Abstract
We incorporate a Markovian signal in the optimal trading framework which was initially proposed by Gatheral et al. (Math. Finance 22:445–474, 2012) and provide results on the existence and uniqueness of an optimal trading strategy. Moreover, we derive an explicit singular optimal strategy for the special case of an Ornstein–Uhlenbeck signal and an exponentially decaying transient market impact. The combination of a meanreverting signal along with a market impact decay is of special interest, since they affect the short term price variations in opposite directions. Later, we show that in the asymptotic limit where the transient market impact becomes instantaneous, the optimal strategy becomes continuous. This result is compatible with the optimal trading framework which was proposed by Cartea and Jaimungal (Appl. Math. Finance 20:512–547, 2013). In order to support our models, we analyse nine months of tickbytick data on 13 European stocks from the NASDAQ OMX exchange. We show that order book imbalance is a predictor of the future price move and has some meanreverting properties. From this data, we show that market participants, especially highfrequency traders, use this signal in their trading strategies.
Keywords
Optimal portfolio liquidation Market impact Optimal stochastic control Predictive signalsMathematics Subject Classification (2010)
93E20 60H30 91G80JEL Classification
C02 C61 G111 Introduction
The financial crisis of 2008/2009 raised concerns about the inventories kept by intermediaries. Regulators and policy makers took advantage of two main regulatory changes (Reg NMS in the US and MiFID in Europe) which were followed by the creation of worldwide trade repositories. They also enforced more transparency on the transactions and hence on market participants’ positions, which pushed the trading processes towards electronic platforms [30, Introduction]. Simultaneously, consumers and producers of financial products asked for less complexity and more transparency.
This tremendous pressure on the business habits of the financial system shifted it from a customised and high margins industry, in which intermediaries could keep large (and potentially risky) inventories, to a mass market industry where logistics have a central role. As a result, investment banks nowadays unwind their risks as fast as possible. In the context of small margins and high velocity of position changes, trading costs are of paramount importance. A major factor of the trading costs is the market impact: the faster the trading rate, the more the buying or selling pressure will move the price in a detrimental way.
Academic efforts to reduce the transaction costs of large trades started with the seminal papers of Almgren and Chriss [6] and Bertsimas and Lo [11]. Both models deal with the trading process of one large market participant (for instance, an asset manager or a bank) who would like to buy or sell a large amount of shares or contracts during a specified duration. The cost minimisation problem turned out to be quite involved, due to multiple constraints on the trading strategies. On the one hand, the market impact (see [8] and references therein) demands to trade slowly, or at least at a pace which takes into account the available liquidity. On the other hand, traders have an incentive to trade rapidly, because they do not want to carry the risk of an adverse price move far away from their decision price.
The importance of optimal trading in the industry generated a lot of variations for the initial meanvariance minimisation of the trading costs (see [17, Chaps. 6 and 7], [25, Chap. 2.3] and [30, Chap. 3] for details). In this paper, we consider the mean–variance minimisation problem in the context of stochastic control (see e.g. [29, 12]). In this approach, some more realistic control variables which are related to order book dynamics and specific stochastic processes for the underlying price can be used (see [26, 32] for related work).
In this paper, we address the question of how to incorporate signals, which are predicting short term price moves, into optimal trading problems. Usually optimal execution problems focus on the tradeoff between market impact and market risk. However, in practice many traders and trading algorithms use short term price predictors. Most of such documented predictors relate to order book dynamics (see e.g. [18] and [16]). They can be divided into two categories: signals which are based on liquidity consuming flows [16], and signals that measure the imbalance of the current liquidity. In [31], an example of how to use liquidity imbalance signals within a very short trading tactic was studied. These two types of signals are closely related, since within short terms, price moves are driven by matching of liquidity supply and demand (i.e., current offers and consuming flows).

Gatheral, Schied and Slynko (GSS) framework [24], in which the market impact is transient and strategies have a fuel constraint, i.e., orders are finished before a given date \(T\);

Cartea and Jaimungal (CJ) framework [13], where the market impact is instantaneous and the fuel constraint on the strategies is replaced by a smooth terminal penalisation.
The main theoretical result of this work deals with the addition of a Markovian signal into the optimal trading problem which was studied in [24]. We argue in Sect. 2.1 that this is modelled mathematically by adding a Markovian drift to the martingale price process. We formulate a cost functional which consists of the trading costs and the risk of holding inventory at each given time. Then we prove that there exists at most one optimal strategy that minimises this cost functional. The optimal strategy is formulated as a solution to an integral equation. We then derive explicitly the optimal strategy for the special case where the signal is an Ornstein–Uhlenbeck process. From the mathematical point of view, this is the first time that a nonmartingale price process is incorporated into an optimal liquidation problem with a decaying market impact. Therefore the results of Theorems 2.3 and 2.4 extend [24, Proposition 2.9 and Theorem 2.11], respectively. Later we show that in the asymptotic regime where the transient market impact becomes instantaneous, the singular optimal strategies which were derived in the GSS framework become continuous. Moreover, we show that the asymptotics of the optimal strategy in the GSS framework coincide with the optimal strategy which is obtained in the CJ framework (see Remark 2.8 and Sect. 3). This benchmark between different trading frameworks provides researchers and practitioners a wider overview when they are facing realistic trading problems.
The use of predictive signals in optimal trading in the context described above is relatively new (see [16]). To the best of our knowledge, this is the first time that a Markovian signal and a transient market impact are confronted. The GSS framework already includes a transient market impact, without using signals. The CJ framework includes only a bounded Markovian signal and not a decaying market impact. Moreover, our results on optimal trading in the GSS framework incorporate a risk aversion term into the cost functional, which was not taken into account in the results of [24].
The main contribution of this work is in providing a new framework for optimal trading, which is an extension of the classical frameworks of [13] and [24], among others. The motivation to use this framework arises from market needs as our data analysis in Sect. 4 suggests. From a theoretical point of view, these models of trading with signals provide some new mathematical challenges. We describe in short two of these challenges.
The optimal strategies that we derive in Theorem 2.4 and Corollary 2.7 (i.e., in the GSS framework) are deterministic, and they use only information on the signal at time 0. One of the challenging questions which remains open is how to optimise the trading costs over strategies which are adapted to the signal’s filtration (see Remark 2.9).
Another contribution of this paper is a statistical analysis of the imbalance signal and its use in actual trading, which we present in Sect. 4. In order to validate our assumptions and theoretical results, we use nine months of real data from Nordic European equity markets (the NASDAQ OMX exchange) to demonstrate the existence of a liquiditydriven signal. We focus the analysis on 13 stocks, accounting for more than 9 billions of transactions. We also show that practitioners are conditioning, at least partly, their trading rate on this signal. Up to 2014, this exchange provided with each transaction the identity of the buyer and the seller. This database was already used for some academic studies; hence the reader can refer to [38, Sect. 2] for more details. We added to these labelled trades a database of Capital Fund Management (CFM) that contains information on the state of the order book just before each transaction. Thanks to this hybrid database, we were able to compute the imbalance of the liquidity just before decisions are taken by participants (i.e., sending market orders which consume liquidity).
We divide most members of the NASDAQ OMX into four classes: global investment banks, institutional brokers, highfrequency market makers and highfrequency proprietary traders (the classification is detailed in the Appendix). Then we compute the average value of the imbalance just before each type of participant takes a decision (see Fig. 4). The conclusion is that some participants condition their trading rate on the liquidity imbalance. Moreover, we provide a few graphs that demonstrate a positive correlation between the state of the imbalance and the future price move. These graphs also provide evidence for the meanreverting nature of the imbalance signal (see Figs. 5–7). In Fig. 9, we present the estimated trading speed of market participants as a function of the average value of the imbalance, within a medium time scale of 10 minutes. The exhibited relation between the trading rate and the signal in this graph is compatible with our theoretical findings.
This paper is structured as follows. In Sect. 2, we introduce a model with market impact decay, a Markovian signal and strategies with a fuel constraint (i.e., in the GSS framework). We provide general existence and uniqueness theorems, and then give an explicit solution for the case of an Ornstein–Uhlenbeck signal. The addition of a signal to the market impact decay is the central ingredient of this section. In Sect. 3, we compare our results from Sect. 2 to the corresponding results in the CJ framework. We show that the optimal strategy in the GSS framework coincides with the optimal strategy in the CJ framework in the asymptotic limit where the transient market impact becomes instantaneous and the signal is an Ornstein–Uhlenbeck process. In Sect. 4, we provide empirical evidence for the predictability of the imbalance signal and its use by different types of market participants. We also perform a statistical analysis which supports our focus on an Ornstein–Uhlenbeck signal in the example which is given in Sect. 2. The last section is dedicated to the proofs of the main results.
2 Model setup and main results
2.1 Model setup and definition of the cost functional
In this section, we define a model which incorporates a Markovian signal into the GSS optimal trading framework. Definitions and results from [24] are used throughout this section.
 (i)
\(t\mapsto X_{t}\) is leftcontinuous and adapted;
 (ii)
\(t\mapsto X_{t}\) has ℙa.s. bounded total variation;
 (iii)
\(X_{0}=x\) and \(X_{t}=0\) ℙa.s. for all \(t>T\).
Remark 1
Note that (2.2) is satisfied for every \(G\in \mathbb{G}\). A characterisation of positive definite kernels (that is, when the inequality (2.4) is not strict but weak) is given in [24, Proposition 2.6].
Remark 2
An important subclass of \(\mathbb{G}\) is the class of bounded, nonincreasing convex functions \(G:(0,\infty ) \rightarrow [0,\infty )\) (see [3, Proposition 2]).
2.2 Results for a Markovian signal
In this section, we introduce our results on the existence and uniqueness of an optimal strategy when the signal is a càdlàg Markov process. As in [24, Sect. 2], we restrict our discussion to deterministic strategies. The minimisation of the cost functional over signaladaptive random strategies is discussed in Remark 2.9.
Theorem 3
Assume that\(G \in \mathbb{G}\). Then there exists at most one minimiser to the cost functional (2.5) in the class\(\Xi (x)\)of admissible strategies.
In our next result, we give a characterisation for the minimiser of the cost functional (2.5).
Theorem 4
A few remarks are in order.
Remark 5
In the special case where the agent does not rely on a signal (i.e., \(I=0\)) and there is zero risk aversion (\(\phi =0\)), Theorems 2.3 and 2.4 coincide with [24, Proposition 2.9 and Theorem 2.11].
Remark 6
Dang [20] studied the case where the risk aversion term in (2.3) is nonzero, but again \(I=0\). In [20, Sect. 4.2], a necessary condition for the existence of an optimal strategy is given when the admissible strategies are deterministic and absolutely continuous. Our condition in (2.6) coincides with Dang’s result in that case. Note, however, that the question whether the condition in [20] is also sufficient and the uniqueness of the optimal strategy remained open even in the special case \(I=0\).
2.3 Result for an Ornstein–Uhlenbeck signal
Corollary 7
Note that \(1b_{0}(T+)=0\) and \(b_{i}(T+)=0\) for \(i=1,2,3\); moreover, the optimal strategy is linear in both \(x\) and \(\iota \).
In Fig. 1, we present some examples of the optimal strategy with the parameters \(\gamma = 0.9\), \(\kappa = 0.1\), \(T = 10\), \(x = 10\). These particular values are compatible with the empirical parameters which are estimated at the end of Sect. 4.2. Arbitrary initial values (−0.5, 0 and \(+0.5\)) are taken for the signal \(\iota \). The special case where \(\iota =0\) gives results similar to Obizhaeva and Wang [33]. The parameter \(\rho \), which controls the market impact decay, cannot be estimated from the data that we have; hence we take two arbitrary but realistic values (1.0 and 2.5). We observe that for large values of \(\rho \), the initial jump in the optimal trading strategy is larger than the corresponding jump in the small \(\rho \) strategies, but the trading speed tends to have less variation. We particularly notice that when the initial signal is in the opposite direction to the trading (\(\iota >0\) for a sell order), the trading starts with purchases as expected, and afterwards the trading speed eventually becomes negative. On the other hand, when the initial signal is in the same direction as the trading, it is optimal to start selling immediately, and most of the inventory is sold before \(T/2\).
In the following remarks, we discuss the result of Corollary 2.7.
Remark 8
Remark 9
(An adaptive version of (2.8))

the optimal strategy \(X^{*}\), limited to the information on the signal at \(t=0\);

an approximate strategy \({\tilde{X}}\) updated at each time \(t\in (0,T)\), which takes into account the whole trajectory of \((I_{t})\);

the optimal strategy which corresponds to a market impact without a decay (as shown in Sect. 3).
Note that in the cost functional \(U\), the timeinconsistency is a result of the transient market impact term. In [36], timeinconsistent optimal liquidation problems were also studied. However, the inconsistency of the problems in [36] arises from the risk aversion term.
Remark 10
(Price manipulation)
Market impact models admit transactiontriggered price manipulations if the expected costs of a sell (buy) strategy can be reduced by intermediate buy (sell) trades (see [3, Definition 1]). Theorem 2.20 in [24] implies that transactiontriggered price manipulations are impossible for the cost functional in (2.6), over the class of admissible strategies, in the case where \(I\equiv 0\) and \(\phi =0\). However, Fig. 1 shows that adding signals to the same market impact model can create optimal strategies which are not monotonically decreasing, and therefore implies a possible price manipulation. It would be very interesting to investigate the conditions on the market impact kernel and the trading signals which ensure that there are no price manipulations. A study of the possible implications of these price manipulations for other market participants is also of major importance.
3 Optimal strategy for temporary market impact
In this section, we study an optimal trading problem that has some common features with the problem introduced in Sect. 2.1. We consider again a price process which incorporates a Markovian signal. The main change in this section is that the market impact in (2.1) is temporary, i.e., the kernel is given by \(G(dt)= \kappa \delta _{0}(dt)\), where \(\delta _{0}\) is Dirac’s delta measure and \(\kappa >0\) is a constant. Note that this type of kernel is not included in the class \(\mathbb{G}\) of kernels introduced in (2.1). The main goal of this section is to show how to incorporate trading signals in the CJ framework [13]. The results we obtain could be compared to the results of Sect. 2 (see Remark 2.8). Recall that we heuristically obtained the optimal strategy when the kernel \(G=G_{\rho }\) ‘converges’ to Dirac’s delta measure as \(\rho \rightarrow \infty \).
In the following example, the fuel constraint on the admissible strategies is replaced with a terminal penalty function. This allows us to consider absolutely continuous strategies as in the framework of Cartea and Jaimungal (see e.g. [14, 15, 16]). We introduce some additional definitions and notations which are relevant to this setting.
Proposition 1
In the following result, we prove that the solution to (3.4) is indeed an optimal control to (3.3).
Proposition 2
Assume that\(\varrho \neq \sqrt{\kappa \phi }\). Then:
The proofs of Propositions 3.1 and 3.2 are given in Sect. 5.2.
In the following remarks, we compare the results of Sects. 2 and 3.
Remark 3
Remark 4
Remark 5
It is important to notice that (2.8) gives the optimal strategy on the time horizon \([0,T]\) in the GSS framework by using only information on the OU signal at \(t=0\). On the other hand, (3.6), which is the optimal trading speed \(r_{t}\) in the CJ framework, is using the information on the signal at time \(t\). A crucial point here is that if one tries to solve repeatedly the control problem in the GSS framework on time intervals \([t,T]\) for any \(t>0\), by using \(I_{t}\) and \(S_{t}\) as an input, the optimal strategy will not necessarily minimise the cost functional (2.3) on \([0,T]\). The reason is that the control problem in (2.3) may be inconsistent. The market impact (and therefore the transaction costs) created on \([0,t]\) affects the cost functional on \([t,T]\) (see Remark 2.9 for more details). Note that this phenomenon does not occur in the instantaneous market impact case (i.e., in the CJ framework).
4 Evidence for the use of signals in trading
In this section, we analyse financial data which is related to the limit order book imbalance. The data analysis in this section is directed to support the models which were introduced in Sects. 2 and 3. In Sect. 4.1, we describe our data base and provide empirical evidence for the use of the imbalance signal, which is a liquiditydriven signal. In Sect. 4.2, we study the statistical properties of the signal and motivate our model from Sect. 2.3 of an Ornstein–Uhlenbeck signal. Finally, in Sect. 4.3, we study the use of this signal during liquidation by different market participants. Note that in Sects. 2 and 3, we also discussed more general signals which are not necessarily liquiditydriven.
Before we start with the detailed analysis of the limit order book imbalance signal, we survey some related work on other processes which are known to affect asset prices and have meanreverting properties. Each of these processes may serve as a signal in the optimal trading framework of Sect. 2.3. We mention these specific examples as they demonstrate predictive signals which are affective at different time scales.
The order flow imbalance has been extensively studied in the literature (see e.g. [18] and references therein). The correlation between the current order flow and the future price move in 10 seconds intervals was studied by Cont et al. in [18]. The meanreverting properties of the order flow were studied by Bechler and Ludkovski in [9] (see also [10]).
Pairstrading refers to the case where two assets \(Q\) and \(P\) are in the same industry or have similar characteristics. In this case, one expects the returns of these two assets to track each other (see e.g. [7, Sect. 1]). Let \(S^{Q}\) and \(S^{P}\) be the price processes of the assets. Then the difference between the weighted returns of \(P\) and \(Q\), \(dJ_{t}:= dS^{P}_{t}/S^{P}_{t}  \beta dS^{Q}_{t}/S^{Q}_{t}\), for a certain constant \(\beta >0\), can be approximated in many cases by a stationary meanreverting process. Hence a trader who wants to liquidate a large amount of asset \(P\), for example, may consider \(J_{t}\) as a trading signal. The typical meanreversion time of such signals may vary between half a day to a month (see [7, Fig. 8]). More examples of trading signals which are used in optimal execution can be found in a presentation by Robert Almgren [5]. In Sect. 4.3, we show that the LOB imbalance signal affects the trading speed of highfrequency proprietary traders in the following 10 minutes time interval.
4.1 The database: NASDAQ OMX trades
The database which is used in this section is made of transactions on the NASDAQ OMX exchange. This exchange used to publish the identity of the buyer and seller of each transaction until 2014. To obtain order book data, we use recordings made by Capital Fund Management (CFM) on the same exchange, which were matched with NASDAQ OMX trades thanks to the timestamp, quantity and price of each trade. On a typical month, the accuracy of such matchings is more than 97% (see Table 2).
The NASDAQ OMX trades were already used for academic studies (see [38] and [31] for details). We study 13 stocks traded on NASDAQ OMX Stockholm from January 2013 to September 2013. The purpose of this section is not to conduct an extensive econometric study on this database; such work deserves a paper of its own. Our goal here is to show qualitative evidence for the existence of the order book imbalance signal and to study how market participants’ decisions depend on its value. The 13 stocks which are used in this section have been selected for this research since highfrequency proprietary traders took part in at least 100’000 trades on each of them during the studied period. More details on the classification of the traders into different classes are given later in this section.
Statistics of the 13 studied stocks. Values and prices are in Swedish krona. The Garman and Klass (GK) volatility is estimated yearly. The table is sorted by the average daily traded value over 180 trading days
Company name (code)  Daily traded value (10^{6})  Average price  Average bid–ask spread  Volatility (GK)  Minimum tick 

Volvo AB (VOLVb.ST)  431.20  94.87  0.057  15.08%  0.05 
Nordea Bank AB (NDA.ST)  384.48  76.09  0.053  15.02%  0.05 
Telefonaktiebolaget LM  
Ericsson (ERICb.ST)  373.20  78.41  0.054  15.20%  0.05 
Hennes & Mauritz AB (HMb.ST)  361.66  232.89  0.112  11.37%  0.10 
Atlas Copco AB (ATCOa.ST)  329.94  175.19  0.110  16.13%  0.10 
Swedbank AB (SWEDa.ST)  313.18  151.97  0.108  15.29%  0.10 
Sandvik AB (SAND.ST)  296.09  90.88  0.067  17.01%  0.05 
SKF AB (SKFb.ST)  255.99  161.11  0.112  16.47%  0.10 
Skandinaviska Enskilda  
Banken AB (SEBa.ST)  221.23  66.85  0.053  15.56%  0.05 
Nokia OYJ (NOKI.ST)  209.77  28.84  0.019  36.89%  0.01 
Telia Co AB (TLSN.ST)  207.09  45.14  0.014  10.13%  0.01 
ABB Ltd (ABB.ST)  179.51  144.35  0.108  11.89%  0.10 
AstraZeneca PLC (AZN.ST)  168.06  318.57  0.127  12.09%  0.10 

global investment banks (GIB);

institutional brokers (IB);

highfrequency market makers (HFMM);

highfrequency proprietary traders (HFPT).
Statistics on labelled trades involving each kind of market participant. Trades count is the sum of trades involving at least one labelled participant. Pct. ident. represents the percentage of trades involving at least one participant out of the four types that we focus on. Pct. LOB matched is the percentage of trades for which we found a matching quote in our LOB database. The averages in the bottom line are calculated over all identified trades
Code  Global banks  HFMM  Instit. brokers  HFPT  Trades count  Pct. ident.  pct. LOB matched 

VOLVb.ST  56.9%  17.1%  10.6%  15.3%  927,467  76.7%  97.4% 
NDA.ST  60.7%  10.6%  9.9%  18.7%  694,509  76.8%  97.4% 
ERICb.ST  57.8%  17.6%  7.7%  16.9%  811,931  81.0%  97.2% 
HMb.ST  58.5%  16.0%  8.9%  16.6%  716,644  76.8%  97.8% 
ATCOa.ST  58.2%  13.7%  10.5%  17.6%  677,981  79.1%  98.0% 
SWEDa.ST  61.2%  12.2%  9.5%  17.2%  600,655  74.6%  97.7% 
SAND.ST  61.0%  15.2%  10.4%  13.4%  701,961  77.4%  96.9% 
SKFb.ST  60.9%  13.8%  10.4%  14.9%  587,088  77.1%  97.0% 
SEBa.ST  61.5%  12.1%  8.8%  17.7%  515,743  75.8%  97.8% 
NOKI.ST  54.5%  8.1%  8.9%  28.5%  710,173  79.6%  99.2% 
TLSN.ST  61.2%  10.0%  10.6%  18.2%  548,602  68.9%  97.8% 
ABB.ST  50.1%  15.6%  5.2%  29.2%  359,067  86.2%  98.1% 
AZN.ST  51.4%  12.8%  9.0%  26.8%  411,118  89.6%  98.8% 
Average  58.3%  13.6%  9.4%  18.7%  –  77.7%  – 
We expect institutional brokers to execute orders for clients without taking additional risks (i.e., act as ‘pure agency brokers’). Such brokers often have mediumsize clients and local asset managers. They do not spend a lot of resources such as technology or quantitative analysts to study the microstructure, and they do not react fast to microscopic events.
Global investment banks can take risks at least on a fraction of their order flow. Most of them already had proprietary trading desks and highfrequency trading activities in 2013 (i.e., during the recording of the data). They usually have large international clients and have the capability to react to changes in the state of the order book.
Highfrequency market makers are providing liquidity on both sides of the order book. They have a very good knowledge on market microstructure. As market makers, we expect them to focus on adverse selection and not to keep large inventories. On the other hand, highfrequency proprietary traders take their own risks in order to earn money, while taking profit of their knowledge of the order book dynamics.

HFMM trade far more with limit orders (73%), than with market orders;

IB use more market orders than limit orders;

on average, HFPT and GIB have balanced order flows.
Descriptive statistics of market participants on an ‘average stock’. All the trades are normalised as if all orders were buy orders. The imbalance is positive when its sign is in the direction of the trade
4.2 The imbalance signal
Results of linear regressions involving the imbalance. The first column is the result of a regression of the price move after 10 trades given the imbalance immediately before the first of these trades. This can also be shown in the slope of Fig. 5. The \(p\)value is very close to zero for all stocks, meaning they are highly significant. The \(R^{2}\) varies between 1% (Nokia) to 16% (Volvo AB and Nordea Bank AB). Other columns are the results of the regression of future imbalance (respectively after 3, 5, 7, 10 and 100 trades) with respect to the imbalance immediately before the first of these trades, given that the imbalance is between −0.5 and 0.5. This regression corresponds to the slopes at the center of Fig. 7. All \(p\)values are significant at more than 99.99%
d price  \(R^{2}\)  Imb. 3 t  Imb. 5 t  Imb. 7 t  Imb. 10 t  Imb. 100 t  

VOLVb.ST  0.58  0.16  0.91  0.72  0.49  0.26  0.03 
NDA.ST  0.58  0.16  0.90  0.71  0.51  0.30  0.04 
ERICb.ST  0.62  0.15  0.93  0.74  0.53  0.30  0.03 
HMb.ST  0.59  0.08  0.84  0.62  0.41  0.21  0.02 
ATCOa.ST  0.60  0.13  0.85  0.58  0.34  0.13  0.02 
SWEDa.ST  0.62  0.14  0.87  0.67  0.45  0.23  0.02 
SAND.ST  0.56  0.15  0.81  0.57  0.37  0.20  0.03 
SKFb.ST  0.59  0.13  0.76  0.49  0.28  0.13  0.01 
SEBa.ST  0.61  0.15  0.91  0.73  0.51  0.28  0.03 
NOKI.ST  0.41  0.01  0.18  0.08  0.05  0.03  0.00 
TLSN.ST  0.54  0.04  0.43  0.22  0.13  0.08  0.02 
ABB.ST  0.59  0.11  0.86  0.61  0.33  0.11  0.03 
AZN.ST  0.64  0.04  0.47  0.20  0.09  0.05  0.02 
Meanreversion of the imbalance
Some numerical values of the model parameters
At a time scale of 35 seconds or 7 trades, \(\gamma \) should be taken close to 0.92 and \(\sigma \) close to 0.22. We also provide an estimator for the instantaneous market impact \(\kappa \) using the empirical average of the midprice move^{1} after a trade times the sign of the trade. Table 9 in the Appendix shows that the average value of \(\kappa \) divided by the average bidask spread is close to 0.1.
4.3 Use of signals by market participants
As previously mentioned, we expect HF proprietary traders, HF market makers and global investment banks to pay more attention to order book dynamics than institutional brokers. However, as market makers, HFMM are expected to earn money by buying and selling when the mid price does not change much (relying on the bid–ask bounce). On the other hand, HFPT are typically alternating between intensive buy and sell phases which are based on price moves.
Our expectations are met in Table 3, where the average imbalance just before a trade is shown for each type of market participant. All the trades in this table are normalised as if all orders were buy orders. The imbalance is positive when its sign is in the direction of the trade, and negative if it is in an opposite direction.

When the transaction is obtained via a market order, the market participant had the opportunity to observe the imbalance before consuming liquidity.

When the transaction is obtained via a limit order, fast participants have the opportunity to cancel their orders to prevent an execution and potential adverse selection.
Table 3 underlines that HF participants and GIB make ‘better choices’ on trading according to the market imbalance. Institutional brokers seems to be the less ‘imbalance aware’ when they decide to trade. This could be explained either by the fact that they invest less in microstructure research, quantitative modelling and automated trading, or because they have less freedom to be opportunistic. Since they act as pure agency brokers, they do not have the choice to retain clients’ orders, and this could prevent them from waiting for the best imbalance to trade.
Strategic behaviour
Once we suspect that some participants take into account the imbalance in their trading decisions, we can look for a relation between the trading rate and the corresponding imbalance for each type of participant. This is motivated by the optimal trading frameworks of the previous sections, where we used the trading rate as a control.
In order to learn more about the relation between the imbalance signal and the trading speed, we compute the imbalanceconditioned trading rates\(R_{+}\) and \(R_{}\) for each type of market participant, during all consecutive intervals of 10 minutes from January 2013 to September 2013 (within the trading hours, i.e., from 9:00 to 17:30). Note that in the following analysis, the signal, time and trading quantities are discrete.
Definition 1

\(\varepsilon (t)\) is the sign of the trade at time \(t\);

\(\bar{\delta }_{\varepsilon (t)\text{sign}(\operatorname{Imb}(t))}( \pm 1)\) is 1 if at time \(t\) the imbalance sign times the sign of the trade is equal to \(\pm 1\), and 0 otherwise;

\(A_{t}\) is the traded amount of the trade at time \(t\);

\(\bar{\delta }_{\mathcal{P}}(t)\) is 1 if the trade at time \(t\) involved a participant of type \(\mathcal{P}\), and 0 otherwise;

\(\bar{\delta }_{{\operatorname{Imb}}(t)}(\iota )\) is 1 if the absolute value of the imbalance at time \(t\) equals \(\iota \), and 0 otherwise;

\(N(\mathcal{T},\mathcal{P},\iota )\) is the number of trades involving participant \(\mathcal{P}\) in \(\mathcal{T}\) when the imbalance equals \(\iota \).

\(R_{+}(\mathcal{T},\mathcal{P}\;\;\iota )\) is an estimate of the amount traded in the direction of the imbalance when the absolute value of the imbalance is \(\iota \), by participants of type \(\mathcal{P}\) during the time interval \(\mathcal{T}\);

\(R_{}(\mathcal{T},\mathcal{P}\;\;\iota )\) is an estimate of the amount traded in the opposite direction of the imbalance when the absolute value of the imbalance is \(\iota \), by participants of type \(\mathcal{P}\) during the time interval \(\mathcal{T}\).

For highfrequency market makers, the higher the imbalance in the order book, the less they trade. This effect does not seem to be related to the direction of their trades. It corresponds to an expected behaviour from market makers.

For highfrequency proprietary traders, the higher the imbalance, the more they trade in a similar direction, and the less they trade in the opposite direction.

Institutional brokers do not seem to be influenced by the imbalance. Additional data analysis shows that they trade more with limit orders when the imbalance is intense; this may drive the price to move in the opposite direction.

The behaviour of global banks seems to be influenced by the imbalance for part of the stocks in our sample.
Towards a theory of strategic use of signals
The analysis in this section suggests that some market participants are using liquiditydriven signals in their trading strategies. The liquidity imbalance, computed from the best bid and ask prices of the order book for medium and largetick stocks, appears to be a good candidate. Moreover, its dynamics exhibit meanreverting properties.
The theory developed in Sects. 2 and 3 can be regarded as a tentative framework to model the behaviour of the following participants. Global investment banks who execute large orders seem to be a typical example for participants who adopt the type of strategies that we model. Highfrequency proprietary traders who are combining slow signals (which may be considered as execution of large orders) along with fast signals could also use our framework. We could moreover hope that thanks to the availability of such frameworks, institutional brokers could optimise their trading and increase the profits for more final investors.
5 Proofs
5.1 Proofs of Theorems 2.3, 2.4 and Corollary 2.7
The proofs of Theorems 2.3 and 2.4 use ideas from the proofs of [24, Proposition 2.9 and Theorem 2.11].
Proof of Theorem 2.3
Proof of Theorem 2.4
Proof of Corollary 2.7
5.2 Proofs of Propositions 3.1 and 3.2
Proof of Proposition 3.1
Proof of Proposition 3.2
(b) Note that an Ornstein–Uhlenbeck process satisfies (3.1). Hence the proof follows immediately from (a). □
Footnotes
 1.
The mid price is the middle of the best bid and best ask prices.
Notes
Acknowledgements
We are very grateful to anonymous referees and to the editors for their careful reading of the manuscript, and for a number of useful comments and suggestions that significantly improved this paper. We also thank Mikko Pakkanen whose useful comments greatly improved the manuscript.
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