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On the game interpretation of a shadow price process in utility maximization problems under transaction costs

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  • Dmitry Rokhlin

Abstract

To any utility maximization problem under transaction costs one can assign a frictionless model with a price process S ∗ , lying in the bid/ask price interval $[\underline{S}, \overline{S}]$ . Such a process S ∗ is called a shadow price if it provides the same optimal utility value as in the original model with bid-ask spread. We call S ∗ a generalized shadow price if the above property is true for the relaxed utility function in the frictionless model. This relaxation is defined as the lower semicontinuous envelope of the original utility, considered as a function on the set $[\underline{S}, \overline{S}]$ , equipped with some natural weak topology. We prove the existence of a generalized shadow price under rather weak assumptions and mark its relation to a saddle point of the trader/market zero-sum game, determined by the relaxed utility function. The relation of the notion of a shadow price to its generalization is illustrated by several examples. Also, we briefly discuss the interpretation of shadow prices via Lagrange duality. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Dmitry Rokhlin, 2013. "On the game interpretation of a shadow price process in utility maximization problems under transaction costs," Finance and Stochastics, Springer, vol. 17(4), pages 819-838, October.
  • Handle: RePEc:spr:finsto:v:17:y:2013:i:4:p:819-838
    DOI: 10.1007/s00780-013-0206-7
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    References listed on IDEAS

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    Cited by:

    1. Yiqing Lin & Junjian Yang, 2016. "Utility maximization problem with random endowment and transaction costs: when wealth may become negative," Papers 1604.08224, arXiv.org, revised Sep 2016.

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    More about this item

    Keywords

    Transaction costs; Utility maximization; Shadow price process; Lower semicontinuous envelope; Saddle point; Duality; 91G10; 91A30; 49K35; G11;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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