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Deep Local Volatility

Author

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  • Marc Chataigner
  • St'ephane Cr'epey
  • Matthew Dixon

Abstract

Deep learning for option pricing has emerged as a novel methodology for fast computations with applications in calibration and computation of Greeks. However, many of these approaches do not enforce any no-arbitrage conditions, and the subsequent local volatility surface is never considered. In this article, we develop a deep learning approach for interpolation of European vanilla option prices which jointly yields the full surface of local volatilities. We demonstrate the modification of the loss function or the feed forward network architecture to enforce (hard constraints approach) or favor (soft constraints approach) the no-arbitrage conditions and we specify the experimental design parameters that are needed for adequate performance. A novel component is the use of the Dupire formula to enforce bounds on the local volatility associated with option prices, during the network fitting. Our methodology is benchmarked numerically on real datasets of DAX vanilla options.

Suggested Citation

  • Marc Chataigner & St'ephane Cr'epey & Matthew Dixon, 2020. "Deep Local Volatility," Papers 2007.10462, arXiv.org.
    • Handle: RePEc:arx:papers:2007.10462
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    References listed on IDEAS

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    1. Hutchinson, James M & Lo, Andrew W & Poggio, Tomaso, 1994. "A Nonparametric Approach to Pricing and Hedging Derivative Securities via Learning Networks," Journal of Finance, American Finance Association, vol. 49(3), pages 851-889, July.
    2. A Itkin, 2019. "Deep learning calibration of option pricing models: some pitfalls and solutions," Papers 1906.03507, arXiv.org.
    3. Garcia, Rene & Gencay, Ramazan, 2000. "Pricing and hedging derivative securities with neural networks and a homogeneity hint," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 93-115.
    4. Damien Ackerer & Natasa Tagasovska & Thibault Vatter, 2019. "Deep Smoothing of the Implied Volatility Surface," Papers 1906.05065, arXiv.org, revised Oct 2020.
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    Cited by:

    1. Samuel N. Cohen & Christoph Reisinger & Sheng Wang, 2022. "Estimating risks of option books using neural-SDE market models," Papers 2202.07148, arXiv.org.
    2. Lukas Gonon & Antoine Jacquier & Ruben Wiedemann, 2024. "Operator Deep Smoothing for Implied Volatility," Papers 2406.11520, arXiv.org, revised Oct 2024.
    3. Christa Cuchiero & Eva Flonner & Kevin Kurt, 2024. "Robust financial calibration: a Bayesian approach for neural SDEs," Papers 2409.06551, arXiv.org, revised Sep 2024.
    4. Samuel N. Cohen & Christoph Reisinger & Sheng Wang, 2021. "Arbitrage-free neural-SDE market models," Papers 2105.11053, arXiv.org, revised Aug 2021.
    5. Marc Chataigner & Areski Cousin & St'ephane Cr'epey & Matthew Dixon & Djibril Gueye, 2022. "Beyond Surrogate Modeling: Learning the Local Volatility Via Shape Constraints," Papers 2212.09957, arXiv.org.
    6. Fred Espen Benth & Nils Detering & Silvia Lavagnini, 2021. "Accuracy of deep learning in calibrating HJM forward curves," Digital Finance, Springer, vol. 3(3), pages 209-248, December.
    7. Zhe Wang & Nicolas Privault & Claude Guet, 2021. "Deep self-consistent learning of local volatility," Papers 2201.07880, arXiv.org, revised Nov 2023.

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