Large deviation principle for Volterra type fractional stochastic volatility models

TL;DR

This paper establishes a large deviation principle for fractional stochastic volatility models with Volterra Gaussian processes, extending previous results to weaker conditions and deriving asymptotic formulas for option pricing and implied volatility.

Contribution

It proves a large deviation principle under weaker assumptions on the volatility function and Gaussian process, including Volterra type processes and self-similarity, with applications to financial option pricing.

Findings

Large deviation principle for models with Volterra Gaussian processes

Asymptotic formulas for option prices and implied volatility

Extension of previous results to weaker regularity conditions

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 H\"{o}lder-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.