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Archil GULISASHVILI (Ohio University) “Gaussian Stochastic Volatility Models: Scaling Regimes, Large Deviations, and Moment Explosions”

May 13, 3:30 pm - 4:30 pm

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, which can be represented as a positive continuous function s of a continuous Gaussian process bB. We call the function s the volatility function, while the process bB is called 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 themodel is of Volterra type. Forde and Zhang established a large deviation principle for the log-price process in a Volterra type Gaussian stochastic volatility model under the assumptions that the function s is globally H¨older continuous and the process bB is fractional Brownian motion. We prove a similar small-noise large deviation principle under significantly weaker restrictions on s and bB. 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 Gaussian stochastic volatility model, and a sample path moderate deviation principle for general Gaussian models. In our work, we provide a unified approach to various scaling regimes associated with Gaussian models. More precisely, large deviation, moderate deviation, and central limit scalings will be discussed. We also study 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 the 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, if the volatility function

 

Details

Date:
May 13
Time:
3:30 pm - 4:30 pm
Event Categories:
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Organizers

Caroline HILLAIRET
Peter TANKOV