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Martingale property and moment explosions in signature volatility models

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  • Eduardo Abi Jaber
  • Paul Gassiat
  • Dimitri Sotnikov

Abstract

We study the martingale property and moment explosions of a signature volatility model, where the volatility process of the log-price is given by a linear form of the signature of a time-extended Brownian motion. Excluding trivial cases, we demonstrate that the price process is a true martingale if and only if the order of the linear form is odd and a correlation parameter is negative. The proof involves a fine analysis of the explosion time of a signature stochastic differential equation. This result is of key practical relevance, as it highlights that, when used for approximation purposes, the linear combination of signature elements must be taken of odd order to preserve the martingale property. Once martingality is established, we also characterize the existence of higher moments of the price process in terms of a condition on a correlation parameter.

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  • Eduardo Abi Jaber & Paul Gassiat & Dimitri Sotnikov, 2025. "Martingale property and moment explosions in signature volatility models," Papers 2503.17103, arXiv.org.
    • Handle: RePEc:arx:papers:2503.17103
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    References listed on IDEAS

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    1. repec:hal:wpaper:hal-03909334 is not listed on IDEAS
    2. Christa Cuchiero & Guido Gazzani & Janka Möller & Sara Svaluto‐Ferro, 2025. "Joint calibration to SPX and VIX options with signature‐based models," Mathematical Finance, Wiley Blackwell, vol. 35(1), pages 161-213, January.
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    6. Julien Guyon & Jordan Lekeufack, 2023. "Volatility is (mostly) path-dependent," Quantitative Finance, Taylor & Francis Journals, vol. 23(9), pages 1221-1258, September.
    7. Eduardo Abi Jaber & Camille Illand & Shaun & Li, 2022. "The quintic Ornstein-Uhlenbeck volatility model that jointly calibrates SPX & VIX smiles," Papers 2212.10917, arXiv.org, revised May 2023.
    8. Eduardo Abi Jaber & Camille Illand & Shaun Xiaoyuan Li, 2023. "The quintic Ornstein-Uhlenbeck volatility model that jointly calibrates SPX & VIX smiles," Post-Print hal-03909334, HAL.
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