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A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options

Author

Listed:
  • Alfred Galichon

    (ECON - Département d'économie (Sciences Po) - Sciences Po - Sciences Po - CNRS - Centre National de la Recherche Scientifique)

  • Pierre Henri-Labordère
  • Nizar Touzi

    (CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

Abstract

We consider the problem of superhedging under volatility uncertainty for an investor allowed to dynamically trade the underlying asset, and statically trade European call options for all possible strikes with some given maturity. This problem is classically approached by means of the Skorohod Embedding Problem (SEP). Instead, we provide a dual formulation which converts the superhedging problem into a continuous martingale optimal transportation problem. We then show that this formulation allows us to recover previously known results about lookback options. In particular, our methodology induces a new proof of the optimality of Azéma–Yor solution of the SEP for a certain class of lookback options. Unlike the SEP technique, our approach applies to a large class of exotics and is suitable for numerical approximation techniques.

Suggested Citation

  • Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," Post-Print hal-03460952, HAL.
    • Handle: RePEc:hal:journl:hal-03460952
    DOI: 10.1214/13-AAP925
    Note: View the original document on HAL open archive server: https://sciencespo.hal.science/hal-03460952
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    References listed on IDEAS

    as
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    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Marcel Nutz & Johannes Wiesel & Long Zhao, 2022. "Limits of Semistatic Trading Strategies," Papers 2204.12251, arXiv.org.
    2. Alfred Galichon, 2021. "The Unreasonable Effectiveness of Optimal Transport in Economics," SciencePo Working papers Main hal-03936221, HAL.
    3. Anton Kolotilin & Roberto Corrao & Alexander Wolitzky, 2022. "Persuasion with Non-Linear Preferences," Papers 2206.09164, arXiv.org, revised Aug 2022.
    4. Maximilien Germain & Huyên Pham & Xavier Warin, 2022. "A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection," Post-Print hal-03498263, HAL.
    5. Julio Backhoff-Veraguas & Gregoire Loeper & Jan Obloj, 2024. "Geometric Martingale Benamou-Brenier transport and geometric Bass martingales," Papers 2406.04016, arXiv.org.
    6. Ariel Neufeld & Antonis Papapantoleon & Qikun Xiang, 2023. "Model-Free Bounds for Multi-Asset Options Using Option-Implied Information and Their Exact Computation," Management Science, INFORMS, vol. 69(4), pages 2051-2068, April.
    7. Tongseok Lim, 2023. "Replication of financial derivatives under extreme market models given marginals," Papers 2307.00807, arXiv.org.
    8. Alessandro Doldi & Marco Frittelli, 2023. "Entropy martingale optimal transport and nonlinear pricing–hedging duality," Finance and Stochastics, Springer, vol. 27(2), pages 255-304, April.
    9. Huy N. Chau & Masaaki Fukasawa & Miklós Rásonyi, 2022. "Super‐replication with transaction costs under model uncertainty for continuous processes," Mathematical Finance, Wiley Blackwell, vol. 32(4), pages 1066-1085, October.
    10. Benjamin Jourdain & Gudmund Pammer, 2023. "An extension of martingale transport and stability in robust finance," Papers 2304.09551, arXiv.org.
    11. Marcel Nutz & Johannes Wiesel & Long Zhao, 2022. "Martingale Schr\"odinger Bridges and Optimal Semistatic Portfolios," Papers 2204.12250, arXiv.org.
    12. Marcel Nutz & Johannes Wiesel & Long Zhao, 2023. "Martingale Schrödinger bridges and optimal semistatic portfolios," Finance and Stochastics, Springer, vol. 27(1), pages 233-254, January.
    13. Tongseok Lim, 2023. "Optimal exercise decision of American options under model uncertainty," Papers 2310.14473, arXiv.org, revised Nov 2023.
    14. Wiesel Johannes & Zhang Erica, 2023. "An optimal transport-based characterization of convex order," Dependence Modeling, De Gruyter, vol. 11(1), pages 1-15, January.
    15. Joshua Zoen-Git Hiew & Tongseok Lim & Brendan Pass & Marcelo Cruz de Souza, 2023. "Geometry of vectorial martingale optimal transport and robust option pricing," Papers 2309.04947, arXiv.org, revised Sep 2023.
    16. Benjamin Jourdain & Gilles Pagès, 2022. "Convex Order, Quantization and Monotone Approximations of ARCH Models," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2480-2517, December.
    17. Marcel Nutz & Johannes Wiesel & Long Zhao, 2023. "Limits of semistatic trading strategies," Mathematical Finance, Wiley Blackwell, vol. 33(1), pages 185-205, January.
    18. Alfred Galichon, 2021. "The Unreasonable Effectiveness of Optimal Transport in Economics," Working Papers hal-03936221, HAL.
    19. Haiyan Liu & Bin Wang & Ruodu Wang & Sheng Chao Zhuang, 2023. "Distorted optimal transport," Papers 2308.11238, arXiv.org.

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