We investigate analytical and numerical challenges arising in the computation of the Fourier-Laplace transform for the Volterra Stein-Stein model, where the volatility is driven by a Volterra-type Gaussian process. A key difficulty stems from the complex square root of a Fredholm determinant, which becomes discontinuous when the determinant crosses the negative real axis. We characterize these crossings and provide a corrected expression for the joint Fourier-Laplace transform of the log-price and integrated variance. Furthermore, we propose a new derivation of the transform by interpreting the joint law of the integrated variance and log-price as the infinite-dimensional limit of a Wishart distribution. This novel approach naturally yields a convergent numerical method, for which we establish a convergence rate. Applying our algorithms to Fourier-based pricing in the rough Stein-Stein model, we achieve a significant increase in accuracy while drastically reducing computational cost compared to existing methods.
This presentation is based on (1) and an ongoing working paper.

(1). Abi Jaber E., Guellil M., ``Complex discontinuities of the square root of Fredholm determinants in the Volterra Stein--Stein model,'', arXiv:2503.02965 (2025).