Modeling of the price microstructure and applications of stochastic control to algorithmic trading.

Summary In this thesis, we deal with the modeling of asset prices in a limit order book and the application of optimal control techniques to algorithmic trading, in particular market making. For assets with small ticks, we develop a market making algorithm in a limit order book where the order arrivals follow a Poisson distribution with mean that decreases exponentially with the distance of the order from the mid-price. Thanks to asymptotic development techniques, we obtain explicit results for a very large class of price models, for which we assume to know only the first two moments. For large-ticket assets, we propose a new model based on a semi-Markovian process, with which we replicate known market phenomena such as mean reversion, large-scale Brownian behavior, and the dependence of the variance estimator on the observation frequency. In this environment, we describe a market making algorithm using optimal control techniques and asymptotic development, reducing the numerical part to the minimum. Finally, we improve the previous model by using VLMC (Variable Length Markov Chains), which allow us to describe the long memory of the price, and, even if we abandon explicit formulas, allow us to obtain interesting applications to algorithmic trading.