Stochastic Volterra volatility models

Financial markets have extremely complex behavior that cannot be fully modeled using classical approaches. In particular, numerous empirical studies show that market volatility exhibits some form of long-range dependence and has time-varying Hölder regularity with prominent periods of “roughness” (i.e. of Hölder order ≈0.1). These two properties are far beyond the capabilities of classical Brownian diffusions and it is challenging to reproduce them simultaneously in one model. In the present thesis, we suggest a novel volatility modeling framework that grasps this unconventional behavior and solves a number of technical problems that are typical for classical stochastic volatility models. Namely, our model comprises the following properties: - flexibility in the noise: the suggested model accepts various drivers – from fractional Brownian motions with different Hurst indices to general Hölder continuous processes – to account for different option pricing phenomenons; - control over the moments of the price: the model ensures the existence of moments of necessary orders for the corresponding price process; - positivity: the volatility process is strictly positive and has inverse moments to ensure reasonable behavior of martingale densities. We also present a variety of associated numerical methods and propose practically feasible algorithms for various applications, such as the pricing of contingent claims (including options with discontinuous payoffs) and mean-square hedging.