  Geometric Asian options are a type of options where the payoff depends on the
geometric mean of the underlying asset over a certain period of time. This
paper is concerned with the pricing of such options for the class of
Volterra-Heston models, covering the rough Heston model. We are able to derive
semi-closed formulas for the prices of geometric Asian options with fixed and
floating strikes for this class of stochastic volatility models. These formulas
require the explicit calculation of the conditional joint Fourier transform of
the logarithm of the stock price and the logarithm of the geometric mean of the
stock price over time. Linking our problem to the theory of affine Volterra
processes, we find a representation of this Fourier transform as a suitably
constructed stochastic exponential, which depends on the solution of a
Riccati-Volterra equation. Finally we provide a numerical study for our results
in the rough Heston model.
