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Sampling Of One-Dimensional Probability Measures In The Convex Order And Computation Of Robust Option Price Bounds

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  • AURÉLIEN ALFONSI

    (Université Paris-Est, Cermics (ENPC), INRIA, F-77455 Marne-la-Vallée, France)

  • JACOPO CORBETTA

    (Université Paris-Est, Cermics (ENPC), INRIA, F-77455 Marne-la-Vallée, France2Zeliade Systems, 56 Rue Jean-Jacques Rousseau, 75001 Paris, France)

  • BENJAMIN JOURDAIN

    (Université Paris-Est, Cermics (ENPC), INRIA, F-77455 Marne-la-Vallée, France)

Abstract

For μ and ν two probability measures on the real line such that μ is smaller than ν in the convex order, this property is in general not preserved at the level of the empirical measures μI = 1 I∑i=1Iδ Xi and νJ = 1 J∑j=1Jδ Yj, where (Xi)1≤i≤I (resp., (Yj)1≤j≤J) are independent and identically distributed according to μ (resp., ν). We investigate modifications of μI (resp., νJ) smaller than νJ (resp., greater than μI) in the convex order and weakly converging to μ (resp., ν) as I,J →∞. According to Kertz & Rösler(1992), the set of probability measures on the real line with a finite first order moment is a complete lattice for the increasing and the decreasing convex orders. For μ and ν in this set, this enables us to define a probability measure μ ∨ ν (resp., μ ∧ ν) greater than μ (resp., smaller than ν) in the convex order. We give efficient algorithms permitting to compute μ ∨ ν and μ ∧ ν (and therefore μI ∨ νJ and μI ∧ νJ) when μ and ν have finite supports. Last, we illustrate by numerical experiments the resulting sampling methods that preserve the convex order and their application to approximate martingale optimal transport problems and in particular to calculate robust option price bounds.

Suggested Citation

  • Aurélien Alfonsi & Jacopo Corbetta & Benjamin Jourdain, 2019. "Sampling Of One-Dimensional Probability Measures In The Convex Order And Computation Of Robust Option Price Bounds," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-41, May.
    • Handle: RePEc:wsi:ijtafx:v:22:y:2019:i:03:n:s021902491950002x
    DOI: 10.1142/S021902491950002X
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    References listed on IDEAS

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    1. Marco Scarsini & Alfred Muller, 2006. "Stochastic order relations and lattices of probability measures," Post-Print hal-00539119, HAL.
    2. David Hobson & Martin Klimmek, 2015. "Robust price bounds for the forward starting straddle," Finance and Stochastics, Springer, vol. 19(1), pages 189-214, January.
    3. Pierre Henry-Labordère & Nizar Touzi, 2016. "An explicit martingale version of the one-dimensional Brenier theorem," Finance and Stochastics, Springer, vol. 20(3), pages 635-668, July.
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    Citations

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    Cited by:

    1. Jonathan Ansari & Eva Lutkebohmert & Ariel Neufeld & Julian Sester, 2022. "Improved Robust Price Bounds for Multi-Asset Derivatives under Market-Implied Dependence Information," Papers 2204.01071, arXiv.org, revised Sep 2023.
    2. Julian Sester, 2023. "On intermediate Marginals in Martingale Optimal Transportation," Papers 2307.09710, arXiv.org, revised Nov 2023.
    3. Benjamin Jourdain & Kexin Shao, 2023. "Non-decreasing martingale couplings," Papers 2305.00565, arXiv.org.
    4. Beatrice Acciaio & Mathias Beiglböck & Gudmund Pammer, 2021. "Weak transport for non‐convex costs and model‐independence in a fixed‐income market," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1423-1453, October.
    5. Sester, Julian, 2024. "A multi-marginal c-convex duality theorem for martingale optimal transport," Statistics & Probability Letters, Elsevier, vol. 210(C).
    6. Julio Backhoff-Veraguas & Daniel Bartl & Mathias Beiglböck & Manu Eder, 2020. "Adapted Wasserstein distances and stability in mathematical finance," Finance and Stochastics, Springer, vol. 24(3), pages 601-632, July.
    7. Wiesel Johannes & Zhang Erica, 2023. "An optimal transport-based characterization of convex order," Dependence Modeling, De Gruyter, vol. 11(1), pages 1-15, January.
    8. 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.
    9. 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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