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Minimax and minimal distance martingale measures and their relationship to portfolio optimization


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  • Thomas Goll

    (Institut für Mathematische Stochastik, Universität Freiburg i. Br., Eckerstraße 1, 79104 Freiburg i. Br., Germany Manuscript)

  • Ludger Rüschendorf

    (Institut für Mathematische Stochastik, Universität Freiburg i. Br., Eckerstraße 1, 79104 Freiburg i. Br., Germany Manuscript)

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    In this paper we give a characterization of minimal distance martingale measures with respect to f-divergence distances in a general semimartingale market model. We provide necessary and sufficient conditions for minimal distance martingale measures and determine them explicitly for exponential Lévy processes with respect to several classical distances. It is shown that the minimal distance martingale measures are equivalent to minimax martingale measures with respect to related utility functions and that optimal portfolios can be characterized by them. Related results in the context of continuous-time diffusion models were first obtained by He and Pearson (1991b) and Karatzas et al. (1991) and in a general semimartingale setting by Kramkov and Schachermayer (1999). Finally parts of the results are extended to utility-based hedging.

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    Bibliographic Info

    Article provided by Springer in its journal Finance and Stochastics.

    Volume (Year): 5 (2001)
    Issue (Month): 4 ()
    Pages: 557-581

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    Handle: RePEc:spr:finsto:v:5:y:2001:i:4:p:557-581

    Note: received: January 2000; final version received: February 2001
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    Cited by:
    1. Tahir Choulli & Christophe Stricker & Jia Li, 2007. "Minimal Hellinger martingale measures of order q," Finance and Stochastics, Springer, vol. 11(3), pages 399-427, July.
    2. Mingxin Xu, 2006. "Risk measure pricing and hedging in incomplete markets," Annals of Finance, Springer, vol. 2(1), pages 51-71, January.
    3. Anastasia Ellanskaya & Lioudmila Vostrikova, 2013. "Utility maximisation and utility indifference price for exponential semi-martingale models with random factor," Papers 1303.1134,
    4. Friedrich Hubalek & Carlo Sgarra, 2006. "Esscher transforms and the minimal entropy martingale measure for exponential Levy models," Quantitative Finance, Taylor & Francis Journals, vol. 6(2), pages 125-145.
    5. Keita Owari, 2011. "On Admissible Strategies in Robust Utility Maximization," Papers 1109.5512,, revised Mar 2012.
    6. Alexander Schied & Ching-Tang Wu, 2005. "Duality theory for optimal investments under model uncertainty," SFB 649 Discussion Papers SFB649DP2005-025, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany, revised Sep 2005.
    7. Tsukasa Fujiwara, 2004. "From the Minimal Entropy Martingale Measures to the Optimal Strategies for the Exponential Utility Maximization: the Case of Geometric Lévy Processes," Asia-Pacific Financial Markets, Springer, vol. 11(4), pages 367-391, December.
    8. Ioannis Karatzas & Constantinos Kardaras, 2008. "The numeraire portfolio in semimartingale financial models," Papers 0803.1877,
    9. Yan, Jun & Gao, Fuqing, 2013. "The minimal entropy martingale measure of a jump process influenced by jump times," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 83-88.
    10. Grzegorz Hara\'nczyk & Wojciech S{\l}omczy\'nski & Tomasz Zastawniak, 2007. "Relative and Discrete Utility Maximising Entropy," Papers 0709.1281,
    11. Gundel, Anne & Weber, Stefan, 2008. "Utility maximization under a shortfall risk constraint," Journal of Mathematical Economics, Elsevier, vol. 44(11), pages 1126-1151, December.
    12. Antonis Papapantoleon, 2008. "An introduction to L\'{e}vy processes with applications in finance," Papers 0804.0482,, revised Nov 2008.
    13. Keita Owari, 2011. "ON ADMISSIBLE STRATEGIES IN ROBUST UTILITY MAXIMIZATION (Forthcoming in "Mathematics and Financial Economics")," CARF F-Series CARF-F-257, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    14. Siddiqi, Hammad, 2013. "Mental Accounting: A Closed-Form Alternative to the Black Scholes Model," MPRA Paper 50759, University Library of Munich, Germany.
    15. Ioannis Karatzas & Constantinos Kardaras, 2007. "The numéraire portfolio in semimartingale financial models," Finance and Stochastics, Springer, vol. 11(4), pages 447-493, October.
    16. Kais Hamza & Fima C. Klebaner & Zinoviy Landsman & Ying-Oon Tan, 2014. "Option Pricing for Symmetric L\'evy Returns with Applications," Papers 1402.1554,
    17. Jan Bergenthum & Ludger Rüschendorf, 2006. "Comparison of Option Prices in Semimartingale Models," Finance and Stochastics, Springer, vol. 10(2), pages 222-249, April.


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