  We consider a stochastic volatility model where the dynamics of the
volatility are described by a linear function of the (time extended) signature
of a primary process which is supposed to be a polynomial diffusion. We obtain
closed form expressions for the VIX squared, exploiting the fact that the
truncated signature of a polynomial diffusion is again a polynomial diffusion.
Adding to such a primary process the Brownian motion driving the stock price,
allows then to express both the log-price and the VIX squared as linear
functions of the signature of the corresponding augmented process. This feature
can then be efficiently used for pricing and calibration purposes. Indeed, as
the signature samples can be easily precomputed, the calibration task can be
split into an offline sampling and a standard optimization. We also propose a
Fourier pricing approach for both VIX and SPX options exploiting that the
signature of the augmented primary process is an infinite dimensional affine
process. For both the SPX and VIX options we obtain highly accurate calibration
results, showing that this model class allows to solve the joint calibration
problem without adding jumps or rough volatility.
