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

Eduardo Abi Jaber () and Maxime Guellil ()
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Eduardo Abi Jaber: CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique - Inria - Institut National de Recherche en Informatique et en Automatique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique
Maxime Guellil: CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique - Inria - Institut National de Recherche en Informatique et en Automatique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique, LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité

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Abstract: 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.

Keywords: Volterra Stein-Stein model; Fredholm determinant; Fourier-Laplace transform; Complex discontinuities; Fourier pricing; Mathematics Subject Classification (2020): 91G20; 45P05 (search for similar items in EconPapers)
Date: 2025-03-04
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