We study fractional stochastic volatility models for the asset price, in which the volatility process is a positive continuous function $\sigma$ of a continuous fractional stochastic process $\widehat{B}$. The main result obtained in the present paper is a generalization of the large deviation principle for the log-price process due to M. Forde and H. Zhang. In their work, Forde and Zhang assume that the function $\sigma$ satisfies a global H\"{o}lder condition and $\widehat{B}$ is fractional Brownian motion, whereas in the present paper, the function $\sigma$ satisfies a very mild condition expressed in terms of a local modulus of continuity, while the process $\widehat{B}$ is a general Volterra type Gaussian process. We establish a small-noise large deviation principle for the log-price in a fractional stochastic volatility model, and under an additional condition of self-similarity of the process $\widehat{B}$, derive a similar large deviation principle in the small-time regime. Using the latter result, we obtain asymptotic formulas for binary options, call and put options, and the implied volatility in the small-maturity small-log-moneyness regime.
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