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An Optimal Transport approach to arbitrage correction: Application to volatility Stress-Tests

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  • Marius Chevallier
  • Stefano De Marco
  • Pierre-Emmanuel L'evy-dit-Vehel

Abstract

We present a method based on optimal transport to remove arbitrage opportunities within a finite set of option prices. The method is notably intended for regulatory stress-tests, which impose to apply important local distortions to implied volatility surfaces. The resulting stressed option prices are naturally associated to a family of signed marginal measures: we formulate the process of removing arbitrage as a projection onto the subset of martingale measures with respect to a Wasserstein metric in the space of signed measures. We show how this projection problem can be recast as an optimal transport problem; in view of the numerical solution, we apply an entropic regularization technique. For the regularized problem, we derive a strong duality formula, show convergence results as the regularization parameter approaches zero, and formulate a multi-constrained Sinkhorn algorithm, where each iteration involves, at worse, finding the root of an explicit scalar function. The convergence of this algorithm is also established. We compare our method with the existing approach by [Cohen, Reisinger and Wang, Appl.\ Math.\ Fin.\ 2020] across various scenarios and test cases.

Suggested Citation

  • Marius Chevallier & Stefano De Marco & Pierre-Emmanuel L'evy-dit-Vehel, 2025. "An Optimal Transport approach to arbitrage correction: Application to volatility Stress-Tests," Papers 2501.12195, arXiv.org.
    • Handle: RePEc:arx:papers:2501.12195
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    References listed on IDEAS

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    1. Samuel N. Cohen & Christoph Reisinger & Sheng Wang, 2020. "Detecting and repairing arbitrage in traded option prices," Papers 2008.09454, arXiv.org.
    2. Cousot, Laurent, 2007. "Conditions on option prices for absence of arbitrage and exact calibration," Journal of Banking & Finance, Elsevier, vol. 31(11), pages 3377-3397, November.
    3. Samuel N. Cohen & Christoph Reisinger & Sheng Wang, 2020. "Detecting and Repairing Arbitrage in Traded Option Prices," Applied Mathematical Finance, Taylor & Francis Journals, vol. 27(5), pages 345-373, September.
    4. Carr, Peter & Madan, Dilip B., 2005. "A note on sufficient conditions for no arbitrage," Finance Research Letters, Elsevier, vol. 2(3), pages 125-130, September.
    5. Mathias Beiglbock & Pierre Henry-Labord`ere & Friedrich Penkner, 2011. "Model-independent Bounds for Option Prices: A Mass Transport Approach," Papers 1106.5929, arXiv.org, revised Feb 2013.
    6. Mark H. A. Davis & David G. Hobson, 2007. "The Range Of Traded Option Prices," Mathematical Finance, Wiley Blackwell, vol. 17(1), pages 1-14, January.
    7. Fengler, Matthias R. & Hin, Lin-Yee, 2015. "Semi-nonparametric estimation of the call-option price surface under strike and time-to-expiry no-arbitrage constraints," Journal of Econometrics, Elsevier, vol. 184(2), pages 242-261.
    8. Mathias Beiglböck & Pierre Henry-Labordère & Friedrich Penkner, 2013. "Model-independent bounds for option prices—a mass transport approach," Finance and Stochastics, Springer, vol. 17(3), pages 477-501, July.
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