Sandwiched Volterra Volatility models: theory and applications

Abstract

This thesis begins with an introduction to mathematical finance, reviewing the historical development of asset-price models and several stylised facts observed in financial markets. We then motivate the study of fractional and rough volatility models, which are driven by fractional Brownian motion and provide a more realistic description of volatility dynamics. The necessary definitions and theoretical results concerning fractional Brownian motion are also presented.

Building on the work of Di Nunno et al. [15], we study the Sandwiched Volterra Volatility (SVV) model. In particular, we reformulate the general SVV framework in a one-dimensional setting, which allows for a more transparent exposition of the model and more detailed proofs of several key results. We further illustrate the model through a number of examples. In addition, we investigate the Malliavin differentiability of both the volatility and asset-price processes within the SVV framework and provide rigorous proofs of the corresponding results.

Finally, we analyse classical rough volatility models in order to characterise those that can be represented within the Sandwiched Volterra Volatility (SVV) framework. We show that most of the models studied fall into this class, with the exception of the Rough Fractional Stochastic Volatility model (for Hurst parameter $H \neq \frac{1}{2}$), the rough Heston model, and the quadratic rough Heston model.