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

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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.

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Acknowledgements

We should like to thank the editor, Prof. Martin Schweizer, an anonymous Associate Editor and two anonymous referees for their valuable comments and suggestions from which the manuscript greatly benefitted.

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  1. Institut für Mathematik, Goethe-Universität Frankfurt, 60054, Frankfurt a.M., Germany

    Christoph Kühn & Alexander Molitor

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  1. Christoph Kühn
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  2. Alexander Molitor
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Correspondence to Christoph Kühn.

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Kühn, C., Molitor, A. Prospective strict no-arbitrage and the fundamental theorem of asset pricing under transaction costs. Finance Stoch 23, 1049–1077 (2019). https://doi.org/10.1007/s00780-019-00403-5

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