Complex discontinuities of the square root of Fredholm determinants in the Volterra Stein-Stein model
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917082184024064 |
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| author | Jaber, Eduardo Abi Guellil, Maxime |
| author_facet | Jaber, Eduardo Abi Guellil, Maxime |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_02965 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complex discontinuities of the square root of Fredholm determinants in the Volterra Stein-Stein model Jaber, Eduardo Abi Guellil, Maxime Mathematical Finance Computational Finance 91G20, 45P05 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. |
| title | Complex discontinuities of the square root of Fredholm determinants in the Volterra Stein-Stein model |
| topic | Mathematical Finance Computational Finance 91G20, 45P05 |
| url | https://arxiv.org/abs/2503.02965 |