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Simulating integrated Volterra square-root processes and Volterra Heston models via Inverse Gaussian

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  • Eduardo Abi Jaber
  • Elie Attal

Abstract

We introduce a novel simulation scheme, iVi (integrated Volterra implicit), for integrated Volterra square-root processes and Volterra Heston models based on the Inverse Gaussian distribution. The scheme is designed to handle $L^1$ kernels with singularities by relying solely on integrated kernel quantities, and it preserves the non-decreasing property of the integrated process. We establish weak convergence of the iVi scheme by reformulating it as a stochastic Volterra equation with a measure kernel and proving a stability result for this class of equations. Numerical results demonstrate that convergence is achieved with very few time steps. Remarkably, for the rough fractional kernel, unlike existing schemes, convergence seems to improve as the Hurst index $H$ decreases and approaches $-1/2$.

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  • Eduardo Abi Jaber & Elie Attal, 2025. "Simulating integrated Volterra square-root processes and Volterra Heston models via Inverse Gaussian," Papers 2504.19885, arXiv.org.
    • Handle: RePEc:arx:papers:2504.19885
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    References listed on IDEAS

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    1. Eduardo Abi Jaber, 2019. "Lifting the Heston model," Quantitative Finance, Taylor & Francis Journals, vol. 19(12), pages 1995-2013, December.
    2. Paul Jusselin & Mathieu Rosenbaum, 2020. "No‐arbitrage implies power‐law market impact and rough volatility," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1309-1336, October.
    3. Eduardo Abi Jaber & Christian Bayer & Simon Breneis, 2024. "State spaces of multifactor approximations of nonnegative Volterra processes," Papers 2412.17526, arXiv.org.
    4. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    5. Christian Bayer & Simon Breneis, 2024. "Efficient option pricing in the rough Heston model using weak simulation schemes," Quantitative Finance, Taylor & Francis Journals, vol. 24(9), pages 1247-1261, September.
    6. Eduardo Abi Jaber, 2024. "Simulation of square-root processes made simple: applications to the Heston model," Papers 2412.11264, arXiv.org, revised Jun 2025.
    7. Ryan McCrickerd, 2019. "On spatially irregular ordinary differential equations and a pathwise volatility modelling framework," Papers 1902.01673, arXiv.org, revised Sep 2021.
    8. Alfonsi, Aurélien, 2025. "Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation," Stochastic Processes and their Applications, Elsevier, vol. 181(C).
    9. Carole Bernard & Zhenyu Cui & Martin Forde & Antoine Jacquier & Don McLeish & Aleksandar Mijatović, 2013. "Correction note for ‘The large-maturity smile for the Heston model’," Finance and Stochastics, Springer, vol. 17(1), pages 223-224, January.
    10. Eduardo Abi Jaber & Nathan De Carvalho, 2023. "Reconciling rough volatility with jumps," Papers 2303.07222, arXiv.org, revised Sep 2024.
    11. Eduardo Abi Jaber & Nathan de Carvalho, 2024. "Reconciling rough volatility with jumps," Post-Print hal-04295416, HAL.
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    Cited by:

    1. Eduardo Abi Jaber, 2024. "Simulation of square-root processes made simple: applications to the Heston model," Post-Print hal-04839193, HAL.

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