A mathematical approach to stock market investing.

Summary The goal of this thesis is to answer the real need to predict future stock price fluctuations. Indeed, the randomness governing these fluctuations constitutes for financial actors, such as market makers, one of the greatest sources of risk. Throughout this study, we highlight the possibility of reducing the uncertainty on future prices by using appropriate mathematical models. This study is made possible thanks to a large financial database and a powerful computational grid made available to us by the Automatic Market Making team of BNP Paribas. In this paper, we only present the results of the research concerning high frequency trading. The results concerning the low-frequency part are of less scientific interest to the academic world and are also confidential. In the first chapter, we present the context and the objectives of this study. We also present the different methods used, as well as the main results obtained. In chapter 2, we focus on the contribution of technological superiority in high frequency trading. For this purpose, we simulate an ultra-fast, omniscient, and aggressive trader, and then we calculate his total gain over 3 years. The gains obtained are very modest and reflect the limited contribution of technology in high frequency trading. In chapter 3, we study the predictability of prices based on order book indicators. Using conditional expectations, we present empirical evidence of statistical dependencies between prices and the different indicators. The importance of these dependencies results from the simplicity of the method, eliminating any risk of overlearning the data. We then focus on the combination of the different indicators by a linear regression and we analyze the different numerical and statistical problems related to this method. Finally, we conclude that prices are predictable for a time horizon of a few minutes and we question the market efficiency hypothesis.In chapter 4, we focus on the price formation mechanism based on the arrival of events in the order book. We classify the orders into twelve types whose statistical properties we analyze. We then study the dependencies between these different types of orders and propose an order book model in line with empirical observations. Finally, we use this model to predict prices and we support the hypothesis of the non-efficiency of markets, suggested in chapter 3.