Does order splitting signal uninformed order flow?

Transcription

1 Does order splitting signal uninformed order flow? Hans Degryse Frank de Jong Vincent van Kervel August 1, 2013 Abstract We study the problem of a large liquidity trader who must trade a fixed amount before a deadline and wishes to minimize the expected cost of trading. We add this trader to the Kyle (1985) framework to endogenize the price impact of trading. Under the assumption that the informed traders have short-lived private information, we show that the predictable component in the order flow only stems from the trades by the large liquidity trader. In turn, the market maker perceives this predictable component as uninformed and does not revise prices, such that the liquidity trader enjoys lower price impacts. We thus show that order splitting is a noisy form of preannouncing trades, i.e., sunshine trading (Admati and Pfleiderer, 1991). This prediction runs in opposite direction to the front-running or predatory trading hypothesis of Brunnermeier and Pedersen (2005). JEL Codes: G10; G11; G14; Keywords: Market microstructure, Kyle model, Order splitting, Optimal execution problem KU Leuven, Tilburg University and CEPR. Tilburg University Corresponding author, VU University Amsterdam. v.l.van.kervel@vu.nl. The authors thank Fabio Feriozzi, Corey Garriott, Peter Hoffmann, Katya Malinova, Andreas Park, Ioanid Rosu and seminar participants at the University of Toronto and the Advances in Algo and HF Trading conference at the University College London, Comments and suggestions are appreciated. All errors are ours. 1

2 1 Introduction In the last decades equity turnover has increased sevenfold, while average order sizes have declined tenfold (Chordia, Roll, and Subrahmanyam, 2011). 1 Among other reasons, this is a consequence of algorithmic trading and the practice of order splitting, where a large quantity to be traded is split up into many small packages which are executed over time. Order splitting is nowadays a standard practice in the investment management industry. This paper studies the optimal execution problem of an institutional investor who must trade a given quantity before a deadline, and wishes to minimize the expected execution costs. The investor has liquidity motives and aims to optimally split up her to-be-traded quantity across periods and trades. We endogenize the price impact parameter by placing this problem in the Kyle (1985) framework, and show that the price impact (Kyle s lambda) is strongly affected by the strategy of this large liquidity trader. Moreover, we find that order splitting is a noisy form of preannouncing trades, i.e., sunshine trading (cf. Admati and Pfleiderer (1991)). In the model, the predictable component in the order flow stems from the large liquidity trader only, which is a noisy signal of her trading interest. This predictable order flow is uninformed and does not affect prices, which reduces the liquidity trader s price impact. Our mechanism at first sight seems counterintuitive as we typically expect traders to hide their trading interest. In particular, our prediction runs in opposite direction of the predatory trading hypothesis by Brunnermeier and Pedersen (2005), where predatory traders (or front-runners ) exploit the liquidity needs of others by trading ahead of them. Our theory thus provides a novel explanation of why a predictable trading strategy may reduce the price impact of trading. The model builds on the multi-period discrete-time adverse selection model of Kyle 1 Average trade sizes have decreased from $80,000 to $7,000, and monthly turnover increased from 6% to 40% of market cap for CRSP stocks in the period (Chordia, Roll, and Subrahmanyam, 2011). 2

3 (1985), where we add a discretionary large liquidity trader who optimally splits up trades over time. The large liquidity trader, the informed trader and the noise traders submit market orders to the competitive market maker, who observes the aggregate order flow and determines the price to clear the market. The market maker is risk neutral and cannot distinguish between trades from informed and uninformed investors. Therefore, the trades by the discretionary large liquidity trader do affect the price, and because the market is anonymous she cannot simply preannounce her trading interest (Admati and Pfleiderer (1991)). 2 The model s key assumption is short-lived private information, meaning that in each period a new informed trader arrives who may only trade in that period. Also, in each period a new innovation in the fundamental value of the asset occurs, such that the asymmetric information does not resolve over time. Our main contribution is that, under these assumptions, order splitting by the discretionary liquidity trader is a noisy form of preannouncing trading interest, i.e., sunshine trading. Our result follows from a general result of the Kyle model that to the market maker, the trades by the informed traders are unpredictable. Then, only the trades by the discretionary liquidity trader cause predictability in the order flow. In equilibrium, the market maker attaches a zero price impact to the predictable component of the order flow. The assumption of short-lived private information creates a separating equilibrium, in the sense that the informed traders cannot mimic the strategy of the discretionary liquidity trader. 3 Effectively, the discretionary liquidity trader sends a credible signal of her uninformed trading interest, and is rewarded by a lower price impact of trading. This result contrasts Brunnermeier and Pedersen (2005), who show that the selling pressure of distressed traders induces predatory traders to also sell and subsequently buy back the asset 2 While an investor could say that he would buy a certain amount of shares in the future, there is no mechanism that forces him to actually to do so, i.e., preannouncement is a non-credible commitment. 3 If the informed traders would also split up trades across periods, then the autocorrelation in the order flow is not strictly uninformed and the equilibrium breaks down. 3

4 at lower prices. Their result requires a lack of liquidity provision in the market, 4 whereas we assume a large group of competitive market makers. 5 We motivate our modeling assumptions for the different types of players as follows. Informed traders have short-lived information, possibly because they are faster in processing public information than other traders. These informed traders use aggressive highfrequency trading strategies with short trading horizons of typically less then a few minutes, such as arbitrage, structural or directional strategies (as defined by the SEC (2010)). 6 The discretionary liquidity trader is an institutional trader with a trading horizon of a day (which is typical according to Campbell, Ramadorai, and Schwartz (2009)), and this horizon is much longer than the few minutes of the informed traders. The market maker in the model could represents a group of high-frequency traders with a passive market making strategy, who predominantly use limit orders to earn the bid-ask spread and liquidity rebates. Brogaard, Hendershott, and Riordan (2012) confirm that high-frequency traders with aggressive strategies are typically informed traders, whereas the high-frequency traders who supply liquidity get adversely selected. The informed traders in the theory of Foucault, Hombert, and Rosu (2012) also have short-lived private information, and are motivated as high-frequency traders who observe news faster than the rest of the market. The degree that the market maker learns about the liquidity traders trading interest depends strongly on the resiliency of the market, i.e., the speed with which prices converge to the fundamental value. The resiliency is determined by the trading aggressiveness of the informed traders, who revert the price impact of the strategic liquidity trader by trading in the opposite direction. Therefore, the net order flow gets close to zero over time in a very resilient market, which impairs the market makers learning about the liquidity trading 4 If the liquidity shock of distressed traders is small, then the predatory traders will in fact compete for liquidity provision (see Proposition 2, p. 1841). 5 See also Carlin, Lobo, and Viswanathan (2007), where predatory trading arises because no market makers exist to absorb order flow. 6 The SEC defines four broad high-frequency trading strategies: passive market making, arbitrage, structural (e.g., trading on latency and the use of flash orders) and directional (trading on fundamentals, momentum and order anticipation). 4

5 interest. The general version of the model allows for a flexible degree of resiliency, but we also consider two special cases. The first is the version with news trading, where the informed traders trade on the innovation in the fundamental value only, rather than the difference between the price and the fundamental value. The trades by the liquidity trader therefore have a permanent price impact, and the market has zero resiliency. This situation is most relevant when the liquidity trader has a very short deadline, such that the price impacts of the individual trades accumulate over time. In the second special case the fundamental value is announced after each period (similar to Admati and Pfleiderer (1988)). In this case the temporary price impact of the current trade has fully disappeared before the liquidity trader submits the next trade. This case corresponds with a perfectly resilient market, and is most realistic when the time between trades is large, or when the market maker may learn from prices of highly correlated assets. In both special cases, the market maker can learn very clearly about the trading interest of the liquidity trader, which strongly reduces the overall execution costs. Our model also contributes to a growing literature that studies the optimal execution problem of a large trader 7 by endogenizing the price impact parameter. The liquidity trader must trade a fixed amount before a deadline, and has a U-shaped optimal execution strategy: the first and last trades are large, and intermediate trades are small. The optimal trade size in each period depends on the following tradeoff. On the one hand, a larger trade increases the expected uninformed order flow in all future rounds, which then receive a zero price impact. On the other hand, a larger trade increases the price of the current and all future rounds, because prices only slowly revert to the fundamental value via informed trading. For the initial trade the first effect is large whereas for the last trade the second effect is small, which generates the U-shape. The price impact parameter (illiquidity) 7 See e.g., Almgren and Chriss (1999, 2000), Engle, Ferstenberg, and Russell (2012), as discussed in the literature section. 5

6 comoves negatively with the quantity traded by the liquidity trader, as uninformed trading reduces the price impact parameter (Kyle, 1985). A more general contribution of the model is that the predictable component of the order flow may explain why resiliency exists in electronic limit order markets, i.e., why liquidity replenishes after a trade. In the model, if the market perceives that a certain trade belongs to a series of liquidity motivated trades, then the liquidity consumed by the trade should be replenished quickly. Similarly, the presence of the predictable component in the order flow may also explain why liquidity on the bid and ask side is asymmetrical at times. 8 We solve the problem numerically, as a closed form solution is not available. The problem has many state variables because the market maker must learn not only about the fundamental value, but also about the trading interest of the discretionary liquidity trader. In addition, the optimization problem of the discretionary liquidity trader is constrained, as she must trade exactly the exogenously given quantity. The model explains several empirical findings. Griffin, Harris, and Topaloglu (2003) estimate a VAR model with five-minute returns, institutional order imbalance and retail order imbalance, and find that the positive autocorrelation of the institutional order imbalance over the preceding 30 minutes does not affect current returns (i.e., is uninformed). In addition, current returns positively predict future institutional order imbalance; which coincides with our theory as the unexpected component of the order flow affects current returns and signals future liquidity trading interest and order imbalance. The predictions of our model also confirm several empirical results of Chordia, Roll, and Subrahmanyam (2002, 2004). In particular, they find that the daily order imbalance is strongly autocorrelated whereas returns have virtually zero autocorrelation, which suggests that predictable order flow is uninformative at the daily level. Heston, Korajczyk, and Sadka (2010) argue that predictable patterns in volume, returns and order imbalance are caused by systematic 8 Van Achter (2008) shows that asymmetric liquidity may result from heterogeneous trading horizons of investors, which affects the decision to place limit or market orders. 6

7 trading patterns of institutional investors. Bessembinder, Carrion, Tuttle, and Venkataraman (2012) test the sunshine trading versus predatory trading hypothesis for an oil futures ETF that rolls over its position on a monthly basis, and find more support for the sunshine trading hypothesis. Our model supports the theoretical results of Obizhaeva and Wang (2005) and Alfonsi, Fruth, and Schied (2010). They also find an optimal U-shaped trading pattern, as a large initial trade creates a price pressure that attracts many new limit orders to the limit order book. Essentially, this mechanism is also present in our model as the informed trades reduce the price pressure caused by the strategic liquidity trader. Novel in our paper is the channel that the expected uninformed order flow has zero impact on prices, and that the resiliency is explicitly modeled in an adverse selection framework. Chordia, Roll, and Subrahmanyam (2004) study a two-period model with a discretionary liquidity trader. They obtain a closed-form solution, but the discretionary trader is limited to trade either in one period only, or to split up his trades equally across both periods. Because trading is restricted to two rounds, they do not obtain the U-shape trading pattern and the liquidity trader cannot update his strategy over time. While their model focusses on the relation between order imbalance and stock returns across days, our model focusses on the optimal intraday trading strategy. 2 Literature review This paper is mostly related to papers that study the optimal execution problem and to extensions of Kyle (1985) that focus on strategic liquidity traders. Our model contributes to the literature on the optimal execution problem, which is the problem of a large liquidity trader who must trade a given quantity before a deadline 7

8 who aims to minimize execution costs (Bertsimas and Lo, 1998). 9 Almgren and Chriss (1999, 2000), Engle, Ferstenberg, and Russell (2012) find the optimal execution strategy in a mean-variance framework. Obizhaeva and Wang (2005) and Alfonsi, Fruth, and Schied (2010) study this problem in a limit order book market, and show that the optimal strategy strongly depends on the resiliency of the book, i.e., the speed with which the limit order book recovers after a trade. Huberman and Stanzl (2005) add transaction costs, which is important as in the continuous time limit the execution cost of the problem of Bertsimas and Lo becomes in fact independent of the actual strategy. 10 They also allow for a timevarying price impact function, which can be obtained in our framework easily as well by changing the variance of the noise trades and the fundamental innovations over time. Easley, Lopez de Prado, and O Hara (2012) show that the order imbalance affects the endogenously determined trading horizon and the price impact function. These papers are partial equilibrium models in the sense that the price dynamics are exogenously determined. We endogenize the price impact function and show how it is affected by order splitting. The paper is closely related to extensions of Kyle (1985) with strategic liquidity traders, i.e., discretionary traders. Admati and Pfleiderer (1988) model a group of discretionary traders who may decide in which period to submit there entire trade, and find that in equilibrium the informed and discretionary traders will trade in the same period. In contrast, we analyze the behavior of a single discretionary liquidity trader, and find that she optimally splits up her trades over time. Subrahmanyam (1995) analyses the role of circuit breakers and extends the Admati and Pfleiderer model to a two-period version, where the discretionary trader is limited to trade either in one period only, or to split up her trades equally across both periods.while in equilibrium the discretionary trader indeed splits up across periods, the informed traders are restricted to trade in the first period only. 11 In 9 See also Kissell, Glantz, and Malamut (2003), Chapter 15 in Hasbrouck (2007). 10 In continuous time, there are infinitely many trading rounds before the fixed deadline, which becomes meaningless. 11 Chordia, Roll, and Subrahmanyam (2004) also make the assumptions of equal order splitting and a single period with informed trading. 8

9 our model a new informed trader arrives every period, which introduces an important dynamic aspect as the price pressure from uninformed trades in the current period affects the informed order flow in the next. Back and Pedersen (1998) extend the Admati and Pfleiderer model by allowing for long-lived private information, and find that market depth and volatility are constant over time. Spiegel and Subrahmanyam (1992) extend the static Kyle model by replacing the noise traders with risk averse and price sensitive liquidity traders who trade for hedging motives. 12 Massoud and Bernhardt (1999) extend the previous model to a two-period version, and find that some results of Kyle get reversed. For example, the price impact becomes steeper over time, because the liquidity traders wish to trade in earlier periods as to avoid the pricing risk in later periods. In Spiegel and Subrahmanyam (1995) risk averse discretionary liquidity traders also trade for hedging motives, and will trade either at the beginning of the day, or later in the same direction as the market makers effectively providing liquidity. Mendelson and Tunca (2004) find that the risk aversion of liquidity traders generally reduces informational efficiency, and that insider trading may improve the welfare of risk averse liquidity traders because they reduce the volatility of prices. This paper also relates to the preannouncement of trading interest (Admati and Pfleiderer, 1991), as order splitting generates a predictable and uninformed component in the order flow. Huddart, Hughes, and Williams (2010) analyze the case where the insider must preannounce his trades, but also has a liquidity motive to trade (e.g., risk sharing). Suboptimal risk sharing follows, because even though an insider might trade for liquidity reasons, the market maker revises prices because the trade may reflects private information. Huddart, Hughes, and Levine (2001) analyze the Kyle model where informed traders must announce their trades after submission, like employees of a corporation need to. In this case, the insider adds noise to his strategy to jam the signal of the market maker. 12 The liquidity traders have hedging motives in Vayanos (1999) too. 9

10 3 Model setup Consider a Kyle (1985) framework, where trading occurs sequentially in a number of batch auctions. Each auction is organized as an anonymous batch market where investors submit market orders. Trading begins at time 0 and ends at time 1, and takes place during t = 1,..., T periods, each of length 1/T. Time 0 represents the beginning of the trading day for example, and time 1 the end. Three players exist in the Kyle model. There is a risk neutral informed trader who observes the fundamental value of the asset and trades to maximize profits. In addition, a group of noise traders trade random amounts every period. Then, a competitive market maker observes the total order flow in a period, and chooses the price and his quantity to clear the market. We deviate from the standard multi-period Kyle model in two important ways. First, we introduce a strategic liquidity trader who must trade a given amount before the deadline at time 1. She is a discretionary liquidity trader who chooses the optimal quantity to trade in each period. 13 The total quantity is drawn from a normal distribution before trading starts, and does not change afterwards. Second, we assume that in every trading round a new informed trader arrives, who observes the fundamental value and may trade only once. Also, in every period an innovation in the fundamental value of the asset occurs. In essence, we assume that private information is short lived, which resembles Foucault, Hombert, and Rosu (2012) where high-frequency traders (HFTs) respond to news (i.e., short-term information) extremely quickly. In this setup the flow of news and asymmetric information is constant over time, which is realistic as news might be generated by trades in correlated assets for example. The traders are all risk neutral and submit market orders. Denote by S N(0, σ 2 S ) the total quantity that the strategic liquidity trader must trade before the deadline, where 13 Our definition differs from Admati and Pfleiderer (1988), where the discretionary liquidity traders may only choose a single period to submit the entire trade. 10

11 a positive value represents a purchase and a negative value a sale. This quantity is exogenously determined and she cannot access other trading venues. In each periods t she submits a fraction f t S, where she chooses the vector f = (f 1,..., f T ) subject to T t=1 f t = 1. Denote by x t the strategically chosen order flow of the informed investor, and u t N(0, σ 2 u) the randomly determined uninformed order flow of non-discretionary noise traders. Then, the total order flow in each period is y t = x t + f t S + u t. (1) The process of the fundamental value is given by v t = v 0 + t j=1 e j, (2) where e j N(0, σe) 2 and IID. An important element in the model is that a trade by the liquidity trader affects the current price, which in turn affects the strategy of informed traders in future periods. In fact, price pressures due to trades in the current period are beneficial to informed traders in future periods as they increase the pricing error. This mechanism does not exist in the models of Bertsimas and Lo (1998), Almgren and Chriss (2000) and Huberman and Stanzl (2005) for example. 3.1 The market maker We restrict attention to the recursive linear equilibrium, where the market maker s pricing function and the strategies of the traders are linear in the information sets. The market maker determines the price P t after observing the current and past order flow, which are summarized in a vector I t = (y 1,..., y t ) that represents his information set. The information set contains the sequence of past order flow (past prices do not contain additional informa- 11

12 tion, as they depend linearly on the order flow). Market efficiency states that the market maker chooses the price such that P t = E(v t I t ). Following Kyle (1985), we conjecture (and verify in equation (18)) that the pricing schedule is P t = P t 1 + λ t (y t E(y t I t 1 )), (3) where E(y t I t 1 ) represents the expected order flow. The price is only affected by the unexpected component of the order flow. Given short lived private information, we show below that E(y t I t 1 ) = f t E(S I t 1 ), i.e., the expected order flow depends only on the market makers expectation of the quantity traded by the strategic liquidity trader. The informed order flow is unpredictable to the market maker, which is a result obtained by Kyle. A proof by contradiction is that if informed trades were autocorrelated to the market maker, he could immediately set a price that reflects this information which eliminates the autocorrelation. In the model, the market maker behaves competitively and incorporates all relevant information to set the price. The strategy of the informed trader is described next. 3.2 The informed trader The problem of the informed trader is identical to the one period version of Kyle (1985). The informed trader observes v t (a perfect signal) and I t 1, and knows the pricing function (3). He chooses x t to maximize his expected profits max x t E[x t (v t P t ) v t, I t 1 ] = max x t [x t (v t P t 1 λ t (E(y t v t, I t 1 ) E(y t I t 1 )))]. (4) 12

13 Given y t = x t + u t + f t S, we have E(y t v t, I t 1 ) E(y t I t 1 ) = x t. We set the first order condition to zero with respect to x t, 0 = E[(v t P t 1 2λ t x t )] x t = β t (v t P t 1 ), with β t = 1 2λ t, (5) where β t represents the aggressiveness of the informed trader in period t. Importantly, the market maker cannot predict the informed order flow, as E(x t I t 1 ) = 0 because E(v t I t 1 ) = E(v t 1 I t 1 ) = P t 1. Note that β t does not depend on the number of trading rounds like in the multi-period version of Kyle, because each insider trades only once. We have fixed the pricing rule and the informed traders strategy for a single period, and we proceed by describing the trading process across periods. 4 Recursive model representation In this section we first present the general version of the model where the informed trader trades with aggressiveness parameter β t on (v t P t 1 ). Then we describe two special cases where she trades with β t on the innovation e t only. The special cases simplify the analysis and will prove useful benchmark models. 4.1 The general model The strategic liquidity trader submits a fraction f t of the total quantity S each period, of which a part is expected and a part unexpected, i.e., reveals new information about her trading interest. In this setup, the market maker needs to learn about the fundamental value of the asset v t and about the quantity S. Learning about S improves the prediction of the uninformed component of the order flow and therefore also the informed component, 13

14 which provides a clearer signal about the true value. The market maker observes the order flow y t, and learns about S and v t by updating the conditional expectations and variances. We assume f is non stochastic and known in equilibrium. The learning process can be derived from a Kalman filter, with state vector X t = S, (6) v t and parameter vector h t = f t. (7) β t The observation equation is y t = h tx t β t P t 1 + u t. (8) The dynamics of the state vector are quite simple here, X t = X t (9) e t Now denote the prediction equations X t t 1 = E(X t I t 1 ), V t t 1 = V ar(x t I t 1 ), (10) and update equations ˆX t = E(X t I t ), ˆVt = V ar(x t I t ). (11) Using the Kalman filter and update equations, we can derive the following results. First, the prediction equations are X t t 1 = ˆX t 1, (12) 14

15 and V t t 1 = ˆV t (13) 0 σe 2 Define the unexpected order flow ỹ t = y t E(y t I t 1 ) = y t f t S t t 1. (14) The updating equations for the expectation and variance of the state vector then are V t t 1 h t ˆX t = X t t 1 + ỹ h tv t t 1 h t + σu 2 t, (15) and ˆV t = V t t 1 V t t 1h t h tv t t 1. (16) h tv t t 1 h t + σu 2 The Kalman filter equations can be solved recursively, starting from ˆV 0 = σ2 S 0, ˆX0 = 0, (17) v 0 0 σ 2 v 0 where σ 2 v 0 is the initial variance of the fundamental value. The market maker uses the unexpected order flow to update the price and the expectation of S according to the parameters λ t and ϕ t respectively, which are given from equation (15) as ϕ t = λ t (f t β t ) V t t 1 V t t 1 f t β t f t + σu 2 β t. (18) Thus, λ t represents the price impact parameter and ϕ t the speed with which the market maker learns about S. Equations (15) and (18) show that the price equals the market makers expectation of the fundamental value, which confirms the conjectured pricing rule 15

16 (3). 4.2 Two special cases In this section we consider a version of the model with news trading, and a version where the fundamental value is announced after each period. In the case of news trading, the informed trader only observes the signal e t and trades with aggressiveness β t. In the case where the fundamental value is announced after each period, the informed trader trades on v t P t 1 = v t v t 1 = e t. 14 In both cases, the market maker learns about e t according to one equation which simplifies the representation. When the informed traders trade on news only, they do not revert the price pressure caused by the strategic liquidity trader. In this case the price impact of the strategic liquidity trader is permanent and the market has zero resiliency. The news trading case is most realistic if the liquidity trader has a relatively short deadline, such that the price impacts of her trades simply accumulate over time. In the case that the fundamental value is announced after each period, the market is perfectly resilient as the price impact of an uninformed trade is immediately reverted. This case would be realistic if the time between trades is relatively long, while the market maker can infer the fundamental value by observing prices of correlated assets. In this situation the liquidity traders temporary price impact fully disappears between the trades. In the general case, the parameter β t of the informed traders strategy β t (v t P t 1 ) represents the degree of resiliency of the market. The special cases where the market has zero perfect resiliency are useful benchmarks. In these cases the state vector and parameter vector are 14 The fundamental value of period t is announced after trading round t, but before the innovation e t+1 has realized. Thus, in each period there is asymmetric information. 16

17 X t = S, h t = f t. (19) The observation and unexpected order flow equations are y t = f t S t + β t e t + u t, (20) ỹ t = y t E(y t I t 1 ) = y t f t S t t 1. (21) The two versions proceed differently News trading In the case of news trading the filter equations are simply V t t 1 = ˆV t 1 and S t t 1 = Ŝt 1. The updating equations are V t t 1 f t Ŝ t = S t t 1 + ỹ f tv t t 1 f t + βt 2 σe 2 + σu 2 t, V t t 1 f t f ˆV t = V t t 1 tv t t 1. f tv t t 1 f t + βt 2 σe 2 + σu 2 After each period the market maker updates the expectation of S, which allows him to form a more precise signal of all the past innovations. For τ 0, the expectation of innovation e t after period t + τ is β t σe 2 E(e t I t+τ ) = (y t f t Ŝ t+τ ). (22) βt 2 σe 2 + σu 2 This recursion is quite simple as only the expectation Ŝt+τ changes each period. The price equals the sum of the expectations of the innovations P t = E(v t I t ) = t n=1 E(e n I t ). (23) 17

18 4.2.2 Announcement of the fundamental value In this case, the market maker updates the expectation and variance of S based on both the order flow and the announcement of the fundamental value. The solutions are in closed form, and given by t Ŝ t = σ2 S i=1 (f i(y i β i e i )) σu 2 + σs 2 t i=1 f, i 2 ˆV t = σ 2 uσ 2 S σ 2 u + σ 2 S t i=1 f 2 i. (24) The updating equation of the price P t is P t = E(v t I t ) = v t 1 + β t σ 2 e β 2 t σ 2 e + σ 2 u + f 2 t ˆV t 1 ỹ t. 4.3 Problem of the strategic liquidity trader The goal of the strategic liquidity trader is to find an optimal strategy f that maximizes the expected trading profits (i.e., minimizes the execution costs). Furthermore, this strategy must be a rational expectations equilibrium, in the sense that the market maker must correctly anticipate the optimal strategy to determine his response, and given the market makers response the strategy must indeed be optimal for the liquidity trader. The strategic liquidity trader knows the realization of the quantity S, which in fact contains two important sources of information. First, although her trades are uninformed, they do affect prices and therefore create a pricing error. This pricing error will be exploited by the informed traders in future periods, and therefore the liquidity trader can predict future order flow and prices. This information is static in the sense that it is predictable before trading starts We will show in the next section that the realization of S does not affect the strategy f t, when we prohibit the liquidity trader to update her strategy over time (see footnote 20). 18

19 Second, after observing any period s order flow the liquidity trader can filter out her own trades, such that the remaining component reveals a more precise signal about the fundamental value (as compared to the market maker). The liquidity trader uses this expectation of the fundamental value to predict the informed order flow in future periods. This information is dynamic and affects the optimal trading strategy depending on the realizations of the noise trades and asset innovations. We allow the strategic liquidity trader to repeat her static optimization every period, which is a (quasi) dynamic solution. 16 She observes the order flow and constructs the expectation and variance of the fundamental value according to the prediction and update equations of the market maker in Section 4.1, with the only difference that the conditional variance of S is zero. 17 Each period she recalculates her optimal trading strategy for the remaining periods. We find numerical solutions for the static and dynamic problem, but only the static solution and the special cases are rational expectations equilibria (REE). In the special cases the informed trader submits β t e t (equation (21)), which is unpredictable to both the market maker and the strategic liquidity trader. Therefore, the liquidity traders additional knowledge of S is not relevant in predicting informed trades, and essentially she does not have any informational advantage. For this reason the static solution is dynamically optimal. The dynamic model is not an REE, as the liquidity trader updates her strategy over time which is unpredictable to the market maker. Specifically, by observing the order flow the liquidity trader forms a more precise estimate of the fundamental value, which allows her to predict future periods informed trades (β t (v t P t 1 )) and price changes. Based on the future price changes she chooses to either accelarete or decelerate her trading program. However, Monte Carlo simulations presented in the results section reveal that the deviations 16 We cannot find a solution with dynamic programming due to the large number of state variables. 17 We omit the equations because the problem is otherwise identical. 19

20 in the dynamic strategy are very small, and have a neglible effect on the expected execution costs. 5 Numerical approach At time 0, the discretionary liquidity trader maximizes expected profits max f 1,...,f T E S 0 s.t. [ T ] f ts(v T P t ), (25) t=1 T t=1 f t = 1. Typically, this type of problem is solved with dynamic programming (see e.g., Bertsimas and Lo (1998)). However, we cannot obtain a closed form solution due to the complexity introduced by the many state variables (the value equation contains high-degree polynomials). Therefore, we use numerical methods to find the solution. We consider a static equilibrium strategy which is fixed before trading starts, and a repeated static strategy that is allowed to change over time. We start by taking numerical values for the exogenous parameters of the model, T, σ 2 u, σ 2 e and σs 2, which are the number of trading rounds, and the variances of the noise trade, the innovations in the fundamental value and the quantity of the strategic liquidity trader. First, the market maker makes a guess of the equilibrium strategy f, which fixes the parameters λ t, β t and ϕ t in the model (equation (18)). Then, we take the expected profit function in equation (25) and iterate the entire problem backwards to period 1 using the recursive equations for the price, the order flow and the Kalman filter. Now we have a single equation that contains the total expected profits as a function of S, f, and the realizations of u 1,..., u T and e 1,..., e T which are zero in expectation. We maximize this function and 20

21 obtain an optimal solution f. 18 Next, we iteratively find a rational expectations equilibrium, where the market maker correctly anticipates f and responds by choosing λ t and ϕ t, and given the response of the market maker the sequence f maximizes the expected trading profits of the liquidity trader. The iteration starts by the market makers initial belief of the sequence f which fixes λ t and ϕ t, for which we maximize the profit function yielding the liquidity traders optimal strategy f. We substitute this solution into the market makers belief, and we recalculate the liquidity traders optimal strategy. We iterate this process until convergence, i.e., until the optimal strategy is arbitrarily close to the market makers expectation of the strategy. 19 Now we have obtained the static solution, which generates fixed values for λ t, β t, ϕ t, ˆV t and V t t 1. Note that the optimal strategy is independent of the realization of S, i.e., the fraction submitted by the liquidity trader each period is independent of the total quantity she must trade. 20 This is a necessary requirement as the market maker cannot observe S, and otherwise would not be able to form a rational expectation of the strategy f. The dynamic solution continues from the static solution. In round 1, the strategic liquidity trader submits f 1 S from the static solution (which now becomes realized), and afterwards observes y 1. Based on y 1 she updates her expectation of v T, and recalculates her optimal strategy yielding f 2,..., f T. The market makers expectation of f from the static solution remains unchanged, as he cannot predict the updates of the liquidity trader. In round 2, the liquidity trader submits f 2 S (which now becomes realized), and afterwards observes y 2. This process is repeated until we reach period T. In each period we find the solution of the sequence f t conditional upon the liquidity traders information set, which is 18 Here we can require that all f t > 0, i.e., that she never sells when S > 0 and never buys when S < 0. For certain parameter values however, it is optimal that some fractions become negative. 19 In the results section, we assume convergence is reached when T i=1 f i f i < 1/100, 000; where f i is the optimized sequence and f i the market makers expectation. 20 The proof is that if we substitute the order flow and price equation into equation (25) and take the FOC with respect to f t (for any t), the S drops out. Intuitively, the per period trading costs f t SP t are quadratic in f t, and minimization of trading costs across periods balances the per period price impact. 21

22 a repeated static optimization. The dynamic solution differs from the static solution. The reason is that over time, the liquidity trader learns from the order flow and updates her entire (expected) future trading strategy. Specifically, she can filter out her own (uninformed) trades from the aggregate order flow, and thus obtains an estimate of the informed order flow and fundamental value which is more precise than that of the market maker. Using this additional information she predicts future informed trades, and decides to accelerate or decelerate her trading program. However, only in the special cases of news trading and announcement of the fundamental value does the dynamic solution coincide with the static solution. 6 Results We first analyze the optimal strategy of the liquidity trader before trading begins, but after the realization of S, T, σu, 2 σe 2 and σs 2. The liquidity trader incorporates in her strategy the impact of her trades on the market makers expectation of S and prices, which in turn affects future periods informed order flow and prices. 6.1 Static solution We solve the model numerically for the base case first, and then change the value of one parameter at a time to analyze the impact of that parameter on the equilibrium outcome. For the base case we use the following parameter values. The number of trading periods T = 5, the strategic liquidity trader must trade the quantity S = 1, which has an unconditional variance of σ 2 S = 3. The volatility of noise trading and innovation in the fundamental value are σ 2 u = 1 and σ 2 e = 1 per trading round. We also set the initial variance of the fundamental value to σ 2 e. We normalize the price to zero, as the initial price level has no impact on the optimal order splitting strategy. 22

23 Figure 1 plots the parameters of interest over time for the base case. The upper panel shows the U-shaped trading strategy f, where the fractions range between 16.2% 25.4%, and the closely corresponding inverted U-shape for the price impact parameter λ t. The variables move in opposite direction as more uninformed trading reduces the price impact (as in Kyle (1985)). The bottom panel shows the total order flow y t, informed order flow x t and the market makers expectation of the uninformed order flow E(S I t ), which are all expectations formed by the liquidity trader before trading starts. As of round 2 the informed trader starts trading in opposite direction to the strategic liquidity trader, which pushes the net order flow closer to zero. Table 1 presents the results for the other parameters (upper panel), and the deviations to the base case (lower panels). Column three shows the parameter ϕ, the speed with which the market maker learns about S from the order flow. Most of the learning on S occurs in the first trading round because f 1 is large (column 5), and then V ar(s I 1 ) reduces from 3 to The expected order flow y 1 is 0.25, which generates a price impact 0.16 cent. Therefore, in period 2 the informed trader sells 0.12 (column 7), which pushes the price partly back towards the fundamental value. In period 2 the liquidity trader buys 0.178, such that the expected net order flow y 2 is The first cell in the last column shows the expected execution cost of a benchmark model where the market maker does not learn about the trading interest of the liquidity trader. Then she trades a fraction 1/T = 0.2 each period, and the model is reduced to a repeated single period Kyle (1985) model. In the benchmark case the expected execution costs are 0.22 and with learning the costs are 0.2, which is a modest reduction of 9%. The main result of the model is that the market maker learns about the trading interest of the uninformed liquidity trader. In particular, the initial large trade increases the market makers expectation of S to E(S I 1 ) = 0.08, such that in period 2 the market maker subtracts the impact of the expected order flow λ 2 f 2 E(S I 1 ) = from the price. The price reduction lasts for all future periods as well, such that the remaining 74% of the total order 23

24 size receives a discount. After round 2, an additional λ 3 f 3 E(S I 2 ) = is subtracted from P 3 and the future prices, and similar amounts in the remaining periods. Summed over all periods, the predictable component of the order flow reduces P 5 by 0.03, which is relevant compared to the total price impact of 0.25 (P 5 ). The predictable component in the order flow reduces prices as it effectively signals the liquidity traders uninformed trading interest, i.e., order splitting is a noisy form of sunshine trading (Admati and Pfleiderer (1991)). The market makers expectation of the uninformed order flow E(S) is small as it does not exceed 0.09 in any period, which is a consequence of the high level of resiliency. The resiliency is determined by the aggressiveness of the informed traders strategy, who reduces the pricing error by half each period. In a resilient market, signalling uninformed trading interest is quite costly to the liquidity trader for two reasons. First, the trading costs in any period f t SP t are quadratic in f t, such that larger trades are relatively more expensive. 21 Second, a larger trade by the liquidity trader in the current period causes a price pressure, which will be followed by sales of the informed traders in future periods. The sales reduce the net order flow in future periods which impairs the market maker s learning about S. The second panel of Table 1 shows a market with low resiliency, i.e., where the informed trader trades with aggressiveness parameter β t = 1/(10λ t ). When resiliency is lower, the market maker can learn much more clearly about S: E 5 (S) = 0.32, compared to 0.09 in the base case. The reason is that the sales by the informed traders in periods 2 to 5 are smaller, such that the net order flow reflects the liquidity traders trades more clearly. Resiliency also has a strong impact on the optimal execution strategy, which has a strong U-shape. The parameter values explain why the optimal trading strategy is U-shaped, which follows from a tradeoff. On the one hand, a larger current trade provides a stronger signal about future trades of the liquidity trader, which then enjoy a lower price impact. On the 21 The reason is that f t affects f t SP t directly, and indirectly via P t. 24

25 other hand, a larger current trade increases both the current and future prices, as trades by informed traders only slowly push the price back towards the fundamental value. The positive effect is particularly important in the first trading period, as a large initial trade generates a clear signal to the market maker about future uninformed trading interest. In the last trading period, the negative effect is small because future prices are then irrelevant to the liquidity trader. The impact of expected uninformed order flow on prices offers a novel explanation for resiliency in electronic limit order markets, i.e., why liquidity replenishes after a trade. If the market perceives that a certain trade belongs to a series of trades and is likely to be uninformed, then the market attaches a low price impact to that trade such that liquidity will be replenished quickly. In this paper we model a batch auction where the resiliency occurs instantaneously, i.e., that λ 2 f 2 E(S I 1 ) is simply subtracted from P 2, but in a limit order market traders need some time to respond. Note that the expected order flow affects the price level, and the slope of the limit order book λ t via the reduction in the conditional variance of S. The mechanism of expected uninformed order flow may also explain why liquidity on the bid and ask side of the order book can be asymmetrical at times. If the market expects substantial order flow from uninformed buyers, the ask side should be more liquid. A trade executed on the bid side is more likely to stem from informed traders, such that the bid side is less liquid. The third panel shows the case where the number of trading periods T = 3. The liquidity trader must trade larger quantities each period, which has a strong effect on the expected price and execution costs. The price effect gets partly mitigated by the fact that the market maker learns more clearly about S. In the bottom panel the unconditional variance of S is higher, σ 2 S = 5. In this case, the market maker can learn more about S, which increases the expected order flow in remaining periods. For this reason, there is also 25

26 more curvature in the U-shape of the sequence of f t. 6.2 Special cases The results for the case of news trading and announcement of the fundamental value are presented in Table 2. In the case of news trading, the informed traders do not revert the price pressure caused by the strategic liquidity trader. This represents a market with permanent price impact and zero resiliency. The upper panel of Table 2 shows that the market maker can form a clearer signal about S due to the absence of resiliency: E 5 (S) = 0.22, compared to 0.09 in the base case. The optimal execution strategy is to trade a slightly increasing fraction each period, but is quite flat in general. The U-shape pattern has disappeared because the permanent price impact makes a large initial trade very costly (as it permanently increases the prices of all future trades). While the level of informed trading and the price impact parameter are lower in each period, the absence of resiliency still increases overall execution costs. Panel 2 shows the case where the fundamental value is announced between trading periods. In this case the market is perfectly resilient and the temporary price impact is zero, which is perhaps reasonable when the time between the trades of the liquidity trader is large. Now the market maker receives a very clear signal about S by observing both the order flow and the fundamental value: E 5 (S) = 0.38, compared to the base case of In this case the execution costs reduce by 20% compared to the case without learning (0.10 instead of 0.08). The optimal execution strategy is similar to that in the news trading case. 26

27 6.3 Dynamic solution The dynamic version of the model repeats the static optimization after each period. The strategic liquidity trader forms expectations of the current and all future parameter values conditional on her information set, and updates her strategy accordingly. Therefore, for each trading round t she constructs T estimates, which gives an T T matrix for every parameter. We are interested in how the optimal strategy depends on the realizations of the order flow. The dynamic results of the model are shown in Table 3. Parameter values in bold font type have realized, whereas the other values are expectations of the strategic liquidity trader. The left block of the table shows the parameters in an environment where the fundamental value rises, as we set e 1 = e 2 = e 3 = 1, e 4 = e 5 = 0 and all noise trades to zero. In the right block we study an environment where only the noise traders purchase, and set u 1 = u 2 = u 3 = 1, u 4 = u 5 = 0 and all innovations in the fundamental value to zero. In the first panels we observe that the realizations of the order flow cause only small dynamic adjustments to the optimal strategy of the strategic liquidity trader. The largest change in the left panel is for example 0.01, E(f 5 I0 S ) minus E(f 5 I1 S ); and in the right panel it is Due to the buying pressure of the informed traders (left panel) and noise traders (right panel), the strategic liquidity trader beliefs the stock is overpriced and therefore postpones her own purchases a little. 22 By postponing her trades, she anticipates that informed traders in future rounds will reduce the pricing error making it cheaper to buy later. The second panel shows the expected and realized prices. The expected prices adjust slowly to the order flow, because the market maker is uncertain whether order flow stems from the informed or uninformed traders. The difference between expected and realized order flow in the next block show that order flow is difficult to predict. If a current 22 The bottom panel of the left block shows that E(v 1 I S 1 ) = while P 1 = 0.665, i.e., the liquidity trader believes the stock is overpriced by