Complex discontinuities of the square root of Fredholm determinants in the Volterra Stein-Stein model

Fourier-based methods are central to option pricing and hedging when the Fourier–Laplace transform of the log-price and integrated variance is available semi-explicitly. This is the case for the Volterra Stein–Stein stochastic volatility model, where the characteristic function is known analytically. However, naive evaluation of this formula can produce discontinuities due to the complex square root of a Fredholm determinant, particularly when the determinant crosses the negative real axis, leading to severe numerical instabilities. We analyze this phenomenon by characterizing the determinant's crossing behavior for the joint Fourier–Laplace transform of integrated variance and log-price. We then derive an expression for the transform to account for such crossings and develop efficient algorithms to detect and handle them. Applied to Fourier-based pricing in the rough Stein–Stein model, our approach significantly improves accuracy while drastically reducing computational cost relative to existing methods.