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Some aspects of the central role of financial market microstructure : Volatility dynamics, optimal trading and market design

Abstract : This thesis is made of three parts. In the first one, we study the connections between the dynamics of the market at the microscopic and macroscopic scales, with a focus on the properties of the volatility. In the second part we deal with optimal control for point processes. Finally in the third part we study two questions of market design.We begin this thesis with studying the links between the no-arbitrage principle and the (ir)regularity of volatility. Using a microscopic to macroscopic approach, we show that we can connect those two notions through the market impact of metaorders. We model the market order flow using linear Hawkes processes and show that the no-arbitrage principle together with the existence of a non-trivial market impact imply that the volatility process has to be rough, more precisely a rough Heston model. Then we study a class of microscopic models where order flows are driven by quadratic Hawkes processes. The objective is to extend the rough Heston model building continuous models that reproduce the feedback of price trends on volatility: the so-called Zumbach effect. We show that using appropriate scaling procedures the microscopic models converge towards price dynamics where volatility is rough and that reproduce the Zumbach effect. Finally we use one of those models, the quadratic rough Heston model, to solve the longstanding problem of joint calibration of SPX and VIX options smiles.Motivated by the extensive use of point processes in the first part of our work we focus in the second part on stochastic control for point processes. Our aim is to provide theoretical guarantees for applications in finance. We begin with considering a general stochastic control problem driven by Hawkes processes. We prove the existence of a solution and more importantly provide a method to implement the optimal control in practice. Then we study the scaling limits of solutions to stochastic control problems in the framework of population modeling. More precisely we consider a sequence of models for the dynamics of a discrete population converging to a model with continuous population. For each model we consider a stochastic control problem. We prove that the sequence of optimal controls associated to the discrete models converges towards the optimal control associated to the continuous model. This result relies on the continuity of the solution to a backward stochastic differential equation with respect to the driving martingale and terminal value.In the last part we address two questions of market design. We are first interested in designing a liquid electronic market of derivatives. We focus on options and propose a two steps method that can be easily applied in practice. The first step is to select the listed options. For this we use a quantization algorithm enabling us to pick the options capturing most of market demand. The second step is to design a make-take fees policy for market makers to incentivize them to set attractive quotes. We formalize this issue as a principal agent problem that we explicitly solve. Finally we look for the optimal auction duration that should be used on a market organized in sequential auctions, the case of auctions with 0 second duration corresponding to the continuous double auctions situation. To do so, we use an agent based model where market takers are competing. We consider that the optimal auction duration is the one leading to the best quality of price formation process. After proving existence of a Nash equilibrium for the competition between market takers we apply our results on stocks market data. We find that for most of the stocks, the optimal auction duration lies between 2 and 10 minutes.
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Submitted on : Monday, August 31, 2020 - 1:21:08 PM
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Paul Jusselin. Some aspects of the central role of financial market microstructure : Volatility dynamics, optimal trading and market design. Probability [math.PR]. Institut Polytechnique de Paris, 2020. English. ⟨NNT : 2020IPPAX025⟩. ⟨tel-02926102⟩

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