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Beyond Surrogate Modeling: Learning the Local Volatility Via Shape Constraints

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  • Marc Chataigner
  • Areski Cousin
  • St'ephane Cr'epey
  • Matthew Dixon
  • Djibril Gueye

Abstract

We explore the abilities of two machine learning approaches for no-arbitrage interpolation of European vanilla option prices, which jointly yield the corresponding local volatility surface: a finite dimensional Gaussian process (GP) regression approach under no-arbitrage constraints based on prices, and a neural net (NN) approach with penalization of arbitrages based on implied volatilities. We demonstrate the performance of these approaches relative to the SSVI industry standard. The GP approach is proven arbitrage-free, whereas arbitrages are only penalized under the SSVI and NN approaches. The GP approach obtains the best out-of-sample calibration error and provides uncertainty quantification.The NN approach yields a smoother local volatility and a better backtesting performance, as its training criterion incorporates a local volatility regularization term.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:2212.09957
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    References listed on IDEAS

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    1. Cousin, Areski & Maatouk, Hassan & Rullière, Didier, 2016. "Kriging of financial term-structures," European Journal of Operational Research, Elsevier, vol. 255(2), pages 631-648.
    2. Marc Chataigner & Stéphane Crépey & Matthew Dixon, 2020. "Deep Local Volatility," Post-Print hal-03910122, HAL.
    3. Jim Gatheral & Antoine Jacquier, 2014. "Arbitrage-free SVI volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 59-71, January.
    4. Johannes Ruf & Weiguan Wang, 2019. "Neural networks for option pricing and hedging: a literature review," Papers 1911.05620, arXiv.org, revised May 2020.
    5. Marc Chataigner & Stéphane Crépey & Matthew Dixon, 2020. "Deep Local Volatility," Risks, MDPI, vol. 8(3), pages 1-18, August.
    6. Bachoc, François & Bevilacqua, Moreno & Velandia, Daira, 2019. "Composite likelihood estimation for a Gaussian process under fixed domain asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    7. Marc Chataigner & St'ephane Cr'epey & Matthew Dixon, 2020. "Deep Local Volatility," Papers 2007.10462, arXiv.org.
    8. Martin Tegn'er & Stephen Roberts, 2019. "A Probabilistic Approach to Nonparametric Local Volatility," Papers 1901.06021, arXiv.org, revised Jan 2019.
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