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On the existence of shadow prices

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Abstract

For utility maximization problems under proportional transaction costs, it has been observed that the original market with transaction costs can sometimes be replaced by a frictionless shadow market that yields the same optimal strategy and utility. However, the question of whether or not this indeed holds in generality has remained elusive so far. In this paper, we present a counterexample which shows that shadow prices may fail to exist. On the other hand, we prove that short selling constraints are a sufficient condition to warrant their existence, even in very general multi-currency market models with possibly discontinuous bid–ask spreads.

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Notes

  1. That is, a consistent price system in the terminology of Schachermayer [31].

  2. Cf. Definition 3.9 below for the formal definition.

  3. In the general multi-currency notation introduced below, this corresponds to [1/π 21,π 12], where π ij denotes the number of units of asset i for which the agent can buy one unit of asset j.

  4. K(Π) contains precisely the solvent portfolios that can be liquidated to zero by trading according to the bid–ask matrix Π and possibly throwing away positive asset holdings.

  5. In the absence of constraints, similar existence results have been established for increasingly general models of the bid–ask spread by [2, 4, 5, 9, 15].

  6. In particular, we do not restrict ourselves to finite variation strategies here.

  7. In fact, J can be seen as the restriction to \(\mathbb{R}^{d}_{+} \setminus \{0\}\) of some other concave function defined on K 0 that allows for negative initial endowment (but forces the agent to make an instantaneous trade at time 0 in that case).

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Acknowledgements

The authors are grateful to Bruno Bouchard, Christoph Czichowsky, Paolo Guasoni, Ioannis Karatzas, Marcel Nutz, Mark P. Owen, and Walter Schachermayer for fruitful discussions, and also acknowledge the constructive comments of two anonymous referees and the editor, Martin Schweizer. The second author thanks the “Chaire Les Particuliers Face aux Risques”, Fondation du Risque (Groupama-ENSAE-Dauphine), the GIP-ANR “Risk” project for their support. The fourth author was partially supported by the National Centre of Competence in Research “Financial Valuation and Risk Management” (NCCR FINRISK), Project D1 (Mathematical Methods in Financial Risk Management), of the Swiss National Science Foundation (SNF).

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Authors and Affiliations

  1. CREST and Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris, France

    Giuseppe Benedetti

  2. Laboratoire Analyse, Géométrie et Applications, Université Paris 13 and CREST, 99, Avenue Jean-Baptiste Clément, 93430, Villetaneuse, France

    Luciano Campi

  3. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Westring 383, 24118, Kiel, Germany

    Jan Kallsen

  4. Departement für Mathematik, and Swiss Finance Institute, ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland

    Johannes Muhle-Karbe

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  1. Giuseppe Benedetti
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  2. Luciano Campi
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  3. Jan Kallsen
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  4. Johannes Muhle-Karbe
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Correspondence to Luciano Campi.

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Benedetti, G., Campi, L., Kallsen, J. et al. On the existence of shadow prices. Finance Stoch 17, 801–818 (2013). https://doi.org/10.1007/s00780-012-0201-4

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