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

Stochastic volatility models based on Gaussian processes, like fractional Brownian motion, are able to reproduce important stylised facts of financial markets such as rich autocorrelation structures, persistence and roughness of sample paths. This is made possible by virtue of the flexibility introduced in the choice of the covariance function of the Gaussian process. The price to pay is that in general, such models are no longer Markovian nor semimartingales, which limits their practical use. We derive, in two different ways, an explicit analytic expression for the joint characteristic function of the log-price and its integrated variance in general Gaussian stochastic volatility models. That analytic expression can be approximated by closed-form matrix expressions. This opens the door to fast approximation of the joint density and pricing of derivatives on both the stock and its realised variance by using Fourier inversion techniques. In the context of rough volatility modelling, our results apply to the (rough) fractional Stein-Stein model and provide the first analytic formulas for option pricing known to date, generalising that of Stein-Stein, Schobel-Zhu and a special case of Heston.