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The robust pricing–hedging duality for American options in discrete time financial markets

Author

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  • Anna Aksamit
  • Shuoqing Deng
  • Jan Obłój
  • Xiaolu Tan
Abstract
We investigate the pricing–hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, a family of European options, only statically. In the first part of the paper, we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a nondominated family of probability measures. Our first insight is that, by considering an enlargement of the space, we can see American options as European options and recover the pricing–hedging duality, which may fail in the original formulation. This can be seen as a weak formulation of the original problem. Our second insight is that a duality gap arises from the lack of dynamic consistency, and hence that a different enlargement, which reintroduces dynamic consistency is sufficient to recover the pricing–hedging duality: It is enough to consider fictitious extensions of the market in which all the assets are traded dynamically. In the second part of the paper, we study two important examples of the robust framework: the setup of Bouchard and Nutz and the martingale optimal transport setup of Beiglböck, Henry‐Labordère, and Penkner, and show that our general results apply in both cases and enable us to obtain the pricing–hedging duality for American options.

Suggested Citation

  • Anna Aksamit & Shuoqing Deng & Jan Obłój & Xiaolu Tan, 2019. "The robust pricing–hedging duality for American options in discrete time financial markets," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 861-897, July.
    • Handle: RePEc:bla:mathfi:v:29:y:2019:i:3:p:861-897
    DOI: 10.1111/mafi.12199
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    Citations

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    Cited by:

    1. Xuwei Yang & Anastasis Kratsios & Florian Krach & Matheus Grasselli & Aurelien Lucchi, 2023. "Regret-Optimal Federated Transfer Learning for Kernel Regression with Applications in American Option Pricing," Papers 2309.04557, arXiv.org, revised Oct 2024.
    2. Bartl, Daniel, 2020. "Conditional nonlinear expectations," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 785-805.
    3. Hölzermann, Julian, 2020. "Pricing Interest Rate Derivatives under Volatility Uncertainty," Center for Mathematical Economics Working Papers 633, Center for Mathematical Economics, Bielefeld University.
    4. Anna Aksamit & Ivan Guo & Shidan Liu & Zhou Zhou, 2021. "Superhedging duality for multi-action options under model uncertainty with information delay," Papers 2111.14502, arXiv.org, revised Nov 2023.
    5. Julian Holzermann, 2020. "Pricing Interest Rate Derivatives under Volatility Uncertainty," Papers 2003.04606, arXiv.org, revised Nov 2021.
    6. 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.
    7. Sester, Julian, 2024. "A multi-marginal c-convex duality theorem for martingale optimal transport," Statistics & Probability Letters, Elsevier, vol. 210(C).
    8. Tongseok Lim, 2023. "Optimal exercise decision of American options under model uncertainty," Papers 2310.14473, arXiv.org, revised Nov 2023.
    9. Erhan Bayraktar & Matteo Burzoni, 2020. "On the quasi-sure superhedging duality with frictions," Finance and Stochastics, Springer, vol. 24(1), pages 249-275, January.
    10. Mun-Chol Kim & Song-Chol Ryom, 2022. "Pathwise superhedging under proportional transaction costs," Mathematics and Financial Economics, Springer, volume 16, number 4, December.
    11. Ariel Neufeld & Julian Sester, 2021. "Model-free price bounds under dynamic option trading," Papers 2101.01024, arXiv.org, revised Jul 2021.

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