This dissertation is divided in two parts. The first topic presented is the functional Ito calculus, introduced by Bruno Dupire in the seminal paper Dupire [2009]. We start with an introductory chapter where the main ideas, definitions and results are discussed. Some original results are also presented. The major original contributions are shown in Chapter 2 and Chapter 3.

More precisely, Chapter 2 is the reproduction of the original work Jazaerli and Saporito [2013]. From its abstract: "Dupire's functional Ito calculus provides an alternative approach to the classical Malliavin calculus for the computation of sensitivities, also called Greeks, of path-dependent derivatives prices. In this paper, we introduce a measure of path-dependence of functionals within the functional Ito calculus framework. Namely, we consider the Lie bracket of the space and time functional derivatives, which we use to classify functionals according to their degree of path-dependence. We then revisit the problem of efficient numerical computation of Greeks for path-dependent derivatives using integration by parts techniques. Special attention is paid to path-dependent functionals with zero Lie bracket, called weakly path-dependent functionals in our classification.".

In Chapter 3, we pursue the functional version of the Meyer-Tanaka Formula for the class of convex functionals. Following the idea of the proof of the classical Meyer-Tanaka formula, we study the mollification of functionals and its convergence properties. As an example, we apply the theory to the running maximum functional.

The second part of this dissertation is devoted to the multiscale stochastic volatility models, introduced by J.-P. Fouque, G. Papanicolaou, R. Sircar, and K. Solna, see for example the lasted book on the subject: Fouque et al. [2011]. This part is divided in two chapters. The first one introduces these multiscale models and discuss their application to Mathematical Finance. In its last section, making use of the functional Ito calculus, we extend these results to path-dependent derivatives. The last chapter is solely based on the original work Fouque, Saporito and Zubelli [2013], where we present a new method to compute the first-order approximation of the price of derivatives on futures in the context of multiscale stochastic volatility models.