We consider the Fourier-Laplace transforms of a broad class of polynomial
Ornstein-Uhlenbeck (OU) volatility models, including the well-known
Stein-Stein, Sch\"obel-Zhu, one-factor Bergomi, and the recently introduced
Quintic OU models motivated by the SPX-VIX joint calibration problem. We show
the connection between the joint Fourier-Laplace functional of the log-price
and the integrated variance, and the solution of an infinite dimensional
Riccati equation. Next, under some non-vanishing conditions of the
Fourier-Laplace transforms, we establish an existence result for such Riccati
equation and we provide a discretized approximation of the joint characteristic
functional that is exponentially entire. On the practical side, we develop a
numerical scheme to solve the stiff infinite dimensional Riccati equations and
demonstrate the efficiency and accuracy of the scheme for pricing SPX options
and volatility swaps using Fourier and Laplace inversions, with specific
examples of the Quintic OU and the one-factor Bergomi models and their
calibration to real market data.