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.