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Archil GULISASHVILI (Ohio University) "Gaussian Stochastic Volatility Models: Scaling Regimes, Large Deviations, and Moment Explosions"
1st Monday of each month
Time: 3:30 pm – 4:30 pm
Date: 13th of May 2019
Place: Room 3105
Archil GULISASHVILI (Ohio University) “Gaussian Stochastic Volatility Models: Scaling Regimes, Large Deviations, and Moment Explosions”
Abstract : In a Gaussian stochastic volatility model, the evolution of volatility is described by a stochastic process that can be represented as a positive continuous function (the volatility function) of a continuous Gaussian process (the volatility 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 called a Volterra type stochastic volatility model. Forde and Zhang established a large deviation principle for the log-price process in a Volterra type model under the assumptions that the volatility function is globally Hölder continuous and the volatility process is fractional Brownian motion. We prove a similar small-noise large deviation principle under significantly weaker restrictions. More precisely, we assume that the volatility function satisfies a mild local regularity condition, while the volatility process is any Volterra type Gaussian process. Moreover, we establish a sample path large deviation principle for the log-price process in a Volterra type model, and a sample path moderate deviation principle for general Gaussian models. In addition, applications are given to the study of the asymptotic behavior of exit probabilities, call pricing functions, and the implied volatility in various mixed scaling regimes. Another problem addressed in our work concerns moment explosions for asset price processes. We prove that for such a process in an uncorrelated Gaussian stochastic volatility model, all the moments of order greater than one explode, provided that the volatility function grows faster than linearly. Partial results are also obtained for correlated models.