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

Summary This thesis is split into three parts. In the first part, we apply the Principal-Agent theory to some problems of market microstructure. First, we build an incentives mechanism to improve the market quality in the context of market-making activity in a lit and a dark pool managed by the same exchange. Then, we adapt the incentives design to the regulation of market-making activity when several market-makers compete in a liquidity platform. We also propose a form of incentives based on the choice of tick sizes on the bid and ask sides of a single asset. Next, we tackle the issue of designing a derivatives market, using a quantization method to select the options listed on the exchange and the Principal-Agent framework to create incentives for an option market-maker. Finally, we develop an incentives mechanism to increase the investment in green bonds, robust to model specification, and outperforming current tax-incentives policies of the governments.The second part of this thesis is dedicated to option market-making in high dimension. We first propose a framework a constant Greek assumption to deal with long-dated options. Then, we propose an approximation of the value function enabling to deal with time-varying Greeks and short-dated options. Finally, we develop a framework for the high-frequency dynamics of the implied volatility surface. Using multidimensional Hawkes processes, we show how this setting can reproduce easily well-known stylized facts such as the skew, smile and term structure of the surface.The last part of this thesis is devoted to optimal trading problems in high dimension. First, we develop a framework to tackle the smart order routing (SOR) problem taking into account non-stationarity of markets. For a large number of venues, we use a deep reinforcement learning approach to compute the optimal controls of the trader. Then, we present a methodology to solve approximately optimal trading problems without using stochastic control theory. We propose a framework in which a myopic agent exhibits approximately an optimal behavior if he uses the gradient of the high-level trajectory as short-term alpha. Finally, we present two new developments on the optimal execution literature. First, we show that we can obtain a closed-form solution for 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 problem of optimal execution of a portfolio of assets, in the context of liquidity stress testing.