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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.
    6. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
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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.
    10. Walter Schachermayer, 2004. "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time," Mathematical Finance, Wiley Blackwell, vol. 14(1), pages 19-48, January.
    11. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, June.
    12. Saul Jacka & Abdelkarem Berkaoui & Jon Warren, 2006. "No-arbitrage and closure results for trading cones with transaction costs," Papers math/0602178, arXiv.org, revised Apr 2008.
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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, revised Aug 2024.
    2. 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.
    3. Erhan Bayraktar & Matteo Burzoni, 2020. "On the quasi-sure superhedging duality with frictions," Finance and Stochastics, Springer, vol. 24(1), pages 249-275, 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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