  We consider a stochastic volatility model where the dynamics of the
volatility are given by a possibly infinite linear combination of the elements
of the time extended signature of a Brownian motion. First, we show that the
model is remarkably universal, as it includes, but is not limited to, the
celebrated Stein-Stein, Bergomi, and Heston models, together with some
path-dependent variants. Second, we derive the joint characteristic functional
of the log-price and integrated variance provided that some infinite
dimensional extended tensor algebra valued Riccati equation admits a solution.
This allows us to price and (quadratically) hedge certain European and
path-dependent options using Fourier inversion techniques. We highlight the
efficiency and accuracy of these Fourier techniques in a comprehensive
numerical study.
