Derivatives and Market (Il)Liquidity∗

Derivatives and Market (Il)liquidity∗

Shiyang Huang† Bart Zhou Yueshen‡ Cheng Zhang§

this version: May 8, 2020

∗ This paper benefits tremendously from discussions with Ulf Axelson, Francisco Barillas, Peter Bossaerts, Georgy Chabakauri, David Cimon, Jared Delisle, Bernard Dumas, Lew Evans, Sergei Gle- bkin, Bruce Grundy, Jianfeng Hu, Christian Julliard, Péter Kondor, Igor Makarov, Ian Martin, Massimo Massa, Mick Swartz, Naveen Gondhi, Andrea Tamoni, Bart Taub, Wing WAC Tham, Jos van Bommel, Grigory Vilkov, Dimitri Vayanos, Ulf von Lilienfeld-Toal, Jiang Wang, Qinghai Wang, Robert Webb, Liyan Yang, and Zhuo Zhong. In addition, comments and feedback are greatly appreciated from participants at conferences and seminars at 9th Erasmus Liquidity Conference, University of New South Wales, University of Melbourne, University of Bristol, and University of Exeter. There are no competing financial interests that might be perceived to influence the analysis, the discussion, and/or the results of this article. † School of Economics and Finance, The University of Hong Kong; [email protected]; K.K. Leung Building, The University of Hong Kong, Pokfulam Road, Hong Kong. ‡ INSEAD; [email protected]; INSEAD, 1 Ayer Rajah Avenue, Singapore 138676. § School of Economics and Finance, Victoria University of Wellington; [email protected]; Ruther- ford House, 23 Lambton Quay, Wellington, New Zealand. Derivatives and Market (Il)liquidity

Abstract

We study how derivatives (with nonlinear payoffs) affect the liquidity of their underlying assets. In a noisy rational expectations equilibrium, informed investors expect low conditional volatility and sell derivatives to others. These derivative trades affect investors’ utility differently, possibly amplifying liquidity risk. As investors delta hedge their derivative positions, price impact in the underlying drops, suggesting improved liquidity, because informed trading is diluted. In contrast, effects on price reversal are ambiguous, depending on investors’ relative delta hedging sensitivity, i.e., the gamma of the derivatives. The model cautions of potential disconnections between illiquidity measures and liquidity risk premium due to derivatives trading.

Keywords: derivatives, options, liquidity risk premium, liquidity measure, price impact, price reversal

(There are no competing ﬁnancial interests that might be perceived to inﬂuence the analysis, the discussion, and/or the results of this article.) 1 Introduction

Investors sometimes receive liquidity shocks, such as hedging needs and/or (time-sensitive) information. They then rush to the market to trade on such liquidity needs, leaving traces in the data: They generate (temporary) price pressure when hedging (Grossman and Miller, 1988) and (permanent) price impacts when speculating on private signals (Kyle, 1985). Anticipating such liquidity shocks, investors require a certain premium when pricing assets. The literature has extensively studied the association between various illiquidity measures and such liquidity risk premium. See, e.g., reviews by Amihud, Mendelson, and Pedersen (2005) and Vayanos and Wang (2013) and the recent issue of Critical Finance Review (2019). This paper examines (il)liquidity and the associated risk premium from a novel angle: How are they affected by the assets’ derivatives trading? Derivatives markets are tremendous. The BIS reports that by the third quarter of 2019, the open interest of all exchange-traded options alone was approximately US$70 trillion. The notional amount outstanding of over-the-counter derivatives exceeded US$600 trillion by the middle of 2019. It is natural to ask whether such significant derivative activity affects the underlying assets’ liquidity and, if so, how. To address this question, we develop a rational expectations equilibrium (REE) model based on the framework of Vayanos and Wang (2012): There is one risky asset in an economy populated by ex ante homogeneous investors. A liquidity shock affects a fraction of randomly selected investors, and the shock has two effects. First, a shocked investor will receive a future endowment correlated with the risky asset. Second, she also observes a private signal about the payoff. To hedge the endowment shock and exploit this private information, shocked investors demand liquidity to trade the risky asset with the other investors, who serve as liquidity suppliers. The ex ante (pre-shock) equilibrium asset price therefore commands two risk premia, one for fundamental risk and another for liquidity (shock) risk. In such a framework, Vayanos and Wang (2012) study how information asymmetry and competition affect various aspects of liquidity risk and market illiquidity. We

1 instead introduce derivatives and study their impacts. Our first finding is that the liquidity demanders always write (or sell) the derivative to the liquidity suppliers. This is because the derivative is valued according to an investor’s expectation of the nonlinear payoff component, conditional on her information set. (The linear part of the payoff, always replicable by the underlying, is redundant.) In such a conditional expectation, the leading term is (approximately) the conditional volatility of the underlying. Since the demanders are more informed, thanks to their private signals, they always face lower conditional volatility— and hence value the derivative to be cheaper—than do the suppliers. (As an extreme example, if a demander has a perfect signal, she knows exactly how much the underlying will pay off and expects zero volatility.) Therefore, our model shows that with information asymmetry, derivatives can serve as a channel for investors to “bet” on the underlying volatility, with the demanders always selling volatility (like writing insurance) to the suppliers. Empirical evidence seems to support this view. For example, Gârleanu, Pedersen, and Poteshman (2009) document that proprietary (more informed) traders mostly write equity options, which can be constructed as spreads and straddles for volatility bets, to market makers. The second result is that derivatives can either exacerbate or alleviate the liquidity risk of the underlying asset. As derivatives affect the trading gains (by allowing the said volatility bets), the investors split “the pie” differently. For example, if there are few suppliers, their per capita trading gain is huge, while the crowd of demanders only receives a negligibly small piece of “the pie”. Therefore, from a pre-shock point of view, the consequence of becoming a demander, i.e., receiving a liquidity shock, is very severe because a demander will be relatively much worse off than a supplier. Anticipating such a utility wedge in the future, investors require a large ex ante risk premium for the liquidity shock. That is, derivatives amplify liquidity risk, and thus increase liquidity risk premium, in this case. If the shock only affects a small population, the reasoning runs in the opposite direction, and the ex ante liquidity risk premium declines. Option is one of the most common derivatives (with nolinear payoffs). A number of empirical

2 works have examined the effect of options listing on the underlying asset’s price. The findings appear mixed. Positive effects have been documented in, e.g., Ben Branch (1981), Conrad (1989), and Detemple and Jorion (1990), while negative effects are seen in Mihov and Mayhew (2000) and Danielsen and Sorescu (2001). Sorescu (2000) finds that the impact on stock prices can be described by a two-regime switching model: a positive price effect is documented before the 1980s, while the opposite is found after. Our model provides a framework to reconcile these mixed findings. The third result is that investors’ derivatives trading also affects their underlying trading. We show that investors adjust their positions in the underlying precisely to delta hedge their derivative holdings. A derivative’s payoff can be thought of as a combination of linear, quadratic, and higher- order functions of the underlying payoff. When an investor acquires a position in the derivative, she gains exposure to such a bundle, even though she only wants the nonlinear part to bet on volatility. As a result, the linear part of the bundle appears as an inventory shock of the underlying, which the investor delta hedges by trading in the underlying market. Importantly, such delta hedging trades can “distort” frequently used empirical measures of illiquidity, such as price impact and price reversal.1 Price impact is defined as the regression coefficient of price returns on (informed investors’ signed) trading volume, à la Kyle (1985) and Amihud (2002). Because delta hedging trades are not information driven, the overall informative portion of the trading volume is smaller with than without derivatives. Therefore, such price impact- based measures would decrease with derivative trading, suggesting improved market liquidity. In other words, the delta hedging trades reduce the information-to-noise ratio in the order flow. Empirically, Damodaran and Lim (1991) show that noise decreases after option listing. More precisely, Kumar, Sarin, and Shastri (1998) show that after introducing options, the adverse- selection component decreases in the bid-ask spread, consistent with our explanation of informed

1 Empirical studies using “price impact” to measure illiquidity include, e.g., Brennan and Subrahmanyam (1996); Acharya and Pedersen (2005); Sadka (2006); and Collin-Dufresne and Fos (2015). Empirical studies using “price reversal” to measure illiquidity include, e.g., Campbell, Grossman, and Wang (1993); Llorente et al. (2002); Pástor and Stambaugh (2003); Hasbrouck (2009); and Bao, Pan, and Wang (2011).

3 trading being diluted by delta hedging. Price reversal is defined as the negative return autocovariance, à la Roll (1984). The idea is that short-run price returns are negatively autocorrelated because the price mean-reverts to the long-run fundamental (Grossman and Miller, 1988). For example, an initial price concession due to strong selling pressure will eventually see a positive return, as the selling pressure dissipates over time. The larger this temporary deviation is, the more negative the autocovariance and the less liquid the market. With derivatives, the suppliers’ and the demanders’ delta hedging trades always differ in sign (because the suppliers are always long and the demanders always short on the derivative). Thus, one group’s delta hedging always contributes to such price reversal, while the other group dampens it. We show that they neutralize one another only in special cases, suggesting that derivatives can either exacerbate or attenuate price reversal, depending on the parameters. In summary, our analysis highlights that derivatives trading can lead to conflicting interpreta- tions of various illiquidity measures and attenuate their association with the liquidity risk premium. These effects could be important in reconciling recent empirical findings. The Amihud (2002) measure, like price impact, is found to associate with stock returns less strongly in recent years (Drienko, Smith, and von Reibnitz, 2019; Harris and Amato, 2019; and Amihud, 2019). Our theory suggests that the rise of derivatives trading during the recent period could contribute to this trend— increased delta hedging volume in underlyings dilutes illiquidity measures such as price impact. The market-wide liquidity measure of Pástor and Stambaugh (2003), related to price reversal, is found to associate with a higher liquidity risk premium in more recent data (Li, Novy-Marx, and Velikov, 2019; Pontiff and Singla, 2019; Pástor and Stambaugh, 2019). This is also consistent with our finding that delta hedging trades exacerbate price pressure and amplify price reversal. Ben-Rephael, Kadan, and Whol (2015) separately study the characteristic liquidity premium and systematic liquidity premium and find that in recent years, both exist only in (small) NASDAQ stocks but not in NYSE or AMEX stocks. We argue that the lack of derivatives trading on relatively small NASDAQ stocks (Mayhew and Mihov, 2004) could explain why their liquidity measures and

4 liquidity premium differ from those of relatively large stocks. The results thus far are qualitative comparisons between an economy with derivatives and one without. To sharpen our empirical predictions, we turn to the most commonly traded derivatives, options, and derive testable implications for the underlying market liquidity. We show that a key determinant of the illiquidity measures, price impact and price reversal, is the (aggregate) moneyness of the options. This is because moneyness governs the net delta hedging: For example, when an out-of-the-money call option is introduced, both liquidity demanders and suppliers know that the call is unlikely to be exercised, but the more informed demanders are more certain about such low moneyness than are the suppliers. As a result, the demanders’ buy delta hedging is less aggressive than the suppliers’ sell delta hedging, and the net delta hedging trade is negative. When moneyness increases, from out-of-money to in-the-money, the above effects reverse: The more informed demanders become more certain that their short positions in the call will likely be exercised; thus, they delta hedge more aggressively by buying the underlying, and such buys exceed the suppliers’ delta hedging sell. The net delta hedging thus turns positive. As option moneyness drives net delta hedging, the illiquidity measures are also affected as a result. For example, we find that in the underlying, the price impact is U-shaped in the moneyness of the option. This is because, as moneyness increases, the demanders’ increased delta hedging buys have two effects. First, as they are not information driven, these buys reduce the information- to-noise ratio and lower price impact. Second, they generate buying pressure that adds to the price impact. The latter effect is negligible when the call is deep out-of-the-money (the demanders need only buy little to delta hedge) but becomes dominant when the call becomes more in-the-money, hence the U-shaped price impact. Our paper contributes to the understanding of links between derivatives and the liquidity of the underlying assets. Chabakauri, Yuan, and Zachariadis (2017) share with our framework the feature of derivatives being “informationally irrelevant;” that is, introducing derivatives does not affect learning or inference about fundamentals. Gao and Wang (2017) study options’ implications

5 for volatility, and their framework also assumes this irrelevance. This feature distinguishes our mechanism from, e.g., Dow (1998), who shows that introducing new securities can worsen market liquidity due to exacerbated adverse selection. For the same reason, the migration of informed trading between the underlying and its derivatives, e.g., Biais and Hillion (1994) and Easley, O’Hara, and Srinivas (1998), is also muted in our model. Cao (1999), Massa (2002), and Huang (2015) study derivatives’ implications for investors’ information acquisition and market quality. Trading dynamics could also aﬀect liquidity as highlighted by Brennan and Cao (1996). Compared to the aforementioned literature, our contribution is the novel channel of how derivatives aﬀect the underlying’s ex ante liquidity risk premium, as well as how investors’ delta hedging drives various illiquidity measures in possibly divergent directions. The remainder of the paper is organized as follows. Section 2 sets up the model and derives the benchmark (no-derivative) equilibrium. In Section 3, we introduce a variance swap to showcase the main forces of the model, as we can obtain analytical expressions for all equilibrium objects. We then turn to options in Section 4, where we rely mainly on numerical analyses, to extrapolate more results. Section 5 then concludes the paper. All proofs are collected in the appendices.

2 Model setup and benchmark

We ﬁrst set up a general model in Section 2.1 following Vayanos and Wang (2012), whose framework suits our purpose to associate liquidity risk with empirical illiquidity measures in the presence of information asymmetry. The general model nests the no-derivative benchmark of Vayanos and Wang (2012), which we replicate in Section 2.2. The purpose is to highlight some key expressions for a comparison with the economy with derivatives in Section 3. We conclude in Section 2.3 with a sharp result that in such a framework, derivatives do not aﬀect investors’ learning.

6 2.1 Model setup

Timeline. There are three dates: t ∈ {0, 1, 2}. At t = 0, homogeneous investors arrive and trade. Between t = 0 and t = 1, a liquidity shock (detailed below) strikes a subset of investors. Then all investors trade again at t = 1. Finally, at t = 2, payoﬀs realize and investors consume. The risk-free rate is normalized to be zero.

Assets. There is a risk-free consumption good in perfectly elastic supply serving as the numéraire.

There is also a risky asset in supply of X¯ > 0 units, each paying oﬀ a random amount of D units of the consumption good. The distribution of D is speciﬁed below. For now, we only need to assume that its unconditional mean D¯ := ED exists. Investors can write a derivative contract, the long side of which receives from the short side a

payoﬀ of f (D) at t = 2. The derivative’s payoﬀ structure f (D) is exogenous, but we endogenously solve for its price and demand.

Investors. There is a continuum of investors of measure one, indexed by i ∈ [0, 1]. They derive constant absolute risk aversion (CARA) utility, with the same risk-aversion parameter α (> 0), over their t = 2 consumption. At t = 0, the investors are homogeneous, endowed with the per capita supply of the risky asset.

Liquidity shock. A liquidity shock hits a proportion of π ∈ (0, 1) investors between t = 0 and t = 1. We follow Vayanos and Wang (2012) and refer to these shocked investors as “liquidity demanders” and the rest (1 − π) as “liquidity suppliers.” Speciﬁcally, the shocked investors trade the risky asset for two reasons: First, they each will receive an amount of (D − D¯ )z units of the consumption good at t = 2, and so they want to hedge this shock at t = 1. The realization of the shock z is private information to the demanders. The suppliers only know the distribution of z, speciﬁed below. Second, information asymmetry is known to be a major impediment to market liquidity. To study this information aspect of illiquidity, we provide the demanders a private signal s := D + ε,

7 where ε is some noise specified below. They therefore demand liquidity to buy or sell the risky asset according to the signal s and profit from such private information. In doing so, they adversely select the uninformed suppliers. We follow Vayanos and Wang (2012) and assume for simplicity that the endowment shock z and the signal s affect the same group of investors (see also, e.g., Biais, Bossaerts, and Spatt, 2010).

Random variable distributions. There are three fundamental random variables in this economy,

{D, z, ε}. They are jointly normal and pairwise independent, with means {D¯, 0, 0} and variances { −1, −1, −1} G0 τz τε . To ensure bounded utility, we assume, following Vayanos and Wang (2012), that

2 −1 −1 < . (1) α G0 τz 1

Intuitively, without this cap, the endowment shock z might be too severe (in ex ante expectation).

Trading. There are two trading rounds, t ∈ {0, 1}. Investors submit demand schedules to maximize their expected utility from consumption at t = 2. The equilibrium asset prices are pinned down via market clearing. We introduce the following notations:

• Prices: {Pt } for the risky asset and {Qt } for the derivative, where t ∈ {0, 1}.

• Demand schedules at t = 0: X0(·) for the risky asset and Y0(·) for the derivative.

• The liquidity demanders at t = 1: X1d (·) for the risky asset and Y1d (·) for the derivative.

• The liquidity suppliers at t = 1: X1s(·) for the risky asset and Y1s(·) for the derivative. The demand schedules are functions of the respective assets’ market clearing prices, conditional on all (other) assets’ current and past prices and the respective investors’ private information and endowment shocks (if any). For notational brevity, these arguments are omitted. The market clearing conditions are:

(2) X0(·) = X¯ and πX1d (·) + (1 − π)X1s(·) = X¯;

(3) Y0(·) = 0 and πY1d (·) + (1 − π)Y1s(·) = 0.

Note that the demanders’ endowment shock (D − D¯ )z does not materialize until t = 2.

8 Equilibrium. There are four prices (two assets, two trading rounds) and six demand schedules in

total (two assets, homogeneous investors at t = 0 and demanders vs. suppliers at t = 1). To fully characterize the equilibrium, we would need to solve for all ten of these endogenous objects, but some shortcuts can be taken: We will not explicitly derive the functional form of X0(·) or Y0(·), as in equilibrium market clearing implies that the t = 0 homogeneous investors must hold the per capita supply, i.e., X0(·) = X¯ and Y0(·) = 0. Further, since there is no demand for the derivative at t = 0 (Y0 = 0), we will not study its price Q0 either. As such, we will only focus on heterogeneous investors’ trading at t = 1, i.e., the demand schedules {X1d (·), Y1d (·), X1s(·), Y1s(·)}, as well as the

three asset prices {P0, P1, Q1}. These objects are suﬃcient for our purpose of studying the risky asset’s liquidity risk premium and illiquidity measures.

2.2 Benchmark

We study a benchmark with no derivatives in this subsection. The purpose is to derive the equilibrium liquidity risk premium and illiquidity measures, so that we can later compare them after we reintroduce derivatives. To study such a benchmark, we switch oﬀ derivatives by setting

f (D) = 0, and the model degenerates to Section 3 of Vayanos and Wang (2012). (As a result, the

derivative becomes redundant with prices Q0 = Q1 = 0 and demand {Y1d, Y1s, Y0} undeﬁned.)

Equilibrium at t = 1. When trading at t = 1, the liquidity demanders have the same information { nd, nd, , } { nd, nd} set of P0 P1 s z , while all suppliers only observe P0 P1 . Standard analysis shows that the CARA investors’ demand schedules are End[D] − Pnd End[D] − Pnd X nd = 1d 1 − z and X nd = 1s 1 , 1d nd[ ] 1s nd[ ] αvar1d D αvar1s D where the subscripts “1s” and “1d” indicate the respective conditional information set at t = 1 while the superscript “nd” emphasizes that these expressions are evaluated in the no-derivative

End[·] = E[· nd, nd, , ] End[·] = E[· nd, nd ] benchmark. Speciﬁcally, 1d P1 P0 s z and 1s P1 P0 (and the same for the conditional variances). Through the market clearing condition (2), we solve for the equilibrium

9 and characterize it in the following proposition:

Proposition 1 (Benchmark equilibrium at t = 1). At t = 1, there is a unique equilibrium. The liquidity demanders’ demand schedule is − nd( , ) = G1d G0 ¯ + G1d G0 − − (4) X1d p;s z D s p z; α G1d G1d the liquidity suppliers’ demand schedule is − − ¯ + ¯ nd( ) = G1s G0 ¯ + G1s G0 · G1p G0D αX − X1s p D p ; α G1s G1s G1 − G0 and the market clears at − nd = G0 ¯ + G1 G0 − α ¯, (5) P1 D η X G1 G1 G1 = nd[ ]−1 = + = nd[ ]−1 = + 2 /( + 2) = where G1d : var1d D G0 τε , G1s : var1s D G0 τε τz τετz α , and G1 :

πG1d + (1 − π)G1s are a demander’s, a supplier’s, and an average investor’s precision of the risky asset payoﬀ by t = 1, respectively; and η := s − α z is the signal about D learned by the τε < < < nd market. Note that G0 G1s G1 G1d because the suppliers only imperfectly infer s from P1 .

The equilibrium characterization is standard, and we brieﬂy explain its intuition. A liquidity demander trades on the diﬀerence between her private valuation (a weighted average between the ¯ nd unconditional mean D and the private signals) and the market price P1 . Her trading aggressiveness,

G1d /α, on this diﬀerence is increasing in her precision G1d and decreasing in her risk aversion α. She also oﬄoads the endowment shock z (hedging). The same interpretation holds for a liquidity

nd supplier, except that 1) she does not see the private signal s but infers it from P1 , and 2) she does nd not have hedging needs. The market clearing price P1 is a weighted average between the expected payoﬀ D¯ and the signal η, and is further adjusted for a risk premium of αX¯ /G1. The signal η is an imperfect inference of the demanders’ private information s.

Equilibrium at t = 0 and liquidity risk premium. The investors are initially homogeneous. Each has probability π (resp. 1−π) of becoming a liquidity demander (resp. supplier) by t = 1. They

10 nd( ) therefore choose a symmetric demand schedule X0 p to maximize expected utility. The market nd clearing condition (2) then yields the equilibrium price P0 , given by the following proposition.

Proposition 2 (Benchmark asset price at t = 0). At t = 0, each investor holds the per capita supply X¯ units of the risky asset. The equilibrium asset price is

nd − πM Pnd = D¯ − αG 1 X¯ − (αΣ)X¯, 0 0 1 − π + πMnd where the parameters Mnd and Σ are given in the proof.

nd nd The interpretation of P0 is similar to that of P1 . Compared to the unconditional expected ¯ nd payoﬀ D, P0 has discounts and there are two components in the discounts. First, investors require a risk premium expressed as the product of an investor’s risk aversion α and the unconditional payoﬀ nd[ ] = −1 = ¯ variance var D G0 , scaled by the per capita holding X0 X. nd Second, there is a “liquidity risk premium.” The fraction πM is the risk-neutral proba- (1−π)+πMnd bility for an investor to receive a liquidity shock (becoming a demander), where Mnd is the ratio of demanders’ expected marginal utility over that of the suppliers. The superscript emphasizes that there is no derivative. For simplicity, Mnd will be referred to as the marginal utility ratio, or MUR,

−1 henceforth. The term Σ is a variance expression, akin to G0 in the ﬁrst discount.

Illiquidity measures. We consider two widely used empirical measures of illiquidity. The first is the price impact of liquidity demanders’ trading, in the spirit of Kyle (1985). It is defined as the sensitivity of price return with respect to liquidity demanders’ signed volume (order flow) at t = 1. From an empiricist’s point of view, this coefficient is obtained by regressing the price return nd − nd ·( nd − nd) = P1 P0 on the order flow π X1d X0 at t 1; that is, cov[Pnd − Pnd, π ·(X nd − X nd)] α G − G 1 (6) λnd := 1 0 1d 0 = 1 0 , [ ·( nd − nd)] 1 − π G − G G var π X1d X0 1 1s 0 where we use the equilibrium derived earlier to obtain the closed-form, right-hand-side expression.

The larger the price impact λnd, the more sensitive the price is to the order ﬂow, implying a less liquid market. A number of empirical papers have used proxies along this line. Glosten and

11 Harris (1988), Brennan and Subrahmanyam (1996), and Sadka (2006) construct regression-based measures to assess the price impact induced by trades. Amihud (2002) calculates a daily ratio of average absolute returns to trading volume and then averages the ratio over a certain period. The second measure is price reversal, originating from Roll (1984) and Grossman and Miller

(1988). The idea is that the risky asset’s price will deviate from its “fundamental” at t = 1, due to the price pressure of demanders’ hedging against the endowment shock z. Such price pressure can be equivalently interpreted as the compensation required by the suppliers to absorb the demanders’ shock z. Eventually (in the long run, t = 2), the price will revert to the fundamental D. An empiricist can compute the negative autocovariance between the two returns nd = − − nd, nd − nd = − G0 − G1s 1 (7) γ cov D P1 P1 P0 1 1 G1 G1 G1s − G0 to measure market illiquidity. The larger γ nd is, the stronger the price pressure at t = 1, hence low liquidity. In this spirit, Roll (1984) links price reversal to the bid-ask spread. Hasbrouck (2009)

derives a Bayesian Gibbs estimator of Roll’s γ . Campbell, Grossman, and Wang (1993) construct a price reversal measure conditional on signed volume. See also Pástor and Stambaugh (2003). Vayanos and Wang (2012) focus on the different roles of these two illiquidity measures and how they are affected by degrees of information asymmetry and imperfect competition. Our focus below turns to the effect of derivatives, which we study later in Sections 3 and 4.

2.3 Information irrelevance of the derivative

We now reintroduce the derivative with some generic payoﬀ f (D) into the economy. At t = 1, the liquidity demanders and suppliers will trade both the underlying risky asset and its derivative.

The demanders’ learning about the fundamental D is unaﬀected, because they observe the private

nd signal s, which is the only source of new information. The suppliers can only learn from P1 in the benchmark (Section 2.2), but they now also observe the derivative price Q1. Do {P1, Q1} aﬀect the

12 suppliers’ learning about D?2 No.

Lemma 1 (Information irrelevance of the derivative). Fix the realizations of the fundamental

random variables {D, ε, z}. Suppose that there is an equilibrium with a derivative of f (D). In

this equilibrium, the suppliers’ posterior distribution of {D, ε, z} conditional on {P1, Q1} is the

nd { } |{ , } same as the posterior in the benchmark equilibrium conditional only on P1 . That is, D P1 Q1 E [ ] = End[ ] [ ] = nd[ ] is normally distributed with 1s D 1s D and var1s D var1s D .

The intuition is as follows. The demanders’ endowment shock z adds noise to the equilibrium prices, preventing the suppliers from perfectly learning the demanders’ private signal s. Such “noise” z ﬁrst enters the demanders’ wealth:

W2d = W0 + (P1 − P0)X0 + (D − P1)X1d + (f (D) − Q1)Y1d + (D − D¯ )z,

from which it is reﬂected in the demanders’ demand X1d and Y1d , jointly with their private signal s. Importantly, we show in the proof of Lemma 1 that the way in which the signal s and the noise z are reﬂected in X1d is exactly the same as in Y1d . As such, for the learning about D, the suppliers

find it equivalent to learn from either X1d or Y1d . In other words, the suppliers find that there is no additional learning from the trading of the derivatives. This result is not unique to our model. Similar features are also seen in, among many other con- tributions, Brennan and Cao (1996), Cao (1999), Huang (2015), Chabakauri, Yuan, and Zachariadis (2017), and Gao and Wang (2017). In particular, Chabakauri, Yuan, and Zachariadis (2017) coin the term “informational irrelevance” and study the conditions for such irrelevance to hold. Such informational irrelevance of derivatives ignores certain aspects of real-world trading. For example, it is well known that due to various frictions in the underlying trading, options can be used to bet on information (see, e.g., Biais and Hillion, 1994; Easley, O’Hara, and Srinivas, 1998). In addition, when the new asset’s payoff itself is informative of the existing asset’s payoff, as in Dow (1998), investors’ learning will be affected. Muting these effects, Lemma 1 highlights the novelty of the

2 It is clear that the t = 0 prices, P0 and Q0, are uninformative about D, as they are determined before the realization of the signal s (and the endowment shock z).

13 channels in our model—they are not about investors’ learning. In fact, it will soon be clear that our channels are orthogonal to the informational role of derivatives. The speciﬁc choice of our model is simply a means to an end. We explore these novel channels next.

3 Introducing variance swaps

In this section, we examine derivatives’ impact on the liquidity risk premium and illiquidity

measures of the underlying. Any derivative payoﬀ f (D) can be generally decomposed into a linear and a nonlinear component. Cao (1999) shows that the linear component is redundant in such frameworks—it is simply a combination of the numéraire and the underlying. To articulate the

economic forces, therefore, we study a variance swap of f (D) = D2, arguably the simplest nonlinear derivative.3 The results and intuition generalize to other nonlinear f (D). Later, in Section 4, we study the eﬀects of options.

3.1 Trading at t = 1

Consider an investor of type j at t = 1, where j = d for a liquidity demander and j = s for a supplier. Her terminal wealth is

(8) W2j = W0 + (P1 − P0)X0 + (D − P1)X1j + (f (D) − Q1)Y1j + (D − D¯ )zj,

2 where zj is her type-speciﬁc endowment shock, with zd = z, zs = 0, and f (D) = D . She chooses −αW2 her demand X1j and Y1j to maximize her conditional expected utility, E1j −e j , taking the

prices {P1 = p, Q1 = q} as given. For the demanders, their X1d and Y1d can also depend on the private signal s and endowment shock z. Thanks to Lemma 1, we know the conditional distribution of D for both the demanders and

3 A variance swap contract typically pays the realized variance of the underlying, e.g., in the form of f (D) = 2 2 (D − P0) . We simplify our exposition by focusing only on the nonlinear component, D . The results in this section are robust to any arbitrary quadratic function of f (D).

14 suppliers. The optimization problems can then be evaluated in closed form, and we obtain the following proposition.

Proposition 3 (Equilibrium at t = 1 with variance swap). There exists a unique equilibrium at t = 1. The liquidity demanders’ demand schedules are 1 −1 X (p, q;s, z) = X nd(p;s, z) − 2pY (p, q;s, z); and Y (p, q;s, z) = q − p2 − G . 1d 1d 1d 1d 2α 1d The suppliers’ demand schedules are 1 −1 X (p, q) = X nd(p) − 2pY (p, q); and Y (p, q) = q − p2 − G . 1s 1s 1s 1s 2α 1s = nd The underlying’s market clears at P1 P1 , the same as in the benchmark (Equation 5). The = 2 + −1 { , , } derivative’s market clears at Q1 P1 G1 . The conditional precision G1d G1s G1 are the same as those deﬁned in Proposition 1.

The formal proof is deferred to the appendix. Here, we explore a few insights revealed by the equilibrium characterization above.

Betting on “volatility.” We begin by investigating investors’ derivative demand Y1j(p, q)—why they trade the derivative. With some rearrangement, it can be seen that ∝ −1 + 2 − = −1 − −1 , Y1j G1j P1 Q1 G1j G1

= 2+ −1 where the equality follows the equilibrium derivative priceQ1 P1 G1 . Recall from Proposition 1 −1 that G1 is the market’s average conditional variance of the underlying payoﬀ D. Therefore, we see that each investor is trading on the diﬀerence between her and the market’s valuations of the

variance of D, as if she engages in a “volatility bet” against the market average. Such a bet is

similar to the underlying trading in the benchmark: adjusting for the future endowment shock zj, nd + ∝ E [ ] − nd nd X1j zj 1j D P1 ; that is, everyone bets her valuation of D against the market average P1 .

The (more informed) demanders write the derivative to the suppliers. In equilibrium,

G − G G − G Y = − 1d 1 < 0 < Y = 1 1s , 1d 2α 1s 2α

15 because G1s < G1 < G1d by construction. That is, a liquidity demander always takes a short position in the derivative, betting that the volatility is low, while a supplier always takes a long position, betting on high volatility. Intuitively, this is because the demanders are more informed than the suppliers, i.e., the demanders’ precision is higher, hence the inequalities above. Our model thus reveals that, apart from other motivations, more informed investors tend to sell “volatility,” as if selling protection or insurance, to the less informed investors. For example, Gârleanu, Pedersen, and Poteshman (2009) document that proprietary traders, who are arguably more informed, write equity options to market makers. To the extent that options can be used to replicate volatility (e.g., through strangles or straddles, etc.), this ﬁnding is consistent with our prediction here.

Trading in the underlying is aﬀected: delta hedging. Proposition 3 shows that investors’ trading in the underlying is aﬀected by the derivative. In particular, compared to the benchmark,

there is a new term of −2pY1j. That is, for every unit of the derivative, the investor trades against it by 2p units of the underlying, where p = P1 is the price of the underlying. This new term turns out to be the investor’s delta hedging. A type-j investor’s terminal utility is u(W2j), where W2j is a function of the underlying payoﬀ D (Equation 8). Therefore, her expected exposure to a small ﬂuctuation in D is h i ∂u(W2j) 0 ∂W2j 0 0 E = E u (W ) =E u (W ) X + z + Y f (D) 1j ∂D 1j 2j ∂D 1j 2j 1j j 1j h i h i 0 0 (9) =E1j u (W2j) X1j + zj + Y1jEˆ 1j f (D) ,

16 where we write4 " # h i 0 0 u (W ) 0 Eˆ ( ) = E 2j ( ) . (10) 1j f D : 1j 0 f D E1j u (W2j)

0 That is, measured in multiples of the expected marginal utility E1j u (W2j) , the investor expects a total exposure to ﬂuctuations in D of h i 0 X1j + zj + Y1jEˆ 1j f (D) .

In words, she holds X1j units of the underlying and will receive an endowment of zj, and her Y1j 0 units of the derivative have a per-unit exposure of Eˆ 1j f (D) , which is precisely the deﬁnition of the “delta” hedging ratio of the derivative.

0 Applied to our current example of variance swaps, the hedging ratio is ∆1j := Eˆ 1j[f (D)] =

Eˆ 1j[2D] = 2P1, and an investor’s delta hedging trade is

− ( , ) = − ( , ) = ( , ) − nd( ). ∆1jY1j P1 Q1 2P1Y1j P1 Q1 X1j P1 Q1 X1j P1

Intuitively, as the investors bet on the volatility through the derivative, they also receive extra exposure to the underlying through the derivative position Y1j. This exposes them to “too much” D, and as such they trade against Y1j to neutralize the exposure. Note that the above analysis is generic for any distribution of D, any utility function (subject to standard regularity), and any (piecewise diﬀerentiable) derivative payoﬀ f (·). We illustrate the robustness in Section 4 with options.

0 ( ) 4 u W2j Note that the ratio Λ1j := 0 is a function of D, is strictly positive, and satisﬁes E1j [Λ1j ] = 1. It therefore E1j [u (W2j )] serves as a Radon-Nikodym derivative and changes the expectation E1j [·] to a risk-neutral pricing measure Eˆ 1j [·]. For example, the ﬁrst-order condition regarding the underlying holding X1j is h i h i h i 0 ∂W2j 0 0 0 E1j u (W2j ) = E1j u (W2j )(D − P1) = 0 =⇒ E1j u (W2j )D = E1j u (W2j ) P1. ∂X1j 0 Dividing both sides by the expected marginal utility E1j u (W2j ) yields the risk-neutral pricing formula P1 = Eˆ 1j [D].

17 = nd A knife-edge result: the underlying price is unaﬀected, i.e., P1 P1 . While all investors delta hedge their derivative positions, the net delta hedging trade (across all investors) is zero: (11) π · X − X nd + (1 − π) X − X nd = −2P · (πY + (1 − π)Y ) = 0. 1d 1d 1s 1s 1 | 1 d {z 1 s} =0, by the derivative’s market clearing As such, the net demand for the underlying asset remains exactly the same as in the benchmark. In

nd other words, there is no price pressure pushing P1 away from the benchmark P1 . However, we would like to emphasize that this is a knife-edge result due to the speciﬁc derivative

payoﬀ f (D) = D2. In particular, both the demanders and the suppliers have the same delta hedging

ratio of ∆1d = ∆1s = 2P1. More generally, however, diﬀerent investors’ hedging ratios will not

always be the same, and the net delta hedging will be nonzero, thus pushing the equilibrium P1

nd away from the benchmark P1 . Such additional price pressure has asset pricing implications and, in particular, aﬀects empirical measures of illiquidity. We will see an example of such in Section 4.

3.2 Liquidity risk premium at t = 0

Just as in the benchmark case, at t = 0, the homogeneous investors hold the per capita asset supply

X0 = X¯ of the risky asset. Similarly, there is no demand for the derivative: Y0 = 0. We therefore

only focus on the underlying price P0.

Proposition 4 (Underlying price at t = 0, with variance swap). In equilibrium, the underlying asset price at t = 0 is − πM P = D¯ − αG 1 X¯ − (αΣ)X¯, 0 0 1 − π + πM

where M is given in the proof and Σ is the same as in the benchmark (Proposition 2).

nd ¯ Compared with the benchmark P0 (Proposition 2), the first two terms—the expected payoff D and −1 ¯ the risk-premium αG0 X—are the same. Introducing the derivative only affects the third term—the liquidity risk premium—and only through the marginal utility ratio (MUR) M in the risk-neutral probability.

18 We show in the proof that M and Mnd diﬀer by a factor of r G −G M G1d − 1d 1s (12) = 2G1 ≷ 1. nd e M G1s Therefore, it is the derivative’s impact on the MUR that determines the liquidity risk premium:

Corollary 1 (Liquidity risk). Introducing a variance swap (f (D) = D2) exacerbates the liquidity > nd − nd < risk if and only if M M . Accordingly, the liquidity risk premium increases, i.e., P0 P0 0, if and only if M − Mnd > 0.

Whether the MUR ratio M > Mnd or M < Mnd is driven by many parameters. As an illustration, we consider π, the pervasiveness of the liquidity shock. New derivatives in general aﬀect the available gains from trade in the market. In the current example, the variance swap allows the demanders to sell volatility, or insurance, to the suppliers. This additional trading gain is split diﬀerently between the demanders and the suppliers, depending on their relative market share. For

example, if there are very few demanders (π close to 0), this small group will be able to extract most of the trading gain, because they are the only writers of the derivative. That is, the derivative

now trades in a sellers’ market. In this case, at t = 0, investors expect a substantial boost in their terminal wealth should they receive a liquidity shock, hence also a lower marginal expected utility.

Relative to the suppliers, this implies a low MUR M. Compared to the benchmark, therefore, for suﬃciently small π, the MUR decreases, i.e., M < Mnd. (When π is large, close to 1, the derivative trades in a buyers’ market, and the opposite follows.)

In fact, the diﬀerence in the MUR M−Mnd is monotonically increasing in shock pervasiveness π. Figure 1(a) shows the pattern. As the liquidity shock becomes more likely, the MUR diﬀerence

increases and crosses zero once at some threshold π ∗ ∈ (0, 1). That is, introducing the derivative alleviates the ex ante liquidity risk if and only if π < π ∗ (the MUR between liquidity demanders and suppliers is reduced). When π > π ∗, the liquidity risk is exacerbated instead. These effects are reflected in how the t = 0 underlying price is affected, as shown in Panel (b). < ∗ > nd When π π , P0 P0 because investors require a smaller risk premium, and vice versa. Note

19 (a) (b)

Impact on the marginal Impact on the underlying 0.03 nd 1.5 P − Pnd × 3 utility ratio, M − M price, 0 0 10

. 0 02 1.0

0.01 0.5

0.00 0.0

−0.01 −0.5 −0.02 Liquidity shock pervasiveness, π Liquidity shock pervasiveness, π −1.0 0.0 0.2 0.4π∗ 0.6 0.8 1.0 0.0 0.2 0.4π∗ 0.6 0.8 1.0

Figure 1: Liquidity risk and liquidity shock. This figure illustrates how the introduction of the derivative of f (D) = D2 affects the equilibrium at t = 0, for various levels of liquidity shock pervasiveness π ∈ (0, 1). Panel (a) plots difference in the marginal utility ratio (MUR), with the derivative less without the derivative, against π. The threshold π ∗ is where the two MUR are equal. Translating this into the ex ante asset nd price, Panel (b) plots the difference in the underlying asset price P0 and P0 against π. The other primitive parameters are set at D¯ = 0.0, X¯ = 0.8, G0 = 1.0, τε = 1.0, τz = 1.0, and α = 0.8.

↓ ↑ = nd that at the two extremes of π 0 and π 1, P0 P0 . This is because in addition to the effect on MUR, π is also the physical probability of receiving a liquidity shock. Recall from Proposition 3 that the risk-neutral probability of a liquidity shock is πM/(1 − π + πM). Therefore, in these two extremes, all investors will be of the same type at t = 1, and there will be no open interest for the derivative (Y1d = Y1s = 0). The prices with and without the derivative therefore converge. The empirical evidence of how derivatives affect their underlying prices is mixed.5 Ben Branch (1981), Conrad (1989), and Detemple and Jorion (1990) find prices of the underlying increase after option listings, while Mihov and Mayhew (2000) and Danielsen and Sorescu (2001) document the

5 While the results so far are based on a particular derivative (a variance swap of f (D) = D2), the intuition generalizes to other derivatives with nolinear payoﬀs. Indeed, in Section 4, we will show that the patterns in Figure 1 remain robust when options are introduced, matching exactly the empirical evidence about option listings cited below.

20 opposite. In particular, Sorescu (2000) finds the effect to be positive before 1981 but negative impact after. The switch coincides in time with the well-known boom of mutual funds in the 1980s. Such mutual funds can be interpreted as the liquidity demanders in our model, as they have private information (at least the active funds do) and are subject to liquidity shocks driven by fund flows.

Thus, the rise of mutual funds—an increase of parameter π—can push the price eﬀect of option listings from positive to negative, as shown in Figure 1(b).

3.3 Measuring illiquidity

The discussion above has focused on how the derivative affects the liquidity risk premium at t = 0. In the empirical literature, this liquidity risk premium, as reflected in the (abnormal) asset returns, is associated with illiquidity measures such as price impact λ and price reversal γ (Section 2.2). Vayanos and Wang (2012) discuss in great detail how these two measures connect to, and might be disconnected from, the liquidity risk premium when market conditions, such as adverse selection and competition, vary. We explore a different aspect: how these two empirical measures are

affected by derivatives. In this subsection, we examine the effect of a variance swap D2 derivative. Defining these two measure as before,

cov[P1 − P0, π · (X1d − X0)] (13) λ := and γ := −cov[D − P1, P1 − P0], var[π · (X1d − X0)] we obtain the following corollary:

Corollary 2 (Illiquidity measures with the f (D) = D2 derivative). Compared to the benchmark, after the derivative of f (D) = D2 is introduced, the underlying asset exhibits a lower price impact and the same price reversal. Mathematically, λ < λnd and γ = γ nd.

Price impact λ. Price impact measures the price elasticity with respect to the liquidity demanders’ nd − trading. Recall from Proposition 3 that compared to the benchmark X1d , a new term of 2P1Y1d

rises in X1d , reﬂecting the liquidity demanders’ delta hedging. Such delta hedging does not reveal

21 additional information. In fact, the informed trading component in X1d remains unchanged as nd in the benchmark X1d . Relatively speaking, therefore, the additional delta hedging reduces the proportion of informed trading relative to the demanders’ overall trading. Consequently, if an empiricist regresses price changes on informed investors’ trading, a lower price impact will obtain after derivatives (such as variance swaps) are introduced.

Price reversal γ . It is not surprising that price reversal remains unaﬀected. This is because, as = nd shown in Proposition 3, the t 1 equilibrium price P1 remains the same as the benchmark P1 . Therefore, the negative autocovariance of price returns is also unchanged. We again emphasize that this is a knife-edge result, due speciﬁcally to the D2 derivative, under which all investors’ delta hedging trades aggregate to zero and the net demand for the underlying asset is the same as in the benchmark (see the discussion on page 18). In Section 4 below, we introduce options as

nd an alternative. There, we show that the underlying price P1 will deviate from the benchmark P1 , tilting it in the direction of investors’ aggregate delta hedging. This new price pressure of aggregate delta hedging makes P1 either exacerbate or alleviate the price reversal γ , depending on the delta hedging sensitivity (i.e., the convexity Γ of the option).

(Dis)Connection with the liquidity risk premium. As can be seen now, after introducing

derivatives, the liquidity risk premium (reflected in the ex ante price P0) can be either amplified or alleviated, depending on the parameters. Illiquidity measures such as price impact (reduce) and price reversal (no effect) also diverge because of the derivative. Such (dis)connection between illiquidity measures and the liquidity risk premium, as an anchoring point of our paper, calls for caution in the interpretation of the empirical results. At a minimum, we argue that properly accounting for derivatives trading is necessary.

22 4 Introducing options

Thus far, we have focused on a special derivative with quadratic payoff f (D) = D2, mainly for its tractability, to formalize the model intuition. To gain additional insights, we turn to options, which are perhaps the most common (nonlinear) derivatives. We first demonstrate that the previous results regarding the liquidity risk premium and illiquidity measures remain robust. We then extract additional empirical predictions based on option-specific features.

Specifically, we introduce a single call option6 for the underlying with f (D) = max{0, D − K}, where K ∈ R is the exogenously specified strike price. The equilibrium is established in Appendix B, where we detail the investors’ optimization, prove equilibrium existence, and characterize the unique equilibrium by a set of nonlinear first-order conditions. Unfortunately, the equilibrium is analytically tractable only up to these first-order conditions. As such, below we discuss the findings and compare them with those of Section 3 mainly through numerical illustrations.

4.1 Trading at t = 1

The two types of investors’ trading of the underlying and the call option is detailed in Appendix B.1. Here, we highlight what remains robust and what diﬀers from Section 3.1.

Betting on volatility. The nonlinearity in the call payoﬀ f (D) = max{0, D −K} provides investors with a vehicle to trade on the underlying volatility. To see this, suppose that the call is (weakly)

out-of-the-money at t = 1, i.e., K ≥ P1. We can then decompose its payoﬀ as (see Lemma 2 in Appendix A)

1 1 (14) max{0, D − K} = |D − P | + 1 − 21{ ≤ ≤ } (D − P ) + 1{ > }(P − K). 2 1 2 P1 D K 1 D K 1

6 The results from this section generalize to arbitrarily many options, either calls or puts. This is because, as will be shown below, these insights are based on the moneyness of the option. When there are multiple calls, our results then speak to the average moneyness of the portfolio of calls. Moreover, the focus on calls is without loss of generality, thanks to put-call parity.

23 1 | − | Note that in expectation, the ﬁrst term, 2 D P1 , is exactly (half of) the underlying asset’s volatility. The same intuition as in Section 3.1 then applies: Because the more informed demanders value volatility less than do the suppliers, all investors trade the option to “bet on volatility;” speciﬁcally,

Y1d < 0 < Y1s .

Figure 2(a) plots an investor’s (risk-neutral) valuation of the asset’s volatility against the option’s

moneyness, deﬁned as P1 − K: The call is in-the-money (ITM) when P1 − K > 0, at-the-money

(ATM) when P1 − K = 0, and out-of-the-money (OTM) when P1 − K < 0. Indeed, a demander always believes that the conditional volatility is lower than does a supplier. As a result, the demanders always sell the option to the suppliers, as shown in Panel (b). Note that only a demander’s equilibrium option holding is plotted in (b) because by market clearing, a suppliers’

holding immediately follows to be positive: Y1s = −πY1d /(1 − π) > 0. Comparing both panels, we see that the demanders write more calls to the suppliers (more negative Y1d ) when their volatility valuations diﬀer more, i.e., either when the call is further away from ATM.

Delta hedging. We next turn to the investors’ delta hedging. A type-j ∈ {s,d} investor in equilibrium holds Y1j units of the call, which also gives her additional underlying exposure. Following Equation (10), the extra exposure, per unit of the option position, is h i 0 ∂ ∆ := Eˆ f (D) = Eˆ max{0, D − K} = Eˆ 1{ > } = Pˆ [D > K], 1j 1j 1j ∂D 1j D K 1j which has the usual interpretation of an option’s delta: the probability of ending up in-the-money.

Therefore, the investor has an incentive to delta hedge against such extra exposure by trading −∆1jY1j units of the underlying, just as in the case of a variance swap (Proposition 3). We numerically verify that this is indeed the case:

( , ) = nd( ) − ( , ). X1j P1 Q1 X1j P1 ∆1jY1j P1 Q1

That is, with a derivative, be it a variance swap or a call option, investors’ underlying trading X1j(·) nd(·) deviates from that in the no-derivative benchmark X1j exactly by the amount of their delta

24 (a) (b) An investor’s risk-neutral A demander’s holding of 2 variance, Eˆ 1 j[(D − P1) ] the call option, Y1d 0.62 0.0

−0.5 0.59

−1.0 0.56 A demander’s A supplier’s −1.5

0.53 −2.0

Moneyness, P1 − K Moneyness, P1 − K 0.50 −2.5 −2 −1 0 1 2 −2 −1 0 1 2 Figure 2: Call option trading at t = 1. This figure describes the equilibrium trading of the call option at t = 1. Panel (a) plots investors’ (risk-neutral) valuation of the underlying volatility in equilibrium. Panel (b) plots a demander’s option position. In both panels, the horizontal axes show a range of moneyness of the call option, defined as the difference between the underlying price and the strike price. The call’s strike is fixed at K = −1.0 and the two state variables—the private signal and the endowment shock—vary to affect the equilibrium price P1 and the moneyness. The other primitive parameters are set at D¯ = 0.0, X¯ = 0.8, G0 = 1.0, τε = 1.0, τz = 1.0, π = 0.5, and α = 0.8.

hedging. Hu (2014) ﬁnds evidence that order ﬂow imbalances in options translate to those in the underlyings through delta hedging.

Demanders and suppliers have diﬀerent delta hedging ratios. The results thus far are qualitatively the same as those seen in Section 3.1. There is a key diﬀerence, however. In the case of a

0 variance swap, the delta hedging ratios are the same for all investors: Eˆ 1j[f (D)] = Eˆ 1j[2D] = 2P1,

∀j ∈ {d, s}. Instead, in the case of a call option here, ∆1d = Pˆ 1d [D > K] , Pˆ 1s[D > K] = ∆1s. Figure 3 illustrates such diﬀerences by plotting various equilibrium objects against the call

option’s moneyness. Panel (a) shows the equilibrium call option price Q1. It has the well-known property of monotonically increasing in the moneyness, with an asymptotic slope equal to zero (one) when the call is extremely OTM (ITM). Panel (b) plots the two types of investors’ delta

25 (a) (b) Q 2.5 Call option price, 1 An investor’s delta hedging ratio, ∆1 j × 100% 100 2.0 A demander’s ∆1d A supplier’s ∆1s 75 1.5

1.0 50

0.5 25

0.0 Moneyness, P1 − K Moneyness, P1 − K 0 −2 −1 0 1 2 −2 −1 0 1 2 (c) (d) Delta hedging ratio Net delta hedging, Impact on the underlying ∆ − ∆ nd 3 . difference, 1d 1s −π∆1dY1d − (1 − π)∆1sY1s price, (P − P ) × 10 0 8 × 1 1 ( 100%, solid) (×103, dashed) 0.8 2

0.4 . 1 0 4

0.0 0 0.0

−1 − . −0.4 0 4

−2 −0.8 − . Moneyness, P1 − K 0 8 Moneyness, P1 − K −2 −1 0 1 2 −2 −1 0 1 2 Figure 3: Equilibrium asset prices at t = 1, with a call option. This figure describes the t = 1 equilibrium with a call option. In all panels, the horizontal axes show a range of moneyness of the call option, defined as the difference between the underlying price and the strike price. The call’s strike is fixed at K = −1.0 and the two state variables—the private signal and the endowment shock—vary to affect the equilibrium price P1 and the moneyness. The other primitive parameters are set at D¯ = 0.0, X¯ = 0.8, G0 = 1.0, τε = 1.0, τz = 1.0, π = 0.5, and α = 0.8.

26 hedging ratios. While at ﬁrst glance the two delta hedging ratios completely overlap, they actually

diﬀer by a small amount. We zoom in on their diﬀerence ∆1d − ∆1s in Panel (c), plotting it with the

solid line (left axis). It can be seen that ∆1d > ∆1s if and only if the call is ITM. The intuition is as follows. A demander always sells the call (betting on low volatility). When

the call is ITM, she knows that she will likely receive a negative shock in the underlying by t = 2 because the call will likely be exercised. To hedge this expected negative inventory shock, the

demander takes a long delta hedge position −∆1dY1d > 0. Conversely, a supplier takes a short delta

hedge position −∆1sY1s < 0. The diﬀerence is that the more informed demanders are “surer” of the moneyness of the call than are the suppliers. Therefore, the demanders’ delta hedging ratio is larger,

∆1d > ∆1s, if the call is ITM. Instead, if the call is OTM, the surer demanders will delta hedge less

than the not-so-sure suppliers, with ∆1d < ∆1s, because the call is unlikely to be exercised. When

the call is ATM, we have the well-known result of ∆1d = ∆1s = 1/2.

Note that when the moneyness becomes extreme, we see that ∆1d −∆1s ↓ 0 again. This is because

the delta of a call is bounded between zero and one: 0 ≤ ∆1j = Pˆ 1j[D > K] ≤ 1. In particular, for a deep ITM (OTM) call, its delta converges to unity (zero), regardless of the information set of the investor. This explains the ﬂattening tails in the solid line in Panel (c).

The net delta hedging. Aggregating across all investors, the net delta hedging trade is

(15) − π∆1dY1d − (1 − π)∆1sY1s,

which is in general nonzero, unlike the case of a variance swap. The dashed line (right axis) in Figure 3(c) plots the net delta hedging trade against the call’s moneyness. It can be seen that it tracks the pattern from the delta hedging ratio diﬀerence (the solid line, left axis). This is unsurprising

because Equation (15) is simply the market clearing condition, πY1d + (1 − π)Y1s = 0, rescaled by the respective investors’ delta hedging ratios. When the call is ITM, the demanders delta hedge (long) more than the suppliers (short), and the net delta hedging becomes positive; vice versa. The analysis thus yields the following novel empirical prediction:

27 P1 − K < 0 P1 − K > 0 OTM-call ITM-put ITM-call OTM-put (a) Option holdings

a demander’s position Y1d < 0 < 0 < 0 < 0

a supplier’s position Y1s > 0 > 0 > 0 > 0 (b) Delta hedging trades * * a demander’s −∆1dY1d > 0 < 0 > 0 < 0 * * a supplier’s −∆1sY1s < 0 > 0 < 0 > 0 (c) Net delta hedging trades

−π∆1dY1d − (1 − π)∆1sY1s < 0 < 0 > 0 > 0

Table 1: Investors’ option holding and delta hedging trades for calls and puts. This table summarizes investors’ option holding and delta hedging trades for four scenarios: out-of-the-money call, in-the-money call, in-the-money put, and out-of-the-money put. Panel (a) shows the equilibrium option positions by the demanders and the suppliers. Panel (b) shows their delta hedging directions. The dominant eﬀect in each scenario (column) is superscripted with ∗, following which Panel (c) shows the signs of the net delta hedging.

Prediction 1 (Net delta hedging and option moneyness). Investors’ net delta hedging against

their volatility bets has the same sign as the moneyness P1 − K. That is, such net delta hedging is positive (negative) when the option is an ITM call or OTM put (OTM call or ITM put).

To depict the robustness of the result to various combinations of option types and moneyness, we summarize the possible scenarios in Table 1. What matters for the net delta hedging trade is whether the (informed) demanders expect a positive or negative cash ﬂow from the option: If the option is in-the-money, as the writer (seller), the surer demanders expect negative t = 2 cash ﬂows from the option and engage in considerable delta hedging, which dominates in the aggregate. If the option is out-of-the-money, the demanders expect to enjoy the positive sales from writing the options and engage in little delta hedging. Therefore, the suppliers’ delta hedging dominates. In any case, the net delta hedging direction is the same with sign[P1 − K]—the moneyness matters.

The underlying price at t = 1. The nonzero net delta hedging is the key diﬀerence from the case of a variance swap, where the delta hedging trades always sum to zero; see the discussion on

28 page 18. Such nonzero price pressure aﬀects the equilibrium underlying price. Figure 3(d) shows

nd the pattern. Relative to the no-derivative benchmark underlying price P1 , P1 increases when the net delta hedging trade shown in Equation (15) is positive. That is, the underlying price is pushed in the direction of the net delta hedging. Such “price pressure” from net delta hedging has impacts on the illiquidity measures, which we study later in Section 4.3. We would like to emphasize that the price pressure due to delta hedging is more general than the specific model studied here. This is because asymmetrically informed investors tend to assign different values to the necessity (urgency) of delta hedging. The more informed agents (the demanders in the model) will delta hedge more aggressively than their less informed counterparties if and only if their derivative positions expect negative cash flows. This intuition is revealed through two modeling ingredients: the trading of the underlying and the option is linked through investors who are asymmetrically informed.

4.2 Liquidity risk premium at t = 0

Section 3.2 shows that the ex ante liquidity risk premium can be either alleviated or worsened after the derivative paying off D2 is introduced (Corollary 1). This result remains robust when options are introduced, and the intuition is the same: With options, investors are now able to trade on the volatility (through the options’ nonlinear payoffs) of the underlying asset. When the liquidity shock pervasiveness π changes, the ex ante marginal utility ratio (MUR) might either increase or decrease, depending on whether the liquidity demanders or the suppliers benefit more.

As an example, Figure 4 qualitatively replicates Figure 1. As π increases, the liquidity shock aﬀects more of the population, the MUR M monotonically increases, and there is a threshold π # only above which M > Mnd. Similarly, there is a threshold π ∗ only above which the liquidity risk premium increases (the underlying price P0 decreases) with the introduction of the call option. Note that unlike Figure 1, the two thresholds π # for MUR and π ∗ for the underlying price are not the same here. This is because in the case of a variance swap (f (D) = D2), only the risk-neutral

29 (a) (b)

Impact on the marginal Impact on the underlying 0.02 nd 0.6 P − Pnd × 3 utility ratio, M − M price, 0 0 10 0.4 0.01 0.2

0.0 0.00 −0.2

−0.4 −0.01 −0.6 Liquidity shock pervasiveness, π Liquidity shock pervasiveness, π −0.02 −0.8 0.0 0.2π# 0.6 0.8 1.0 0.0 0.2π∗ 0.4 0.6 0.8 1.0

Figure 4: Liquidity risk and liquidity shock, with one call option. This ﬁgure qualitatively replicates the patterns shown in Figure 1. The diﬀerence is that here we only introduce a single call option at strike price K = −1.0. The other primitive parameters are set at D¯ = 0.0, X¯ = 0.8, G0 = 1.0, τε = 1.0, τz = 1.0, and α = 0.8.

probability of receiving a liquidity shock, πM/(1 − π + πM), is aﬀected by π. In general, the

parameter π could also aﬀect other equilibrium components in the ex ante underlying price P0, the closed-form solution of which we unfortunately do not obtain.

4.3 Measuring illiquidity

We now turn to the illiquidity measures for the underlying asset. Figure 5 plots the two focal

measures, price impact λ and price reversal γ (deﬁned in Equation 13), against various moneyness values of the call.

Price impact. Figure 5(a) shows a qualitatively similar result to that seen in Section 3.3: The

price impact is lower with a derivative, i.e., λ < λnd. The key driver of the lower price impact is

30 that the liquidity demanders now delta hedge their volatility bets. Such additional hedging trades are not informed speculation, making the demanders’ trades less “toxic” overall, thereby reducing price impact. The literature has documented such reduction; see, e.g., Kumar, Sarin, and Shastri (1998). Further, after introducing options, the bid-ask spread in the underlying typically also decreases (Damodaran and Lim, 1991; Fedenia and Grammatikos, 1992; and Kumar, Sarin, and Shastri, 1995). In particular, Kumar, Sarin, and Shastri (1998) find that it is the adverse-selection component of the bid-ask spread that decreases, consistent with our intuition of informed trading being diluted by delta hedging. Recent evidence by Hu (2017) is also consistent with our channel whereby uninformed trading becomes dominant (relative to informed trading) after option listing. It should be noted that the existing empirical literature typically argues that the price impact or bid-ask spread decreases because informed investors migrate from the underlying to the options (see, e.g., Easley, O’Hara, and Srinivas, 1998). While the prediction is the same, our proposed channel of uninformed delta hedging is different because in our model, derivatives are purposefully made informationally irrelevant. To examine our channel, we propose to investigate how price impact is affected. Figure 5(a) depicts the patterns regarding how the underlying’s price impact changes with respect to the

moneyness of the call. Starting from the left side of the graph, where the call is deep OTM (K is very large), we see that λ ≈ λnd. This is intuitive because the deep OTM call is valueless. Its impact on the equilibrium, compared to the no-derivative benchmark, is close to nothing.

As the moneyness increases, we see that λ first decreases and then increases. This is because there are two components of the price return P1 − P0, and they are affected differently by the demanders’ trading X1d − X0. To see them, rewrite the price return as

P − P = (E [D] − E [D]) + ((P − E [D]) − (P − E [D])), 1 0 | 1 {z 0 } | 1 1 {z 0 0 } (i) change in the fundamental (ii) change in the price pressure

where we essentially nonparametrically decompose a price Pt into (i) the time-t fundamental Et [D]

and the remainder Pt − Et [D], which we refer to as (ii) price pressure. Therefore, the price impact

31 (a) (b) λ × 3 7 Price impact Price reversal γ 10 γ with a call option 79.6 nd γ in the benchmark

λ with a call option nd . 5 λ in the benchmark 79 4

79.2

3 79.0

. Average moneyness, E[P1] − K 78 8 Average moneyness, E[P1] − K 1 −4 −2 0 2 4 −4 −2 0 2 4

Figure 5: Illiquidity measures, with vs. without a call option. This figure shows how illiquidity measures are affected by a call option. Panel (a) shows the price impact λ (with the call, the solid line) and λnd (no derivative, dashed), while Panel (b) shows the price reversal γ (with the call, solid) and γ nd (no derivative, dashed). In both panels, the horizontal axis is the average moneyness of the introduced call option, calculated by varying the strike price K, solving the underlying price P1 at t = 1, and then taking expectation of the difference, i.e., E[P1] − K. The other parameters are set at π = 0.50, D¯ = 0.0, X¯ = 0.8, G0 = 1.0, τε = 1.0, τz = 1.0, and α = 0.8.

λ is more precisely the sum of the impact on (i) and that on (ii).7 As the call becomes more ITM, the demanders know that the call they write is more likely to be exercised eventually, and they delta hedge more. Such increases in delta hedging trades are uninformative (Lemma 1) and only increase the volume of the demanders’ underlying trading. Therefore, the price impact in (i) is lower—there is more trading but the fundamental price is unaﬀected. However, such delta hedging trades create increasingly larger price pressure in (ii). In particular, when the call option becomes deep ITM, component (ii) of price pressure dominates,

7 In the empirical market microstructure literature, trades’ impact on (i) and (ii) are referred to as the permanent and the transitory price impact, respectively, because changes in (i) are driven by information revealed in trades and therefore persist, while changes in (ii) will mean-revert to zero in the long run. See, e.g., Glosten and Harris (1988).

32 thus driving λ higher, resulting in the U-shape seen in Figure 5(a). The following empirical prediction summarizes the discussion thus far.

Prediction 2 (Price impact and option moneyness). The price impact λ of the underlying is U-shaped in the option moneyness.

While the discussion has focused only on a call option, the prediction above does not distinguish between calls and puts, thanks to put-call parity. (While the moneyness ﬂips between a call and a put of the same strike, the predicted U-shape still remains U-shaped after the ﬂip.)

Price reversal. In Section 3.3, the price reversal is unaﬀected by the variance swap of f (D) = D2. Such a knife-edge result no longer holds here. Figure 5(b) shows that when the moneyness of the call

option is moderate (roughly between -1.7 and +1.7), the price reversal γ is exacerbated (γ > γ nd); however, when either deep ITM or OTM, γ is alleviated (γ < γ nd). Two features of the model can explain this pattern. First, the demanders’ delta hedging trade,

−∆1dY1d , always ampliﬁes price pressure, while that of the suppliers −∆1sY1s has a dampening

eﬀect. To see this, consider a small positive increase in the equilibrium price P1, which makes the call option more ITM. As such, both types of investors would like to delta hedge more than before,

as their ∆1j increases (Figure 3b). In particular, the demanders delta hedge more by buying more of the underlying, thus amplifying the small initial price increase, while the suppliers sell more, dampening the initial price increase. Whether the net eﬀect is amplifying or dampening depends

on whether ∆1d is more or less sensitive than ∆1s. That is, the sensitivity of the delta hedging ratios—the gamma, Γ—matters.

This leads to the second feature: The demanders’ ∆1d is more sensitive than the suppliers’ ∆1s only when the call option’s moneyness is moderate. This can be seen by zooming in on Figure 3(b):

∆1d is steeper than ∆1s for moderate moneyness but ﬂatter when the call is deep ITM or OTM.

Figure 6 plots the two types of investors’ average delta hedging sensitivity E[Γ1d ] and E[Γ1s] in Panel (a) and their diﬀerence in Panel (b).

33 (a) (b) 2 An investor’s average delta hedging Gamma difference, E[Γ1d − Γ1s] × 10 sensitivity, E[Γ1 j] A demander’s 3 A supplier’s 4

2 2

0 1

−2 Average moneyness, E[P1] − K Average moneyness, E[P1] − K 0 −4 −2 0 2 4 −4 −2 0 2 4

Figure 6: Investors’ delta hedging sensitivity, gamma. This figure plots investors’ average delta hedging sensitivity to fluctuations in the underlying price P1, i.e., their gamma. Panel (a) jointly plots both the demanders’ and the suppliers’ gammas. Panel (b) plots their difference, namely the demanders less the suppliers’ gamma. The other parameters are set at π = 0.50, D¯ = 0.0, X¯ = 0.8, G0 = 1.0, τε = 1.0, τz = 1.0, and α = 0.8.

By combining these two features, we see that the price reversal γ (relative to the no-derivative case γ nd) is driven by the delta hedging sensitivity, or gamma, of the call option. Compare Figures 5(b) and 6(b). It can be seen that the price reversal is exacerbated precisely when the demanders’ delta hedging is more sensitive (Γ1d > Γ1s), i.e., for moderate moneyness of the call. In this region, the demanders’ amplifying delta hedging trades respond to price ﬂuctuations more than the suppliers’ dampening delta hedging trades. We summarize the discussion in the following empirical prediction.

Prediction 3 (Price reversal and option moneyness). Compared to the no-derivative benchmark, the underlying’s price reversal with an option is higher only when the option’s moneyness is moderate (not too OTM or too ITM).

34 Barbon and Buraschi (2019) provide consistent evidence. They approximate dealers’ (liquidity suppliers’) aggregate gamma imbalance and ﬁnd that when it is negative, there is larger intraday momentum, i.e., larger price reversal.

Taken together: different measures, different effects. By comparing the two panels in Figure 5, we see that the two illiquidity measures are affected very differently by the call option: Price impact

λ < λnd, while price reversal γ ≶ γ nd depending on the moneyness of the call. This is because the two measures reﬂect diﬀerent aspects of the very broad notion of “illiquidity.” First, illiquidity can arise from pure information asymmetry (Kyle, 1985; Glosten and Milgrom, 1985). Second, illiquidity can arise from inventory cost and risk-sharing motives (Roll, 1984; Grossman and Miller,

1988). In the current model setting, the price impact λ mainly reﬂects the suppliers’ risk aversion to demanders’ private information, while the price reversal γ mainly captures the transitory price pressure due to risk sharing. In particular, the decrease in price impact (λ < λnd) is due to the decline in the proportion of informed trading in demanders’ total trading—lower illiquidity of information asymmetry. The increase in price reversal (γ > γ nd) is due to the additional (delta) hedging—higher illiquidity of risk sharing. These results suggest that derivatives might disconnect the empirical illiquidity measures from the actual liquidity risk premium in asset prices:

(1) ex ante MUR might either increase or decrease, i.e., M ≶ Mnd; (2) illiquidity measures would potentially disagree, with λ < λnd but γ ≶ γ nd. As such, it is important that the trading in the derivatives be properly accounted for when analyzing liquidity risk and market illiquidity. Our analysis also highlights that by properly controlling for the moneyness of the derivative, one can generate sharper predictions regarding these illiquidity measures.

35 5 Conclusion

This paper studies how derivatives (with nonlinear payoffs) affect their underlying securities’ liquidity. Through a rational expectations equilibrium model, we examine such derivatives’ implications for the liquidity risk premium and for empirical illiquidity measures. The key element of the model is a liquidity shock that randomly strikes a fraction of the otherwise homogeneous population. The shocked investors then demand liquidity for both information and hedging reasons, while the rest provide liquidity to them. We contrast equilibrium outcomes with and without derivatives. In terms of the liquidity risk premium, we find that derivatives have ambiguous effects. Intro- ducing derivatives allows investors to trade on some nonlinear structures of the underlying’s future payoff, creating additional trading gains. However, the split of such trading gains between liquidity demanders and suppliers depends on model parameters. In general, derivatives affect the wedge between liquidity demanders’ and suppliers’ ex ante marginal expected utilities. As such, before the shock, investors adjust—sometimes amplify—the risk-neutral probability of receiving the liquidity shock and might demand higher liquidity risk premia. In terms of empirical illiquidity measures, we find seemingly “contradictory” messages, depending on the specific measure chosen. The key channel is investors’ delta hedging of their derivative positions in the underlying. Such new hedging trades dilute the informed trading in the total trading volume, lowering price impacts—hence yielding better liquidity. On the other hand, the equilibrium price is now more subject to the price pressure of delta hedging trades, sometimes amplifying but sometimes dampening the price reversal. Taken together, the results of our model emphasize the potential disconnect between assets’ liquidity risk premium and their empirical illiquidity measures because of the trading of derivatives. Empirical studies associating the two sides should carefully control for the activity in the assets’ derivative markets.

36 Appendix

A Proofs

Lemma 1

Proof. Clearly, the demanders’ learning is unaffected by the derivative: D|s still normally distributed −1 G0 G1d −G0 with var1 [D] = G and E1 [D] = D¯ + s. We need to show that the suppliers do not d 1d d G1d G1d learn more than they do in the benchmark. That is, fixing the same realizations of {D, ε, z}, with or without the derivative, the suppliers hold the same posterior that D is normally distributed with nd −1 nd G0 G1s −G0 var1 [D] = var [D] = G and E1 [D] = E [D] = D¯ + η. s 1s 1s s 1s G1s G1s Consider an arbitrary investor in this economy with type-j ∈ {d, s}. Her terminal wealth is given in Equation (8). She chooses her demand X1j and Y1j to maximize her conditional expected −αW2 utility at t = 1, i.e., U1j := E1j −e j , taking both prices {P1, Q1} as given. It is well-known that such an optimization problem is strictly concave.8 Therefore, if an equilibrium exists, it must be interior and the first-order conditions must hold: ∂U1j − = E α · (D − P )e αW2j = 0; and ∂X 1j 1 (A.1) 1j ∂U 1j −αW2j = E1j α · (f (D) − Q1)e = 0. ∂Y1j Recalling that from a demander’s point of view, D is conditionally normal. Using the conditional density, her above first-order conditions can be rearranged and simplified to a two-equation-two- unknown system: ¹ G 2 − 1d − ¯ − ˆ+ α −α f (D)Y1d 2 D D η G X1d (A.2) 0 = (D − P1)· e · e 1d dD; and R ¹ G 2 − 1d − ¯ − ˆ+ α −α f (D)Y1d 2 D D η G X1d (A.3) 0 = (f (D) − Q1)· e · e 1d dD. R

∂2 8 U1j = Formally, the Hessian H of the optimization problem is a symmetric 2-by-2 matrix, with diagonal terms ∂ 2 X1j − ∂2U − E − 2 · ( − )2 αW2j = E [− ( )2] 1j = E − 2 · ( ( ) − )2 αW2j = E [− ( )2] 1j α D P1 e : 1j a D and 2 1j α f D Q1 e : 1j b D , both strictly ∂Y1j ∂2 U1j 2 −αW negative. The oﬀ-diagonal term is = E1 −α · (D − P1)(f (D) − Q1)e 2j = E1 [−a(D)b(D)]. It follows that ∂X1j Y1j j j = [ , ]> ∈ R2 > = E [− ( )2 2 − ( ) ( ) − ( )2 2] = E [−( ( ) + ( ) )2] < for any nonzero x x1 x2 , x Hx 1j a D x1 2a D b D x1x2 b D x2 1j a D x1 b D x2 0, i.e., the Hessian is negative deﬁnite. Therefore, the optimization problem is strictly convex.

37 where τ α τ ηˆ := ε s − D¯ − z = ε η − D¯ τε + G0 τε τε + G0 is informationally equivalent to what suppliers can infer in the benchmark (c.f., Equation A.5). Now turn to the suppliers. In equilibrium, they know that Equations (A.2) and (A.3) must hold.

They can condition on the equilibrium prices {P1, Q1}. They also observe the demanders’ demand

realizations {X1d, Y1d }. This is because in equilibrium, the suppliers know their own demand

{X1s, Y1s } and through the market clearing conditions πX1d +(1−π)X1s = X¯ and πY1d +(1−π)Y1s = 0,

the suppliers can thus infer perfectly the realizations of {X1d, Y1d }. Knowing {P1, Q1, X1d, Y1d }, therefore, from any supplier’s perspective, each equation in the system (A.2) and (A.3) has one and only one unknown, ηˆ. Therefore, they can infer, at best, ηˆ.

We further show below that given {P1, X1d }, the ﬁrst line of Equation (A.2) has unique solution of ηˆ. Hence, the suppliers can fully back out ηˆ, learning exactly the same as they do in the benchmark. (Equation A.3 is redundant for learning, in this sense.) To prove the uniqueness of the solution, we begin by writing G 2 − 1d − ¯ − ˆ+ α α −α f (D)Y1d 2 D D η G X1d a(D;η ˆ) := e · e 1d and b(D;η ˆ) := D − D¯ − ηˆ + X1d, G1d ∂a(D;η ˆ) = ( ) ( ) so that ∂ηˆ G1da D;η ˆ b D;η ˆ and Equation (A.2) can be rearranged as ∫ ( ) R Da D;η ˆ dD (A.4) P = ∫ . 1 ( ) R a D;η ˆ dD Note that this is an equation of ηˆ and the suppliers try to solve ηˆ from it. To show the uniqueness of the solution, take derivative with respect to ηˆ on the right-hand side to get ¹ ¹ ¹ ¹ G1d ∫ DabdD adD − DadD abdD , 2 R R R R R adD where we omit the arguments of a(·) and b(·) for notation simplicity. Note that b can be rewritten α as b = D − c where c := D¯ + ηˆ − X1 is independent of D conditional on ηˆ. Plug in b = D − c G1d d and simplify to get " # ¹ ¹ ¹ 2 G1d 2 ∫ D adD adD − DadD . 2 R R R R adD ∫ 2 ∫ ∫ ( ) ( ) ≤ ( )2 ( )2 2 = Cauchy-Schwarz inequality has that R f x д x dx R f x dx R д x dx. Note D a

38 √ √ D a · a always holds because a > 0. Applying the above Cauchy-Schwarz inequality with √ √ f = D a and д = a yields ¹ 2 ¹ ¹ DadD ≤ D2adD adD. R R R Therefore, the right-hand side of Equation (A.4) is monotone increasing in ηˆ. That is, so long the equilibrium exists, suppliers can exactly infer the ηˆ.9 Since ηˆ is informationally equivalent to what suppliers can learn without the call options (as in the benchmark), there is no additional information revealed. □

Lemma 2

Lemma 2 (Decomposition of a call). Suppose the underlying price at t = 1 is P1. The t = 2

payoﬀ of an out-of-the-money call option with strike K ≥ P1 can be decomposed into 1 1 max{0, D − K} = |D − P | + 1 − 21{ ≤ ≤ } (D − P ) + 1{ > }(P − K); 2 1 2 P1 D K 1 D K 1

and that of an in-the-money call with K ≤ P1 can be decomposed into 1 1 max{0, D − K} = |D − P | + 1 + 21{ ≤ ≤ } (D − P ) + 1{ > }(P − K). 2 1 2 K D P1 1 D K 1

Proof. Consider the out-of-the-money call with K ≥ P1. 1 1 1 1 max{0, D − K} = |D − K | + (D − K) = |V − (K − P )| + (V − (K − P )) 2 2 2 1 2 1

where V := D − P1 as a shorthand notation. Compare |V − (K − P1)| to |V |:  K − P , if V < 0  1 | − ( − )| − | | = − + ( − ), ≤ ≤ − . V K P1 V  2V K P1 if 0 V K P1  −( − ), > −  K P1 if V K P1 | − ( − )| = | | + 1 ( − ) + 1 (− + − ) − 1 ( − ) Therefore, V K P1 V {V <0} K P1 {0≤V ≤K−P1} 2V K P1 {V >K−P1} K P1 . Substituting into the call’s payoﬀ expression and simplifying gives the expression stated in the lemma. The proof for the decomposition of the in-the-money call repeats the above steps and is omitted. □

9 More rigorously, one needs to prove the existence of the solution to Equation (A.4) in solving for ηˆ. This is trivial, because Equation (A.4) is a rewriting of the ﬁrst-order condition (A.1). Since the equilibrium is assumed to exist, the ﬁrst-order condition necessarily holds.

39 Propositions 1 and 2

Proof. The two propositions (and their proofs) correspond to Propositions 3.1-3.3 in Vayanos and Wang (2012). We highlight some notation diﬀerences below. The three constants a, b, and c in

Equation (3.1) of Vayanos and Wang (2012) can be expressed in our notation as a = D¯ − αX¯ /G1, b = 1 − G0/G1, and c = α/(G1d − G0). These then help verify our Proposition 1 as a replication

of their Propositions 3.1 and 3.2. The stated P0 in our Proposition 2 has the same form as the one

in their Proposition 3.3, where our αΣ replaces their ∆1. That is, our Σ can be expressed in their notation as α 2bσ 2(σ 2 + σ 2)σ 2 Σ = ϵ z + ( − )2 − 2 2 2 1 ∆0 1 π α σ σz 2 = −1 2 = −1 2 = −1 where σ G0 , σz τz , σϵ τε , and ∆0 is given in Their equation (3.7a) and can be ( − )2 equivalently written in our notation as ∆ = 1 G0 G1 G1s . Finally, the marginal utility ratio 0 2 G1 −G0 2 π s G1 Mnd in Proposition 2 can be expressed as s α 1 + π 2∆ Mnd = exp ∆ θ¯2 0 2 + ( − )2 − 2 2 2 2 1 ∆0 1 π α σ σz ¯ = ¯ 2 = −1 2 = −1 where θ X, σ G0 , σz τz , and ∆2 is given in their equation (3.7c). In our notation, ∆2 can be written as h i ( − )2 3 −1 −2 + G0 G1 G1d α τ G 1 − 2 z 0 G1d G0 G ∆ = 1 . 2 + ( − )2 − 2 −1 −1 1 ∆0 1 π α τz G0 We also highlight how the suppliers learn in this equilibrium. The only source of new infor- nd mation for the suppliers is the risky asset’s price P1 , whose closed-form solution is spelled out in Equation (5). It can be seen that the only “learnable” component is a linear combination of the private signal s and the endowment shock z: α α (A.5) η := s − z = D + ε − z. τε τε That is, the suppliers are only learning from the above noisy signal η, which is a linear combination of all three random variables in this economy. Given this, we have varnd[ ] = var[ nd ] = −1 1s D D P1 G1s G G −G and End[D] = E[D Pnd ] = 0 D¯ + 1s 0 η. □ 1s 1 G1s G1s

40 Proposition 3

Proof. Consider a type-j investor. Her terminal wealth W2j is given by Equation (8). Lemma 1 ensures that she holds the same posterior distribution for D with or without the derivative. In particular, D remains conditionally normal. Evaluating the expected utility (e.g., Lemma A.1 of

Marín and Rahi, 1999), we get (writing W1 = W0 + (p − P0)X0): 1 −αW2j E1j −e = − p exp α −W1 + X1jp + zjD¯ + Y1jq 1 + 2αvar [D]Y 1j 1j ! α 2var [D] X + z + 2Y E [D] 2 · − ( + )E [ ] − E2 [ ] + 1j 1j j 1j 1j . exp α X1j zj 1j D αY1j 1j D 2 1 + 2αvar1j[D]Y1j

The ﬁrst-order condition with respect to X1j yields

E1j[D] − p X1j = − zj − 2pY1j . αvar1j[D]

−αW2 Plug this expression back to E1j[−e j ] and evaluate the ﬁrst-order condition with respect to Y1j to get: = 1 1 − 1 . Y1j 2 2α q − p var1j[D]

Finally, clearing the market with (2) and (3) yields the equilibrium prices p = P1 and q = Q1 as stated in the proposition. (The utility maximization problem is a strictly concave one. Hence, the above solution implied by the ﬁrst-order conditions are unique.) □

Proposition 4 −αW2 Proof. The proof of Proposition 3 gives an investor’s expected utility at t = 1, E1j −e j ,

taking p = P1 and q = Q1 as given. Consider a demander (j = d) ﬁrst. Expanding with

W1 = W0 + (P1 − P0)X0 and E1d [D] with s and z gives r − − 2 G G1 G1d − G1d ¯ + G1d G0 ( − ¯ )− −αW2d 1d 2G −α·(W0+(P1−P0)X0+(P1−D¯ )z) 2 D G s D P1 E1d −e = − · e 1 · e · e 1d G1 = nd where P1 P1 can be further written as a linear combination of s and z. Taking the expectation of the above over {s, z} yields the “interim” utility U0d of a demander; that is, the expected utility after the type realizes but before the signal and the endowment shock are observed: r !− 1 − 2 2 2 2 G G1 G1d − G G α α 1d 2G αW0d 0 1d U0d = − · e 1 1 + 1 − 1 + − , G1 G1d − G0 G1 τετz G0τz

41 where " # −1 2 2 2 α α 2 α G0 G1d α α W0d :=W0 + (D¯ − P0)X0 − X0X¯ + X¯ − 1 + 1 − 1 + − G0 2G0 2 G1d − G0 G1 τετz G0τz ( 2 2 G1 − G0 1 1 α 2 · + 1 + X0 − X¯ G1 G0 τε τετz " " # #) 2 2 2 2 1 1 α G0 G0 G0 G1d G0 2 + + 2 1 − 1 − X0X¯ + 1 − − 1 − X¯ . G0 τε τz G1 G1d G1d G1 G1

Note that condition (1) ensures U0d is well-deﬁned; in particular, the term inside the brackets is always positive. Similarly, the interim utility U0s of liquidity suppliers can be derived as r !− 1 − 2 2 G G1 G1s − G G 1s 2G αW0s 0 1s U0s = − · e 1 1 + 1 − , G1 G1s − G0 G1 where " # −1 α αG G G 2 W = W + (D¯ − P )X − X 2 + 1s 1 + 0 1 − 1s · X − X¯ 2. 0s 0 0 0 2G 0 2 G − G G 0 0 2G1 1s 0 1

At t = 0, investors choose X0 to maximize

πU0d + (1 − π)U0s .

The ﬁrst-order condition, together with the market clearing condition X0 = X¯, leads to · ¯ − − −1 ¯ − ¯ + ( − ) ¯ − − −1 ¯ = , π D p αG0 X αΣX M 1 π D p αG0 X 0 where r s − G1s G1 + 2 d G1d α 2 1 π ∆0 = 2G1 exp ¯ , M e ∆2X 2 2 G1s 2 1 + ∆0(1 − π) − α /(τzG0) and ∆0 and ∆2 are the same coeﬃcients as given in the proof of Propositions 2 and 1. (Note that

the second-order conditions are satisﬁed as well as both U0d and U0s are monotone transformation

of quadratic terms in X0.) It can be seen that the above ﬁrst-order condition is linear in the market

clearing price p, which then uniquely solves the equilibrium P0 stated in the proposition. □

42 Corollary 1

Proof. Based on Proposition 2 and 4, the diﬀerence between the two underlying prices at time 0 is nd nd πM πM P0 − P = −αΣX¯ − . 0 1 − π + πM 1 − π + πMnd < nd / nd > □ Clearly, P0 P0 if and only if M M 1.

Corollary 2

Proof. By market clearing, we have π(X1d − X¯) = −(1 − π)(X1s − X¯ ), where X1s is given by Proposition 3. Therefore, [ − , ·( − )] −cov P − P , (1 − π) X − X¯ ( − ) =cov P1 P0 π X1d X0 = 1 0 1s = 1 G1 G0 α . λ [ ·( − )] − ( − ) var π X1d X0 var (1 − π)(X1s − X¯) 1 π G1 G1s G1 nd Compared to Equation (6) in the benchmark, we have λ/λ = G0/G1 < 1. = nd From Proposition 3, P1 is unchanged after the introduction of options, P1 P1 . As a result, the price reversal γ is unaﬀected (see Equation 7). □

B Characterizing the equilibrium with ﬁnite calls

We introduce n (n < ∞) call options, each with (ﬁnite) strike price Ki, i ∈ {1, 2, ..., n}. That is, the i-th call pays max{0, D − Ki } at t = 2. The setting resembles the one studied by Gao and Wang (2017), who focus on the implications of options on volatility but, diﬀerent from the current setting, do not feature an ex ante period of t = 0 to study liquidity risk.

B.1 Trading at t = 1

To begin with, Lemma 1 ensures that these additional call options do not reveal new information.10 Therefore, both the demanders and the suppliers have the same posterior distribution of D as in the benchmark.

10 Lemma 1 only proves the information redundancy result for a single arbitrary derivative. It can be easily generalized by noting (1) that the demanders’ learning is unaffected; (2) that the suppliers can infer from the market clearing conditions the exact quantities of the demanders’ demand in all assets; (3) that the suppliers effectively learn from the demanders’ first-order conditions; and (4) that the demanders’ first-order condition for the underlying always reveals the same information as in the benchmark.

43 Next, we consider investors’ optimization. A type-j ∈ {d, s} investor’s terminal wealth is Õn + W2j = W1 + (D − P1)X1j + (D − Ki) − Q1i Y1ji + (D − D¯ )zj, i=1

where W1 = W0 + (P1 − P0)X0; Q1i is the price of the call with strike Ki (at t = 1); Y1ji is the = = investor’s holding of that call; zj is her endowment shock (zd z and zs 0). She chooses her −αW2 demand X1j and {Y1ji } to maximizes U1j := E1j −e j .

We first establish the uniqueness of the optimal demand, taking all prices P1 and {Q1i } as given. This is because the optimization problem is strictly concave: 2 ∂ U1j − = E − 2 ·( − )2 αW2j < 0 2 1j α D P1 e ∂X1j ∂2 h i U1j + 2 − = E − 2 · ( − ) − αW2j < 0, ∀ ∈ {1, ..., }. 2 1j α D Ki Q1i e i n ∂Y1ji Therefore, the first-order conditions (if the solution exists) are sufficient for the global optimum: ∂U1j − = E α · (D − P )e αW2j = 0; and ∂X 1j 1 (B.1) 1j ∂U 1j + −αW2j = E1j α · (D − Ki) − Q1i e = 0, ∀i ∈ {1, ..., n}. ∂Y1ji While not analytically tractable, it is clear that the first-order conditions (B.1) always have a solution

(existence): Consider the extreme choices of X1j and its implication for U1j for example. When

|X1j | → ∞, U1j → −∞ because with infinite holding X1j, the uncertainty associated with the risky payoff D becomes infinity. Such infinite payoff risk makes the investor suffer from infinite = −∞ risk. Given that the optimization is strictly concave and that limX1j →∞ U1j , there exists a unique X1j that maximizes U1j. The same argument applies to option holdings {Y1ji }.

We have shown that the optimal demand is a unique function of the prices P1 and {Q1i }. These n + 1 prices are pinned down via the n + 1 market clearing conditions:

πX1d + (1 − π)X1s − X¯ = 0; and πY1di + (1 − π)Y1si = 0, ∀i ∈ {1, ..., n}. or, deﬁning F : Rn+1 7→ Rn+1:     πX + (1 − π)X − X¯ 0  1d 1s     + ( − )     πY1d1 1 π Y1s1  0 (B.2) F(P1, Q1i, ..., Q1n) :=  .  = ..  .  .          πY1dn + (1 − π)Y1sn  0

44 It turns out that the prices pinned down by F(·) = 0 are also unique. We prove this by showing that the Jacobian matrix of F(·)

 dX dX dX   dX dX dX   1d 1d ··· 1d   1s 1s ··· 1s   dP1 dQ11 dQ1n   dP1 dQ11 dQ1n   dY1d1 dY1d1 ··· dY1d1   dY1s1 dY1s1 ··· dY1s1   dP1 dQ11 dQ1   dP1 dQ11 dQ1  (B.3) π ·  . . . . n  + (1 − π) . . . . n   . . . . .   . . . . .       dY dY dY   d d d   1dn 1dn ··· 1dn   Y1sn Y1sn ··· Y1sn  dP1 dQ11 dQ1n dP1 dQ11 dQ1n is negative deﬁnite. To do so, consider the underlying asset for example (the argument is generic for

any of the n +1 securities). The ﬁrst-order condition (B.1) is an implicit function of the demand X1j and all n + 1 prices. By implicit function theorem, we have − − dX ∂2U / ∂X ∂P E αe αW2j − α 2 ·(D − P )X e αW2j 1j = − 1j 1j 1 = 1j 1 1j . 2 2 2 2 −αW dP1 ∂ U1j/∂X1j E1j −α ·(D − P1) e 2j

Clearly, the denominator is negative (it is the second-order derivative of U1j with respect to X1j). The numerator can be further simpliﬁed: − − − − E αe αW2j − α 2 ·(D − P )X e αW2j = E αe αW2j − αX E α ·(D − P )e αW2j > 0. 1j 1 1j 1j 1j | 1 j {z 1 } =0, by the ﬁrst-order condition (B.1)

Therefore, we have dX1j/dP1 < 0 in equilibrium, which is an intuitive result that the demand for a security strictly decreases with its price, for both j ∈ {d, s}. Again by implicit function theorem, ∀i ∈ {1, ..., n}, we also have − dX ∂2U / ∂X ∂Q E −α 2 ·(D − P )Y e αW2j 1j = − 1j 1j 1i = 1j 1 1ji 2 2 2 2 −αW dQ1i ∂ U1j/∂X1j E1j −α ·(D − P1) e 2j where, like before, the numerator can be further simpliﬁed as 2 −αW2j −αW2j E1j −α ·(D − P1)Y1jie = −αY1jiE1j α ·(D − P1)e = 0 following the ﬁrst-order condition (B.1). Therefore, dX1j/dQ1i = 0, ∀i ∈ {1, ..., n}.

Following the same steps as above, for each call option i, we have dY1ji/dP1 = 0, dY1ji/dQ1i < 0,

and dY1ji/dQ1l = 0 for l , i. Therefore, the Jacobian (B.3) has all off-diagonal terms equal to zero and all diagonal terms strictly negative. It is negative definite, implying a unique solution to F(·) = 0. This unique solution of prices also implies that investors’ demand, pinned down by the first-order conditions (B.1), is unique. To sum up, we have characterized the t = 1 equilibrium under any arbitrary set of (call) options in terms of investors’ optimal demand (B.1) and market clearing (B.2). We have also shown that the solution to the equation system (B.1) and (B.2) is unique.

45 B.2 The underlying price at t = 0

The investors are ex ante homogeneous at t = 0. Their expected utility is πU0d + (1 − π)U0s,

where U0j = E[U1j] is a type-j investor’s expected utility just after the type realization (but not yet

observing the private signal s or the endowment shock z, if j = d). Note that U0j is a function of

an investor’s t = 0 underlying demand X0 and of the equilibrium price p0, which the competitive investors take as given. (Clearly, by market clearing, the demand for options is all zero at t = 0 as

the investors are homogeneous.) Therefore, the first-order condition regarding X0 must hold: ∂U ∂U π 0d + (1 − π) 0s = 0, ∂X0 ∂X0 h i ∂U 0 0 − where, following the analysis for t = 1, we have 0j = E U · (P − p ) , where U = αe αW2j . ∂X0 1j 1 0 1j Rearranging the above first-order condition yields the usual pricing equation: E 0 + ( − ) 0 πU1dP1 1 π U1sP1 p = 0 0 . 0 E + ( − ) πU1d 1 π U1s 0 Note that U1j depends on the unknown demand X0 in general. Therefore, the above pricing equation needs to be solved by finding a fixed-point {P0, X0} at which it holds together with the market clearing condition. In the current model, this is particularly simple as the market clearing

condition immediately pins down the equilibrium demand X0 = X¯. The above pricing equation can then be evaluated easily by numerically solving the expectations in the numerator and the denominator.

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