Analisi Stocastica nei Modelli Rough per la Volatilità

My thesis deals with the econometric issue of volatility modeling in derivatives pricing and the mathematical analysis of the volatility process. My aim is to analyze thoroughly the mathematical tools used in the paper "Volatility is Rough" by Gatheral, Jaisson, and Rosenbaum \cite{0}, the main reference for this work. The article opens a new chapter in the volatility modeling literature characterizing the volatility process as a rough fractionally integrated process. The authors specify the Rough Fractional Stochastic Volatility model, which is able to reproduce very accurately the dynamics of the volatility process, despite not accounting for the long memory property in the strict sense. A deep study of both the literature that preceeds and that follows the article enables to better explain some of the results obtained, to fill some mathematical gaps left in the article, and to investigate the future developments of this new volatility modeling framework.