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Stochastic volatility models based on Gaussian processes, like fractionalBrownian motion, are able to reproduce important stylized facts of financialmarkets such as rich autocorrelation structures, persistence and roughness ofsample paths. This is made possible by virtue of the flexibility introduced inthe choice of the covariance function of the Gaussian process. The price to payis that, in general, such models are no longer Markovian nor semimartingales,which limits their practical use. We derive, in two different ways, an explicitanalytic expression for the joint characteristic function of the log-price andits integrated variance in general Gaussian stochastic volatility models. Suchanalytic expression can be approximated by closed form matrix expressions. Thisopens the door to fast approximation of the joint density and pricing ofderivatives on both the stock and its realized variance using Fourier inversiontechniques. In the context of rough volatility modeling, our results apply tothe (rough) fractional Stein--Stein model and provide the first analyticformulae for option pricing known to date, generalizing that of Stein--Stein,Sch{\"o}bel-Zhu and a special case of Heston.

The characteristic function of Gaussian stochastic volatility models: an analytic expression