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A Wiener Chaos Approach to Martingale Modelling and Implied Volatility Calibration

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  • Pere Diaz-Lozano
  • Thomas K. Kloster

Abstract

Calibration to a surface of option prices requires specifying a suitably flexible martingale model for the discounted asset price under a risk-neutral measure. Assuming Brownian noise and mean-square integrability, we construct an over-parameterized model based on the martingale representation theorem. In particular, we approximate the terminal value of the martingale via a truncated Wiener--chaos expansion and recover the intermediate dynamics by computing the corresponding conditional expectations. Using the Hermite-polynomial formulation of the Wiener chaos, we obtain easily implementable expressions that enable fast calibration to a target implied-volatility surface. We illustrate the flexibility and expressive power of the resulting model through numerical experiments on both simulated and real market data.

Suggested Citation

  • Pere Diaz-Lozano & Thomas K. Kloster, 2026. "A Wiener Chaos Approach to Martingale Modelling and Implied Volatility Calibration," Papers 2602.16232, arXiv.org.
    • Handle: RePEc:arx:papers:2602.16232
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    References listed on IDEAS

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    1. Jim Gatheral & Radov{s} Radoiv{c}i'c, 2023. "A generalization of the rational rough Heston approximation," Papers 2310.09181, arXiv.org, revised Feb 2024.
    2. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
    3. Omar El Euch & Mathieu Rosenbaum, 2019. "The characteristic function of rough Heston models," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 3-38, January.
    4. Eduardo Abi Jaber & Louis-Amand G'erard, 2024. "Signature volatility models: pricing and hedging with Fourier," Papers 2402.01820, arXiv.org, revised Jun 2025.
    5. Torben G. Andersen & Nicola Fusari & Viktor Todorov, 2015. "Parametric Inference and Dynamic State Recovery From Option Panels," Econometrica, Econometric Society, vol. 83(3), pages 1081-1145, May.
    6. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    7. Eduardo Abi Jaber & Louis-Amand Gérard, 2025. "Signature volatility models: pricing and hedging with Fourier," Post-Print hal-04435238, HAL.
    8. Masaaki Fukasawa, 2011. "Asymptotic analysis for stochastic volatility: martingale expansion," Finance and Stochastics, Springer, vol. 15(4), pages 635-654, December.
    9. Stefan Gerhold & Christoph Gerstenecker & Arpad Pinter, 2019. "Moment explosions in the rough Heston model," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 575-608, December.
    10. 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.
    11. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 1997. "Empirical Performance of Alternative Option Pricing Models," Journal of Finance, American Finance Association, vol. 52(5), pages 2003-2049, December.
    12. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv.
    13. Christa Cuchiero & Eva Flonner & Kevin Kurt, 2024. "Robust financial calibration: a Bayesian approach for neural SDEs," Papers 2409.06551, arXiv.org, revised Jun 2025.
    14. Hideharu Funahashi & Masaaki Kijima, 2014. "An Extension of the Chaos Expansion Approximation for the Pricing of Exotic Basket Options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(2), pages 109-139, April.
    15. Giulia Di Nunno & Kęstutis Kubilius & Yuliya Mishura & Anton Yurchenko-Tytarenko, 2023. "From Constant to Rough: A Survey of Continuous Volatility Modeling," Mathematics, MDPI, vol. 11(19), pages 1-35, October.
    16. Jim Gatheral & Radoš Radoičić, 2024. "A generalization of the rational rough Heston approximation," Quantitative Finance, Taylor & Francis Journals, vol. 24(2), pages 329-335, January.
    17. Dias, Fabio S. & Peters, Gareth W., 2021. "Option pricing with polynomial chaos expansion stochastic bridge interpolators and signed path dependence," Applied Mathematics and Computation, Elsevier, vol. 411(C).
    18. Paolo Guasoni & Mikl'os R'asonyi & Walter Schachermayer, 2008. "Consistent price systems and face-lifting pricing under transaction costs," Papers 0803.4416, arXiv.org.
    19. Julien Guyon & Scander Mustapha, 2023. "Neural Joint S&P 500/VIX Smile Calibration," Post-Print hal-03932780, HAL.
    20. 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.
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