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How Non-Arbitrage, Viability and Num\'eraire Portfolio are Related

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  • Tahir Choulli
  • Jun Deng
  • Junfeng Ma
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    Abstract

    This paper proposes two approaches that quantify the exact relationship among the viability, the absence of arbitrage, and/or the existence of the num\'eraire portfolio under minimal assumptions and for general continuous-time market models. Precisely, our first and principal contribution proves the equivalence among the No-Unbounded-Profit-with-Bounded-Risk condition (NUPBR hereafter), the existence of the num\'eraire portfolio, and the existence of the optimal portfolio under an equivalent probability measure for any "nice" utility and positive initial capital. Herein, a 'nice" utility is any smooth von Neumann-Morgenstern utility satisfying Inada's conditions and the elasticity assumptions of Kramkov and Schachermayer. Furthermore, the equivalent probability measure ---under which the utility maximization problems have solutions--- can be chosen as close to the real-world probability measure as we want (but might not be equal). Without changing the underlying probability measure and under mild assumptions, our second contribution proves that the NUPBR is equivalent to the "{\it local}" existence of the optimal portfolio. This constitutes an alternative to the first contribution, if one insists on working under the real-world probability. These two contributions lead naturally to new types of viability that we call weak and local viabilities.

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    File URL: http://arxiv.org/pdf/1211.4598
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1211.4598.

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    Date of creation: Nov 2012
    Date of revision: Jun 2014
    Handle: RePEc:arx:papers:1211.4598

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    1. Ralf Korn & Frank Oertel & Manfred Schäl, 2003. "Notes and Comments: The numeraire portfolio in financial markets modeled by a multi-dimensional jump diffusion process," Decisions in Economics and Finance, Springer, vol. 26(2), pages 153-166, November.
    2. Marco Frittelli, 2000. "The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 39-52.
    3. Hardy Hulley & Martin Schweizer, 2010. "M6 - On Minimal Market Models and Minimal Martingale Measures," Research Paper Series 280, Quantitative Finance Research Centre, University of Technology, Sydney.
    4. repec:fth:inseep:9513 is not listed on IDEAS
    5. Dirk Becherer, 2001. "The numeraire portfolio for unbounded semimartingales," Finance and Stochastics, Springer, vol. 5(3), pages 327-341.
    6. Jouini Elyes & Kallal Hedi, 1995. "Martingales and Arbitrage in Securities Markets with Transaction Costs," Journal of Economic Theory, Elsevier, vol. 66(1), pages 178-197, June.
    7. Yuri M. Kabanov & Christophe Stricker, 2002. "On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper," Mathematical Finance, Wiley Blackwell, vol. 12(2), pages 125-134.
    8. Kallal, Hedi & Jouini, Elyès, 1995. "Martingales and arbitrage in securities markets with transaction costs," Economics Papers from University Paris Dauphine 123456789/5630, Paris Dauphine University.
    9. Freddy Delbaen & Peter Grandits & Thorsten Rheinländer & Dominick Samperi & Martin Schweizer & Christophe Stricker, 2002. "Exponential Hedging and Entropic Penalties," Mathematical Finance, Wiley Blackwell, vol. 12(2), pages 99-123.
    10. Kramkov, D.O., 1994. "Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets," Discussion Paper Serie B 294, University of Bonn, Germany.
    11. Tahir Choulli & Christophe Stricker, 2005. "Minimal Entropy-Hellinger Martingale Measure In Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 15(3), pages 465-490.
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    Cited by:
    1. Erhan Bayraktar & Xiang Yu, 2013. "On the Market Viability under Proportional Transaction Costs," Papers 1312.3917, arXiv.org.
    2. Claudio Fontana, 2013. "Weak and strong no-arbitrage conditions for continuous financial markets," Papers 1302.7192, arXiv.org, revised May 2014.

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