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On the Weak Error for Local Stochastic Volatility Models

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  • Peter K. Friz
  • Benjamin Jourdain
  • Thomas Wagenhofer
  • Alexandre Zhou

Abstract

Local stochastic volatility refers to a popular model class in applied mathematical finance that allows for "calibration-on-the-fly", typically via a particle method, derived from a formal McKean-Vlasov equation. Well-posedness of this limit is a well-known problem in the field; the general case is largely open, despite recent progress in Markovian situations. Our take is to start with a well-defined Euler approximation to the formal McKean-Vlasov equation, followed by a newly established half-step-scheme, allowing for good approximations of conditional expectations. In a sense, we do Euler first, particle second in contrast to previous works that start with the particle approximation. We show weak order one for the Euler discretization, plus error terms that account for the said approximation. The case of particle approximation is discussed in detail and the error rate is given in dependence of all parameters used.

Suggested Citation

  • Peter K. Friz & Benjamin Jourdain & Thomas Wagenhofer & Alexandre Zhou, 2025. "On the Weak Error for Local Stochastic Volatility Models," Papers 2506.10817, arXiv.org.
    • Handle: RePEc:arx:papers:2506.10817
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    References listed on IDEAS

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    1. Aitor Muguruza, 2019. "Not so Particular about Calibration: Smile Problem Resolved," Papers 1909.13366, arXiv.org.
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    4. Christian Bayer & Denis Belomestny & Oleg Butkovsky & John Schoenmakers, 2024. "A reproducing kernel Hilbert space approach to singular local stochastic volatility McKean–Vlasov models," Finance and Stochastics, Springer, vol. 28(4), pages 1147-1178, October.
    5. Martin Forde & Antoine Jacquier, 2011. "Small-Time Asymptotics for an Uncorrelated Local-Stochastic Volatility Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 18(6), pages 517-535, April.
    6. Mathias Beiglböck & George Lowther & Gudmund Pammer & Walter Schachermayer, 2024. "Faking Brownian motion with continuous Markov martingales," Finance and Stochastics, Springer, vol. 28(1), pages 259-284, January.
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