Imperfect Competition

Transcription

1 Market Making with Asymmetric Information, Inventory Risk and Imperfect Competition Hong Liu Yajun Wang June 16, 2013 Abstract Existing microstructure literature cannot explain the empirical evidence that bid-ask spreads can decrease with information asymmetry and ignores either information asymmetry or inventory risks. We develop a market making model that highlights the asset pricing impact of market makers capability of making offsetting trades in markets where both information asymmetry and inventory risk are significant (e.g., Over-the-Counter markets). We solve equilibrium bid/ask prices and depths, and equilibrium trading volume in closed-form. We show that this model can help explain the bid-ask spread puzzle and also propose a new measure of information asymmetry. JEL Classification Codes: D42, D53, D82, G12, G18. Keywords: Illiquidity, Bid-Ask Spread, Asymmetric Information, Imperfect Competition. We thank Wen Chen, George Constantinides, Douglas Diamond, Phil Dybvig, Diego García (AFA discussant), David Hirshleifer, Nengjiu Ju, Pete Kyle, Mark Loewenstein, Anna Obizhaeva, Maureen O Hara, Avanidhar Subrahmanyam (Duke/UNC Asset Pricing Conference discussant), Mark Van Achter, S. Viswanathan, Jiang Wang and seminar participants at the 2010 CICF, 2010 Duke/UNC Asset Pricing Conference, 2010 SIF, 2011 AFA, Erasmus University, Michigan State University, Tilburg University, University of Luxembourg and Washington University in St. Louis for helpful comments. Olin Business School, Washington University in St. Louis and CAFR, liuh@wustl.edu. Robert H. Smith School of Business, University of Maryland, ywang22@rhsmith.umd.edu.

2 It is widely documented that bid-ask spreads can decrease with information asymmetry (e.g., Brooks (1996), Huang and Stoll (1997), Chordia, Roll, and Subrahmanyam (2001), Acker, Stalker and Tonks (2002), Acharya and Johnson (2007)). For example, Brooks (1996) finds a negative relationship between bid-ask spreads and information asymmetry around earnings and dividends announcements. Similarly, Acker, Stalker and Tonks (2002) find that bid-ask spreads start to narrow about two weeks before earnings announcements. Acharya and Johnson (2007) show that in the credit default swap (CDS) market, spreads can also be lower with greater information asymmetry. In contrast, extant asymmetric information models predict that as information asymmetry increases, bid-ask spreads always increase (e.g., Copeland and Galai (1983), Glosten and Milgrom (1985), Easley and O Hara (1987), Admati and Pfleiderer (1988), Wang (1994)). In addition, most of the rational expectations models in market microstructure literature assume a zero expected profit condition for each trade of a market maker, implying a zero expected profit from acquired inventory. However, this is inconsistent with the liquidity provision mandate for a designated market maker and ignores her definitive advantage of offsetting her trades on the other side. As shown by the existing empirical literature (e.g., Sofianos (1993), Shachar (2012)), market makers tend to make offsetting trades to avoid any significant inventory position, and make their profit mostly from bid-ask spreads. With the improvement in market transparency and liquidity, the time between orders of opposite directions becomes shorter and thus the capability of making offsetting trades has become even more important in affecting a market maker s trading and pricing decision. Furthermore, existing literature either ignores information asymmetry (e.g., Garman (1976), Stoll (1978), and Ho and Stoll (1981)) or abstracts away a market maker s inventory risk (e.g., Kyle (1985), Glosten and Milgrom (1985)). However, both information asymmetry and inventory risk are important determinants of market prices and market liquidity for many relatively illiquid financial markets such as over-the-counter (OTC) markets and securities markets in developing countries, which are generally characterized by great information asymmetry, high inventory risk, and significant market power and risk aversion of market makers (see, e.g., Garman (1976), Lyons (1995), Green, Hollifield, and Schürhoff (2007), Gravelle (2010), Ang, Shtauber, and Tetlock (2011)). Motivated by these unsatisfactory features of the existing literature, we develop a market making model that highlights the impact of market makers capability of making offsetting trades on their trading and pricing decisions in a market where both information asymmetry and inventory risk are 1

3 significant. We show that this model can help explain the puzzle that bid-ask spreads may decrease with information asymmetry. In addition, consistent with empirical evidence, market makers make profit from bid-ask spreads and in order to reduce inventory risk they are willing to lose money on a particular trade in expectation. Specifically, we consider a one-period setting with three types of risk averse investors: informed investors, uninformed investors, and a market maker who is also uninformed. On date 0, all investors optimally choose how to trade a risk-free asset and a risky security (e.g., OTC stocks, corporate bonds, and derivatives) to maximize their expected constant absolute risk averse (CARA) utility from the terminal wealth on date 1 and all may be endowed with some shares of the risky security but not the risk-free asset. On date 0 informed investors can observe a private signal about the date 1 payoff of the security before trading and thus they have trading demand motivated by the private information. They are also subject to a liquidity shock that is realized on date 0 before trading. We model the liquidity shock as a random endowment of a nontradable asset whose payoff is correlated with that of the security. Accordingly, informed investors also have trading demand motivated by the liquidity needs for hedging. Different from the existing literature, we assume that there is a public signal about the realized liquidity shock that everyone can observe on date 0 before trading (e.g., macroeconomic news that is related with liquidity shocks). There is a continuum of the informed and the uninformed and thus neither the informed nor the uninformed trade strategically. Due to high search costs, informed and uninformed investors must trade through the market maker. Following Kyle (1989), we assume that the market maker knows the demand schedules of the informed and the uninformed investors. Different from Kyle (1989), however, the demand schedules are dependent on bid and ask prices (rather than a single price). The market maker determines how much to buy at the bid and how much to sell at the ask taking into account the price impact implied by the demand schedules. The equilibrium bid and ask prices are determined by the market clearing conditions at the bid and at the ask, i.e., the total amount the market maker buys (sells) at the bid (ask) is equal to the total amount other investors sell (buy). In equilibrium, the risk-free asset market also clears. As shown later, the equilibrium outcome is equivalent to the solution to a Nash bargaining game between investors and the market maker where the market maker has all the bargaining power. 1 Although this model incorporates many important features in these markets, such as asymmetric information, inventory risk, imperfect competition, and risk aversion, 1 We also solve the case where both investors and the market maker have bargaining power. The qualitative results are the same. For example, the equilibrium bid-ask spread is still proportional to the absolute value of the reservation price difference between the informed and the uninformed. 2

4 and allows both bid/ask prices and depths as well as all demand schedules to be endogenous, the model is still tractable. Indeed, we solve the equilibrium bid and ask prices, bid and ask depths, trading volume, and inventory levels in closed-form even when investors have different risk aversion, different inventory levels, different liquidity shocks, different resale values of the risky asset and heterogeneous private information. We show that in equilibrium, the bid-ask spread and market trading volume are all positively proportional to the absolute value of the reservation price difference between the informed and the uninformed. 2 Intuitively, the greater the reservation price difference, the greater the total gain to trades, the more the informed and the uninformed trade, and the more the market maker can extract by charging a higher spread. This implies that in contrast to the literature on portfolio selection with transaction costs (e.g., Davis and Norman (1990), Liu (2004)), bid-ask spreads can be positively correlated with trading volume. More importantly, we show that not only ex post bid -ask spreads (i.e., spreads given realizations of signals and liquidity shocks) but also expected bid-ask spreads can be smaller with asymmetric information. The result on ex post spreads follows because asymmetric information can reduce the reservation price difference between the informed and the uninformed. Unlike noise traders in most rational expectations models, the uninformed in our model rationally revise their reservation price upon observing market prices. The reservation price difference between the informed and the uninformed is proportional to the uninformed s estimate (i.e., conditional expectation) of the informed s liquidity shock plus the premium required for the estimation risk, because if the uninformed knew the informed s realized liquidity shock, the uninformed would be able to infer from market prices the exact signal about the security payoff that the informed observed and thus would have the same reservation price as the informed. With asymmetric information, the estimation risk premium required by the uninformed and the uninformed s overestimation or underestimation of the liquidity shock relative to the true realized value may reduce the reservation price difference and hence may decrease the spread. For the intuition on the result on expected spreads, consider the special case where the estimation risk premium is zero (e.g., when the initial endowment of the risky asset is zero), for expositional simplicity. On average, the uninformed s estimation of the informed s liquidity shock is correct, which implies that the expected reservation price difference is zero in this special case. Because the spread is proportional to the absolute value of the reservation price difference, the expected spread can be viewed as the value of a straddle (a call+ a put) on the 2 The reservation price is the critical price such that an investor buys (sells) the security if and only if the ask (bid) is lower (higher) than this critical price. 3

5 reservation price difference and thus increases in the volatility of the reservation price difference. With asymmetric information, the information about the liquidity shock is less precise, and thus the volatility of the conditional expectation of the liquidity shock is smaller, so is the volatility of the reservation price difference, and so is the expected spread, because the variance of conditional expectation of any random variable decreases as the conditioning information becomes less precise. 3 While information asymmetry has been a focal point of most of the existing information-based models, there is still no good measure of information asymmetry as far as we know. Changes in a good measure of information asymmetry should change information asymmetry, but not the aggregate information quality (measured by the precision of security payoff distribution conditional on all the information in the economy), because both information asymmetry and information quality can affect economic variables of interest (e.g., prices, liquidity). For example, a candidate information asymmetry measure may be the precision of the private signal about the security payoff. However, as the private signal becomes more precise, the aggregate information quality also increases. In contrast, we use the variance of the public signal about the liquidity shock as a measure of information asymmetry. Because informed investors observe their liquidity shock perfectly, this variance does not change the aggregate information quality. On the other hand, because market prices reveal the combined demand from information and liquidity shock, as this variance decreases, the uninformed can better estimate the security payoff and thus information asymmetry decreases. For example, when this variance decreases to 0, the uninformed can perfectly observe the liquidity shock, can then fully back out the private signal of the informed and thus the model becomes the one with symmetric information. In the other extreme, as this variance increases to infinity, it becomes equivalent to the standard asymmetric information case where there is no public signal about the liquidity shock and thus achieves the maximum information asymmetry, ceteris paribus. 4 We also solve in closed-form a generalized model where multiple market makers engage in oligopolistic competition, investors can have different risk aversion, different inventory levels, different liquidity shocks, different resale values of the security and heterogeneous private information. There are eight types of equilibria characterized by the trading directions of investors, e.g., some investors may choose not to trade in equilibrium, and both the informed and the uninformed can trade in the same directions. As expected, competition decreases bid-ask spreads. Some empirically testable implications from the generalized model include: average bid-ask spreads are more 3 In the extreme case where the conditioning information has infinite variance, the conditional expectation reduces to the unconditional expectation which is a constant and thus has zero variance. 4 This implies one additional benefit from introducing the public signal: our model nests both the symmetric information case and the standard asymmetric information case. 4

6 sensitive to a market maker s inventory level in less active markets; as a market maker s inventory increases, average bid and ask prices decrease, average ask depth increases and average bid depth decreases. Different from inventory-based models, our model takes into account the impact of information asymmetry on bid and ask prices and inventory levels, which is especially important for less liquid markets such as OTC markets. In addition, we also allow market makers to vary bid and ask depths to better manage inventory risk. 5 In contrast to information-based (rational expectations) models, our model recognizes that market makers can make offsetting trades, can profit from bidask spreads and can face significant inventory risk. In addition, market makers in our model may be willing to lose money from a particular trade in expectation (at bid or at ask) in equilibrium especially when they have high initial inventory. 6 On the other hand, when market makers have low initial inventory they make positive expected profit from inventory carried to date 1 because of the required inventory risk premium, consistent with the findings of Hendershott, Moulton, and Seasholes (2007). The remainder of the paper proceeds as follows. In Section 1 we briefly describe the OTC markets and discuss additional related literature. We present the model in Section 2. In Section 3 we derive the equilibrium. In Section 4 we provide some comparative statics on asset prices, illiquidity, and welfare. We present, solve and discuss a generalized model in Section 5. We conclude in Section 6. All proofs are provided in Appendix A. In Appendix B, we present results on price impact and the rest of the results of Theorem 2 for the generalized model. 1. Over-the-Counter Markets and Related Literature Most types of government and corporate bonds, a wide range of derivatives (e.g., CDS and interest rate swaps), securities lending and repurchase agreements, currencies, and penny stocks are traded in the OTC markets. 7 In almost all of these markets, investors only trade with designated dealers 5 Madhavan and Sofianos (1998) find that market makers mainly adjust quote depths to manage inventories. 6 In the special case where market makers are extremely risk averse and thus do not hold any inventories across trading periods, the potential date 1 payoff of the stock is irrelevant for market makers pricing or trading decision. This special case bears some similarity to high frequency market makers who only carry any significant inventory for at most very short period of time (typically less than 1 hour) and private information about the fundamentals of the security is thus less relevant. 7 The Nasdaq stock market was traditionally also a dealer market before the regulatory reforms implemented in Some of the empirical studies we cite (e.g., Huang and Stoll (1997) ) use before-1997 Nasdaq data. 5

7 (market makers) through bilateral negotiation. 8 Nash bargaining has been widely used by the existing literature to model bilateral negotiation in OTC markets (e.g., Duffie, Garleanu, and Pedersen (2005), Gofman (2011), Atkeson, Eisfeldt, and Weill (2013)). As shown right after the proof of Theorem 3 in Appendix A, our modeling approach where the market maker maximizes her utility taking into the price impact of her trades implied by the demand schedules of other investors is equivalent to the solution to a Nash bargaining game between investors and the market maker where the market maker has all the bargaining power. Dealers in OTC markets face significant information asymmetry and inventory risk, and therefore, they frequently engage in offsetting trades within a short period of time with customers or with other dealers when their inventory level deviates significantly from desired targets (e.g., Acharya and Johnson (2007), Shachar (2012)). The cost of searching for a counterparty can be significant in some OTC markets for some investors, which motivates many studies to use search-based or network-based models for OTC markets (e.g., Duffie, Gârleanu, and Pedersen (2005), Vayanos and Wang (2007), Vayanos and Weill (2008)). On the other hand, the search cost is relatively small for central dealers and frequent traders. Central dealers have better access to customers and peripheral dealers, while frequent traders usually go back to the same small subset of dealers (e.g., Afonso et al. (2011), Li and Schürhoff (2011)). In addition, the search cost has been significantly reduced in some other markets by the regulatory effort to improve market transparency. For example, after the introduction of the Transaction Reporting and Compliance Engine (TRACE) in July 2002 in the US corporate bond markets, for TRACE-eligible bond trades investors receive information about bond identification, the date and time of execution, and the price and yield. A fixed-income trader at an investment company, referring to the post-trace environment, was quoted in Vames (2003) as saying, You don t have to go to three or four different people to find out where something is trading.... [W]hen you have access to (TRACE) information, you have a better idea where things are before you make your first call. The reduction of search cost resulted in large trading volume in the U.S. corporate bond market. For example, Shulman (2004) report that on a typical day approximately $20 billion par value of corporate bonds turn over in approximately 25,000 transactions. While we do not explicitly model the search cost, the assumption that investors can only trade through market makers and only with a small number of market makers can be viewed as a result of significant costs of searching for other market makers. In addition, the generalized model can indirectly capture some additional costs (e.g., from searching) for liquidation of inventory on 8 For example, Li and Schürhoff (2011) show that dealers intermediate 94% of the trades in the municipal bond market, with most of the intermediated trades representing customer-dealer-customer transactions. 6

8 date 1. For example, when search cost is high, search takes a long time, and the resale value of the security is highly uncertain, one can model this as a low mean and high volatility distribution for the resale value of the security acquired by market makers. This is clearly just a reduced form, but likely indirectly captures the first order effect of these features. For example, when search cost is high and the uncertainty about the resale value of the security is large, market makers charge a higher premium for the security on date 0 and the bid-ask spread increases in a search model (e.g., Duffie, Gârleanu, and Pedersen (2005, 2007)). With a lower mean and higher volatility for the resale value, our model can generate the same qualitative result. More importantly, explicitly modeling searching would not change our main results such as bid-ask spreads and trading volume increase with the magnitude of the reservation price difference and bid-ask spreads can decrease with information asymmetry, because after a successful search of a counterparty, traders face the same optimization problems as what we model. In our model we assume that buy and sell orders arrive simultaneously and the market maker faces one period of inventory risk. This is clearly a simplification because orders typically arrive sequentially (possibly due to search costs) and market makers can carry inventory for multiple periods. Ideally one should use a dynamic model to capture the possible non-synchronous offsetting trades of a market maker and how a market maker s inventory evolves. However, such a dynamic model with risk averse market makers who maximize expected terminal utility becomes intractable and more importantly, unlikely yields qualitatively different results. For example, in such a dynamic model, a market maker will still be willing to lose money in expectation on a particular trade if she expects to be able to unwind it later at a better price and the spread will still be positively related to the expected magnitude of the reservation price difference by the same intuition as for our model. In addition, spreads can still decrease with information asymmetry, because even when orders arrive sequentially and thus a market maker needs to wait a period of time for the offsetting trades, as long as she has a reasonable estimate of the next order, she will choose qualitatively the same trading strategy. The key for the bid-ask spreads result is the ability of a market maker to make offsetting trades and the fact that information asymmetry can reduce the reservation price difference between the buyer and the seller. Whether the buyer and the seller arrive at the same time is not critical. For the effect of inventory risk on asset pricing, as shown by Stoll (1978), a one period model generates the same qualitative results as a dynamic model. For example, even in a dynamic model, the market maker will still trade off inventory risk with the profit from the spread at time 0 and require a risk premium for holding any inventory. 7

9 There is a large literature on dealership markets. For example, Amihud and Mendelson (1980) consider the problem of a price-setting monopolistic market-maker in a dealership market where the stochastic demand and supply follow independent Poisson processes. Duffie, Gârleanu, and Pedersen (2005) analyze market making in over-the-counter markets using a search model for OTC markets where investors can trade directly with other investors (instead of trading only through dealers). Neither of these papers examines the effect of asymmetric information on market prices and market liquidity, and thus they only apply to OTC markets where information asymmetry is insignificant. Our model is also related to Diamond and Verrecchia (1981), Subrahmanyam (1991), Diamond and Verrecchia (1991), Dennert (1993), Naik, Neuberger, and Viswanathan (1999), Back and Baruch (2004), Vayanos and Wang (2012), and Rosu (2010). Diamond and Verrecchia (1981) analyze a rational expectations equilibrium model of a competitive security market in which traders possess independent pieces of information about the return of a risky asset. Subrahmanyam (1991) finds that increasing the precision of private information intensifies competition between risk averse informed investors and thus can increase market liquidity. Diamond and Verrecchia (1991) show that reducing information asymmetry can increase liquidity and security prices may be nonmonotonic in information asymmetry because of the potential exit of market makers. In all these three papers, market makers post a single price, the trading needs of some of the uninformed investors (i.e., noise investors ) are exogenous and thus do not respond to price changes. Therefore if market makers were allowed to post bid and ask prices, then in contrast to our predictions, the bid-ask spread would always be increasing in information asymmetry. Dennert (1993) examines how price competition among risk neutral market makers affects market quality, assuming informed traders are risk neutral and have unlimited capacity. He shows that an increase in the number of market makers induces more insider trading and may decrease market quality. Naik, Neuberger, and Viswanathan (1999) examine whether full and prompt disclosure of public-trade details improves the welfare of a risk-averse investor in a two-stage dealership market. Similar to Diamond and Verrecchia (1991), market makers post a single price which is the conditional expected payoff of the security. Back and Baruch (2004) solve a version of the Glosten-Milgrom model with a single informed investor, in which the informed investor chooses his trading times optimally. As in the original Glosten-Milgrom model, they find that the bid-ask spread is greater with asymmetric information. Vayanos and Wang (2012) examine how liquidity and asset prices are affected by market imperfections such as asymmetric information, leverage constraints, and transaction costs. They find that imperfections may decrease expected returns, and can have opposite impact on 8

10 market illiquidity. In contrast to our model, Vayanos and Wang (2012) assume a zero bid-ask spread and transaction costs are exogenous. Rosu (2010) finds that bid-ask spread can decrease with the fraction of informed investors because the competition among them intensifies (similar to the effect of signal precision on competition in Subrahmanyam (1991)) and they can trade with impatient investors who are assumed to always submit market orders. Another strand of literature studies the effect of illiquidity on portfolio choice and asset pricing (e.g., Constantinides (1986), Vayanos (1998), Liu and Loewenstein (2002), Lo, Mamaysky and Wang (2004), Liu (2004), Acharya and Pedersen (2005)). In this literature, illiquidity is generally modeled as exogenous transaction costs and therefore the fundamental question of what affects illiquidity is largely unanswered. This paper is also related to the double-auction literature (e.g., Kyle (1989), Vives (2011), Rostek and Weretka (2012)). Vives (2011) considers the problem of a finite number of sellers competing in schedules to supply a fixed elastic demand, where the cost of each seller is random. Rostek and Weretka (2012) present a model of a uniform-price double auction with an arbitrary number of traders, cast in a linear-normal setting. The main difference from other strategic models of information aggregation is that they permit shock structures with heterogeneous correlations in values. Different from most of the double auction models, some agents in our model (i.e., market makers) serve a dual role: buyers in one market and sellers in another ( one at the bid, the other at the ask). Our solution shows how this dual role of some participants affects the equilibrium outcome in these markets. 2. The model We consider a one period setting with trading dates 0 and 1. There are a continuum of identical informed investors with mass N I, a continuum of identical uninformed investors with mass N U, and designated market makers (M) who are also uninformed. They can trade one risk-free asset and one risky security on date 0 and date 1 to maximize their expected constant absolute risk aversion (CARA) utility from their wealth on date 1. There is a zero net supply of the risk-free asset, which also serves as the numeraire and thus the risk-free interest rate is normalized to 0. The total supply of the security is N θ 0 shares where N = N I + N U + and the date 1 payoff of each share Ṽ N( V, σv 2 ) becomes public on date 1, where V is a constant, σ V > 0, and N denotes the normal distribution. The aggregate risky asset endowment is N i θ shares for type i {I, U, M} investors. No investor is endowed with any risk-free asset. 9

11 On date 0, informed investors observe a private signal ŝ = Ṽ V + ε (1) about the payoff Ṽ, where ε is independently normally distributed with mean zero and variance σ 2 ε. 9 In addition to the security, every informed investor is also subject to a liquidity shock that is modeled as a random endowment of ˆXI N(0, σx 2 ) units of a non-tradable risky asset on date 0, with ˆX I realized on date 0 and only directly known to informed investors. 10 The non-tradable asset has a per-unit payoff of Ñ N(0, σn 2 ) that has a covariance of σ V N with Ṽ and is realized and becomes public on date 1. The correlation between the non-tradable asset and the security results in a liquidity demand for the risky asset to hedge the non-tradable asset payoff. In a model with private information sources such as a private signal ŝ and a private liquidity shock (e.g., Grossman and Stiglitz (1980), Wang (1994), O Hara (2003), and Vayanos and Wang (2012)), assuming that all investors who are subject to liquidity shock also observe ŝ is only for simplicity: even if they do not observe ŝ, they can infer it perfectly from the equilibrium price, because the equilibrium price is an invertible function of the weighted sum of ŝ and the private liquidity shock. In this type of models asymmetric information can therefore exist only if some investors who do not have any liquidity shock do not observe ŝ either and are thus uninformed, as in Vayanos and Wang (2012) for example. We assume that these investors are all uninformed for simplicity. 11 In contrast to the existing literature that assume uninformed investors cannot observe any signal about liquidity demands (e.g., those from noise traders) while the informed can perfectly infer the liquidity demands from their private information about the security and the equilibrium asset price, we assume that there is a public signal Ŝ x = ˆX I + ˆη (2) about the liquidity shock ˆX I that all investors (i.e., the uninformed, market makers, and the informed) can observe, where ˆη is independently normally distributed with mean zero and variance 9 Throughout this paper, bar variables are constants, tilde random variables are realized on date 1 and hat random variables are realized on date The random endowment can represent any shock in the demand for the security, such as a liquidity shock or a change in the needs for rebalancing an existing portfolio or a change in a highly illiquid asset. 11 Alternatively, one can view an informed investor as a broker who combines information motivated trades and liquidity motivated trades. The assumption that all informed traders have the same information is only for simplicity. Our main results still hold when they have differential information (see the generalized model in Section 5). 10

12 ση This public signal represents public news about liquidity demand determinants, such as fund flows, macroeconomic conditions and regulation shocks. 13 While this additional signal Ŝ x is not critical for our main results (e.g., spread can be smaller with asymmetric information), it allows us to model different degrees of information asymmetry in one unified setting. For example, the case where ση 2 = 0 implies that the uninformed and market makers can perfectly observe ˆX I from the public signal and thus can in turn perfectly infer the private signal ŝ from the equilibrium security price, as will be shown later. Therefore, the case where ση 2 = 0 represents the symmetric information case. The case where ση 2 =, on the other hand, implies that the signal Ŝx is useless and thus corresponds to the asymmetric information case as modeled in the standard literature, i.e., there is no public news about liquidity demand. In addition, as shown later, the variance ση 2 can serve as a measure of the degree of information asymmetry. All trades must go through designated market makers (dealers) whose market making costs are assumed to be Specifically, I and U investors sell to market makers at the bid B or buy from them at the ask A or do not trade at all. Given that there is a continuum of informed and uninformed investors, we assume that they are price takers. Following Kyle (1989), we assume that market makers know the demand schedules of the informed and uninformed investors before deciding on how to trade. Different from Kyle (1989), however, the demand schedules depend on both the bid and ask prices rather than a single trading price. For each i {I, U, M}, investors of type i are identical both before and after realizations of signals on date 0 and thus adopt the same trading strategy. Let I i represent a type i investor s information set on date 0 for i {I, U, M}. For i {I, U}, a type i investor s problem is to choose the (signed) demand schedule θ i (A, B) to solve max E[ e δ W i I i ], (3) where W i = θ i B θ+ i A + ( θ + θ i )Ṽ + ˆX i Ñ, (4) 12 While this public signal can also be observed by the informed, it is useless to them because they can already perfectly observe it privately. 13 As another example, in a companion paper where we consider two independent securities that are both correlated with liquidity demands, the equilibrium price of the other security can serve as a public signal about the liquidity demand. 14 Assuming zero market making cost is only for better focus and expositional simplicity. Market making cost is considered in an earlier version, where a potential market maker must pay a fixed market-making utility cost on date 0 to become a market maker. We show that no results in this paper are altered by this fixed cost. 11

13 ˆX U = 0, δ > 0 is the absolute risk-aversion parameter, x + := max(0, x), and x := max(0, x). Since I and U investors buy from market makers at ask and sell to them at bid, we can view these trades occur in two separate markets: the ask market and the bid market. In the ask market, market makers are the suppliers and other investors are demanders and the opposite is true in the bid market. To simplify the exposition of our main results, in the rest of the main text we focus on the monopoly case, i.e., there is a single market maker ( = 1). In the proof of the main results and the generalized model in Section 5, however, we provide closed-form solutions for the general case with 1 to show the impact of competition. Facing the demand curve in the ask market and the supply curve in the bid market, the monopolist market maker can either directly choose bid and ask prices or directly choose how much to buy at bid (bid depth) and how much to sell at ask (ask depth). With a single market maker, these two approaches are equivalent because given bid and ask prices, bid and ask depths are uniquely determined by the supply and demand curves through market clearing, and vise versa. To be consistent with the more general model shown in Appendix A where we consider multiple market makers engaging in Cournot competition, we also use the latter approach to solve for the equilibrium in this monopoly case. Let α and β be ask depth and bid depth respectively. 15 Given the demand schedules of the informed and the uninformed (θi (A, B) and θ U (A, B)), the bid price B(β) (i.e., the inverse supply function in the bid market) and the ask price A(α) (i.e., the inverse demand function in the ask market) can be determined by the following security market clearing conditions at the bid and ask prices. 16 α = i=i, U N i θ i (A, B) +, β = i=i, U N i θ i (A, B), (5) where the left-hand sides represent the sale and purchase by the market maker respectively and the right-hand sides represent the total purchases and sales by other investors respectively. Then the designated market maker M s problem is to choose ask depth α 0 and bid depth β 0 to solve ] max E [ e δ W M I M, (6) 15 To help remember, Alpha denotes Ask depth and Beta denotes Bid depth. 16 The risk-free asset market will be automatically cleared by the Walras law. A buyer s (seller s) trade only depends on ask A (bid B). So A only depends on α and B only depends on β. 12

14 where W M = αa(α) βb(β) + ( θ + β α)ṽ. (7) Note that different from other investors, a market maker takes into account the price impact of her own trades, i.e., recognizing that both A and B will be affected by her trades. This leads to our definition of an equilibrium: Definition 1 An equilibrium (θ I (A, B), θ U (A, B), A, B, α, β ) is such that 1. given any A and B, θi (A, B) solves a type i investor s Problem (3) for i {I, U}; 2. given θ I (A, B) and θ U (A, B), α and β solve the market maker s Problem (6); 3. A := A(α ) and B := B(β ) clear both the risky security and the risk-free asset markets, where A(α) and B(β) solve Equation (5). 3. The equilibrium In this section, we solve the equilibrium bid and ask prices, bid and ask depth and trading volume in closed form. Given A and B, the optimal demand schedule for a type i investor for i {I, U} is θi (A, B) = P R i A δvar[ṽ I i] A < P R i, 0 B P R i A, (8) B P R i δvar[ṽ I i] B > P R i, where P R i = E[Ṽ I i] δcov[ṽ, Ñ I i] ˆX i δvar[ṽ I i] θ (9) is the reservation price of a type i investor (i.e., the critical price such that non-market-makers buy (sell, respectively) the security if and only if the ask price is lower (the bid price is higher, respectively) than this critical price). Equation (9) also shows that if there is no aggregate risk ( θ = 0, e.g., for derivatives whose net supply is zero), the reservation prices only depend on the 13

15 expected payoff and hedging premium (i.e., the extra cost an investor is willing to pay for hedging, represented by δcov[ṽ, Ñ I i] ˆX i, ), but not on payoff conditional variance. Because the informed know exactly {ŝ, ˆX I } while equilibrium prices A and B and the public signal Ŝx are only noisy signals about {ŝ, ˆX I }, the information set of the informed in equilibrium is I I = {ŝ, ˆX I }, (10) which implies that E[Ṽ I I] = V + ρ I ŝ, Var[Ṽ I I] = ρ I σ 2 ε, Cov[Ṽ, Ñ I I] = (1 ρ I )σ V N, (11) where Equation (9) then implies that ρ I := σ2 V σv 2 +. (12) σ2 ε P R I = V + Ŝ δρ Iσ 2 ε θ, (13) where Ŝ := ρ Iŝ+h ˆX I and h = δ(1 ρ I )σ V N represents the hedging premium per unit of liquidity shock. While ŝ and ˆX I both affect the informed investor s demand and thus the equilibrium prices, other investors can only infer the value of Ŝ from market prices because the joint impact of ŝ and ˆX I on market prices is only in the form of Ŝ. In addition to Ŝ, other investors can also observe the public signal Ŝx about the liquidity shock ˆX I. Thus we conjecture that the equilibrium prices A and B depend on both Ŝ and Ŝx. Accordingly, the information sets for the uninformed investors and the market maker are 17 I U = I M = {Ŝ, Ŝx}. (14) Then the conditional expectation and conditional variance of Ṽ for the uninformed and the market maker are respectively E[Ṽ I U] = V + ρ U Ŝ hρ U ρ X Ŝ x, (15) 17 Note that uninformed only need to observe their own trading price, i.e., A or B, not both A and B. For OTC markets, investors may not be able to observe trading prices by others, although with improving transparency, this has also become possible in some markets (e.g., TRACE system in the bond market). 14

16 V ar[ṽ I U] = ρ U ρ I σ 2 ε + ρ U ρ I h 2 ρ X σ 2 η, (16) where ρ X := σ2 X σx 2 +, ρ U := σ2 η ρ I σ 2 V ρ I σ 2 V + h2 ρ X σ 2 η 1. (17) It follows that the reservation price for a U investor and the market maker is PU R = PM R = V ) + ρ U (Ŝ hρx Ŝ x δ ρ U ( ρ 2 ρ I σε 2 + h 2 ρ X σ 2 ) η θ. (18) I Let RP denote the difference in the reservation prices of I and U investors. We then have ( RP := PI R PU R = (1 ρ U ) Ŝ + ρ IσV 2 ) hση 2 Ŝ x + δρ I σv 2 θ. (19) Let ν := Var[Ṽ I U] Var[Ṽ I I] = ρ U + ρ Uρ X h 2 σ 2 η ρ 2 I σ2 ε 1 be the ratio of the security payoff conditional variance of the uninformed to that of the informed, and N := νn I + N U + 1 N be the information weighted total population. The following theorem provides the equilibrium bid and ask prices and equilibrium security demand. 18 Theorem 1 1. The equilibrium bid and ask prices are A := A(α ) = PU R + νn I 2 ( RP + ) RP +, N B := B(β ) = PU R + νn I 2 ( RP ) RP, N Because all utility functions are strictly concave, all budget constraints are linear in the amount invested in the security and both the informed and the uninformed are price takers, there is a unique solution to the problem of each informed and each uninformed given the bid and ask prices. Because the inverse demand and supply functions implied by the market clearing conditions are linear in market depths and when a market maker trades a strictly positive amount with both the informed and the uninformed, there is a unique solution to her utility maximization problem (which already takes into account the market clearing conditions). This implies that there is a unique equilibrium when all investors trade in equilibrium. When some investors do not trade in equilibrium, there are multiple equilibria because either bid or ask would not be unique (see Theorem 2). 15

17 and we have A > P > B, where P = νn I N P R I + N U N P R U + 1 N P R M (20) is the equilibrium price of a perfect competition equilibrium where the market maker is also a price taker. The bid-ask spread is A B = RP 2 = 1 2 (1 ρ U) Ŝ + ρ IσV 2 Ŝ x + δρ I σv 2 θ. (21) hσ 2 η 2. The equilibrium quantities demanded are θ I = N U ( N + 1 ) RP δvar[ṽ I I], θ U = νn I 2 ( N + 1 ) RP δvar[ṽ I U] ; (22) the equilibrium ask and bid depths are respectively α = N I (θ I) + + N U (θ U) +, (23) β = N I (θ I ) + N U (θ U), (24) which implies that the equilibrium trading volume is α + β = N I(N U + 1) N + 1 ( RP δvar[ṽ I I] ). (25) As shown right after the proof of Theorem 3 ( the version that allows for multiple market makers, 1) in Appendix A, the above equilibrium can be reinterpreted as the solution to a Nash bargaining game between investors and the market maker where the market maker has all the bargaining power. 19 In a nutshell, in the Nash bargaining game, the market maker and an investor bargain over the trading amount with the trading price determined by the trading amount and the optimal demand schedule of the investor. Therefore, the Nash bargaining game where the market maker has all the bargaining power is to choose the trading amount to maximize the 19 We also solve the case where the investor also has bargaining power. Main qualitative results are the same. For example, both spreads and trading volume are still proportional to the reservation price difference. 16

18 market maker s expected utility given the demand schedule of the investor, and thus yields exactly the same outcome as our solution above. Part 1 of Theorem 1 implies that in equilibrium both bid and ask prices are determined by the reservation price of the uninformed and the reservation price difference between the informed and the uninformed. In addition, given the public signal Ŝx, all investors can indeed infer Ŝ from observing their trading prices as conjectured, because of the one-to-one mapping between the two. 20 Furthermore, Part 1 shows that the equilibrium bid-ask spread is equal to the absolute value of the reservation price difference between the informed and the uninformed, divided by 2 (more generally, + 1). 21 Equation (22) implies that I investors buy and U investors sell if and only if I investors have a higher reservation price than U investors. Because the market maker has the same reservation price as the U investors, in the net she trades in the same direction as U investors. In the standard literature on portfolio choice with transaction costs (e.g., Liu and Loewenstein (2002), Liu (2004)), it is well established that as the bid-ask spread increases, investors reduce trading volume to save on transaction costs and thus trading volume and bid-ask spread move in the opposite directions in these models. In contrast, Theorem 1 implies that bid-ask spreads and trading volume can move in the same direction, because both trading volume and bid-ask spread increase with RP. Lin, Sanger and Booth (1995) find that trading volume and effective spreads are positively correlated at the beginning and the end of the day. Chordia, Roll, and Subrahmanyam (2001) find that the effective bid-ask spread is positively correlated with trading volume. Our model suggests that these positive correlations may be caused by changes in the valuation difference of investors. The net order size is the magnitude of the difference between the total buy order size and total sell order size and is equal to α β by market clearing condition (5). Theorem 1 implies that net order size and bid-ask spread are positively correlated even in the symmetric information case (σ 2 η 0, as shown in Proposition 1), because both increase with RP. A typical justification of this positive correlation (e.g., Easley and O Hara (1987, 1992)) is that as the net order size increases, the adverse selection effect of information asymmetry increases and thus the bid-ask spread increases. We offer an alternative explanation: it may be that changes in both the net order 20 In our model, the market maker can infer how much informed investors are trading. However, she does not know how much of the informed investors trades is due to information on the security s payoff or how much is due to the hedging demand. This is similar to the set-up of Glosten (1989) and Vayanos and Wang (2012). 21 The generalized model implies as the number of market makers increases, both bid and ask converge to the competitive market equilibrium price P. 17

19 size and the spread are driven by changes in the reservation price difference. Intuitively, as the reservation price difference increases, the spread increases, a market maker is willing to sell or buy more in the net and thus the net order size also increases. Next we provide the essential intuition for the results in Theorem 1 through graphical illustrations. Suppose PI R > PU R and thus I investors buy and U investors sell. The market clearing condition (5) implies that the inverse demand and supply functions faced by the market maker are respectively A = PI R k I α, B = PU R + k U β, where k i = δvar[ṽ I i] N i, i = I, U. We plot the above inverse demand and supply functions and equilibrium spreads in Figure 1 (a). Similarly, in Figure 1 (b), we plot the case where the informed sell and the uninformed buy. Figure 1 shows that as a market maker buys (sells) more at the bid (ask), the bid (ask) price goes up (down). Facing the inverse demand and supply functions, a monopolist market maker optimally trades off the prices and quantities. Similar to the results of classical Cournot competition models of multiple firms who compete through choosing the amount of output of a homogeneous product, the bid and ask spread is equal to the absolute value of the reservation price difference RP, divided by the number of market makers plus one (which is 2 for the monopoly case). In addition, as implied by Theorem 1, Figure 1 illustrates that the difference between P R I (P R U ) and the ask (bid) price is also proportional to the reservation price difference RP. Therefore the trading amount of both I and U investors and thus the aggregate trading volume all increase with RP. The shaded areas represent the profits (min(α, β )(A B )) the market maker makes from the bid-ask spread at time 0. In contrast to the standard rational expectations literature which assumes zero expected profit for each trade (e.g., Glosten and Milgrom (1985)), Theorem 1 implies that a market maker may lose money in expectation on a particular trade. For example, suppose RP > 0 (which implies that the informed buy at the ask and the uninformed sell at the bid), the per share expected profit of the market maker from the trade at the bid (not including the profit from the spread) is equal to E[Ṽ I M] B = δvar[ṽ I M] θ N Iν 2( N RP, (26) + 1) which can be negative if RP is large, in which case the market maker on average loses to the 18

20 = MM s Profit from Bid-Ask Spread ( ) = + (a) The Informed Buy and the Uninformed Sell, MM s Profit from Bid-Ask Spread = = +, (b) The Uninformed Buy and the Informed Sell Figure 1: Inverse Demand/Supply Functions and Bid/Ask Spreads. 19

21 uninformed and makes money from the informed. The market maker is willing to buy from the uninformed in anticipation of a loss from this trade because she can sell the purchased shares at a higher price (i.e., ask). Because of the hedging benefit, the informed may be willing to buy from the market maker in anticipation of a loss from this purchase. This same intuition applies to a dynamic setting where orders arrive sequentially. For example, seeing an order to sell at the bid, if the market maker expects that she will be able to unwind part of her purchase later at a higher price, she would be willing to accommodate the sell order even in anticipation of a loss for this purchase. This suggests that using a dynamic model does not change these qualitative results, while making the analysis less tractable. Theorem 1 and Equation (26) imply when RP < 0, a market maker buys in the net and she makes positive expected profit from inventory carried over if she does not have any initial inventory (i.e., θ = 0), because of the required inventory risk premium. This is consistent with the findings of Hendershott, Moulton, and Seasholes (2007). In the following proposition, we show that the symmetric information case where all investors observe the same signal ŝ as the informed and the standard asymmetric information case where there is no public signal about the liquidity shock are both special cases of the asymmetric information case we consider above. 22 Proposition 1 1. As σ 2 η 0, we have (a) Var[Ṽ I U] Var[Ṽ I I] and E[Ṽ I U] E[Ṽ I I], therefore the asymmetric information converges to the symmetric information case where all investors observe the signal ŝ; (b) In addition, ν 1, N N, ρ X 1, ρ U 1, Ŝx ˆX I, RP δ(1 ρ I )σ V N ˆXI ; 2. As σ 2 η, we have (a) Var[Ṽ I U] converges to Var[Ṽ Ŝ] = ρ Uρ I (σ 2 ε + h 2 σ 2 X /ρ2 I ) and E[Ṽ I U] converges to E[Ṽ Ŝ] = V + ρ U Ŝ, therefore the asymmetric information case where there is a public signal about ˆXI converges to the standard asymmetric information case where there is no such public signal. (b) In addition, ρ X 0, ρ X σ 2 η σ 2 X, RP (1 ρ U ) (Ŝ + δρi σv 2 θ ), 22 The proof is straightforward from direct computation and thus omitted. 20