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Adapted Wasserstein distances and stability in mathematical finance

Author

Listed:
  • Julio Backhoff-Veraguas

    (University of Vienna
    University of Twente)

  • Daniel Bartl

    (University of Vienna)

  • Mathias Beiglböck

    (University of Vienna)

  • Manu Eder

    (University of Vienna)

Abstract

Assume that an agent models a financial asset through a measure ℚ with the goal to price/hedge some derivative or optimise some expected utility. Even if the model ℚ is chosen in the most skilful and sophisticated way, the agent 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 with respect to the 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 (SIAM J. Optim. 20:1406–1420, 2009, SIAM J. Optim. 22:1–23, 2012, Multistage Stochastic Optimization, 2014). 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 Beiglböck & Manu Eder, 2020. "Adapted Wasserstein distances and stability in mathematical finance," Finance and Stochastics, Springer, vol. 24(3), pages 601-632, July.
    • Handle: RePEc:spr:finsto:v:24:y:2020:i:3:d:10.1007_s00780-020-00426-3
    DOI: 10.1007/s00780-020-00426-3
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    References listed on IDEAS

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    Cited by:

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    3. Bingyan Han, 2022. "Distributionally robust risk evaluation with a causality constraint and structural information," Papers 2203.10571, arXiv.org, revised Apr 2023.
    4. 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.
    5. Benjamin Jourdain & Gudmund Pammer, 2023. "An extension of martingale transport and stability in robust finance," Papers 2304.09551, arXiv.org.
    6. 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.
    7. 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.
    8. Beatrice Acciaio & Daniel Krv{s}ek & Gudmund Pammer, 2024. "Multicausal transport: barycenters and dynamic matching," Papers 2401.12748, arXiv.org.
    9. Nicolas Boursin & Carl Remlinger & Joseph Mikael & Carol Anne Hargreaves, 2022. "Deep Generators on Commodity Markets; application to Deep Hedging," Papers 2205.13942, arXiv.org.
    10. Daniel Krv{s}ek & Gudmund Pammer, 2024. "General duality and dual attainment for adapted transport," Papers 2401.11958, arXiv.org.
    11. 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.
    12. John Armstrong & Andrei Ionescu, 2023. "Gamma Hedging and Rough Paths," Papers 2309.05054, arXiv.org, revised Mar 2024.
    13. Julio Backhoff-Veraguas & Xin Zhang, 2023. "Dynamic Cournot-Nash equilibrium: the non-potential case," Mathematics and Financial Economics, Springer, volume 17, number 1, June.
    14. Erhan Bayraktar & Bingyan Han, 2023. "Fitted Value Iteration Methods for Bicausal Optimal Transport," Papers 2306.12658, arXiv.org, revised Nov 2023.
    15. Marlon Moresco & M'elina Mailhot & Silvana M. Pesenti, 2023. "Uncertainty Propagation and Dynamic Robust Risk Measures," Papers 2308.12856, arXiv.org, revised Feb 2024.
    16. 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.
    17. Daniel Bartl & Johannes Wiesel, 2022. "Sensitivity of multiperiod optimization problems in adapted Wasserstein distance," Papers 2208.05656, arXiv.org, revised Jun 2023.
    18. Beatrice Acciaio & Julio Backhoff & Gudmund Pammer, 2022. "Quantitative Fundamental Theorem of Asset Pricing," Papers 2209.15037, arXiv.org, revised Jan 2024.
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    More about this item

    Keywords

    Hedging; Utility maximisation; Optimal transport; Causal optimal transport; Wasserstein distance; Sensitivity; Stability;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium

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