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Adapted Wasserstein Distances and Stability in Mathematical Finance

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  • Julio Backhoff-Veraguas
  • Daniel Bartl
  • Mathias Beiglbock
  • Manu Eder

Abstract

Assume that an agent models a financial asset through a measure Q with the goal to price / hedge some derivative or optimize some expected utility. Even if the model Q is chosen in the most skilful and sophisticated way, she is left with the possibility that Q does not provide an "exact" description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of Q? If we measure proximity with the usual Wasserstein distance (say), the answer is NO. Models which are similar w.r.t. Wasserstein distance may provide dramatically different information on which to base a hedging strategy. Remarkably, this can be overcome by considering a suitable "adapted" version of the Wasserstein distance which takes the temporal structure of pricing models into account. This adapted Wasserstein distance is most closely related to the nested distance as pioneered by Pflug and Pichler \cite{Pf09,PfPi12,PfPi14}. It allows us to establish Lipschitz properties of hedging strategies for semimartingale models in discrete and continuous time. Notably, these abstract results are sharp already for Brownian motion and European call options.

Suggested Citation

  • Julio Backhoff-Veraguas & Daniel Bartl & Mathias Beiglbock & Manu Eder, 2019. "Adapted Wasserstein Distances and Stability in Mathematical Finance," Papers 1901.07450, arXiv.org, revised May 2020.
    • Handle: RePEc:arx:papers:1901.07450
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    References listed on IDEAS

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    3. Michael Kupper & Max Nendel & Alessandro Sgarabottolo, 2023. "Risk measures based on weak optimal transport," Papers 2312.05973, arXiv.org.
    4. Erhan Bayraktar & Leonid Dolinskyi & Yan Dolinsky, 2020. "Extended weak convergence and utility maximisation with proportional transaction costs," Finance and Stochastics, Springer, vol. 24(4), pages 1013-1034, October.
    5. Nicolas Boursin & Carl Remlinger & Joseph Mikael, 2022. "Deep Generators on Commodity Markets Application to Deep Hedging," Risks, MDPI, vol. 11(1), pages 1-18, December.
    6. Bingyan Han, 2022. "Distributionally robust risk evaluation with a causality constraint and structural information," Papers 2203.10571, arXiv.org, revised Apr 2023.
    7. Daniel Bartl & Johannes Wiesel, 2022. "Sensitivity of multiperiod optimization problems in adapted Wasserstein distance," Papers 2208.05656, arXiv.org, revised Jun 2023.
    8. Beatrice Acciaio & Mathias Beiglboeck & Gudmund Pammer, 2020. "Weak Transport for Non-Convex Costs and Model-independence in a Fixed-Income Market," Papers 2011.04274, arXiv.org, revised Aug 2023.
    9. Benjamin Jourdain & Gudmund Pammer, 2023. "An extension of martingale transport and stability in robust finance," Papers 2304.09551, arXiv.org.
    10. Julio Backhoff-Veraguas & Gudmund Pammer, 2019. "Stability of martingale optimal transport and weak optimal transport," Papers 1904.04171, arXiv.org, revised Dec 2020.
    11. Beatrice Acciaio & Julio Backhoff-Veraguas & Junchao Jia, 2020. "Cournot-Nash equilibrium and optimal transport in a dynamic setting," Papers 2002.08786, arXiv.org, revised Nov 2020.
    12. Cohen, Asaf & Saha, Subhamay, 2021. "Asymptotic optimality of the generalized cμ rule under model uncertainty," Stochastic Processes and their Applications, Elsevier, vol. 136(C), pages 206-236.
    13. Beatrice Acciaio & Daniel Krv{s}ek & Gudmund Pammer, 2024. "Multicausal transport: barycenters and dynamic matching," Papers 2401.12748, arXiv.org.
    14. Beatrice Acciaio & Julio Backhoff & Gudmund Pammer, 2022. "Quantitative Fundamental Theorem of Asset Pricing," Papers 2209.15037, arXiv.org, revised Jan 2024.
    15. Nicolas Boursin & Carl Remlinger & Joseph Mikael & Carol Anne Hargreaves, 2022. "Deep Generators on Commodity Markets; application to Deep Hedging," Papers 2205.13942, arXiv.org.
    16. Park, Kyunghyun & Wong, Hoi Ying & Yan, Tingjin, 2023. "Robust retirement and life insurance with inflation risk and model ambiguity," Insurance: Mathematics and Economics, Elsevier, vol. 110(C), pages 1-30.
    17. Daniel Krv{s}ek & Gudmund Pammer, 2024. "General duality and dual attainment for adapted transport," Papers 2401.11958, arXiv.org.
    18. Beatrice Acciaio & Anastasis Kratsios & Gudmund Pammer, 2022. "Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer," Papers 2201.13094, arXiv.org, revised Mar 2023.
    19. John Armstrong & Andrei Ionescu, 2023. "Gamma Hedging and Rough Paths," Papers 2309.05054, arXiv.org, revised Mar 2024.
    20. Julio Backhoff-Veraguas & Xin Zhang, 2023. "Dynamic Cournot-Nash equilibrium: the non-potential case," Mathematics and Financial Economics, Springer, volume 17, number 1, June.

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