Large Deviation Principle for Volterra Type Fractional Stochastic Volatility Models

Abstract

We study fractional stochastic volatility models in which the volatility process is a positive continuous function $\sigma$ of a continuous Gaussian process $\widehat{B}$. Forde and Zhang established a large deviation principle for the log-price process in such a model under the assumptions that the function $\sigma$ is globally Hölder continuous and the process $\widehat{B}$ is fractional Brownian motion. In the present paper, we prove a similar small-noise large deviation principle under weaker restrictions on $\sigma$ and $\widehat{B}$. We assume that $\sigma$ satisfies a mild local regularity condition, while the process $\widehat{B}$ is a Volterra type Gaussian process. Under an additional assumption of the self-similarity of the process $\widehat{B}$, we derive a large deviation principle in the small-time regime. As an application, we obtain asymptotic formulas for binary options, call and put pricing functions, and the implied volatility in certain mixed regimes.