms in the lowest leverage (LM) quintile. Conditional upon being in the lowest LM quintile, the long-short OVR strategy results in positive hedge returns in all but one year while again increasing the mean return and decreasing the variance relative to the unconditional OVR strategy. Together, the 13 Because the Column (2) results rely upon taking positions among equities with high short sale costs, the reported results are not intended to reflect the actual returns achieved through implementation. 30

32 results of Table 7 demonstrate a robust association between OVR and return which is stronger when short sale costs are high or option leverage is low. In addition to the above results pertaining to OVR, we also examine what the ratio of call volume and put volume tells us about future equity returns. First, Empirical Prediction 4 states that ratio of call and put volume is a positive predictor of future return skewness. Second, Prediction 5 states that the ratio of call and put volume carries no predictive power for future returns. The results of Table 8 confirm both of these predictions. For each firm-month, we define the relative call-put volume ratio, RVR, as the ratio of total call volume to total put volume. Specifically, the call-put volume ratio is defined as: RVR i,m = VLC i,m VLP i,m, (13) where VLC i,m is the total call volume for firm i in month m and VLP i,m is defined analogously for puts. Firms are sorted into deciles based on RVR, where the tenth (first) decile corresponds to high (low) levels of call volume relative to put volume. For each calendar month, we calculate the cross-section skewness of monthly returns within each decile portfolio of RVR. 14 Panel A of Table 8 contains the results of regressing portfolio skewness on the RVR decile rank. In column (1), the coefficient on the RVR decile rank is positive with a t-statistic of 2.31, indicating that RVR is positively associated with future return skewness. The relationship between RVR and return skewness holds after controlling for the lagged skewness of a given portfolio and is consistent with the model prediction that higher values of RVR reflect informed traders with moderately negative news or extremely good news. 14 We use cross-sectional skewness because the model only predicts the skewness of firm value conditional on the type of option volume. The trading process by which equity prices converge to this value, what we would be measuring with time-series skewness based on daily returns, may or may not have the same skewness as the final firm value. Cross-sectional skewness also provides us much more data, approximately 150 firms in each decile as opposed to 20 days per month. 31

33 Panel B of Table 8 presents risk-adjusted alphas by deciles RVR. We find no evidence of return predictability associated with RVR, consistent with the idea that informed traders transact in both calls and puts in response to both positive and negative news. Collectively, the evidence in Table 8 is consistent with our model s equilibrium predictions that informed traders buy puts for extremely bad news, sell calls for moderate bad news, sell puts for moderate good news, and buy calls for extremely good news. III. Additional Analyses In this section, we provide additional evidence that relative option volume reflects private information by examining whether OVR provides predictive power for the sign and magnitude of future quarterly earnings surprises. We assemble a new dataset from four sources. The OptionMetrics, Compustat industrial quarterly file, CRSP daily stock file, and IBES consensus file provide information on option volume, quarterly firm attributes, equity prices, and earnings surprises, respectively. We apply the same sample restrictions outlined in Section 2. The intersection of these four databases results in a final sample consisting of 46,670 firm/quarter observations. To the extent that informed traders gravitate toward options ahead of negative news, we predict that OVR is negatively associated with the resulting earnings surprise. For each earnings announcement, we measure OVR on the calendar month that directly precedes it. For example, we match earnings announcements that occur in July with OVR measured in June of the same year. This empirical design directly mimics the main analyses used in Section 2 by using OVR in month m 1 to predict returns in month m, except here we focus the analysis on the prediction of earnings news and earnings-announcement window returns revealed in month m. We use three variables to capture the news released at earnings announcements. First, SURPRISE is measured as the firm s actual EPS minus the consensus EPS forecasts immediately prior to 32

34 the announcement, scaled by the beginning of quarter price. Second, as an alternative measure of earnings surprise, we also measure standardized unexplained earnings (SUE), defined as the realized EPS minus EPS from four-quarters prior, divided by its standard deviation over the prior eight quarters. Finally, CAR( 1,+1) equals three-day cumulative market-adjusted returns during the earnings announcement window from t 1 to t + 1, where day t is the earnings announcement date. Column (1) of Table 9 demonstrates that the prior calendar month s OVR decile is predictive of negative earnings surprises. The negative relation between relative option volume and earnings surprises is consistent with the negative OVR-return relationship reflecting informed trade. Column (2) produces analogous results where SUE is the dependent variable. The coefficient on OVR decile is with a t-statistic of -4.43, indicating that OVR is negatively associated with earnings innovations. Column (3) presents the regression results when the announcement window abnormal returns, CAR( 1, +1), is the main dependent variable. The coefficient on OVR remains negative and statistically significant incremental to the firm s momentum decile, size decile, and book-to-market decile, which is consistent with relative option volume reflecting private information about future asset values corresponding to the announcement of earnings news. As an example of the economic significance, a change in the OVR decile from 1 to 10 reduces market adjusted return by 76.5 (= ) basis points in the 3 day announcement window (all else equal). Collectively, the results in Table 9 are consistent with relative trading volume reflecting private information which is gradually impounded into prices in response to earnings news. 33

35 IV. Conclusion We provide theoretical and empirical evidence that the concentration of trading volume in option markets, relative to equity markets, is a negative cross-sectional signal for future returns. For each firm-month, we measure the amount of trading volume in option markets relative to equity markets, which we call OVR. Stocks in the lowest decile of OVR outperform the highest decile by 1.17% per month on a factor-adjusted basis in the month following the portfolio formation. We offer a simple explanation for this finding, specifically that it results from how informed traders choose between trading in equity and option markets in the presence of short sale costs. We explicitly model the capital allocation and price-setting processes in a multimarket setting and develop novel predictions regarding information transmission across markets. In equilibrium, short sale costs and capital constraints cause informed traders to trade more frequently in option markets when in possession of a negative signal, thus predicting that the concentration of volume in options markets, relative to equity markets, is indicative of negative private information. By empirically documenting that OVR is a negative cross-sectional signal for future equity returns, our results are consistent with market frictions preventing equity prices from immediately reflecting the information in the option volume signal. The model also predicts that the option volume ratio is a stronger signal when short sale costs are higher and when option leverage is lower, and that while volume differences across calls and puts predict future return skewness, they do not predict the sign of returns, all of which we confirm in the data. We find that portfolio alphas associated with OVR are monotonically increasing with the cost of short selling, which is consistent our model s prediction that informed traders with negative signals shift their capital allocation toward option markets to avoid short sale costs in equity markets. Similarly, conditional on low average leverage traded in options, 34

36 sorting stocks by deciles of relative option volume results in an average hedge annual return of 21.3% and produces positive hedge returns in 12 out of 13 years in our sample. Finally, consistent with the option volume ratio reflecting private information, we find that OVR predicts the sign and magnitude of earnings surprises and abnormal returns at quarterly earnings announcements. 35

37 Appendix A: Simultaneous Equations The full set of simultaneous equations that characterize the equilibrium are: q b s a s = κ q s s = q b s q b c a c = κ q s c = ϕq s s q b p a p = κ q s c = ϕq s s (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) a s = µ + b s = µ a c = b c = a p = b p = θ(φ( k 5 σ ε ) φ( k 6 σ ε )) θ(φ( k 5 σ ε ) Φ( k 6 σ ε )) + (1 θ)γ 1 σ ε θ(φ( k 2 σ ε ) φ( k 1 σ ε )) θ(φ( k 2 σ ε ) Φ( k 1 σ ε )) + (1 θ)γ 2 σ ε θ k 6 φ(ε)c (ε,σ η )dε + (1 θ)γ 3 φ(0) θ(1 Φ( k 6 σ ε )) + (1 θ)γ 3 θ k 3 k 2 φ(ε)c (ε,σ η )dε + (1 θ)γ 4 φ(0) θ(φ( k 3 σ ε ) Φ( k 2 σ ε )) + (1 θ)γ 4 σ 2 + σ 2 ε η σ 2 + σ 2 ε η θ k 1 φ(ε)p(ε,σ η )dε + (1 θ)γ 5 φ(0) σ 2 ε + σ 2 η θ(φ( k 1 σ ε )) + (1 θ)γ 5 θ k 5 k 4 φ(ε)p(ε,σ η )dε + (1 θ)γ 6 φ(0) θ(φ( k 5 σ ε ) Φ( k 4 σ ε )) + (1 θ)γ 6 σ 2 + σ 2 ε η (A.7) (A.8) (A.9) (A.10) (A.11) (A.12) 36

38 q b p (P(k 1,σ η ) a p ) = q s s (b s (1 ρ) µ k 1 )) q s s (b s (1 ρ) µ k 2 ) = q s c (b c C (k 2,σ η )) q s c (b c C (k 3,σ η )) = 0 (A.13) (A.14) (A.15) 0 = q s p (b p P(k 4,σ η )) (A.16) q s p (b p P(k 5,σ η )) = q b s (a s µ k 5 ) q b s (a s µ k 5 ) = q b c (a c C (k 5,σ η )) (A.17) (A.18) Equations (A.1) (A.6) ensure that the trade quantities satisfy the budget constraint. For example, (A.1) ensures that the amount of a stock transaction a s q b s equals the budget κ. Equations (A.7) (A.12) are the zero profit conditions for the market-maker. Equation (A.7), for example, ensures that the ask price for a stock is exactly the expectation of Ṽ conditional on a stock trade. Computing this conditional expectation in the case of options trades (equations (A.9) (A.12)) requires integrating the value function for options C and P. These functions are the mean value of a call and put, respectively, conditional on the signal ε and the standard deviation of η. Specifically, ε ε C (ε,σ η ) E( C ε = ε) = Φ ε + φ (A.19) ε ε P(ε,σ η ) E( P ε = ε) = Φ ε + φ σ η σ η σ η σ η σ η σ η (A.20) When solving these equations numerically, we estimate the integrals using a Reimann sum. Finally, equations (A.13) through (A.18) ensure informed traders are indifferent between the two neighboring portfolios at the cutoff points. For example, (A.13) ensures they are indifferent between buying puts and shorting stock given the signal k 1. These equations cannot be solved in closed form due to the nonlinearity of C and P, however we can prove some general results directly from the simultaneous equations without needing a closed form solution. Throughout, we assume the exogenous parameters are chosen so that there exists a set of equilibrium parameters satisfying (A.1) through (A.18) as well as k 1 < k 2 < k 3 k4 < k 5 < k 6. For some parameters no such equilibrium exists, typically because the informed trader never finds it optimal to trade stock so k 2 = k 3 ; we focus on the case when informed traders use stock because our goal is to model the impact of short sale costs, which are only relevant when informed traders use equity. We do consider parametrizations where the informed trader chooses to trade every signal, meaning k 3 = k 4. In this case, we have one fewer free parameter and we need to replace equations (A.15) and (A.16) with a the single equation: q s c (b c C (k 3,σ η )) = q s p (b p P(k 3,σ η )) (A.21) 37

39 Appendix B: Proofs Result 1. When uninformed demand satisfies γ 1 = γ 2 and γ 3 = γ 4 = γ 5 = γ 6, in equilibrium E(Ṽ option trade) E(Ṽ equity trade). We obtain a strict inequality when ρ > 0. Proof. Denote k as the equilibrium cutoff points and a 1 i and b i as the prices in the case that ρ = 0. Here, the problem is completely symmetric and we have that k 1 = k 6, and k 2 = k 5. Therefore, we get E(ε ε (k 5, k 6 )) = E(ε ε (k 1, k 2 )) which yields E(Ṽ option trade) = E(Ṽ equity trade). When ρ > 0, we show that k 1 > k 6, and k 2 < k 5. Assuming informed traders can trade at the quantities and prices available when ρ = 0, but that they consider short sale cost ρ > 0 when choosing their cutoff points, the informed trader prefers buying puts to shorting stock at k since 1 q b p ( p(k 1,σ n ) = a p ) = q s s (b s µ k 1 ) < q s s(b s (1 ρ) µ k 1 ) (B.1) Similarly, informed traders prefers writing calls to shorting stock at k 2 because q s c (b c C (k 2,σ n ) = a p ) = q s s (b s µ k 1 ) > q s s(b s (1 ρ) µ k 2 ) (B.2) However, because short sale costs do not appear in equations (A.15) (A.18), they remain satisfied by k... 3 k. The best response to ρ = 0 prices is therefore k 6 1 > k, k 1 2 < k, k 2 i = k i 3. i If the informed trader uses the cutoff strategy k i, the market-maker needs to adjust its bidask prices in order to satisfy the zero profit condition. In particular, the informed trader less frequently shorts stock and more frequently buys puts and writes calls, meaning the market maker should increase b s, increase a p, and decrease b c. Continuing this sequential thinking, the informed trader would respond to the price changes by decreasing k 1 and increasing k 2, partially reversing their initial change from the ρ = 0 case. The process continues ad infinitum, marketmakers adjusting spreads and informed traders adjusting their cutoff points, until equilibrium is reached. But because there are uninformed traders, each adjustment in this process is smaller than the previous, and therefore the dominant effect is the first response of the informed traders, namely a reduced range of signals for which informed traders short stock. Informed traders new equilibrium strategy k ρ therefore satisfies k ρ > i 2 kρ and 4 kρ > 1 k ρ, which implies that E(ε ε (k 5 4, k 5 )) > E(ε ε (k 1, k 2 )), yielding the desired result that E(Ṽ option trade) E(Ṽ equity trade). Result 2. Given the same assumptions as Result 1, the scaled difference in conditional means D E(Ṽ stock trade) E(Ṽ option trade) is weakly increasing in the short sale cost ρ. µ Proof. Since the short sale cost ρ has no impact on µ, an equivalent statement is that d E(Ṽ stock trade) E(Ṽ option trade) 38

40 is weakly increasing in ρ. Say we have 0 < ρ 1 < ρ 2, and that k (1) are the equilibrium cutoff i points in the informed traders strategy when the short sale cost is ρ 1 while k (2) are the cutoffs i when short sale costs are ρ 2. As long as informed traders still short equity for some non-empty range of signals (k, 2 k ), by the same line of reasoning used in the proof of Result 1 we have that 3 when ρ = ρ 2, the range of signals for which the informed trader shorts equity shrinks and d strictly increases. We state that d is only weakly increasing in ρ because when ρ 1 is sufficiently large that the informed trader never shorts equity, increasing short sale costs to ρ 2 has no impact on the informed trader s strategy and therefore no impact d. Result 3. Holding the signal strength r = σ ε σ η E(Ṽ option trade) E(Ṽ equity trade) D is decreasing in the leverage in options as measured by λ = µ constant, the scaled difference in conditional means µ 2 σ 2 ε +σ2 η φ ωε ωσ Proof. First we show that D is invariant to changes in µ as long as σ 2 and σ 2 change proportionally so that λ remains the same. ε η Suppose that parameters k i, a i, and b i constitute an equilibrium for the case µ = µ 0, σ ε = σ r, and σ η = σ, where r is the signal strength ratio. We show that k = ωk i i, a = ωa i i, and b = ωb i i are an equilibrium for the case µ = µ 0, σ = ωσ r, and σ = σω, holding all other parameters ε η fixed. We call the first the "unprimed" case and the second the "primed" case. In order to verify we have an equilibrium, we need to assure all 18 simultaneous equations are satisfied. The first six equations are met provided that we set q s = κ = κ = q b s, and similarly b a ωa s s ω q = q s s, s s ω q c = q b c, b ω q c = q s c, s ω q p = q b p, and b ω q p = q s p ωε. Noting that C (ωε,ωσ) = Φ ωσ + s ω ωσ ωσ = ωc (ε,σ), and similarly P(ωε,ωσ) = ωp(ε,σ), the market-maker s zero profit conditions (A.7) through (A.12) are trivially satisfied since the cutoff points are always scaled by σ ε = σ r. The informed traders indifference equations (A.13) through (A.18) are satisfied as well, for example we calculate for k 1 : q b p(p(k 1,σ η ) a p ) = q b p ω /(P(ωk 1,ωσ η ) ωa p ) = q b p (P(k 1,σ η ) a p ) =q s s(b s (1 ρ) µ k 1 ) = q s s ω (ωb s (1 ρ) ωµ ωk 1 ) =q s s(b s (1 ρ) µ k 1 ) (B.3) Now consider the expectation of Ṽ conditional on a specific trade. Denoting expectations and probabilities using the primed parameters E and P respectively, we have that:. 39

41 E (Ṽ trade i) =µ + P (informed trade i)e ( ε k i 1 < ε < k i ) =ωµ + P(informed trade i)e ( ε ωk i 1 < ε < ωk i ) =ωµ + P(informed trade i)e(ω ε k i 1 < ε < k i ) =ωe(ṽ trade i) (B.4) Therefore, we have that D is the same in the primed and unprimed case: D = (E (Ṽ option trade) E (Ṽ stock trade)) µ ωe(ṽ option trade) ωe(ṽ stock trade) = ωµ =D (B.5) We can therefore focus on the change in D as σ changes, since changes a change from µ to cµ is equivalent to a change from σ to σ by the above. We show that as σ increases (and λ decreases), c D increases. To achieve this, we show that for when ρ > 0, the following inequality holds for all exogenous parameters permitting an equilibrium: σ ( k 5 (σ) σ k 4 (σ) σ ) > σ ( k 2 (σ) σ k 1 (σ) σ ) (B.6) where k i (σ) is the equilibrium cutoff point k i when the volatility parameter is σ. This inequality implies that, holding other parameters fixed, the range of signals (scaled to the standard normal) for which informed traders buy stock increases with σ faster than the range for which they sell stock. Because D is increasing in the difference between these two signal ranges, we can conclude from (a) that D itself increases in σ. All that remains is to show (a). To simplify notation, we call the normalized cutoff point j i = k i and continue to use σ = σ σ ε η and σ r = σ ε. Consider a set of exogenous parameters and corresponding equilibrium. Together they must satisfy equation (13) above, and therefore: q s s (b s µ j 1 σ r ) =q b p (P( j 1 σ r,σ) a p ) j 1 σ r =b s µ q b p q s s (P( j 1 σ r,σ) a p ) (B.7) 40

42 Following the reasoning in the proof of Result 1, we must consider which of the terms in the above change with σ even if the other parameters do not change. The informed trader will want to adjust their cutoff points in response to a new σ because the leverage in options has changed, so the cutoff points j i experience a first order impact. Changes in σ also impact the mean value of the option, so the quote prices for options markets must change even if the informed traders do not change their strategy. The zero-profit bid and ask prices are a i = σa i and b i = σb i, where a i and b i do not vary with σ. The value functions for options, P( j 1 σ r,σ) and C ( j 1 σ r,σ) are also directly proportional to σ and therefore are directly impacted by changed in σ. The buy quantity for puts q b p is set according to q b p = κ a p = κ σa p. The stock prices and quantities, b s and q s s do not change directly with σ because the mean value of the asset does not depend on σ. Denoting those terms that change directly with σ as functions of σ, we have: b s µ (q b p (σ)) j 1 (σ)σ r = q s s (P( j 1 (σ)σ r,σ) a p (σ)) σ ( j 1 (σ)σ r ) = σ q b p (σ)σ ( P( j 1 (σ)σ r,σ) )) a σ p ) (B.8) q s s We compute that q b p (σ)σ = 0, P( j 1 (σ)σ r,σ) = Φ( εr ) j σ q s s σ i +φ( ε), and P( j j 1 1 σ r,σ) = Φ( εr )σ r. Therefore, we get: σ ( j 1 (σ)σ r ) = q b p (σ)σ Φ( j 1 r )σ r j 1 q s s σ σ r j 1 σ + j 1 r = q b p (σ)σ Φ( j 1 r )σ r j 1 q s s σ j 1 σ = j 1 σ(1 Φ( j 1 r ) q b p q s s ) (B.9) 41

43 Similar calculations yield: j 2 σ = j 2 r + ψσ(b c C ( j 2,σ)) σ r (ψφ( j 2 r )) j 5 σ = ψφ(b p P( j 5,σ)) k 5 r σ r (1 ψφ( j 5 r )) j 6 σ = j 6 σ (Φ( j 6 r ) q b c 1) q s s (B.10) (B.11) (B.12) Noting that j 2 > 0, that j 2 increases with j σ σ 2, and that j 5 = ( j 2 ) σ σ j 2 = j 4, we have that j 5 > j 2. σ σ Similarly, j 6 > 0, j 6 increases with j σ σ 6, and j 6 = ( j 1 ) σ σ j 1 = j 6 so we get that j 6 > j 1. σ σ Adding these together, we get exactly inequality (a), which completes the proof. Result 4. Equity value has positive skewness conditional on a call trade and negative skewness conditional on a put trade. Proof. The third centralized moment (skewness before standardization) of Ṽ conditional on a specific trade, defining ˆV i E(Ṽ trade i), is: E((Ṽ ˆV i ) 3 trade i) = E((Ṽ ˆV i ) 3 informed trade i)p(informed trade i) Furthermore, the third centralized moment of Ṽ conditional on a call trade of unknown direction is: E((Ṽ ˆV call ) 3 call trade) = E((Ṽ ˆV call ) 3 informed buy call)p(informed buy call trade) + E((Ṽ ˆV call ) 3 informed sell call)p(informed sell call trade) We can use the fact that the third centralized moment of a standard normal truncated below at a and above at b is: φ(a)(a 2 + 2) φ(b)(b 2 + 2) (Φ(b) Φ(a)) to compute the above conditional skewness once we have a numeric solution. As is natural given the form of the informed traders strategy, in all numeric examples we have solved the conditional skewness is positive for a call trade and negative for a put trade. However, working directly from the simultaneous equations, the complexity of the conditional skewness formulas together with the non-linearity of the equations satisfied by the cutoff points k i has thus far made it impossible for us to show Result 4 in generality. 42

44 Result 5. The mean equity value conditional on a call (or put) trade may be above or below the unconditional mean depending on the parameters. Proof. Using the parametrization described in Section 1C, the mean equity value conditional on a call trade is 30.4, while conditional on a put trade it is On the other hand, if we change the leverage in option sells ϕ to 25, the mean equity value conditional on a call trade is 29.8, while conditional on a put trade it is The determining factor here is whether option buys or sells have more leverage. If option buys have more leverage (as is the case in Section 1C), relatively more informed option buys occur than option sells, meaning that on average call trades are good news and put trades are bad. If option sells have more leverage (as is the case when ϕ = 25), relatively more informed option sells occur than option buys, meaning that on average call trades are bad news and put trades are good. We cannot observe the direction of volume in the data, so it would be difficult to determine in practice whether option buys or sells have greater leverage, which in turn makes it difficult to build a trading strategy from this prediction of the model. 43

45 References Amin, K. I., & Lee, C. M. C. (1997). Option trading and earnings news dissemination. Contemporary Accounting Research, 14, Bakshi, G., Cao, C., & Chen, Z. (2000). Do call prices and the underlying stock always move in the same direction? Review of Financial Studies, 13, 549. Black, F. (1975). Fact and Fantasy in the Use of Options. Financial Analysts Journal, 31, Cremers, M., & Weinbaum, D. (2010). Deviations from put-call parity and stock return predictability. Journal of Financial and Quantitative Analysis, Forthcoming. Diamond, D. W., & Verrecchia, R. E. (1987). Constraints on short-selling and asset price adjustment to private information. Journal of Financial Economics, 18, Easley, D., O Hara, M., & Srinivas, P. S. (1998). Option volume and stock prices: evidence on where informed traders trade. The Journal of Finance, 53, Glosten, P. R., & Milgrom, L. R. (1985). Bid, ask and transaction prices in a specialist market with heterogeneously informed traders. Journal of Financial Economics, 14, Lee, C. M. C., & Swaminathan, B. (2000). Price momentum and trading volume. Journal of Finance, 55, Leland, H. E. (1985). Option pricing and replication with transactions costs. Journal of Finance, 40, Manaster, S., & Rendleman Jr, R. J. (1982). Option prices as predictors of equilibrium stock prices. Journal of Finance, 37, Nagel, S. (2005). Short sales, institutional investors and the cross-section of stock returns. Journal of Financial Economics, 78, Pan, J., & Poteshman, A. M. (2006). The information in option volume for future stock prices. Review of Financial Studies, 19, Roll, R., Schwartz, E., & Subrahmanyam, A. (2009). Options trading activity and firm valuation. Journal of Financial Economics, 94, Roll, R., Schwartz, E., & Subrahmanyam, A. (2010). O/S: The relative trading activity in options and stock. Journal of Financial Economics, 96, Stephan, J. A., & Whaley, R. E. (1990). Intraday price change and trading volume relations in the stock and stock option markets. Journal of Finance, 45,

46 Zhang, X., Zhao, R., & Xing, Y. (2010). What Does Individual Option Volatility Smirk Tell Us About Future Equity Returns? Journal of Financial and Quantitative Analysis, Forthcoming. 45

47 Figure 1: Informed Traders Profits from Each Trade in the Numeric Example Figure 1 displays the informed traders profits from each available trade as a function of the expected asset value conditional on their signal. This profit depends on the equilibrium prices and quantities, along with the cutoff points in the informed traders optimal strategy, for a numeric example. The exogenous parameter values in the example are: the mean equity value µ = 30, the shock observed by the informed traders ε has variance σ 2 ε = 5, the unobserved shock η has variance σ2 = 4, a fraction θ = 1/10 of all traders η receive this signal, uninformed traders are 20 times as likely to use stock than options, so γ 1 = γ 2 = 10/21 and γ 3 = γ 4 = γ 5 = γ 6 = 1/84, traders can write ϕ = 2.5 options for every share they can short, the capital constraint κ = 100, and the short sale cost ρ = 1%. 46

48 Table 1: Model Solution for a Numeric Example Panel A presents the bid and ask prices for each asset, along with their unconditional mean value, for a numeric example of our model. Panel B presents the informed traders equilibrium trading strategy as a function of their signal and also gives the equilibrium trade quantities. The exogenous parameter values in the example are: the mean equity value µ = 30, the shock observed by the informed traders ε has variance σ 2 ε = 6, the unobserved shock η has variance σ2 = 3, a fraction θ = 1/20 of all traders receive η this signal, uninformed traders are 20 times as likely to use stock than options, so γ 1 = γ 2 = 10/21 and γ 3 = γ 4 = γ 5 = γ 6 = 1/84, traders can write ϕ = 2.5 options for every share they can short, the capital constraint κ = 100, and the short sale cost ρ = 1%. Panel A. Equilibrium bid-ask spreads for a numeric example Stock Call Put Ask price Mean asset value Bid price Panel B. Informed Traders Equilibrium Strategy Trade Range of signals Fraction of signals Buy q b p = 45.7 put options ε < % Sell q s s = 3.3 shares of stock 0.96 < ε < % Sell q s c = 8.3 call options 0.76 < ε < % Sell q s p = 8.3 put options 0.02 < ε < % Buy q b s = 3.3 shares of stock 0.63 < ε < % Buy q b c = 45.7 call options 0.99 < ε 16.1 % 47

49 Table 2: Descriptive Statistics By Year Panel A provides sample size information and descriptive statistics of OVR i,m (shown as a percentage), where OVR i,m equals the ratio of option volume to equity volume of firm i in month m as outlined in Section 2. Panel B gives average firm characteristics by decile of OVR. LMC is the log of market capitalization of the firm and LBM is the log of the firm s book-to-market ratio measured at the firm s last quarterly announcement date. VLC (VLP) equals the total call (put) contract volume traded in a given month; each contract represents 100 shares. OPVOL equals the sum of VLC and VLP. EQVOL equals the total equity volume traded, in units of 100 shares. Panel A: Sample Characteristics and OVR Descriptive Statistics by Year Firms Firm-Months MEAN P25 MEDIAN P75 SKEW ,094 7, ,425 10, ,609 11, ,719 12, ,800 14, ,662 12, ,564 12, ,497 12, ,676 13, ,791 15, ,917 16, ,062 18, ,996 18, ALL 175, Panel B: Firm Characteristics by Deciles of OVR OVR LMC LBM VLC VLP OPVOL EQVOL 1 (Low) , , , , , , , ,674 1,380 4, , ,124 2,240 6, , ,806 3,659 10, , ,318 6,227 17, , ,018 10,637 28, , ,881 18,692 48, , (High) ,186 45, , ,223 High-Low ,664 44, , ,745 48

50 Table 3: Cumulative Abnormal Returns By Deciles of Option Volume Ratio Panel A presents the time-series average equal-weighted returns by deciles of OVR i,m, where OVR i,m equals the ratio of option volume to equity volume of firm i in month m as outlined in Section 2. Decile portfolios are formed at the conclusion of each month. Deciles range from 1 to 10 with the highest (lowest) values located in the 10th (1st) decile. MOMEN equals the cumulative market adjusted returns measured over the 6 months leading up to portfolio formation. RET(0) is the return in the portfolio formation month and RET(1) is the market-adjusted return in the first month following portfolio formation. Panel B presents pooled cross-sectional regression results from regressing RET(1) on deciles OVR, MOMEN, LBM and LMC. LMC is the log of market capitalization of the firm and LBM is the log of the firm s bookto-market ratio measured at the firm s last quarterly announcement date. In Panel B, year fixed effects are included and standard errors are clustered at the monthly level. The resulting t-statistics are shown in parentheses. *** indicates the coefficient is significant at the 1% level. Panel A: Market-Adjusted Returns by Deciles of OVR MOMEN RET(0) RET(1) 1 (Low) (High) High-Low *** 3.003*** *** t-stat High-Low Panel B: Regression Results Using Deciles of OVR Dep. Variable: RET(1) (1) (2) (3) Intercept 0.391*** *** (3.10) (-0.39) (-3.12) Decile(OVR) *** *** *** (-8.16) (-9.20) (-9.05) Decile(MOMEN) 0.113*** 0.118*** (7.49) (7.79) Decile(LMC) 0.068*** (4.95) Decile(LBM) 0.042*** (3.01) Adj-RSquared Year Fixed Effects? Yes Yes Yes 49

51 Table 4: One-Month Five-Factor Adjusted Returns by Deciles of OVR Panel A presents portfolio alphas for by lagged deciles of OVR i,m, where OVR i,m equals the ratio of option volume to equity volume of firm i in month m. Decile portfolios are formed at the conclusion of each month. Deciles range from 1 to 10 with the highest (lowest) values located in the 10th (1st) decile. A k-month lag indicates that the portfolio is formed at the conclusion of month m k and the returns are realized in month m. Portfolio alphas are calculated using a Fama-French three-factor model, the Carhart momentum factor, and the Pastor-Stambaugh liquidity factor. Panel B is defined analogously for OVR, where OVR equals the difference between OVR in the portfolio formation month and the average over prior six months, scaled by this average. Panel A: Five-Factor Alphas by Deciles of OVR Lag: One-Month Lag Two-Month Lag Three-Month Lag Alpha t-stat Alpha t-stat Alpha t-stat 1 (Low) (High) Low-High Annualized Panel B: Five-Factor Alphas by Deciles of OVR Lag: One-Month Lag Two-Month Lag Three-Month Lag Alpha t-stat Alpha t-stat Alpha t-stat 1 (Low) (High) Low-High Annualized

52 Table 5: Transition Matrices by Deciles of OVR Panel A provides the pooled month-to-month transition probabilities by deciles of OVR i,m, where OVR i,m equals the ratio of option volume to equity volume of firm i in month m. Decile portfolios are formed at the conclusion of each month. Deciles range from 1 to 10 with the highest (lowest) values located in the 10th (1st) decile. Panel B re-examines five-factor portfolio alphas corresponding to each OVR decile where the sample is partioned by whether the firm changed OVR deciles relative to the preceding month. The left (right) half of Panel B contains the portfolio alphas among firms whose OVR decile is unchanged from (identical to) the preceding month. Panel A: Month-to-Month OVR Decile Transition Matrix 1 (Low) (High) 1 (Low) (High) Avg. Diag Panel B: Five-Factor Alphas by Deciles of OVR and Decile Change Indicator OVR Decile Same as OVR Decile Changed Preceding Month from Preceding Month Alpha t-stat Alpha t-stat 1 (Low) (Low) (High) (High) Low-High Low-High Annualized Annualized

53 Table 6: Portfolio Alphas by Quintiles of Short Sale Costs and Leverage Panel A presents portfolio alphas sorted by quintiles of residual institutional ownership (RI). RI is obtained from cross-sectional regressions as detailed in Nagel (2005), where lower values of RI correspond to higher short sale costs, and vice versa. Portfolio alphas are obtained from initiating an equal-weighted long-short position in the extreme OVR i,m deciles. OVR i,m equals the ratio of option volume to equity volume of firm i in month m. Decile portfolios are formed at the conclusion of each month. Deciles range from 1 to 10 with the highest (lowest) values located in the 10th (1st) decile. Portfolio alphas are calculated using a Fama-French three-factor model, the Carhart momentum factor, and the Pastor-Stambaugh liquidity factor. Panel B is defined analogously for quintiles of LM i,m calculated as the open-interest-weighted average λ of firm i in month m. Panel A: Five-Factor OVR Alphas by Quintiles of Residual Institutional Ownership Residual Institutional Ownership High Short Low Short Sale Costs Sale Costs High-Low RI(1) RI(2) RI(3) RI(4) RI(5) Short Sale Costs Alpha t-statistic (3.20) (2.46) (2.76) (1.86) (1.49) (2.29) Annualized Panel B: Five-Factor OVR Alphas by Quintiles of Option Leverage (LM) Option Leverage (LM) Low Leverage High Leverage Low-High LM(1) LM(2) LM(3) LM(4) LM(5) Leverage Alpha t-statistic (4.07) (1.98) (0.22) (0.04) -(1.75) (4.40) Annualized

54 Table 7: Cumulative Hedge Returns by Year The table presents cumulative annual returns to three strategies assuming monthly portfolio rebalancing for each year in the sample. Column (1) contains to the equal-weighted long-short position in the extreme OVR i,m deciles. OVR i,m equals the ratio of option volume to equity volume of firm i in month m. Decile portfolios are formed at the conclusion of each month. Deciles range from 1 to 10 with the highest (lowest) values located in the 10th (1st) decile. In addition to OVR i,m deciles, firms are independently sorted into quintiles of residual institutional ownership and option leverage. Column (2) contains cumulative longshort returns for firms in the lowest quintile of residual institutional ownership (RI). RI is obtained from cross-sectional regressions as detailed in Nagel (2005), where lower values of RI correspond to higher short sale costs, and vice versa. Column (3) contains analogous results for firms in the lowest leverage (LM) quintile, where LM i,m is calculated as the open-interest-weighted average lambda of firm i in month m. (1) (2) (3) High-Low OVR High-Low OVR High-Low OVR [Full Sample] [High Short Sale Costs] [Low Leverage] Average Std.Dev

55 Table 8: Future Return Characteristics by Deciles of Call-Put Volume Ratio The dependent variable in Panel A is SKEW, defined as the cross-sectional skewness of monthly returns in the month following portfolio formation. SKEW is calculated each calendar month and for each decile of RVR i,m. RVR i,m equals the ratio of total call volume to total put volume of firm i in month m. Decile portfolios are formed at the conclusion of each month. Deciles range from 1 to 10 with the highest (lowest) values located in the 10th (1st) decile. Year fixed effects are included and standard errors are clustered at the monthly level. The resulting t-statistics are shown in parentheses. *** indicates the coefficient is significant at the 1% level. Panel B contains portfolio alphas corresponding to deciles of RVR i,m calculated using a Fama-French three-factor model, the Carhart momentum factor, and the Pastor- Stambaugh liquidity factor. Panel A: Regression Results of Monthly Cross-Sectional Skewness Dependent Variable: SKEW (1) (2) Intercept 0.280*** 0.213** (3.25) (2.51) Decile(RVR) 0.022** 0.020** (2.31) (2.16) Lag(SKEW) 0.217*** (7.47) Adj-RSquared Year Fixed Effects? Yes Yes Panel B: Five-Factor Alphas by Deciles of RVR Alpha t-stat 1 (Low) (High) Low-High Annualized

56 Table 9: Earnings Surprises and Earning Announcement Returns The sample for Table 9 consists of 46,670 quarterly earnings announcements during the 1996 through 2008 sample window. Each measure of earnings news is regressed on deciles of OVR i,m from the prior calendar month. OVR i,m equals the ratio of option volume to equity volume of firm i in month m. Decile portfolios are formed at the conclusion of each month. Deciles range from 1 to 10 with the highest (lowest) values located in the 10th (1st) decile. Columns 1, 2, and 3 contain the regression results where the dependent variables are SURPRISE, SUE, and BHAR(-1,+1), respectively. SURPRISE equals the firm s actual EPS minus the consensus EPS forecasts immediately prior to the announcement, scaled by the beginning of quarter price. SUE equals the standard unexplained earnings, calculated as realized EPS minus EPS from four-quarters prior, divided by its standard deviation over the prior eight quarters. CAR( 1, +1) is the cumulative market-adjusted return during the three-day window surrounding the announcement date. LMC is the log of market capitalization of the firm and LBM is the log of the firm s book-to-market ratio measured at the firm s last quarterly announcement date. Year fixed effects are included and standard errors are two-way clustered by firm and quarter. The resulting t-statistics are shown in parentheses. ***, **, * indicates the coefficient is significant at the 1%, 5%, 10% level, respectively. (1) (2) (3) Dependent Variable SURPRISE SUE CAR(-1,+1) Intercept *** *** (-5.42) (-5.17) (-0.20) Decile(OVR) *** *** *** (-2.77) (-4.43) (-5.08) Decile(MOMEN) 0.021*** 0.129*** (12.78) (17.29) (0.25) Decile(LBM) 0.009*** *** 0.095*** (5.43) (-4.40) (4.23) Decile(LMC) *** *** (-3.36) (-8.12) (1.56) R-square Year Fixed Effects? Yes Yes Yes 55