Essays on Quantization in Financial Mathematics

This thesis is devoted to the study of some applications of quantization to Financial Mathematics, especially to option pricing and calibration of financial data. Quantization is a technique that comes originally from numerical probability, and consists in approximating random variables and stochastic processes taking infinitely many values, with a discrete version of them, in order to simplify the quadrature algorithms for the computation of expected values. The purpose of this thesis is to show the great flexibility that quantization can have in the area of numerical probability and option pricing. In the literature, often there are ad hoc methods for a particular type of model or derivative, but no general framework seems to exist. Finite difference methods are heavily affected by the curse of dimensionality, while Monte Carlo methods need intense computational effort in order to have good precision, and are not designed for calibration purposes. Quantization can give an alternative methodology for a broad class of models and deriva- tives. The aim of the thesis is twofold: first, the extension of the literature about quantization to a broad class of models, namely local and stochastic volatility models, affine, pure jumps and polynomial processes, is an interesting theoretical exercise in itself. In fact, every time we deal with a different model we have to take in consideration the properties of the process and therefore the quantization algorithm must be adapted. Second, it is important to consider the computational results of the new types of quantization introduced. Indeed, the algorithms that we have developed turn out to be fast and numerically stable, and these aspects are very relevant, as we can overcome some of the issues present in literature for other types of approach. The first line of research deals with a technique called Recursive Marginal Quantization. Introduced in Pagès and Sagna (2015), this methodology exploits the conditional distribution