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Minimax theorems for American options without time-consistency


  • Denis Belomestny

    () (University of Duisburg–Essen
    National University Higher School of Economics)

  • Tobias Hübner

    () (University of Duisburg–Essen)

  • Volker Krätschmer

    () (University of Duisburg–Essen)

  • Sascha Nolte

    () (University of Duisburg–Essen)


Abstract In this paper, we give sufficient conditions guaranteeing the validity of the well-known minimax theorem for the lower Snell envelope. Such minimax results play an important role in the characterisation of arbitrage-free prices of American contingent claims in incomplete markets. Our conditions do not rely on the notions of stability under pasting or time-consistency and reveal some unexpected connection between the minimax result and path properties of the corresponding process of densities. We exemplify our general results in the case of families of measures corresponding to diffusion exponential martingales.

Suggested Citation

  • Denis Belomestny & Tobias Hübner & Volker Krätschmer & Sascha Nolte, 2019. "Minimax theorems for American options without time-consistency," Finance and Stochastics, Springer, vol. 23(1), pages 209-238, January.
  • Handle: RePEc:spr:finsto:v:23:y:2019:i:1:d:10.1007_s00780-018-0378-2
    DOI: 10.1007/s00780-018-0378-2

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    References listed on IDEAS

    1. Amarante, Massimiliano, 2014. "A characterization of exact non-atomic market games," Journal of Mathematical Economics, Elsevier, vol. 54(C), pages 59-62.
    2. Bayraktar, Erhan & Yao, Song, 2011. "Optimal stopping for non-linear expectations--Part I," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 185-211, February.
    3. Erhan Bayraktar & Song Yao, 2009. "Optimal Stopping for Non-linear Expectations," Papers 0905.3601,, revised Jan 2011.
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    More about this item


    Minimax; Lower Snell envelope; Time-consistency; Nearly sub-Gaussian random fields; Metric entropies; Simons’ lemma;

    JEL classification:

    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty


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