LARUELLE Sophie

This thesis is divided into three parts. In the first part, we apply Principal-Agent theory to some market microstructure problems. First, we develop an incentive policy to improve the quality of market liquidity in the context of market-making activity in a bed and a dark pool managed by the same exchange. We then adapt this incentive design to the regulation of market-making activity when several market-makers compete on a platform. We also propose a form of incentive based on the choice of asymmetric tick sizes for buying and selling an asset. We then address the issue of designing a derivatives market, using a quantization method to select the options listed on the platform, and Principal-Agent theory to create incentives for an options market-maker. Finally, we develop an incentive mechanism robust to the model specification to increase investment in green bonds.The second part of this thesis is devoted to high-dimensional options market-making. The second part of this paper is devoted to the market-making of high-dimensional options. Assuming constant Greeks, we first propose a model to deal with long-maturity options. Then we propose an approximation of the value function to handle non-constant Greeks and short maturity options. Finally, we develop a model for the high frequency dynamics of the implied volatility surface. Using multidimensional Hawkes processes, we show how this model can reproduce many stylized facts such as the skew, the smile and the term structure of the surface.The last part of this thesis is devoted to optimal trading problems in high dimension. First, we develop a model for optimal trading of stocks listed on several platforms. For a large number of platforms, we use a deep reinforcement learning method to compute the optimal trader controls. Then, we propose a methodology to solve trading problems in an approximately optimal way without using stochastic control theory. We present a model in which an agent exhibits approximately optimal behavior if it uses the gradient of the macroscopic trajectory as a short-term signal. Finally, we present two new developments on the optimal execution literature. First, we show that we can obtain an analytical solution to the Almgren-Chriss execution problem with geometric Brownian motion and quadratic penalty. Second, we propose an application of the latent order book model to the optimal execution problem of a portfolio of assets, in the context of liquidity stress tests.

Recent empirical analyses have revealed the existence of the Zumbach effect. This discovery led to the development of the quadratic Hawkes process, adapted to reproduce this effect. Since this model does not relate to the price formation process, we extended it to the order book with a generalized quadratic Hawkes process (GQ-Hawkes). Using market data, we show that there is a Zumbach-like effect that decreases future liquidity. Microfounding the Zumbach effect, it is responsible for a potential destabilization of financial markets. Moreover, the exact calibration of a QM-Hawkes process tells us that markets are at the edge of criticality. This empirical evidence has therefore prompted us to analyze an order book model constructed with a Zumbach-type coupling. We therefore introduced the Santa Fe quadratic model and proved numerically that there is a phase transition between a stable market and an unstable market subject to liquidity crises. Thanks to a finite size analysis we were able to determine the critical exponents of this transition, belonging to a new universality class. Not being analytically solvable, this led us to introduce simpler models to describe liquidity crises. Putting aside the microstructure of the order book, we obtain a class of spread models where we have computed the critical parameters of their transitions. Even if these exponents are not those of the Santa Fe quadratic transition, these models open new horizons to explore the spread dynamics. One of them has a nonlinear coupling that reveals a metastable state. This elegant alternative scenario does not need critical parameters to obtain an unstable market, even if the empirical evidence is not in its favor. Finally, we looked at order book dynamics from another angle: reaction-diffusion. We modeled a liquidity that reveals itself in the order book with a certain frequency. Solving this model in equilibrium reveals that there is a stability condition on the parameters beyond which the order book empties completely, corresponding to a liquidity crisis. By calibrating it on market data, we were able to qualitatively analyze the distance to this unstable region.

The main objective of this thesis is to understand the various aspects of market impact. It consists of four chapters in which market impact is studied in different contexts and at different scales. The first chapter presents an empirical study of the market impact of limit orders in European equity markets. In the second chapter, we have extended the methodology presented for the equity markets to the options markets. This empirical study has shown that our definition of an options meta-order allows us to recover all the results highlighted in the equity markets. The third chapter focuses on market impact in the context of derivatives valuation. This chapter attempts to bring a microstructure component to the valuation of options by proposing a theory of market impact disturbances during the re-hedging process. In the fourth chapter, we explore a fairly simple model for metaorder relaxation. Metaorder relaxation is treated in this section as an informational process that is transmitted to the market. Thus, starting from the point of departure that at the end of the execution of a meta-order the information carried by it is maximal, we propose an interpretation of the relaxation phenomenon as being the result of the degradation of this information at the expense of the external noise of the market.

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 extends the link between stochastic approximation (SA) theory and randomized urn models developed in Laruelle, Pagès (2013), and their applications to clinical trials introduced in Bai, HU (1999,2005) and Bai, Hu, Shen (2002). We no longer assume that the drawing rule is uniform among the balls of the urn (which contains d colors), but can be reinforced by a function f. This is a way to model risk aversion. Firstly, by considering that f is concave or convex and by reformulating the dynamics of the urn composition as an SA algorithm with remainder, we derive the a.s. convergence and the asymptotic normality (Central Limit Theorem, CLT) of the normalized procedure by calling upon the so-called ODE and SDE methods. An in-depth analysis of the case d=2 exhibits two different behaviors: A single equilibrium point when f is concave, and when f is convex, a transition phase from a single attracting equilibrium to a system with two attracting and one repulsive equilibrium points. The last setting is solved using results on non-convergence toward noisy and noiseless ``traps" in order to deduce the a.s. convergence toward one of the attracting points. Secondly, the special case of a Polya urn (when the addition rule is the identity matrix) is analyzed, still using result from SA theory about ``traps''. Finally, these results are applied to a function with regular variation and to an optimal asset allocation in Finance.

This thesis is composed of four chapters.The first chapter is a preliminary description of the Factset Ownership database. The first chapter is a preliminary description of the Factset Ownership database. We give a statistical description of the database and present some stylized facts characterizing the portfolio structure of financial institutions and investment funds, as well as the market capitalization of the companies listed in the database.The second chapter proposes a method for statistically evaluating the similarity between pairs of financial institutions' portfolios. The second chapter proposes a method to statistically evaluate the similarity between pairs of portfolios of financial institutions. Since a statistically significant pair leads to the creation of a similarity link between these two entities, we are able to project an originally bi-partite network (between financial institutions and firms) into a mono-partite network (between institutions only) in order to study the evolution of its structure over time. Indeed, from an economic point of view, it is suspected that similar investment motives constitute an important risk factor of financial contagion that can be at the origin of bankruptcies with significant systemic consequences.The third chapter focuses on the collective behavior of investment fund managers and, in particular, on the way in which the structure of the portfolio of these funds optimally takes into account, on average, transaction costs in the presence of weak investment constraints. This phenomenon, where in many situations the median or average of a group of people's estimates is surprisingly close to the true value, is known as the wisdom of the crowd.The fourth chapter is devoted to the simultaneous study of market data. We use over 6.7 billion trades from the Thomson-Reuters Tick History database, and portfolio data from the FactSet Ownership database. We study the tick-to-tick dynamics of the order book as well as the aggregate action, i.e. on a much larger time scale, of investment funds. In particular, we show that the long memory of the sign of market orders is much shorter in the presence of the action, absolute or directional, of investment funds. Conversely, we explain to what extent a stock characterized by a weak memory is subject to directional trading due to the action of investment funds.

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This paper presents the link between stochastic approximation and multi-arm clinical trials based on randomized urn models investigated in Bai et al. (J. Multivar. Anal. 81(1):1–18, 2002) where the urn updating depends on the past performances of the treatments. We reformulate the dynamics of the urn composition, the assigned treatments and the successes of assigned treatments as standard stochastic approximation (SA) algorithms with remainder. Then, we derive the a.s. convergence of the normalized procedure under less stringent assumptions by calling upon the ODE and a new asymptotic normality result (Central Limit Theorem CLT) by calling upon the SDE methods.

No summary available.

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.

This paper presents the link between stochastic approximation and clinical trials based on randomized urn models investigated in Bai and Hu (1999,2005) and Bai, Hu and Shen (2002). We reformulate the dynamics of both the urn composition and the assigned treatments as standard stochastic approximation (SA) algorithms with remainder. Then, we derive the a.s. convergence and the asymptotic normality (CLT) of the normalized procedure under less stringent assumptions by calling upon the ODE and SDE methods. As a second step, we investigate a more involved family of models, known as multi-arm clinical trials, where the urn updating depends on the past performances of the treatments. By increasing the dimension of the state vector, our SA approach provides this time a new asymptotic normality result.

This thesis deals with the analysis of stochastic algorithms and their application in Finance. The first part presents a convergence result for stochastic algorithms where innovations verify averaging assumptions with a certain speed. We apply it to different types of innovations and illustrate it on examples mainly motivated by Finance. We then establish a "universal" speed of convergence result in the framework of equi-separated innovations and compare our results with those obtained in the i-framework. I. D. The second part is devoted to applications. We first present an optimal allocation problem applied to dark pools. The execution of the maximum desired quantity leads to the construction of a constrained stochastic algorithm studied in the innovation framework i. I. D. and averaging innovations. The next chapter presents a constrained stochastic optimization algorithm with projection to find the best placement distance in an order book by minimizing the execution cost of a given quantity. We then study the implementation and calibration of parameters in financial models by stochastic algorithm and illustrate these 2 techniques with examples of application on Black-Scholes, Merton and pseudo-CEV models. The last chapter deals with the application of stochastic algorithms in the framework of random urn models used in clinical trials. Using the ODE and DHS methods, we recover the convergence and speed results of Bai and Hu under weaker assumptions on the generating matrices.