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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, that is, the difference of the insider's and the less informed trader's expected logarithmic utility is described in terms of the information drift, that is, 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," Papers math/0503013, arXiv.org, revised May 2006.
  • Handle: RePEc:arx:papers:math/0503013
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    Cited by:

    1. Albina Danilova & Michael Monoyios & Andrew Ng, 2009. "Optimal investment with inside information and parameter uncertainty," Papers 0911.3117, arXiv.org, revised Feb 2010.
    2. Huy N. Chau & Andrea Cosso & Claudio Fontana, 2018. "The value of informational arbitrage," Papers 1804.00442, arXiv.org.
    3. Buckley, Winston S. & Long, Hongwei, 2015. "A discontinuous mispricing model under asymmetric information," European Journal of Operational Research, Elsevier, vol. 243(3), pages 944-955.
    4. repec:spr:finsto:v:22:y:2018:i:1:d:10.1007_s00780-017-0345-3 is not listed on IDEAS
    5. Ankirchner, Stefan, 2008. "On filtration enlargements and purely discontinuous martingales," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1662-1678, September.
    6. Peter Imkeller & Nicolas Perkowski, 2015. "The existence of dominating local martingale measures," Finance and Stochastics, Springer, vol. 19(4), pages 685-717, October.
    7. Kardaras, Constantinos, 2010. "The continuous behavior of the numéraire portfolio under small changes in information structure, probabilistic views and investment constraints," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 331-347, March.
    8. Constantinos Kardaras, 2009. "Num\'{e}raire-invariant preferences in financial modeling," Papers 0903.3736, arXiv.org, revised Nov 2010.
    9. Constantinos Kardaras & Jan Obłój & Eckhard Platen, 2017. "The Numéraire Property And Long-Term Growth Optimality For Drawdown-Constrained Investments," Mathematical Finance, Wiley Blackwell, vol. 27(1), pages 68-95, January.
    10. Constantinos Kardaras, 2008. "The continuous behavior of the numeraire portfolio under small changes in information structure, probabilistic views and investment constraints," Papers 0804.2912, arXiv.org, revised Nov 2009.
    11. Kardaras, Constantinos, 2010. "Numéraire-invariant preferences in financial modeling," LSE Research Online Documents on Economics 44993, London School of Economics and Political Science, LSE Library.
    12. Dario Gasbarra & Jos'e Igor Morlanes & Esko Valkeila, 2011. "Initial Enlargement in a Markov chain market model," Papers 1108.2623, arXiv.org, revised Aug 2011.
    13. Song, Shiqi, 2016. "Drift operator in a viable expansion of information flow," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2297-2322.
    14. Edward Hoyle & Andrea Macrina & Levent A. Menguturk, 2017. "Markov-Modulated Information Flows," Papers 1708.06948, arXiv.org.
    15. repec:eee:phsmap:v:494:y:2018:i:c:p:465-483 is not listed on IDEAS
    16. Ngoc Huy Chau & Wolfgang Runggaldier & Peter Tankov, 2016. "Arbitrage and utility maximization in market models with an insider," Papers 1608.02068, arXiv.org, revised Sep 2016.

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