Rough Volatility: pushing the boundaries of quantitative modelling past the Markovian era

Abstract

Rough volatility models have brought a breeze of fresh air into financial modelling, which historically has been mostly tied to Markovian models. Alas, this novelty comes with a number of challenges and questions that require new techniques to be answered. The present thesis first explores the numerical implementation of rough volatility models, both to provide convergence results and to ensure a competitive implementation. Our findings conclude that fractional calculus and Hölder spaces provide the right framework to set the theoretical grounds of numerical schemes related to rough volatility models and fractional diffusions in general. In addition, we explore the use of deep learning methods to leverage the speed of numerical schemes and provide state-of-the-art calibration techniques that make rough volatility models as simple and efficient to implement as the Black-Scholes formula. The second part of the thesis explores the theoretical properties of rough volatility models regarding volatility products such as VIX and realized variance options. We explore short-time asymptotics using two different techniques: Malliavin Calculus and Large Deviations Principle. Remarkably, we find that both approaches complement each other and provide a very accurate description of the short-time implied volatility smile. Building upon these short-time asymptotics, we further provide approximation formulas for the at-the-money implied volatility and its skew. Finally, in the third part of the thesis we cross the bridge between the physical P and risk-neutral Q measures. We provide sufficient conditions for such change of measure to be well defined and explore the numerical estimation of the market risk premium process in rough volatility models.