In this paper, we study Gaussian stochastic volatility models. The evolution of volatility in such a model is described by a stochastic process, which can be represented as a nonnegative continuous function of a continuous Gaussian process. If the volatility process exhibits fractional features, then the model is called a Gaussian fractional stochastic volatility model. Important examples of fractional volatility processes are fractional Brownian motion, the Riemann-Liouville fractional Brownian motion, and the fractional Ornstein-Uhlenbeck process. If the volatility process admits a Volterra type representation, then the model is of Volterra type. In the paper, we provide a unified approach to various scaling regimes associated with Gaussian models. More precisely, large deviation, moderate deviation, and central limit scalings are introduced in the paper. We prove a sample path large deviation principle for the log-price process in a Volterra type Gaussian stochastic volatility model and a sample path moderate deviation principle for the same process in a Gaussian stochastic volatility model. We also study the asymptotic behavior of call pricing functions and the implied volatility in mixed scaling regimes. It is also shown that tail estimates arising in the large deviation theory are locally uniform. Another problem addressed in the paper concerns moment explosions for the asset price process in a Gaussian stochastic volatility model. We prove that for the asset price process in an uncorrelated Gaussian stochastic volatility model with the volatility function growing faster than linearly, all the moments of order greater than one are infinite. Moment explosions are also characterized in the case of negatively, where \rho is the correlation coefficient, are shown for } correlated models, and partial results are obtained for positively correlated ones .
In this paper, we study Gaussian stochastic volatility models. The evolution of volatility in such a model is described by a stochastic process, which can be represented as a nonnegative continuous function of a continuous Gaussian process. If the volatility process exhibits fractional features, then the model is called a Gaussian fractional stochastic volatility model. Important examples of fractional volatility processes are fractional Brownian motion, the Riemann-Liouville fractional Brownian motion, and the fractional Ornstein-Uhlenbeck process. If the volatility process admits a Volterra type representation, then the model is of Volterra type. In the paper, we provide a unified approach to various scaling regimes associated with Gaussian models. More precisely, large deviation, moderate deviation, and central limit scalings are introduced in the paper. We prove a sample path large deviation principle for the log-price process in a Volterra type Gaussian stochastic volatility model and a sample path moderate deviation principle for the same process in a Gaussian stochastic volatility model. We also study the asymptotic behavior of call pricing functions and the implied volatility in mixed scaling regimes. It is also shown that tail estimates arising in the large deviation theory are locally uniform. Another problem addressed in the paper concerns moment explosions for the asset price process in a Gaussian stochastic volatility model. We prove that for the asset price process in an uncorrelated Gaussian stochastic volatility model with the volatility function growing faster than linearly, all the moments of order greater than one are infinite. Explosions of all the moments of order \gamma>(1-\rho^2)^{-1, where \rho is the correlation coefficient, are shown for } correlated models, and partial results are obtained for the case where \gamma=(1-\rho^2)^{-1 .
R
Moment explosions are also characterized in the case of negatively
Explosions of all the moments of order \gamma>(1-\rho^2)^{-1
R
positively correlated ones
the case where \gamma=(1-\rho^2)^{-1