LEHALLE Charles Albert

Using a large database of US institutional investors’ trades in the equity market, this paper explores the effect of simultaneous executions on trading cost. We design a Bayesian network modelling the inter-dependencies between investors’ transaction costs, stock characteristics (bid-ask spread, turnover and volatility), meta-order attributes (side and size of the trade) and market pressure during execution, measured by the net order flow imbalance of investors meta-orders. Unlike standard machine learning algorithms, Bayesian networks are able to account for explicit inter-dependencies between variables. They also prove to be robust to missing values, as they are able to restore their most probable value given the state of the world. Order flow imbalance being only partially observable (on a subset of trades or with a delay), we show how to design a Bayesian network to infer its distribution and how to use this information to estimate transaction costs. Our model provides better predictions than standard (OLS) models. The forecasting error is smaller and decreases with the investors' order size, as large orders are more informative on the aggregate order flow imbalance (R2 increases out-of-sample from -0.17% to 2.39% for the smallest to the largest decile of order size). Finally, we show that the accuracy of transaction costs forecasts depends heavily on stock volatility, with a coefficient of 0.78.

Anticipating client needs is crucial for any company - this is especially true for investment banks such as BNP Paribas Corporate and Institutional Banking given their role in the financial markets. This thesis focuses on the problem of predicting future customer interests in the financial markets, with a particular emphasis on the development of ad hoc algorithms designed to solve specific problems in the financial world.This manuscript consists of five chapters, divided as follows:- Chapter 1 presents the problem of predicting future customer interests in the financial markets. The purpose of this chapter is to provide the reader with all the keys necessary for a good understanding of the rest of this thesis. These keys are divided into three parts: a highlighting of the datasets available to us for solving the future interest prediction problem and their characteristics, a non-exhaustive overview of the algorithms that can be used to solve this problem, and the development of metrics to evaluate the performance of these algorithms on our datasets. This chapter closes with the challenges that can be encountered when designing algorithms to solve the problem of predicting future interests in finance, challenges that will be, in part, solved in the following chapters: - Chapter 2 compares some of the algorithms introduced in Chapter 1 on a dataset from BNP Paribas CIB, and highlights the difficulties encountered when comparing algorithms of different nature on the same dataset, as well as some ways to overcome these difficulties. This comparison puts into practice classical recommendation algorithms only considered from a theoretical point of view in the previous chapter, and allows us to acquire a more detailed understanding of the different metrics introduced in chapter 1 through the analysis of the results of these algorithms. Chapter 3 introduces a new algorithm, Experts Network, i.e., a network of experts, designed to solve the problem of heterogeneous behavior of investors in a given market through an original neural network architecture, inspired by research on expert mixtures. In this chapter, this new methodology is used on three distinct datasets: a synthetic dataset, an open access dataset, and a dataset from BNP Paribas CIB. Chapter 4 also introduces a new algorithm, called History-augmented collaborative filtering, which proposes to augment the classical matrix factorization approaches with the help of the interaction histories of the considered customers and products. This chapter continues the study of the dataset studied in Chapter 2 and extends the introduced algorithm with many ideas. Specifically, this chapter adapts the algorithm to address the cold start problem, i.e., the inability of a recommender system to provide predictions for new users, as well as a new application case on which this adaptation is tried.- Chapter 5 highlights a collection of ideas and algorithms, both successful and unsuccessful, that have been tried in the course of this thesis. This chapter closes with a new algorithm combining the ideas of the algorithms introduced in chapters 3 and 4.

The thesis deals with numerical schemes for Markovian decision problems (MDPs), partial differential equations (PDEs), backward stochastic differential equations (SRs), as well as reflected backward stochastic differential equations (SRDEs). The thesis is divided into three parts.The first part deals with numerical methods for solving MDPs, based on quantization and local or global regression. A market-making problem is proposed: it is solved theoretically by rewriting it as an MDP. and numerically by using the new algorithm. In a second step, a Markovian embedding method is proposed to reduce McKean-Vlasov type probabilities with partial information to MDPs. This method is implemented on three different McKean-Vlasov type problems with partial information, which are then numerically solved using numerical methods based on regression and quantization.In the second part, new algorithms are proposed to solve MDPs in high dimension. The latter are based on neural networks, which have proven in practice to be the best for learning high dimensional functions. The consistency of the proposed algorithms is proved, and they are tested on many stochastic control problems, which allows to illustrate their performances.In the third part, we focus on methods based on neural networks to solve PDEs, EDSRs and reflected EDSRs. The convergence of the proposed algorithms is proved and they are compared to other recent algorithms of the literature on some examples, which allows to illustrate their very good performances.

This thesis consists of two interrelated parts. In the first part, we empirically study the behavior of high-frequency traders on European financial markets. In the second part, we use the results obtained to build new multi-agent models. The main objective of these models is to provide regulators and trading platforms with innovative tools to implement microstructure relevant rules and to quantify the impact of various participants on market quality.In the first part, we perform two empirical studies on unique data provided by the French regulator. We have access to all orders and trades of CAC 40 assets, at the microsecond scale, with the identities of the actors involved. We begin by comparing the behavior of high-frequency traders to that of other players, particularly during periods of stress, in terms of liquidity provision and trading activity. We then deepen our analysis by focusing on liquidity consuming orders. We study their impact on the price formation process and their information content according to the different categories of flows: high-frequency traders, participants acting as clients and participants acting as principal.In the second part, we propose three multi-agent models. Using a Glosten-Milgrom approach, our first model constructs the entire order book (spread and volume available at each price) from the interactions between three types of agents: an informed agent, an uninformed agent and market makers. This model also allows us to develop a methodology for predicting the spread in case of a change in the price step and to quantify the value of the priority in the queue. In order to focus on an individual scale, we propose a second approach where the specific dynamics of the agents are modeled by nonlinear Hawkes-type processes that depend on the state of the order book. In this framework, we are able to compute several relevant microstructure indicators based on individual flows. In particular, it is possible to classify market makers according to their own contribution to volatility. Finally, we introduce a model where liquidity providers optimize their best bid and offer prices according to the profit they can generate and the inventory risk they face. We then theoretically and empirically highlight an important new relationship between inventory and volatility.

Using a large database of the US institutional investors’ trades, this paper revisits the question of anomalies-based portfolio transaction costs. The real costs paid by large investors to implement the well-identified size, value, and momentum anomalies are lower than what has been documented in the previous studies. We find that the average investor pays an annual transaction cost of 17bps for size, 24bps for value, and 274bps for momentum. The three strategies generate statistically significant returns of respectively 5.21%, 2.79% and 2.77% after accounting for transaction costs. When the market impact is taken into account, transaction costs reduce substantially the profitability of the well-known anomalies for large portfolios, however, these anomalies remain profitable for average size portfolios. The break-even capacities in terms of fund size are $ 206 billion for size, $ 16.1 billion for value and $ 310 million for momentum.

In this paper we propose a mathematical framework to address the uncertainty emergingwhen the designer of a trading algorithm uses a threshold on a signal as a control. We rely ona theorem by Benveniste and Priouret to deduce our Inventory Asymptotic Behaviour (IAB)Theorem giving the full distribution of the inventory at any point in time for a well formulatedtime continuous version of the trading algorithm.Since this is the first time a paper proposes to address the uncertainty linked to the use of athreshold on a signal for trading, we give some structural elements about the kind of signals thatare using in execution. Then we show how to control this uncertainty for a given cost function.There is no closed form solution to this control, hence we propose several approximation schemesand compare their performances.Moreover, we explain how to apply the IAB Theorem to any trading algorithm drivenby a trading speed. It is not needed to control the uncertainty due to the thresholding of asignal to exploit the IAB Theorem. it can be applied ex-post to any traditional trading algorithm.

This paper goes beyond the optimal trading Mean Field Game model introduced by Pierre Cardaliaguet and Charles-Albert Lehalle in [Cardaliaguet, P. and Lehalle, C.-A., Mean field game of controls and an application to trade crowding, Mathematics and Financial Economics (2018)]. It starts by extending it to portfolios of correlated instruments. This leads to several original contributions: first that hedging strategies naturally stem from optimal liquidation schemes on portfolios. Second we show the influence of trading flows on naive estimates of intraday volatility and correlations. Focussing on this important relation, we exhibit a closed form formula expressing standard estimates of correlations as a function of the underlying correlations and the initial imbalance of large orders, via the optimal flows of our mean field game between traders. To support our theoretical findings, we use a real dataset of 176 US stocks from January to December 2014 sampled every 5 minutes to analyze the influence of the daily flows on the observed correlations. Finally, we propose a toy model based approach to calibrate our MFG model on data.

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

The main objective of this thesis is to understand the interactions between financial agents and the order book. We consider in the first chapter the control problem of an agent trying to take into account the available liquidity in the order book in order to optimize the placement of a unit order. Our strategy reduces the risk of adverse selection. Nevertheless, the added value of this approach is weakened in the presence of latency: predicting future price movements is of little use if agents' reaction time is slow.In the next chapter, we extend our study to a more general execution problem where agents trade non-unitary quantities in order to limit their impact on the price. In the third chapter, we build on the previous approach to solve this time market making problems rather than execution problems. This allows us to propose relevant strategies compatible with the typical actions of market makers. Then, we model the behavior of directional high frequency traders and institutional brokers in order to simulate a market where our three types of agents interact optimally with each other.We propose in the fourth chapter an agent model where the flow dynamics depend not only on the state of the order book but also on the market history. To do so, we use generalizations of nonlinear Hawkes processes. In this framework, we are able to compute several relevant indicators based on individual flows. In particular, it is possible to classify market makers according to their contribution to volatility.To solve the control problems raised in the first part of the thesis, we have developed numerical schemes. Such an approach is possible when the dynamics of the model are known. When the environment is unknown, stochastic iterative algorithms are usually used. In the fifth chapter, we propose a method to accelerate the convergence of such algorithms.The approaches considered in the previous chapters are suitable for liquid markets using the order book mechanism. However, this methodology is not necessarily relevant for markets governed by specific operating rules. To address this issue, we propose, first, to study the behavior of prices in the very specific electricity market.

Trend-following strategies have met with strong interest from institutional investors in recent years, due in particular to their good performance during the economic and financial crisis of 2008. The years 2016 to 2018 have reshuffled the deck, with performance deemed poor by many clients. This thesis focuses on the different characteristics of trend-following strategies, namely performance, risk and execution costs, and proposes new ways to approach these topics. Chapter 1 explains the difference in performance between hedge fund styles by confirming the presence of trends within the CTA and Global Macro strategies. The insurance nature of this strategy is confirmed within all types of hedge funds. Chapter 2 proposes a new decomposition of the risk associated with trend-following strategies into a common component and a specific component. The extraction of a systematic risk factor and its addition to the standard factor models allows us to better explain the performance of hedge fund styles in a different way than in chapter 1. Finally, chapter 3 addresses the issue of the execution of a trend following strategy. The cost paid by the investor, i.e. the cost associated with managing the portfolio, is not only a function of the individual liquidity of the assets handled but also depends on the allocation decisions made by the manager to meet the fund's performance and risk objectives.

This thesis is composed of three chapters that deal with impulse control problems. In the first chapter, we introduce a general framework for impulse control with uncertainty. Knowing an a priori law on unknown parameters, we explain how it should evolve and integrate it to the optimal control problem. We characterize the solution through a quasivariational parabolic equation that can be solved numerically and give examples of applications to finance. In the second chapter, we introduce an impulse control problem with uncertainty in an actuarial setting. An (re)insurer faces natural catastrophes and can issue CAT bonds to reduce the risk taken. We again characterize the optimal control problem through a numerically solvable quasi-variational parabolic equation and give some application examples. In the last chapter, we propose a model of the price through a completely endogenous order book. We solve impulse optimal control problems (order placement) of rational economic agents that we gather on a same market.

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

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The objective of this thesis is a quantitative study of the different mathematical problems that arise in algorithmic trading. Due to the strong applied character of this work, we are not only interested in the mathematical rigor of our results, but we also want to understand this research work in the context of the different steps that are part of the practical implementation of the tools that we develop. e.g. model interpretation, parameter estimation, computer implementation etc.From the scientific point of view, the core of our work is based on two techniques borrowed from the world of optimization and probability: stochastic control and stochastic approximation. In particular, we present original academic results for the high frequency market-making problem and the portfolio liquidation problem using limit-orders. Similarly, we solve the market-making problem using a forward optimization approach, which is innovative in the optimal trading literature as it opens the door to machine learning techniques. From a practical point of view, this thesis seeks to create a bridge between academic research and the financial industry. Our results are constantly considered from the perspective of their practical implementation. Thus, a large part of our work is focused on studying the different factors that are important to understand when transforming our quantitative techniques into industrial value: understanding the microstructure of the markets, stylized facts, data processing, model discussions, limitations of our scientific framework etc.

The price formation process is at the heart of all financial mathematics models. First approximated by a Brownian motion (cf. [Karatzas & Shreve, 1998]), then by integrating jumps (see [Shiryaev, 1999]) as long as long time scales were involved, the taking into account of the volumes exchanged came later: - In an econometric framework (see for example [Tauchen & Pitts, 1983]) to try to better explain the dynamics of prices - in the framework of theoretical studies of general equilibria where the "auction game" is modelled (as in [Kyle, 1985], [Ho & Stoll, 1983] or [Glosten & Milgrom, 1985]) As regulation pushes more and more exchanges towards electronic markets, very large databases are now available. They contain not only the actual transactions, but also the declarations of interest of all participants. This has opened the door to empirical studies (such as [Lillo et al. , 2003]) that provide interesting avenues for new families of models (see for example [Bacry & Muzy, 2013]). Properly modeling the price formation process helps guide regulators, who try to encourage trading in electronic markets (as they are more easily trac ̧able). It also allows for the development of optimal trading techniques, which, when used by investors and financial intermediaries, will minimize the disruption of prices due to trading intensity.

This paper deals with a stochastic order-driven market model with waiting costs, for order books with heterogenous traders. Offer and demand of liquidity drives price formation and traders anticipate future evolutions of the order book. The natural framework we use is mean field game theory, a class of stochastic differential games with a continuum of anonymous players. Several sources of heterogeneity are considered including the mean size of orders. Thus we are able to consider the coexistence of Institutional Investors and High Frequency Traders (HFT). We provide both analytical solutions and numerical experiments. Implications on classical quantities are explored: order book size, prices, and effective bid/ask spread. According to the model, in markets with Institutional Investors only we show the existence of inefficient liquidity imbalances in equilibrium, with two symmetrical situations corresponding to what we call liquidity calls for liquidity. During these situations the transaction price significantly moves away from the fair price. However this macro phenomenon disappears in markets with both Institutional Investors and HFT, although a more precise study shows that the benefits of the new situation go to HFT only, leaving Institutional Investors even with higher trading costs.

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High Frequency finance has recently evolved from statistical modeling and analysis of financial data – where the initial goal was to reproduce stylized facts and develop appropriate inference tools – toward trading optimization, where an agent seeks to execute an order (or a series of orders) in a stochastic environment that may react to the trading algorithm of the agent (market impact, invoentory). This context poses new scientific challenges addressed by the minisymposium OPSTAHF.

Recent regulatory changes, known as Reg NMS in the United States or MiFID in Europe, together with the effects of the financial crisis (mainly its impact on liquidity), induced major changes on market microstructure in two main aspects: • the fragmentation of the liquidity around several trading venues, with the appearance of newcomers in Europe like Chi-X, BATS Europe, or Turquoise, some of them being not regulated or “dark". • the rise of a new type of agents, the high frequency traders, liable for 40% to 70% of the transactions. These two effects are linked since the high frequency traders, being the main clients of the trading venues, have an implicit impact on the products offered by these venues. Combining a survey of recent academic findings and empirical evidences, this paper presents what we consider to be the key elements to understand the stakes of these changes, and also provides potential clues to mitigate some of them. A first section is dedicated to exposes the recent modifications in market microstructure. The second one explains the role of the price formation process and how, interacting with liquidity supply and demand, high frequency traders can reshape it. The next section discloses the various strategies used by these new market participants and their profitability. A final section discusses recent tools designed in order to assess and control the high frequency trading activity.

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Considering that a trader or a trading algorithm interacting with markets during continuous auctions can be modeled by an iterating procedure adjusting the price at which he posts orders at a given rhythm, this paper proposes a procedure minimizing his costs. We prove the a.s. convergence of the algorithm under assumptions on the cost function and give some practical criteria on model parameters to ensure that the conditions to use the algorithm are fulfilled (using notably the co-monotony principle). We illustrate our results with numerical experiments on both simulated data and using a financial market dataset.

Market makers continuously set bid and ask quotes for the stocks they have under consideration. Hence they face a complex optimization problem in which their return, based on the bid-ask spread they quote and the frequency they indeed provide liquidity, is challenged by the price risk they bear due to their inventory. In this paper, we consider a stochastic control problem similar to the one introduced by Ho and Stoll and formalized mathematically by Avellaneda and Stoikov. The market is modeled using a reference price S_t following a Brownian motion, arrival rates of buy or sell liquidity-consuming orders depend on the distance to the reference price S_t and a market maker maximizes the expected utility of its PnL over a short time horizon. We show that the Hamilton-Jacobi-Bellman equations can be transformed into a system of linear ordinary differential equations and we solve the market making problem under inventory constraints. We also provide a spectral characterization of the asymptotic behavior of the optimal quotes and propose closed-form approximations.

We study the optimal liquidation problem using limit orders. If the seminal literature on optimal liquidation, rooted to Almgren-Chriss models, tackles the optimal liquidation problem using a trade-off between market impact and price risk, it only answers the general question of the liquidation rhythm. The very question of the actual way to proceed is indeed rarely dealt with since most classical models use market orders only. Our model, that incorporates both price risk and non-execution risk, answers this question using optimal posting of limit orders. The very general framework we propose regarding the shape of the intensity generalizes both the risk-neutral model presented of Bayraktar and Ludkovski and the model developed in Gueant, Lehalle and Fernandez-Tapia, restricted to exponential intensity.

The aim of this work is to present new results concerning the use of a particular class of formal neural networks (the Piecewise Affine Perceptrons: PAP) in the context of closed loop optimal control. The main results obtained are: several properties of PAPs, concerning the nature of the functions they can emulate, a constructive representation theorem for piecewise affine continuous functions, which allows to explicitly construct a PAP from a collection of affine functions, a set of heuristics for learning the parameters of a perceptron in a closed loop and in an optimal control framework, theoretical results concerning the stability of PAPs used as controllers. The last part is devoted to applications of these results to the automatic construction of car engine combustion controllers, which have led to the filing of two patents by Renault.

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