Incorporating signals into optimal trading

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 mean-variance 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).

As mentioned earlier, one of the major influencers on transaction costs is the market impact. Empirical studies have shown that the influence of the market impact is transient, that is, it decays within a short time period after each trade (see [8] and references therein). In this paper, we focus on two frameworks which take into account different types of market impact: Note that [24] is not the only framework with market impact decay. This kind of dynamics was originally introduced in [33] and reused in [2] as well as in some other papers. We decided to focus on these two frameworks since they are extensively used in the financial literature. The model and analysis which are developed in this paper could be applied also to other optimal trading frameworks.

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 non-martingale 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).

An interesting phenomenon which arises from our results is that the optimal strategies may be not monotonic once we take into account trading signals (see Fig. 1). This implies that price manipulations triggered by trading strategies are possible. Another challenge is to establish conditions on the market impact kernel function and on the signal that prevent price manipulations (see Remark 2.10).

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 liquidity-driven 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, high-frequency market makers and high-frequency 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 mean-reverting 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.

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 \(\rho \rightarrow \infty \).

We continue to assume that \(I\) is a càdlàg Markov process as in the beginning of Sect. 2, but we add the assumption that for some constant \(C(T)>0\). For the sake of simplicity, we assume that \(M=\sigma ^{P}W\) so that where \((W_{t})_{t\geq 0}\) is a Brownian motion and \(\sigma ^{P}>0\) is a constant.

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.

Let \(\mathcal{V}\) denote the class of progressively measurable control processes \(r=(r_{t})_{t\geq 0}\) for which \(\int _{0}^{T}|r_{t}|dt< \infty \)\(P\)-a.s. For any \(x \geq 0\), we define Here \(X^{r}_{t}\) is the amount of inventory held by the trader at time \(t\). We often suppress the dependence of \(X\) on \(r\) to ease the notation. The price process, which is affected by the linear instantaneous market impact, is given by where \(\kappa >0\). Note that \(S_{t}\) here corresponds to (2.1) when \(G(dt)=\kappa \delta _{0}(dt)\). The investor’s cash \(\mathcal{C}_{t}\) satisfies with \(\mathcal{C}_{0} =c\).

For the sake of consistency with earlier work of Cartea and Jaimungal in [14, 15, 16], we define the liquidation problem as a maximisation of the difference between the cash and the risk aversion. As mentioned earlier, the fuel constraint on the admissible strategies is replaced by a penalty function, which is given by \(X_{T}(P_{T}-\varrho X_{T})\), where \(\varrho >0\) is a constant. The cost functional is given by where \(\phi \geq 0\) is a constant and \(E_{t,\iota ,c,x,p}\) represents the expectation conditioned on \(I_{t}=\iota , \mathcal{C}_{t} =c, X_{t}=x, P_{t}=p\). The value function is Note that this control problem could be easily transformed to a minimisation of the trading costs and risk aversion as in Sect. 2.

Let \(\mathcal{L}^{I}\) be the generator of the process \(I\). Then the corresponding HJB equation is with the terminal condition Let \(E_{t,\iota }\) represent the expectation conditional on \(I_{t}=\iota \). In the following result, we derive a solution to (3.4). The proof of Proposition 3.1 follows the same lines as the proof of [16, Proposition 1].

In the following result, we prove that the solution to (3.4) is indeed an optimal control to (3.3).

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.

In Fig. 2, we simulate the optimal inventory \(X^{*}\) which corresponds to the optimal trading speed \(r^{*}\) from Proposition 3.2 (b). In the black solid line, we present the optimal inventory in the case where there is no signal. In this case, the optimal strategy is deterministic. The red region in Fig. 2 is a ‘heat map’ of 1000 realisations of the optimal inventory \(X^{*}\). The parameters of the signal (2.7) are \(\gamma =0.1\), \(\sigma =0.1\) and \(I_{0}=0\). We also set \(T=10\), \(\kappa =0.5\), \(\phi =0.1\), \(X_{0}=10\) and \(\varrho = 10\) in the cost functional (3.3). We observe that the random strategies are a perturbation of the classical deterministic optimal strategy. In Fig. 3, we present the value function (3.5) at \(t=0\) under the same assumptions as in Fig. 2, that is, assuming that \(I\) is an OU process and that the model parameters are similar. More precisely, we plot \(V(0,\iota ,c,x,p)-(c-xp)\); hence we omit constants which do not contribute to the behaviour of the model. We observe that the revenue which corresponds to the optimal sell strategy \(r^{*}\) is affected by the direction and value \(\iota \) of the signal. The revenue of a sell strategy when the signal is positive, which indicates a potential price increase, is higher than with negative signal scenarios.

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 liquidity-driven 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 liquidity-driven.

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 mean-reverting 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 mean-reverting properties of the order flow were studied by Bechler and Ludkovski in [9] (see also [10]).

Pairs-trading 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 mean-reverting 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 mean-reversion 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 high-frequency proprietary traders in the following 10 minutes time interval.