Three Essays in Financial Mathematics

The following PhD Thesis consists of three chapters. In these chapters I describe the projects we developed during my PhD. The first chapter, starting from the preliminary results of my Master’s Thesis, presents the study of the problems of consistency and existence of finite-dimensional realizations for Heath-Jarrow-Morton multi-curve interest-rate models. We generalize to the multi-curve setting the geometric approach at the basis of the results obtained by T. Bjork and coauthors in the single-curve framework. Hence, we propose a calibration algorithm based on the theoretical results we proved, applying it to Euribor market data. The second chapter deals with the analysis of a financial market populated by agents who access different amount of information. We study the problem of equilibrium price formation, obtained balancing the demand and supply of a single asset. To do so, we adopt a mean-field approach. We prove the existence of a mean-field equilibrium price, showing that it fills the information gap between a more informed major agent and a group of less informed standard agents. Moreover, we prove that, in the context of a market populated by finitely many standard agents and a major agent, the mean-field equilibrium price satisfies a weak version of the balance between demand and supply. In the third chapter, we provide several results towards a foundamental theorem of asset pricing for statistical arbitrage opportunities. We study a result present in the literature, that establishes a characterization of the absence of statistical arbitrage opportunities for markets defined on finite probability spaces. For this result a counterexample is provided in a recent paper. Actually, since this counterexample does not disprove the original statement, we confirm the latest in a rigorous mathematical setting. Therefore, we focus on market models defined on more general probability spaces and we show that the characterization proved for the finite markets is still guaranteed, under some additional and tailor-made assumptions.