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Computational aspects of robust optimized certainty equivalents and option pricing

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  • Daniel Bartl
  • Samuel Drapeau
  • Ludovic Tangpi

Abstract

Accounting for model uncertainty in risk management and option pricing leads to infinite‐dimensional optimization problems that are both analytically and numerically intractable. In this article, we study when this hurdle can be overcome for the so‐called optimized certainty equivalent (OCE) risk measure—including the average value‐at‐risk as a special case. First, we focus on the case where the uncertainty is modeled by a nonlinear expectation that penalizes distributions that are “far” in terms of optimal‐transport distance (e.g. Wasserstein distance) from a given baseline distribution. It turns out that the computation of the robust OCE reduces to a finite‐dimensional problem, which in some cases can even be solved explicitly. This principle also applies to the shortfall risk measure as well as for the pricing of European options. Further, we derive convex dual representations of the robust OCE for measurable claims without any assumptions on the set of distributions. Finally, we give conditions on the latter set under which the robust average value‐at‐risk is a tail risk measure.

Suggested Citation

  • Daniel Bartl & Samuel Drapeau & Ludovic Tangpi, 2020. "Computational aspects of robust optimized certainty equivalents and option pricing," Mathematical Finance, Wiley Blackwell, vol. 30(1), pages 287-309, January.
  • Handle: RePEc:bla:mathfi:v:30:y:2020:i:1:p:287-309
    DOI: 10.1111/mafi.12203
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    Citations

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    Cited by:

    1. Michael Kupper & Max Nendel & Alessandro Sgarabottolo, 2023. "Risk measures based on weak optimal transport," Papers 2312.05973, arXiv.org.
    2. Yu Feng & Ralph Rudd & Christopher Baker & Qaphela Mashalaba & Melusi Mavuso & Erik Schlögl, 2021. "Quantifying the Model Risk Inherent in the Calibration and Recalibration of Option Pricing Models," Risks, MDPI, vol. 9(1), pages 1-20, January.
    3. Haktanır, Elif & Kahraman, Cengiz, 2023. "Intuitionistic fuzzy risk adjusted discount rate and certainty equivalent methods for risky projects," International Journal of Production Economics, Elsevier, vol. 257(C).
    4. Max Nendel & Alessandro Sgarabottolo, 2022. "A parametric approach to the estimation of convex risk functionals based on Wasserstein distance," Papers 2210.14340, arXiv.org, revised Aug 2024.
    5. Carole Bernard & Silvana M. Pesenti & Steven Vanduffel, 2024. "Robust distortion risk measures," Mathematical Finance, Wiley Blackwell, vol. 34(3), pages 774-818, July.
    6. Daniel Bartl & Stephan Eckstein & Michael Kupper, 2020. "Limits of random walks with distributionally robust transition probabilities," Papers 2007.08815, arXiv.org, revised Apr 2021.
    7. Junichi Imai, 2022. "A Numerical Method for Hedging Bermudan Options under Model Uncertainty," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 893-916, June.
    8. Kim, Sojung & Weber, Stefan, 2022. "Simulation methods for robust risk assessment and the distorted mix approach," European Journal of Operational Research, Elsevier, vol. 298(1), pages 380-398.
    9. Bingyan Han, 2022. "Distributionally robust risk evaluation with a causality constraint and structural information," Papers 2203.10571, arXiv.org, revised Aug 2024.
    10. Weiwei Li & Dejian Tian, 2023. "Robust optimized certainty equivalents and quantiles for loss positions with distribution uncertainty," Papers 2304.04396, arXiv.org.
    11. Anders Max Reppen & Halil Mete Soner, 2023. "Deep empirical risk minimization in finance: Looking into the future," Mathematical Finance, Wiley Blackwell, vol. 33(1), pages 116-145, January.
    12. Daniel Bartl & Ludovic Tangpi, 2020. "Non-asymptotic convergence rates for the plug-in estimation of risk measures," Papers 2003.10479, arXiv.org, revised Oct 2022.
    13. Fuhrmann, Sven & Kupper, Michael & Nendel, Max, 2021. "Wasserstein Perturbations of Markovian Transition Semigroups," Center for Mathematical Economics Working Papers 649, Center for Mathematical Economics, Bielefeld University.
    14. Zhi Chen & Weijun Xie, 2021. "Sharing the value‐at‐risk under distributional ambiguity," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 531-559, January.
    15. Jiarui Chu & Ludovic Tangpi, 2021. "Non-asymptotic estimation of risk measures using stochastic gradient Langevin dynamics," Papers 2111.12248, arXiv.org, revised Feb 2023.
    16. Sojung Kim & Stefan Weber, 2020. "Simulation Methods for Robust Risk Assessment and the Distorted Mix Approach," Papers 2009.03653, arXiv.org, revised Jan 2022.
    17. Marcelo Righi, 2024. "Robust convex risk measures," Papers 2406.12999, arXiv.org, revised Oct 2024.

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