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An optimal transport approach for the multiple quantile hedging problem

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  • Cyril Bénézet

    (ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise, LaMME - Laboratoire de Mathématiques et Modélisation d'Evry - ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise - UEVE - Université d'Évry-Val-d'Essonne - Université Paris-Saclay - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Jean-François Chassagneux

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité, UPCité - Université Paris Cité)

  • Mohan Yang

    (ADIA - Abu Dhabi Investment Authority)

Abstract

We consider the multiple quantile hedging problem, which is a class of partial hedging problems containing as special examples the quantile hedging problem (Föllmer & Leukert 1999) and the PnL matching problem (introduced in Bouchard & Vu 2012). In complete non-linear markets, we show that the problem can be reformulated as a kind of Monge optimal transport problem. Using this observation, we introduce a Kantorovitch version of the problem and prove that the value of both problems coincide. In the linear case, we thus obtain that the multiple quantile hedging problem can be seen as a semi-discrete optimal transport problem, for which we further introduce the dual problem. We then prove that there is no duality gap, allowing us to design a numerical method based on SGA algorithms to compute the multiple quantile hedging price.

Suggested Citation

  • Cyril Bénézet & Jean-François Chassagneux & Mohan Yang, 2023. "An optimal transport approach for the multiple quantile hedging problem," Working Papers hal-04175061, HAL.
    • Handle: RePEc:hal:wpaper:hal-04175061
    Note: View the original document on HAL open archive server: https://hal.science/hal-04175061v1
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