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The Measure Preserving Martingale Sinkhorn Algorithm

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  • Benjamin Joseph
  • Gregoire Loeper
  • Jan Obloj

Abstract

We contribute to the recent studies of the so-called Bass martingale. Backhoff-Veraguas et al. (2020) showed it is the solution to the martingale Benamou-Brenier (mBB) problem, i.e., among all martingales with prescribed initial and terminal distributions it is the one closest to the Brownian motion. We link it with semimartingale optimal transport and deduce an alternative way to derive the dual formulation recently obtained in Backhoff-Veraguas et al. (2023). We then consider computational methods to compute the Bass martingale. The dual formulation of the transport problem leads to an iterative scheme that mirrors to the celebrated Sinkhorn algorithm for entropic optimal transport. We call it the measure preserving martingale Sinkhorn (MPMS) algorithm. We prove that in any dimension, each step of the algorithm improves the value of the dual problem, which implies its convergence. Our MPMS algorithm is equivalent to the fixed-point method of Conze and Henry-Labordere (2021), studied in Acciaio et al. (2023), and performs very well on a range of examples, including real market data.

Suggested Citation

  • Benjamin Joseph & Gregoire Loeper & Jan Obloj, 2023. "The Measure Preserving Martingale Sinkhorn Algorithm," Papers 2310.13797, arXiv.org, revised May 2024.
  • Handle: RePEc:arx:papers:2310.13797
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    References listed on IDEAS

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    1. Bruno Bouchard & G. Loeper & Y. Zou, 2017. "Hedging of covered options with linear market impact and gamma constraint," Post-Print hal-01611790, HAL.
    2. Bruno Bouchard & G Loeper & Y Zou, 2016. "Almost-sure hedging with permanent price impact," Post-Print hal-01133223, HAL.
    3. Bruno Bouchard & G Loeper & Y Zou, 2017. "Hedging of covered options with linear market impact and gamma constraint," Post-Print hal-01247523, HAL.
    4. Ivan Guo & Grégoire Loeper & Shiyi Wang, 2022. "Calibration of local‐stochastic volatility models by optimal transport," Mathematical Finance, Wiley Blackwell, vol. 32(1), pages 46-77, January.
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    Cited by:

    1. Julio Backhoff-Veraguas & Gregoire Loeper & Jan Obloj, 2024. "Geometric Martingale Benamou-Brenier transport and geometric Bass martingales," Papers 2406.04016, arXiv.org.
    2. Beatrice Acciaio & Antonio Marini & Gudmund Pammer, 2023. "Calibration of the Bass Local Volatility model," Papers 2311.14567, arXiv.org.
    3. Mathias Beiglbock & Gudmund Pammer & Lorenz Riess, 2024. "Change of numeraire for weak martingale transport," Papers 2406.07523, arXiv.org.

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