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From Bachelier to Dupire via optimal transport

Author

Listed:
  • Mathias Beiglböck

    (Universität Wien)

  • Gudmund Pammer

    (ETH Zürich)

  • Walter Schachermayer

    (Universität Wien)

Abstract

Famously, mathematical finance was started by Bachelier in his 1900 PhD thesis where – among many other achievements – he also provided a formal derivation of the Kolmogorov forward equation. This also forms the basis for Dupire’s (again formal) solution to the problem of finding an arbitrage-free model calibrated to a given volatility surface. The latter result has rigorous counterparts in the theorems of Kellerer and Lowther. In this survey article, we revisit these hallmarks of stochastic finance, highlighting the role played by some optimal transport results in this context.

Suggested Citation

  • Mathias Beiglböck & Gudmund Pammer & Walter Schachermayer, 2022. "From Bachelier to Dupire via optimal transport," Finance and Stochastics, Springer, vol. 26(1), pages 59-84, January.
    • Handle: RePEc:spr:finsto:v:26:y:2022:i:1:d:10.1007_s00780-021-00466-3
    DOI: 10.1007/s00780-021-00466-3
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    References listed on IDEAS

    as
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    1. Thomas J. Sargent & John Stachurski, 2022. "Economic Networks: Theory and Computation," Papers 2203.11972, arXiv.org, revised Jul 2022.
    2. Beatrice Acciaio & Antonio Marini & Gudmund Pammer, 2023. "Calibration of the Bass Local Volatility model," Papers 2311.14567, arXiv.org.

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    More about this item

    Keywords

    Bachelier; Dupire’s formula; Kellerer’s theorem; Optimal transport; Martingales; Peacocks;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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