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

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Author Info
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)
Abstract

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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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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Find related papers by JEL classification:
G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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  1. 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. [Downloadable!]
  2. Mingxin Xu, 2006. "Risk measure pricing and hedging in incomplete markets," Annals of Finance, Springer, vol. 2(1), pages 51-71, January. [Downloadable!] (restricted)
    Other versions:
  3. Grzegorz Hara\'nczyk & Wojciech S{\l}omczy\'nski & Tomasz Zastawniak, 2007. "Relative and Discrete Utility Maximising Entropy," Quantitative Finance Papers 0709.1281, arXiv.org. [Downloadable!]
  4. 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. [Downloadable!] (restricted)
  5. 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. [Downloadable!] (restricted)
  6. Ioannis Karatzas & Constantinos Kardaras, 2008. "The numeraire portfolio in semimartingale financial models," Quantitative Finance Papers 0803.1877, arXiv.org. [Downloadable!]
  7. Ioannis Karatzas & Constantinos Kardaras, 2007. "The numéraire portfolio in semimartingale financial models," Finance and Stochastics, Springer, vol. 11(4), pages 447-493, October. [Downloadable!] (restricted)
  8. Antonis Papapantoleon, 2008. "An introduction to L\'{e}vy processes with applications in finance," Quantitative Finance Papers 0804.0482, arXiv.org, revised Nov 2008. [Downloadable!]
  9. Jan Bergenthum & Ludger Rüschendorf, 2006. "Comparison of Option Prices in Semimartingale Models," Finance and Stochastics, Springer, vol. 10(2), pages 222-249, April. [Downloadable!] (restricted)
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