## Publications

26 results found

Li M, Lehalle C-A, 2021, Do word embeddings really understand loughran-McDonald's polarities?, Publisher: arXiv

In this paper we perform a rigorous mathematical analysis of the word2vecmodel, especially when it is equipped with the Skip-gram learning scheme. Ourgoal is to explain how embeddings, that are now widely used in NLP (NaturalLanguage Processing), are influenced by the distribution of terms in thedocuments of the considered corpus. We use a mathematical formulation to shedlight on how the decision to use such a model makes implicit assumptions on thestructure of the language. We show how Markovian assumptions, that we discuss,lead to a very clear theoretical understanding of the formation of embeddings,and in particular the way it captures what we call frequentist synonyms. Theseassumptions allow to produce generative models and to conduct an explicitanalysis of the loss function commonly used by these NLP techniques. Moreover,we produce synthetic corpora with different levels of structure and showempirically how the word2vec algorithm succeed, or not, to learn them. It leadsus to empirically assess the capability of such models to capture structures ona corpus of around 42 millions of financial News covering 12 years. That for,we rely on the Loughran-McDonald Sentiment Word Lists largely used on financialtexts and we show that embeddings are exposed to mixing terms with oppositepolarity, because of the way they can treat antonyms as frequentist synonyms.Beside we study the non-stationarity of such a financial corpus, that hassurprisingly not be documented in the literature. We do it via time series ofcosine similarity between groups of polarized words or company names, and showthat embedding are indeed capturing a mix of English semantics and joineddistribution of words that is difficult to disentangle.

Lehalle C-A, Neuman E, Shlomov S, 2021, Phase transitions in Kyle's model with market maker profit incentives, Publisher: arXiv

We consider a stochastic game between three types of players: an insidetrader, noise traders and a market maker. In a similar fashion to Kyle's model,we assume that the insider first chooses the size of her market-order and thenthe market maker determines the price by observing the total order-flowresulting from the insider and the noise traders transactions. In addition tothe classical framework, a revenue term is added to the market maker'sperformance function, which is proportional to the order flow and to the sizeof the bid-ask spread. We derive the maximizer for the insider's revenuefunction and prove sufficient conditions for an equilibrium in the game. Then,we use neural networks methods to verify that this equilibrium holds. We showthat the equilibrium state in this model experience interesting phasetransitions, as the weight of the revenue term in the market maker'sperformance function changes. Specifically, the asset price in equilibriumexperience three different phases: a linear pricing rule without a spread, apricing rule that includes a linear mid-price and a bid-ask spread, and ametastable state with a zero mid-price and a large spread.

Leal L, Laurière M, Lehalle C-A, 2021, Learning a functional control for high-frequency finance, Publisher: arXiv

We use a deep neural network to generate controllers for optimal trading onhigh frequency data. For the first time, a neural network learns the mappingbetween the preferences of the trader, i.e. risk aversion parameters, and theoptimal controls. An important challenge in learning this mapping is that inintraday trading, trader's actions influence price dynamics in closed loop viathe market impact. The exploration--exploitation tradeoff generated by theefficient execution is addressed by tuning the trader's preferences to ensurelong enough trajectories are produced during the learning phase. The issue ofscarcity of financial data is solved by transfer learning: the neural networkis first trained on trajectories generated thanks to a Monte-Carlo scheme,leading to a good initialization before training on historical trajectories.Moreover, to answer to genuine requests of financial regulators on theexplainability of machine learning generated controls, we project the obtained"blackbox controls" on the space usually spanned by the closed-form solution ofthe stylized optimal trading problem, leading to a transparent structure. Formore realistic loss functions that have no closed-form solution, we show thatthe average distance between the generated controls and their explainableversion remains small. This opens the door to the acceptance of ML-generatedcontrols by financial regulators.

Mounjid O, Lehalle C-A, 2020, Improving reinforcement learning algorithms: towards optimal learning rate policies, Publisher: arXiv

This paper investigates to what extent one can improve reinforcement learningalgorithms. Our study is split in three parts. First, our analysis shows thatthe classical asymptotic convergence rate $O(1/\sqrt{N})$ is pessimistic andcan be replaced by $O((\log(N)/N)^{\beta})$ with $\frac{1}{2}\leq \beta \leq 1$and $N$ the number of iterations. Second, we propose a dynamic optimal policyfor the choice of the learning rate $(\gamma_k)_{k\geq 0}$ used in stochasticapproximation (SA). We decompose our policy into two interacting levels: theinner and the outer level. In the inner level, we present the\nameref{Alg:v_4_s} algorithm (for "PAst Sign Search") which, based on apredefined sequence $(\gamma^o_k)_{k\geq 0}$, constructs a new sequence$(\gamma^i_k)_{k\geq 0}$ whose error decreases faster. In the outer level, wepropose an optimal methodology for the selection of the predefined sequence$(\gamma^o_k)_{k\geq 0}$. Third, we show empirically that our selectionmethodology of the learning rate outperforms significantly standard algorithmsused in reinforcement learning (RL) in the three following applications: theestimation of a drift, the optimal placement of limit orders and the optimalexecution of large number of shares.

Bucci F, Mastromatteo I, Eisler Z,
et al., 2019, Co-impact: crowding effects in institutional trading activity, *QUANTITATIVE FINANCE*, Vol: 20, Pages: 193-205, ISSN: 1469-7688

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Lehalle C-A, Neuman E, 2019, Incorporating signals into optimal trading, *Finance and Stochastics*, ISSN: 1432-1122

Optimal trading is a recent field of research which was initiated by Almgren, Chriss, Bertsimas and Lo in the late 90's. Its main application is slicing large trading orders, in the interest of minimizing trading costs and potential perturbations of price dynamics due to liquidity shocks. The initial optimization frameworks were based on mean-variance minimization for the trading costs. In the past 15 years, finer modelling of price dynamics, more realistic control variables and different cost functionals were developed. The inclusion of signals (i.e. short term predictors of price dynamics) in optimal trading is a recent development and it is also the subject of this work.We incorporate a Markovian signal in the optimal trading framework which was initially proposed by Gatheral, Schied, and Slynko [21] 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 mean-reverting 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 were 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 [10].In order to support our models, we analyse nine months of tick by tick data on 13 European stocks from the NASDAQ OMX exchange. We show that orderbook imbalance is a predictor of the future price move and it has some mean-reverting properties. From this data we show that market participants, especially high frequency traders, use this signal in their trading strategies.

Huang W, Lehalle C-A, Rosenbaum M, 2015, How to predict the consequences of a tick value change? Evidence from the Tokyo Stock Exchange pilot program

The tick value is a crucial component of market design and is often considered the most suitable tool to mitigate the effects of high frequency trading. The goal of this paper is to demonstrate that the approach introduced in Dayri and Rosenbaum (2015) allows for an ex ante assessment of the consequences of a tick value change on the microstructure of an asset. To that purpose, we analyze the pilot program on tick value modifications started in 2014 by the Tokyo Stock Exchange in light of this methodology. We focus on forecasting the future cost of market and limit orders after a tick value change and show that our predictions are very accurate. Furthermore, for each asset involved in the pilot program, we are able to define (ex ante) an optimal tick value. This enables us to classify the stocks according to the relevance of their tick value, before and after its modification.

Bacry E, Iuga A, Lasnier M, et al., 2014, Market impacts and the life cycle of investors orders

In this paper, we use a database of around 400,000 metaorders issued byinvestors and electronically traded on European markets in 2010 in order tostudy market impact at different scales. At the intraday scale we confirm a square root temporary impact in the dailyparticipation, and we shed light on a duration factor in $1/T^{\gamma}$ with$\gamma \simeq 0.25$. Including this factor in the fits reinforces the squareroot shape of impact. We observe a power-law for the transient impact with anexponent between $0.5$ (for long metaorders) and $0.8$ (for shorter ones).Moreover we show that the market does not anticipate the size of themeta-orders. The intraday decay seems to exhibit two regimes (though hard toidentify precisely): a "slow" regime right after the execution of themeta-order followed by a faster one. At the daily time scale, we show pricemoves after a metaorder can be split between realizations of expected returnsthat have triggered the investing decision and an idiosynchratic impact thatslowly decays to zero. Moreover we propose a class of toy models based on Hawkes processes (theHawkes Impact Models, HIM) to illustrate our reasoning. We show how the Impulsive-HIM model, despite its simplicity, embeds appealingfeatures like transience and decay of impact. The latter is parametrized by aparameter $C$ having a macroscopic interpretation: the ratio of contrarianreaction (i.e. impact decay) and of the "herding" reaction (i.e. impactamplification).

Azencott R, Beri A, Gadhyan Y,
et al., 2014, Real-time market microstructure analysis: online transaction cost analysis, *QUANTITATIVE FINANCE*, Vol: 14, Pages: 1167-1185, ISSN: 1469-7688

Guéant O, Lehalle C-A, Fernandez-Tapia J, 2013, Dealing with the inventory risk: a solution to the market making problem, *Mathematics and Financial Economics*, Vol: 7, Pages: 477-507, ISSN: 1862-9679

Guéant O, Lehalle C-A, Fernandez-Tapia J, 2012, Optimal Portfolio Liquidation with Limit Orders, *SIAM Journal on Financial Mathematics*, Vol: 3, Pages: 740-764

Bińkowski M, Lehalle C-A, Endogeneous Dynamics of Intraday Liquidity

In this paper we investigate the endogenous information contained in fourliquidity variables at a five minutes time scale on equity markets around theworld: the traded volume, the bid-ask spread, the volatility and the volume atfirst limits of the orderbook. In the spirit of Granger causality, we measurethe level of information by the level of accuracy of linear autoregressivemodels. This empirical study is carried out on a dataset of more than 300stocks from four different markets (US, UK, Japan and Hong Kong) from a periodof over five years. We discuss the obtained performances of autoregressive (AR)models on stationarized versions of the variables, focusing on explaining theobserved differences between stocks. Since empirical studies are often conducted at this time scale, we believe itis of paramount importance to document endogenous dynamics in a simpleframework with no addition of supplemental information. Our study can hence beused as a benchmark to identify exogenous effects. On the other hand, mostoptimal trading frameworks (like the celebrated Almgren and Chriss one), focuson computing an optimal trading speed at a frequency close to the one weconsider. Such frameworks very often take i.i.d. assumptions on liquidityvariables; this paper document the auto-correlations emerging from real data,opening the door to new developments in optimal trading.

Lehalle C-A, Mouzouni C, A Mean Field Game of Portfolio Trading and Its Consequences On Perceived Correlations

This paper goes beyond the optimal trading Mean Field Game model introducedby Pierre Cardaliaguet and Charles-Albert Lehalle in [Cardaliaguet, P. andLehalle, C.-A., Mean field game of controls and an application to tradecrowding, Mathematics and Financial Economics (2018)]. It starts by extendingit to portfolios of correlated instruments. This leads to several originalcontributions: first that hedging strategies naturally stem from optimalliquidation schemes on portfolios. Second we show the influence of tradingflows on naive estimates of intraday volatility and correlations. Focussing onthis important relation, we exhibit a closed form formula expressing standardestimates of correlations as a function of the underlying correlations and theinitial imbalance of large orders, via the optimal flows of our mean field gamebetween traders. To support our theoretical findings, we use a real dataset of176 US stocks from January to December 2014 sampled every 5 minutes to analyzethe influence of the daily flows on the observed correlations. Finally, wepropose a toy model based approach to calibrate our MFG model on data.

Guo X, Lehalle C-A, Xu R, Transaction Cost Analytics for Corporate Bonds

The electronic platform has been increasingly popular for executing largecorporate bond orders by asset managers, who in turn have to assess the qualityof their executions via Transaction Cost Analysis (TCA). One of the challengesin TCA is to build a realistic benchmark for the expected transaction cost andto characterize the price impact of each individual trade with given bondcharacteristics and market conditions. Taking the viewpoint of retail investors, this paper presents an analyticalmethodology for TCA of corporate bond trading. Our analysis is based on theTRACE Enhanced dataset; and starts with estimating the initiator of a bondtransaction, followed by estimating the bid-ask spread and the mid-pricedynamics. With these estimations, the first part of our study is to identifykey features for corporate bonds and to compute the expected average tradingcost. This part is on the time scale of weekly transactions, and is by applyingand comparing several regularized regression models. The second part of ourstudy is using the estimated mid-price dynamics to investigate the amplitude ofits price impact and the decay pattern of individual bond transaction. Thispart is on the time scale of each transaction of liquid corporate bonds, and isby applying a transient impact model to estimate the price impact kernel usinga non-parametric method. Our benchmark model allows for identifying abnormal transactions and forenhancing counter-party selections. A key discovery of our study is the priceimpact asymmetry between customer-buy orders and consumer-sell orders.

March HD, Lehalle C-A, Optimal trading using signals

In this paper we propose a mathematical framework to address the uncertaintyemergingwhen the designer of a trading algorithm uses a threshold on a signalas a control. We rely ona theorem by Benveniste and Priouret to deduce ourInventory Asymptotic Behaviour (IAB)Theorem giving the full distribution of theinventory at any point in time for a well formulatedtime continuous version ofthe trading algorithm.Since this is the first time a paper proposes to addressthe uncertainty linked to the use of athreshold on a signal for trading, wegive some structural elements about the kind of signals thatare using inexecution. Then we show how to control this uncertainty for a given costfunction.There is no closed form solution to this control, hence we proposeseveral approximation schemesand compare their performances.Moreover, weexplain how to apply the IAB Theorem to any trading algorithm drivenby atrading speed. It is not needed to control the uncertainty due to thethresholding of asignal to exploit the IAB Theorem; it can be applied ex-postto any traditional trading algorithm.

Lehalle C-A, Mounjid O, Rosenbaum M, Optimal liquidity-based trading tactics

We consider an agent who needs to buy (or sell) a relatively small amount ofasset over some fixed short time interval. We work at the highest frequencymeaning that we wish to find the optimal tactic to execute our quantity usinglimit orders, market orders and cancellations. To solve the agent's controlproblem, we build an order book model and optimize an expected utility functionbased on our price impact. We derive the equations satisfied by the optimalstrategy and solve them numerically. Moreover, we show that our optimal tacticenables us to outperform significantly naive execution strategies.

Geeraert S, Lehalle C-A, Pearlmutter B, et al., Mini-symposium on automatic differentiation and its applications in the financial industry

Automatic differentiation is involved for long in applied mathematics as analternative to finite difference to improve the accuracy of numericalcomputation of derivatives. Each time a numerical minimization is involved,automatic differentiation can be used. In between formal derivation andstandard numerical schemes, this approach is based on software solutionsapplying mechanically the chain rule to obtain an exact value for the desiredderivative. It has a cost in memory and cpu consumption. For participants offinancial markets (banks, insurances, financial intermediaries, etc), computingderivatives is needed to obtain the sensitivity of its exposure to well-definedpotential market moves. It is a way to understand variations of their balancesheets in specific cases. Since the 2008 crisis, regulation demand to computethis kind of exposure to many different case, to be sure market participantsare aware and ready to face a wide spectrum of configurations. This paper showshow automatic differentiation provides a partial answer to this recentexplosion of computation to perform. One part of the answer is astraightforward application of Adjoint Algorithmic Differentiation (AAD), butit is not enough. Since financial sensitivities involves specific functions andmix differentiation with Monte-Carlo simulations, dedicated tools andassociated theoretical results are needed. We give here short introductions totypical cases arising when one use AAD on financial markets.

Cardaliaguet P, Lehalle C-A, Mean Field Game of Controls and An Application To Trade Crowding

In this paper we formulate the now classical problem of optimal liquidation(or optimal trading) inside a Mean Field Game (MFG). This is a noticeablechange since usually mathematical frameworks focus on one large trader in frontof a "background noise" (or "mean field"). In standard frameworks, theinteractions between the large trader and the price are a temporary and apermanent market impact terms, the latter influencing the public price. In thispaper the trader faces the uncertainty of fair price changes too but not only.He has to deal with price changes generated by other similar marketparticipants, impacting the prices permanently too, and acting strategically.Our MFG formulation of this problem belongs to the class of "extended MFG", wehence provide generic results to address these "MFG of controls", beforesolving the one generated by the cost function of optimal trading. We provide aclosed form formula of its solution, and address the case of "heterogenouspreferences" (when each participant has a different risk aversion). Last butnot least we give conditions under which participants do not need toinstantaneously know the state of the whole system, but can "learn" it dayafter day, observing others' behaviors.

Lehalle C-A, Mounjid O, Limit Order Strategic Placement with Adverse Selection Risk and the Role of Latency

This paper is split in three parts: first we use labelled trade data toexhibit how market participants accept or not transactions via limit orders asa function of liquidity imbalance; then we develop a theoretical stochasticcontrol framework to provide details on how one can exploit his knowledge onliquidity imbalance to control a limit order. We emphasis the exposure toadverse selection, of paramount importance for limit orders. For a participantbuying using a limit order: if the price has chances to go down the probabilityto be filled is high but it is better to wait a little more before the trade toobtain a better price. In a third part we show how the added value ofexploiting a knowledge on liquidity imbalance is eroded by latency: being ableto predict future liquidity consuming flows is of less use if you have notenough time to cancel and reinsert your limit orders. There is thus a rationalfor market makers to be as fast as possible as a protection to adverseselection. Thanks to our optimal framework we can measure the added value oflatency to limit orders placement. To authors' knowledge this paper is the first to make the connection betweenempirical evidences, a stochastic framework for limit orders including adverseselection, and the cost of latency. Our work is a first stone to shed light onthe roles of latency and adverse selection for limit order placement, within anaccurate stochastic control framework.

Lachapelle A, Lasry J-M, Lehalle C-A, et al., Efficiency of the Price Formation Process in Presence of High Frequency Participants: a Mean Field Game analysis

This paper deals with a stochastic order-driven market model with waitingcosts, for order books with heterogenous traders. Offer and demand of liquiditydrives price formation and traders anticipate future evolutions of the orderbook. The natural framework we use is mean field game theory, a class ofstochastic differential games with a continuum of anonymous players. Severalsources of heterogeneity are considered including the mean size of orders. Thuswe are able to consider the coexistence of Institutional Investors and HighFrequency Traders (HFT). We provide both analytical solutions and numericalexperiments. Implications on classical quantities are explored: order booksize, prices, and effective bid/ask spread. According to the model, in marketswith Institutional Investors only we show the existence of inefficientliquidity imbalances in equilibrium, with two symmetrical situationscorresponding to what we call liquidity calls for liquidity. During thesesituations the transaction price significantly moves away from the fair price.However this macro phenomenon disappears in markets with both InstitutionalInvestors and HFT, although a more precise study shows that the benefits of thenew situation go to HFT only, leaving Institutional Investors even with highertrading costs.

Laruelle S, Lehalle C-A, Pagès G, Optimal posting price of limit orders: learning by trading

Considering that a trader or a trading algorithm interacting with marketsduring continuous auctions can be modeled by an iterating procedure adjustingthe price at which he posts orders at a given rhythm, this paper proposes aprocedure minimizing his costs. We prove the a.s. convergence of the algorithmunder assumptions on the cost function and give some practical criteria onmodel parameters to ensure that the conditions to use the algorithm arefulfilled (using notably the co-monotony principle). We illustrate our resultswith numerical experiments on both simulated data and using a financial marketdataset.

Laruelle S, Lehalle C-A, Pagès G, Optimal split of orders across liquidity pools: a stochastic algorithm approach

Evolutions of the trading landscape lead to the capability to exchange thesame financial instrument on different venues. Because of liquidity issues, thetrading firms split large orders across several trading destinations tooptimize their execution. To solve this problem we devised two stochasticrecursive learning procedures which adjust the proportions of the order to besent to the different venues, one based on an optimization principle, the otheron some reinforcement ideas. Both procedures are investigated from atheoretical point of view: we prove a.s. convergence of the optimizationalgorithm under some light ergodic (or "averaging") assumption on the inputdata process. No Markov property is needed. When the inputs are i.i.d. we showthat the convergence rate is ruled by a Central Limit Theorem. Finally, themutual performances of both algorithms are compared on simulated and real datawith respect to an "oracle" strategy devised by an "insider" who knows a priorithe executed quantities by every venues.

Huang W, Lehalle C-A, Rosenbaum M, Simulating and analyzing order book data: The queue-reactive model

Through the analysis of a dataset of ultra high frequency order book updates,we introduce a model which accommodates the empirical properties of the fullorder book together with the stylized facts of lower frequency financial data.To do so, we split the time interval of interest into periods in which a wellchosen reference price, typically the mid price, remains constant. Within theseperiods, we view the limit order book as a Markov queuing system. Indeed, weassume that the intensities of the order flows only depend on the current stateof the order book. We establish the limiting behavior of this model andestimate its parameters from market data. Then, in order to design a relevantmodel for the whole period of interest, we use a stochastic mechanism thatallows for switches from one period of constant reference price to another.Beyond enabling to reproduce accurately the behavior of market data, we showthat our framework can be very useful for practitioners, notably as a marketsimulator or as a tool for the transaction cost analysis of complex tradingalgorithms.

Labadie M, Lehalle C-A, Optimal starting times, stopping times and risk measures for algorithmic trading: Target Close and Implementation Shortfall, *Journal of Investment Strategies*, Vol: 3

We derive explicit recursive formulas for Target Close (TC) andImplementation Shortfall (IS) in the Almgren-Chriss framework. We explain howto compute the optimal starting and stopping times for IS and TC, respectively,given a minimum trading size. We also show how to add a minimum participationrate constraint (Percentage of Volume, PVol) for both TC and IS. We also studyan alternative set of risk measures for the optimisation of algorithmic tradingcurves. We assume a self-similar process (e.g. Levy process, fractionalBrownian motion or fractal process) and define a new risk measure, thep-variation, which reduces to the variance if the process is a brownian motion.We deduce the explicit formula for the TC and IS algorithms under aself-similar process. We show that there is an equivalence between selfsimilarmodels and a family of risk measures called p-variations: assuming aself-similar process and calibrating empirically the parameter p for thep-variation yields the same result as assuming a Brownian motion and using thep-variation as risk measure instead of the variance. We also show that p can beseen as a measure of the aggressiveness: p increases if and only if the TCalgorithm starts later and executes faster. Finally, we show how the parameterp of the p-variation can be implied from the optimal starting time of TC, andthat under this framework p can be viewed as a measure of the joint impact ofmarket impact (i.e. liquidity) and volatility.

Lehalle C-A, Market Microstructure Knowledge Needed for Controlling an Intra-Day Trading Process

A great deal of academic and theoretical work has been dedicated to optimalliquidation of large orders these last twenty years. The optimal split of anorder through time (`optimal trade scheduling') and space (`smart orderrouting') is of high interest \rred{to} practitioners because of the increasingcomplexity of the market micro structure because of the evolution recently ofregulations and liquidity worldwide. This paper translates into quantitativeterms these regulatory issues and, more broadly, current market design. Itrelates the recent advances in optimal trading, order-book simulation andoptimal liquidity to the reality of trading in an emerging global network ofliquidity.

Guéant O, Lehalle C-A, General Intensity Shapes in Optimal Liquidation

The classical literature on optimal liquidation, rooted in Almgren-Chrissmodels, tackles the optimal liquidation problem using a trade-off betweenmarket impact and price risk. Therefore, it only answers the general questionof the optimal liquidation rhythm. The very question of the actual way toproceed with liquidation is then rarely dealt with. Our model, thatincorporates both price risk and non-execution risk, is an attempt to tacklethis question using limit orders. The very general framework we propose tomodel liquidation generalizes the existing literature on optimal posting oflimit orders. We consider a risk-adverse agent whereas the model of Bayraktarand Ludkovski only tackles the case of a risk-neutral one. We consider verygeneral functional forms for the execution process intensity, whereas Gu\'eantet al. is restricted to exponential intensity. Eventually, we link theexecution cost function of Almgren-Chriss models to the intensity function inour model, providing then a way to see Almgren-Chriss models as a limit ofours.

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