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The Shannon Information of Filtrations and the Additional Logarithmic Utility of Insiders

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  • Stefan Ankirchner
  • Steffen Dereich
  • Peter Imkeller

Abstract

The background for the general mathematical link between utility and information theory investigated in this paper is a simple financial market model with two kinds of small traders: less informed traders and insiders, whose extra information is represented by an enlargement of the other agents’ filtration. The expected logarithmic utility increment, i.e. the difference of the insider’s and the less informed trader’s expected logarithmic utility is described in terms of the information drift, i.e. the drift one has to eliminate in order to perceive the price dynamics as a martingale from the insider’s perspective. On the one hand, we describe the information drift in a very general setting by natural quantities expressing the probabilistic better informed view of the world. This on the other hand allows us to identify the additional utility by entropy related quantities known from information theory. In particular, in a complete market in which the insider has some fixed additional information during the entire trading interval, its utility increment can be represented by the Shannon information of his extra knowledge. For general markets, and in some particular examples, we provide estimates of maximal utility by information inequalities.

Suggested Citation

  • Stefan Ankirchner & Steffen Dereich & Peter Imkeller, 2005. "The Shannon Information of Filtrations and the Additional Logarithmic Utility of Insiders," SFB 649 Discussion Papers SFB649DP2005-030, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    • Handle: RePEc:hum:wpaper:sfb649dp2005-030
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    References listed on IDEAS

    as
    1. Axel Grorud & Monique Pontier, 1998. "Insider Trading in a Continuous Time Market Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 1(03), pages 331-347.
    2. Amendinger, Jürgen & Imkeller, Peter & Schweizer, Martin, 1998. "Additional logarithmic utility of an insider," Stochastic Processes and their Applications, Elsevier, vol. 75(2), pages 263-286, July.
    3. Imkeller, Peter & Pontier, Monique & Weisz, Ferenc, 2001. "Free lunch and arbitrage possibilities in a financial market model with an insider," Stochastic Processes and their Applications, Elsevier, vol. 92(1), pages 103-130, March.
    4. José Mª Corcuera & Peter Imkeller & Arturo Kohatsu & David Nualart, 2003. "Additional utility of insiders with imperfect dynamical information," Economics Working Papers 675, Department of Economics and Business, Universitat Pompeu Fabra.
    5. Martin Schweizer & Dirk Becherer & Jürgen Amendinger, 2003. "A monetary value for initial information in portfolio optimization," Finance and Stochastics, Springer, vol. 7(1), pages 29-46.
    6. Duffie, Darrell & Huang, Chi-fu, 1986. "Multiperiod security markets with differential information : Martingales and resolution times," Journal of Mathematical Economics, Elsevier, vol. 15(3), pages 283-303, June.
    7. Amendinger, Jürgen & Imkeller, Peter & Schweizer, Martin, 1998. "Additional logarithmic utility of an insider," SFB 373 Discussion Papers 1998,25, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
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    Cited by:

    1. Stefan Ankirchner, 2005. "Utility duality under additional information: conditional measures versus filtration enlargements," SFB 649 Discussion Papers SFB649DP2005-029, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.

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    More about this item

    Keywords

    enlargement of filtration; logarithmic utility; utility maximization; heterogeneous information; insider model; Shannon information; information difference; entropy; differential entropy;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

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