Quantitative finance at the microstructure scale : algorithmic trading and regulation.

Summary 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.