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 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 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. Welder condition and the process }\widehat{B}  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 .
We study fractional stochastic volatility models for asset prices , in which the volatility process is a positive continuous function \sigma of a continuous fractional stochastic process B. The main results obtained in the present paper are generalizations of small-noise and small-time large deviation principles for the log-price process due to Forde and Zhang. It is assumed in the present paper that the function \sigma satisfies a relatively mild condition expressed in terms of its local modulus of continuity in L^2 , while the process \widehat{B} is a Volterra type Gaussian process. The assumptions used by Forde and Zhang are more restrictive. It is supposed in their work that the function \sigma satisfies the global H\"{older condition and the process }\widehat{B} is fractional Brownian motion. In the present paper, we establish a small-noise large deviation principle for the log-price in a fractional stochastic volatility model, and under an additional assumption of the self-similarity of the process \widehat{B}, derive a similar 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 .
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