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Robust estimation of superhedging prices

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  • Jan Obloj
  • Johannes Wiesel

Abstract

We consider statistical estimation of superhedging prices using historical stock returns in a frictionless market with d traded assets. We introduce a plugin estimator based on empirical measures and show it is consistent but lacks suitable robustness. To address this we propose novel estimators which use a larger set of martingale measures defined through a tradeoff between the radius of Wasserstein balls around the empirical measure and the allowed norm of martingale densities. We establish consistency and robustness of these estimators and argue that they offer a superior performance relative to the plugin estimator. We generalise the results by replacing the superhedging criterion with acceptance relative to a risk measure. We further extend our study, in part, to the case of markets with traded options, to a multiperiod setting and to settings with model uncertainty. We also study convergence rates of estimators and convergence of superhedging strategies.

Suggested Citation

  • Jan Obloj & Johannes Wiesel, 2018. "Robust estimation of superhedging prices," Papers 1807.04211, arXiv.org, revised Apr 2020.
  • Handle: RePEc:arx:papers:1807.04211
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    References listed on IDEAS

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    1. Friedrich Hubalek & Walter Schachermayer, 1998. "When Does Convergence of Asset Price Processes Imply Convergence of Option Prices?," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 385-403, October.
    2. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Quantitative Finance, Taylor & Francis Journals, vol. 10(6), pages 593-606.
    3. Rüdiger Kiesel & Robin Rühlicke & Gerhard Stahl & Jinsong Zheng, 2016. "The Wasserstein Metric and Robustness in Risk Management," Risks, MDPI, vol. 4(3), pages 1-14, August.
    4. Krätschmer, Volker & Schied, Alexander & Zähle, Henryk, 2012. "Qualitative and infinitesimal robustness of tail-dependent statistical functionals," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 35-47, January.
    5. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Post-Print hal-00413729, HAL.
    6. Takahashi, Rinya, 1987. "Normalizing constants of a distribution which belongs to the domain of attraction of the Gumbel distribution," Statistics & Probability Letters, Elsevier, vol. 5(3), pages 197-200, April.
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    Cited by:

    1. Daniel Bartl & Samuel Drapeau & Jan Obloj & Johannes Wiesel, 2020. "Sensitivity analysis of Wasserstein distributionally robust optimization problems," Papers 2006.12022, arXiv.org, revised Nov 2021.

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