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An optimal transport approach for the multiple quantile hedging problem

Author

Listed:
  • Cyril B'en'ezet

    (ENSIIE, LaMME)

  • Jean-Franc{c}ois Chassagneux

    (LPSM)

  • Mohan Yang

    (ADIA)

Abstract

We consider the multiple quantile hedging problem, which is a class of partial hedging problems containing as special examples the quantile hedging problem (F{\"o}llmer \& 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 B'en'ezet & Jean-Franc{c}ois Chassagneux & Mohan Yang, 2023. "An optimal transport approach for the multiple quantile hedging problem," Papers 2308.01121, arXiv.org.
    • Handle: RePEc:arx:papers:2308.01121
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    References listed on IDEAS

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    1. Gabriel Peyré & Marco Cuturi, 2017. "Computational Optimal Transport," Working Papers 2017-86, Center for Research in Economics and Statistics.
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    3. Bruno Bouchard & Thanh Nam Vu, 2012. "A Stochastic Target Approach for P&L Matching Problems," Mathematics of Operations Research, INFORMS, vol. 37(3), pages 526-558, August.
    4. Ludovic Moreau, 2011. "Stochastic target problems with controlled loss in jump diffusion models," Post-Print hal-00515522, HAL.
    5. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    6. Bruno Bouchard & Ngoc-Minh Dang, 2013. "Generalized stochastic target problems for pricing and partial hedging under loss constraints—application in optimal book liquidation," Finance and Stochastics, Springer, vol. 17(1), pages 31-72, January.
    7. Bruno Bouchard & Jean-François Chassagneux & Géraldine Bouveret, 2016. "A backward dual representation for the quantile hedging of Bermudan options," Post-Print hal-01069270, HAL.
    8. Ying Jiao & Olivier Klopfenstein & Peter Tankov, 2017. "Hedging under multiple risk constraints," Finance and Stochastics, Springer, vol. 21(2), pages 361-396, April.
    9. Philippe Briand & Romuald Elie & Ying Hu, 2018. "BSDEs with mean reflection," Post-Print hal-01318649, HAL.
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