GUEANT Olivier

In most over-the-counter (OTC) markets, a small number of market makers provide liquidity to other market participants. More precisely, for a list of assets, they set prices at which they agree to buy and sell. Market makers face therefore an interesting optimization problem: they need to choose bid and ask prices for making money while mitigating the risk associated with holding inventory in a volatile market. Many market-making models have been proposed in the academic literature, most of them dealing with single-asset market making whereas market makers are usually in charge of a long list of assets. The rare models tackling multiasset market making suffer however from the curse of dimensionality when it comes to the numerical approximation of the optimal quotes. The goal of this paper is to propose a dimensionality reduction technique to address multiasset market making by using a factor model. Moreover, we generalize existing market-making models by the addition of an important feature: the existence of different transaction sizes and the possibility for the market makers in OTC markets to answer different prices to requests with different sizes.

In most OTC markets, a small number of market makers provide liquidity to other market participants. More precisely, for a list of assets, they set prices at which they agree to buy and sell. Market makers face therefore an interesting optimization problem: they need to choose bid and ask prices for making money while mitigating the risk associated with holding inventory in a volatile market. Many market making models have been proposed in the academic literature, most of them dealing with single-asset market making whereas market makers are usually in charge of a long list of assets. The rare models tackling multi-asset market making suffer however from the curse of dimensionality when it comes to the numerical approximation of the optimal quotes. The goal of this paper is to propose a dimensionality reduction technique to address multi-asset market making by using a factor model. Moreover, we generalize existing market making models by the addition of an important feature: the existence of different transaction sizes and the possibility for the market makers in OTC markets to answer different prices to requests with different sizes.

In corporate bond markets, which are mainly OTC markets, market makers play a central role by providing bid and ask prices for a large number of bonds to asset managers from all around the globe. Determining the optimal bid and ask quotes that a market maker should set for a given universe of bonds is a complex task. Useful models exist, most of them inspired by that of Avellaneda and Stoikov. These models describe the complex optimization problem faced by market makers: proposing bid and ask prices in an optimal way for making money out of the difference between bid and ask prices while mitigating the market risk associated with holding inventory. While most of the models only tackle one-asset market making, they can often be generalized to a multi-asset framework. However, the problem of solving numerically the equations characterizing the optimal bid and ask quotes is seldom tackled in the literature, especially in high dimension. In this paper, our goal is to propose a numerical method for approximating the optimal bid and ask quotes over a large universe of bonds in a model \`a la Avellaneda-Stoikov. Because we aim at considering a large universe of bonds, classical finite difference methods as those discussed in the literature cannot be used and we present therefore a discrete-time method inspired by reinforcement learning techniques. More precisely, the approach we propose is a model-based actor-critic-like algorithm involving deep neural networks.

This thesis is a study of various optimal portfolio allocation problems where the rate of appreciation, called drift, of the Brownian motion of asset dynamics is uncertain. We consider an investor with a belief about drift in the form of a probability distribution, called a priori. Uncertainty about drift is taken into account by a Bayesian learning approach that updates the a priori probability distribution of drift. The thesis is divided into two independent parts. The first part contains two chapters: the first one develops the theoretical results, and the second one contains a detailed application of these results on market data. The first part of the thesis is devoted to the Markowitz portfolio selection problem in the multidimensional case with drift uncertainty. This uncertainty is modeled via an arbitrary a priori law that is updated using Bayesian filtering. We first transform the Bayesian Markowitz problem into a standard auxiliary control problem for which dynamic programming is applied. Then, we show the existence and uniqueness of a regular solution to the associated semi-linear partial differential equation (PDE). In the case of an a priori Gaussian distribution, the multidimensional solution is explicitly computed. Moreover, we study the quantitative impact of learning from the progressively observed data, by comparing the strategy that updates the drift estimate, called learning strategy, to the one that keeps it constant, called nonlearning strategy. Finally, we analyze the sensitivity of the learning gain, called information value, to different parameters. We then illustrate the theory with a detailed application of the previous results to historical market data. We highlight the robustness of the added value of learning by comparing the optimal learning and non-learning strategies in different investment universes: indices of different asset classes, currencies and smart beta strategies. The second part deals with a discrete time portfolio optimization problem. Here, the investor's objective is to maximize the expected utility of the terminal wealth of a portfolio of risky assets, assuming an uncertain drift and a maximum drawdown constraint satisfied. In this section, we formulate the problem in the general case, and we numerically solve the Gaussian case with the constant relative risk aversion (CRRA) utility function, via a deep learning algorithm. Finally, we study the sensitivity of the strategy to the degree of uncertainty surrounding the drift estimate and empirically illustrate the convergence of the unlearned strategy to a constrained Merton problem, without short selling.

The literature on continuous-time stochastic optimal control seldom deals with the case of discrete state spaces. In this paper, we provide a general framework for the optimal control of continuous-time Markov chains on finite graphs. In particular, we provide results on the long-term behavior of value functions and optimal controls, along with results on the associated ergodic Hamilton-Jacobi equation.

In this article, we tackle the problem of a market maker in charge of a book of options on a single liquid underlying asset. By using an approximation of the portfolio in terms of its vega, we show that the seemingly high-dimensional stochastic optimal control problem of an option market maker is in fact tractable. More precisely, when volatility is modeled using a classical stochastic volatility model—e.g. the Heston model—the problem faced by an option market maker is characterized by a low-dimensional functional equation that can be solved numerically using a Euler scheme along with interpolation techniques, even for large portfolios. In order to illustrate our findings, numerical examples are provided.

In this article, we tackle the problem of a market maker in charge of a book of options on a single liquid underlying asset. By using an approximation of the portfolio in terms of its vega, we show that the seemingly high-dimensional stochastic optimal control problem of an option market maker is in fact tractable. More precisely, when volatility is modeled using a classical stochastic volatility model -- e.g. the Heston model -- the problem faced by an option market maker is characterized by a low-dimensional functional equation that can be solved numerically using a Euler scheme along with interpolation techniques, even for large portfolios. In order to illustrate our findings, numerical examples are provided.

When firms want to buy back their own shares, they have a choice between several alternatives. If they often carry out open market repurchase, they also increasingly rely on banks through complex buyback contracts involving option components, e.g. accelerated share repurchase contracts, VWAP-minus profit-sharing contracts, etc. The entanglement between the execution problem and the option hedging problem makes the management of these contracts a difficult task that should not boil down to simple Greek-based risk hedging, contrary to what happens with classical books of options. In this paper, we propose a machine learning method to optimally manage several types of buyback contract. In particular, we recover strategies similar to those obtained in the literature with partial differential equation and recombinant tree methods and show that our new method, which does not suffer from the curse of dimensionality, enables to address types of contract that could not be addressed with grid or tree methods.

When firms want to buy back their own shares, they have a choice between several alternatives. If they often carry out open market repurchase, they also increasingly rely on banks through complex buyback contracts involving option components, e.g. accelerated share repurchase contracts, VWAP-minus profit-sharing contracts, etc. The entanglement between the execution problem and the option hedging problem makes the management of these contracts a difficult task that should not boil down to simple Greek-based risk hedging, contrary to what happens with classical books of options. In this paper, we propose a machine learning method to optimally manage several types of buyback contract. In particular, we recover strategies similar to those obtained in the literature with partial differential equation and recombinant tree methods and show that our new method, which does not suffer from the curse of dimensionality, enables to address types of contract that could not be addressed with grid or tree methods.

The literature on continuous-time stochastic optimal control seldom deals with the case of discrete state spaces. In this paper, we provide a general framework for the optimal control of continuous-time Markov chains on finite graphs. In particular, we provide results on the long-term behavior of value functions and optimal controls, along with results on the associated ergodic Hamilton-Jacobi equation.

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.

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.

This paper presents several models addressing optimal portfolio choice, optimal portfolio liquidation, and optimal portfolio transition issues, in which the expected returns of risky assets are unknown. Our approach is based on a coupling between Bayesian learning and dynamic programming techniques that leads to partial differential equations. It enables to recover the well-known results of Karatzas and Zhao in a framework à la Merton, but also to deal with cases where martingale methods are no longer available. In particular, we address optimal portfolio choice, portfolio liquidation, and portfolio transition problems in a framework à la Almgren–Chriss, and we build therefore a model in which the agent takes into account in his decision process both the liquidity of assets and the uncertainty with respect to their expected return.

In corporate bond markets, which are mainly OTC markets, market makers play a central role by providing bid and ask prices for bonds to asset managers. Determining the optimal bid and ask quotes that a market maker should set for a given universe of bonds is a complex task. The existing models, mostly inspired by the Avellaneda-Stoikov model, describe the complex optimization problem faced by market makers: proposing bid and ask prices for making money out of the difference between them while mitigating the market risk associated with holding inventory. While most of the models only tackle one-asset market making, they can often be generalized to a multi-asset framework. However, the problem of solving the equations characterizing the optimal bid and ask quotes numerically is seldom tackled in the literature, especially in high dimension. In this paper, we propose a numerical method for approximating the optimal bid and ask quotes over a large universe of bonds in a model à la Avellaneda–Stoikov. As classical finite difference methods cannot be used in high dimension, we present a discrete-time method inspired by reinforcement learning techniques, namely, a model-based deep actor-critic algorithm.

In most illiquid markets, there is no obvious proxy for the market price of an asset. The European corporate bond market is an archetypal example of such an illiquid market where mid-prices can only be estimated with a statistical model. In this OTC market, dealers/market makers only have access, indeed, to partial information about the market. In real time, they know the price associated with their trades on the dealer-to-dealer (D2D) and dealer-to-client (D2C) markets, they know the result of the requests for quotes (RFQ) they answered, and they have access to composite prices (e.g., Bloomberg CBBT). This paper presents a Bayesian method for estimating the mid-price of corporate bonds by using the real-time information available to a dealer. This method relies on recent ideas coming from the particle filtering/sequential Monte Carlo literature.

By using variational techniques, we provide an optimal payoff written on a given random variable for hedging – in the sense of minimizing the Expected Shortfall at a given threshold – a payoff written on another random variable. In numerous financially relevant examples, our result leads to optimal payoffs in closed form. From a theoretical viewpoint, our result is also useful for providing bounds to the classical Expected Shortfall minimization problem with given financial instruments.

Mean field game theory was introduced in 2006 by Jean-Michel Lasry and Pierre-Louis Lions. It allows the study of game theory in certain configurations where the number of players is too large to hope for a practical resolution. We study the theory of mean-field games on graphs based on the work of Olivier Guéant which we will extend to more general Hilbertian forms. We will also study the links between K-means and mean-field games, which will in principle allow us to propose new algorithms for K-means using numerical resolution techniques specific to mean-field games. Finally, we will study a mean-field game, namely the "meeting start time" problem by extending it to situations where agents can choose between two meetings. We will study analytically and numerically the existence and multiplicity of solutions of this problem.

Whether it be to take advantage of stock undervaluation or in order to distribute part of their profits to shareholders, firms may buy back their own shares. One of the ways they do this is by including accelerated share repurchases as part of their repurchase programs. We study the pricing and optimal execution strategy of an accelerated share repurchase contract with a fixed notional. In such a contract the firm pays a fixed notional F to the bank and receives in exchange a number of shares corresponding to the ratio of F to the average stock price over the purchase period (the duration of this period being decided upon by the bank). From a mathematical point of view, the problem is related to both optimal execution and exotic option pricing.

Market makers provide liquidity to other market participants: they propose prices at which they stand ready to buy and sell a wide variety of assets. They face a complex optimization problem with static and dynamic components: they need indeed to propose bid and offer/ask prices in an optimal way for making money out of the difference between these two prices (their bid-ask spread), while mitigating the risk associated with price changes -- because they seldom buy and sell simultaneously, and therefore hold long or short inventories which expose them to market risk. In this paper, (i) we propose a general modeling framework which generalizes (and reconciles) the various modeling approaches proposed in the literature since the publication of the seminal paper ``High-frequency trading in a limit order book'' by Avellaneda and Stoikov, (ii) we prove new general results on the existence and the characterization of optimal market making strategies, (iii) we obtain new closed-form approximations for the optimal quotes, (iv) we extend the modeling framework to the case of multi-asset market making, and (v) we show how the model can be used in practice in the specific case of the corporate bond market and for two credit indices.

The ad-trading desks of media-buying agencies are increasingly relying on complex algorithms for purchasing advertising inventory. In particular, Real-Time Bidding (RTB) algorithms respond to many auctions -- usually Vickrey auctions -- throughout the day for buying ad-inventory with the aim of maximizing one or several key performance indicators (KPI). The optimization problems faced by companies building bidding strategies are new and interesting for the community of applied mathematicians. In this article, we introduce a stochastic optimal control model that addresses the question of the optimal bidding strategy in various realistic contexts: the maximization of the inventory bought with a given amount of cash in the framework of audience strategies, the maximization of the number of conversions/acquisitions with a given amount of cash, etc. In our model, the sequence of auctions is modeled by a Poisson process and the \textit{price to beat} for each auction is modeled by a random variable following almost any probability distribution. We show that the optimal bids are characterized by a Hamilton-Jacobi-Bellman equation, and that almost-closed form solutions can be found by using a fluid limit. Numerical examples are also carried out.

For the last two decades, most financial markets have undergone an evolution toward electronification. The market for corporate bonds is one of the last major financial markets to follow this unavoidable path. Traditionally quote-driven (i.e., dealer-driven) rather than order-driven, the market for corporate bonds is still mainly dominated by voice trading, but a lot of electronic platforms have emerged. These electronic platforms make it possible for buy-side agents to simultaneously request several dealers for quotes, or even directly trade with other buy-siders. The research presented in this article is based on a large proprietary database of requests for quotes (RFQ) sent, through the multi-dealer-to-client (MD2C) platform operated by Bloomberg Fixed Income Trading, to one of the major liquidity providers in European corporate bonds. Our goal is (i) to model the RFQ process on these platforms and the resulting competition between dealers, and (ii) to use our model in order to implicit from the RFQ database the behavior of both dealers and clients on MD2C platforms.

For the last two decades, most financial markets have undergone an evolution toward electronification. The market for corporate bonds is one of the last major financial markets to follow this unavoidable path. Traditionally quote-driven (i.e., dealer-driven) rather than order-driven, the market for corporate bonds is still mainly dominated by voice trading, but a lot of electronic platforms have emerged. These electronic platforms make it possible for buy-side agents to simultaneously request several dealers for quotes, or even directly trade with other buy-siders. The research presented in this article is based on a large proprietary database of requests for quotes (RFQ) sent, through the multi-dealer-to-client (MD2C) platform operated by Bloomberg Fixed Income Trading, to one of the major liquidity providers in European corporate bonds. Our goal is (i) to model the RFQ process on these platforms and the resulting competition between dealers, and (ii) to use our model in order to implicit from the RFQ database the behavior of both dealers and clients on MD2C platforms.

Be it for taking advantage of stock undervaluation or in order to distribute part of their profits to shareholders, firms may buy back their own shares. One of the way they proceed is by including Accelerated Share Repurchases (ASR) as part of their repurchase programs. In this article, we study the pricing and optimal execution strategy of an ASR contract with fixed notional. In such a contract the firm pays a fixed notional F to the bank and receives, in exchange, a number of shares corresponding to the ratio between F and the average stock price over the purchase period, the duration of this period being decided upon by the bank. From a mathematical point of view, the problem is related to both optimal execution and exotic option pricing.

In this article, we develop a general CARA framework to study optimal execution and to price block trades. We prove existence and regularity results for optimal liquidation strategies and we provide several differential characterizations. We also give two different proofs that the usual restriction to deterministic liquidation strategies is optimal. In addition, we focus on the important topic of block trade pricing and we therefore give a price to financial (il)liquidity. In particular, we provide a closed-form formula for the price a block trade when there is no time constraint to liquidate, and a differential characterization in the time-constrained case.

In this article, we develop a general CARA framework to study optimal execution and to price block trades. We prove existence and regularity results for optimal liquidation strategies and we provide several differential characterizations. We also give two different proofs that the usual restriction to deterministic liquidation strategies is optimal. In addition, we focus on the important topic of block trade pricing and we therefore give a price to financial (il)liquidity. In particular, we provide a closed-form formula for the price a block trade when there is no time constraint to liquidate, and a differential characterization in the time-constrained case.

No summary available.

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 Guéant, Lehalle and Fernandez-Tapia, restricted to exponential intensity.

In spite of the growing consideration for optimal execution issues in the financial mathematics literature, numerical approximations of optimal trading curves are almost never discussed. In this article, we present a numerical method to approximate the optimal strategy of a trader willing to unwind a large portfolio. The method we propose is very general as it can be applied to multi-asset portfolios with any form of execution costs, including a bid-ask spread component, even when participation constraints are imposed. Our method, based on convex duality, only requires Hamiltonian functions to have C^{1,1} regularity while classical methods require additional regularity and cannot be applied to all cases found in practice.

In this thesis, we consider two types of applications of feedback effects in finance. These effects come into play when market participants execute sequences of trades or take part in chain reactions, which generate peaks of activity. The first part presents a dynamic optimal execution model in the presence of an exogenous stochastic market order flow. We start from the benchmark model of Obizheva and Wang, which defines an optimal execution framework with a mixed price impact. We add an order flow modeled using Hawkes processes, which are jump processes with a self-excitation property. Using stochastic control theory, we determine the optimal strategy analytically. Then we determine the conditions for the existence of Price Manipulation Strategies, as introduced by Huberman and Stanzl. These strategies can be excluded if the self-excitation of the order flow exactly offsets the price resilience. In a second step, we propose a calibration method for the model, which we apply on high frequency financial data from CAC40 stock prices. On these data, we find that the model explains a non-negligible part of the price variance. An evaluation of the optimal strategy in backtesting shows that it is profitable on average, but that realistic transaction costs are sufficient to prevent price manipulation. Then, in the second part of the thesis, we focus on the modeling of intraday volatility. In the literature, most of the backward-looking volatility models focus on the daily time scale, i.e., on day-to-day price changes. The objective here is to extend this type of approach to shorter time scales. We first present an ARCH-type model with the particularity of taking into account separately the contributions of past intraday and night-time returns. A calibration method for this model is studied, as well as a qualitative interpretation of the results on US and European stock returns. In the next chapter, we further reduce the time scale considered. We study a high-frequency volatility model, the idea of which is to generalize the Hawkes process framework to better reproduce some empirical market characteristics. In particular, by introducing quadratic feedback effects inspired by the QARCH discrete time model we obtain a power law distribution for volatility as well as time skewness.

This paper presents a general existence and uniqueness result for mean field games equations on graphs. In particular, our setting allows to take into account congestion effects of almost any form. These general congestion effects are particularly relevant in graphs in which the cost to move from one node to another may for instance depend on the proportion of players in both the source node and the target node. Existence is proved using a priori estimates and a fixed point argument à la Schauder. We propose a new criterion to ensure uniqueness in the case of Hamiltonian functions with a complex (non-local) structure. This result generalizes the discrete counterpart of existing uniqueness results.

In this article, we consider a specific optimal execution problem associated to accelerated share repurchase contracts. When firms want to repurchase their own shares, they often enter such a contract with a bank. The bank buys the shares for the firm and is paid the average market price over the execution period, the length of the period being decided upon by the bank during the buying process. Mathematically, the problem is new and related to both option pricing (Asian and Bermudan options) and optimal execution. We provide a model, along with associated numerical methods, to determine the optimal stopping time and the optimal buying strategy of the bank.

In this article we consider the pricing and (partial) hedging of a call option when liquidity matters, that is either for a large nominal or for an illiquid underlying. In practice, as opposed to the classical assumptions of a price-taker agent in a frictionless market, traders cannot be perfectly hedged because of execution costs and market impact. They face indeed a trade-off between mishedge errors and hedging costs that can be solved using stochastic optimal control. Our framework is inspired from the recent literature on optimal execution and permits to account for both execution costs and the lasting market impact of our trades. Prices are obtained through the indifference pricing approach and not through super-replication. Numerical examples are provided using PDEs, along with comparison with the Black model.

No summary available.

1 Introduction 2 MFG on graphs: setup 3 Existence result 4 Uniqueness result 5 Potential games and Master equation.

This paper presents a general existence and uniqueness result for mean field games equations on graphs ($$\mathcal{G}$$G-MFG). In particular, our setting allows to take into account congestion effects of almost any form. These general congestion effects are particularly relevant in graphs in which the cost to move from one node to another may for instance depend on the proportion of players in both the source node and the target node. Existence is proved using a priori estimates and a fixed point argument a la Schauder. We propose a new criterion to ensure uniqueness in the case of Hamiltonian functions with a complex (non-local) structure. This result generalizes the discrete counterpart of uniqueness results obtained in Lasry and Lions (C. R. Acad. Sci. Paris 343(10):679---684, 2006).

In this article, we develop a liquidation model in which the trader is constrained to liquidate a portfolio at a constant participation rate. Considering the functional forms usually used by practitioners, we obtain a closed-form expression for the optimal participation rate and for the liquidity premium a trader should quote to buy a large block. We also show that the difference in terms of liquidity premium between the constant participation rate case and the usual Almgren-Chriss-like case never exceeds 15%.

In this article, we develop a liquidation model in which the trader is constrained to liquidate a portfolio at a constant participation rate. Considering the functional forms usually used by practitioners, we obtain a closed-form expression for the optimal participation rate and for the liquidity premium a trader should quote to buy a large block. We also show that the difference in terms of liquidity premium between the constant participation rate case and the usual Almgren-Chriss-like case never exceeds 15%.

If optimal liquidation using VWAP strategies has been considered in the literature, it has never been considered in the presence of permanent market impact and only rarely with execution costs. Moreover, only VWAP strategies have been studied and no pricing of guaranteed VWAP contract is provided. In this article, we develop a model to price guaranteed VWAP contracts in the most general framework for market impact. Numerical applications are also provided.

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.

Mean field games models describing the limit of a large class of stochastic differential games, as the number of players goes to infinity, have been introduced by J.-M. Lasry and P.-L. Lions. We use a change of variables to transform the mean field games (MFG) equations into a system of simpler coupled partial differential equations, in the case of a quadratic Hamiltonian. This system is then used to exhibit a monotonic scheme to build solutions of the MFG equations. Effective numerical methods based on this constructive scheme are presented and numerical experiments are carried out.

The agency problem between an investor and his mutual funds managers has long been studied in the economic literature. Because the very business of mutual funds managers is not only to manage money but also, and rather, to increase the money under management, one of the numerous agency problems is the implicit incentive induced by the relationship between inflows and performance. If the consequences of incentives, be they implicit or explicit – as for compensation schemes of individual asset managers – are well known in terms of risk-shifting when the incentives are linked to a benchmark, the very fact that the mutual fund market is a tournament does not seem to be modeled properly in the literature. In this paper, we propose a mean field games model to quantify the risk-shifting induced by a tournament-like competition between mutual funds.

Introduced by J. -M. Lasry and P.-L. Lions, mean-field game theory simplifies interactions between economic agents using an approach inspired by physical theories. Economic applications are presented concerning the labor market, asset management, population distribution problems, and growth theory. The models presented use mean-field game theory in various forms, sometimes static, often dynamic, with discrete or continuous state space and in a deterministic or stochastic environment. Various notions of stability are discussed, including the notion of educative stability, which has inspired numerical methods of resolution. Indeed, we present numerical methods that allow to obtain solutions to both stationary and dynamic problems, while abstracting from the forward/backward structure, which is a priori problematic from a numerical point of view. In the margin of mean-field game theory, the problem of appropriate discount rates to deal with sustainable development problems is addressed. We discuss the notion of ecological rate introduced by R. Guesnerie and provide new non-asymptotic properties, notably continuity.