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Prospective strict no-arbitrage and the fundamental theorem of asset pricing under transaction costs

Author

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  • Christoph Kühn

    (Goethe-Universität Frankfurt)

  • Alexander Molitor

    (Goethe-Universität Frankfurt)

Abstract

In discrete-time markets with proportional transaction costs, Schachermayer (Math. Financ. 14:19–48, 2004) showed that robust no-arbitrage is equivalent to the existence of a strictly consistent price system. In this paper, we introduce the concept of prospective strict no-arbitrage that is a variant of the strict no-arbitrage property from Kabanov et al. (Finance Stoch. 6:371–382, 2002). The prospective strict no-arbitrage condition is slightly weaker than the robust no-arbitrage condition, and it implies that the set of portfolios attainable from zero initial endowment is closed in probability. A weak version of prospective strict no-arbitrage turns out to be equivalent to the existence of a consistent price system. In contrast to the fundamental theorem of asset pricing of Schachermayer (Math. Financ. 14:19–48, 2004), the consistent frictionless prices may lie on the boundary of the bid–ask spread. On the technical level, a crucial difference to Schachermayer (Math. Financ. 14:19–48, 2004) and Kabanov et al. (Finance Stoch. 7:403–411, 2003) is that we prove closedness without having at hand that the null-strategies form a linear space.

Suggested Citation

  • Christoph Kühn & Alexander Molitor, 2019. "Prospective strict no-arbitrage and the fundamental theorem of asset pricing under transaction costs," Finance and Stochastics, Springer, vol. 23(4), pages 1049-1077, October.
    • Handle: RePEc:spr:finsto:v:23:y:2019:i:4:d:10.1007_s00780-019-00403-5
    DOI: 10.1007/s00780-019-00403-5
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    References listed on IDEAS

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    1. Dimitri De Vallière & Yuri Kabanov & Christophe Stricker, 2007. "No-arbitrage criteria for financial markets with transaction costs and incomplete information," Finance and Stochastics, Springer, vol. 11(2), pages 237-251, April.
    2. (**), Christophe Stricker & (*), Miklós Rásonyi & Yuri Kabanov, 2002. "No-arbitrage criteria for financial markets with efficient friction," Finance and Stochastics, Springer, vol. 6(3), pages 371-382.
    3. Yuri Kabanov, 2009. "Markets with Transaction Costs. Mathematical Theory," Post-Print hal-00488168, HAL.
    4. Saul Jacka & Abdelkarem Berkaoui & Jon Warren, 2008. "No arbitrage and closure results for trading cones with transaction costs," Finance and Stochastics, Springer, vol. 12(4), pages 583-600, October.
    5. Schachermayer, W., 1992. "A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time," Insurance: Mathematics and Economics, Elsevier, vol. 11(4), pages 249-257, December.
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    8. Bruno Bouchard, 2006. "No-arbitrage in Discrete-time Markets with Proportional Transaction Costs and General Information structure," Finance and Stochastics, Springer, vol. 10(2), pages 276-297, April.
    9. Bruno Bouchard, 2006. "No-arbitrage in Discrete-time Markets with Proportional Transaction Costs and General Information structure," Finance and Stochastics, Springer, vol. 10(2), pages 276-297, April.
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    Cited by:

    1. Christoph Kuhn, 2023. "The fundamental theorem of asset pricing with and without transaction costs," Papers 2307.00571, arXiv.org.
    2. Erhan Bayraktar & Matteo Burzoni, 2020. "On the quasi-sure superhedging duality with frictions," Finance and Stochastics, Springer, vol. 24(1), pages 249-275, January.
    3. René Carmona, 2022. "The influence of economic research on financial mathematics: Evidence from the last 25 years," Finance and Stochastics, Springer, vol. 26(1), pages 85-101, January.

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    More about this item

    Keywords

    Proportional transaction costs; Arbitrage; Fundamental theorem of asset pricing;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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