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A Fundamental Theorem of Asset Pricing for Large Financial Markets

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  • Irene Klein

Abstract

We formulate the notion of "asymptotic free lunch" which is closely related to the condition "free lunch" of Kreps (1981) and allows us to state and prove a fairly general version of the fundamental theorem of asset pricing in the context of a large financial market as introduced by Kabanov and Kramkov (1994). In a large financial market one considers a sequence ("S"-super-"n") "n"=1 -super-∞ of stochastic stock price processes based on a sequence (Ω-super-"n", "F"-super-"n", ("F" "t" -super-"n") "t" is an element of "I"-super-"n" , P-super-"n") "n"=1 -super-∞ of filtered probability spaces. Under the assumption that for all "n" is an element of N there exists an equivalent sigma-martingale measure for "S"-super-"n", we prove that there exists a "bicontiguous" sequence of equivalent sigma-martingale measures if and only if there is no asymptotic free lunch (Theorem 1.1). Moreover we present an example showing that it is not possible to improve Theorem 1.1 by replacing "no asymptotic free lunch" by some weaker condition such as "no asymptotic free lunch with bounded" or "vanishing risk." Copyright Blackwell Publishers, Inc..

Suggested Citation

  • Irene Klein, 2000. "A Fundamental Theorem of Asset Pricing for Large Financial Markets," Mathematical Finance, Wiley Blackwell, vol. 10(4), pages 443-458.
  • Handle: RePEc:bla:mathfi:v:10:y:2000:i:4:p:443-458
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    Citations

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    Cited by:

    1. Miklós Rásonyi, 2004. "Arbitrage pricing theory and risk-neutral measures," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 27(2), pages 109-123, December.
    2. Miklós Rásonyi, 2016. "On Optimal Strategies For Utility Maximizers In The Arbitrage Pricing Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(07), pages 1-12, November.
    3. Michał Baran, 2007. "Asymptotic pricing in large financial markets," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(1), pages 1-20, August.
    4. Soumik Pal, 2016. "Exponentially concave functions and high dimensional stochastic portfolio theory," Papers 1603.01865, arXiv.org, revised Mar 2016.
    5. Scott Robertson & Konstantinos Spiliopoulos, 2014. "Indifference pricing for Contingent Claims: Large Deviations Effects," Papers 1410.0384, arXiv.org, revised Feb 2016.
    6. Miklos Rasonyi, 2015. "Maximizing expected utility in the Arbitrage Pricing Model," Papers 1508.07761, arXiv.org, revised Mar 2017.
    7. Winslow Strong, 2014. "Fundamental theorems of asset pricing for piecewise semimartingales of stochastic dimension," Finance and Stochastics, Springer, vol. 18(3), pages 487-514, July.
    8. Christa Cuchiero & Irene Klein & Josef Teichmann, 2014. "A new perspective on the fundamental theorem of asset pricing for large financial markets," Papers 1412.7562, arXiv.org, revised Sep 2015.
    9. repec:spr:compst:v:66:y:2007:i:1:p:1-20 is not listed on IDEAS
    10. Micha{l} Barski, 2015. "Asymptotic pricing in large financial markets," Papers 1512.06582, arXiv.org.
    11. Oleksii Mostovyi, 2014. "Utility maximization in the large markets," Papers 1403.6175, arXiv.org, revised Oct 2014.
    12. De Donno, M. & Guasoni, P. & Pratelli, M., 2005. "Super-replication and utility maximization in large financial markets," Stochastic Processes and their Applications, Elsevier, vol. 115(12), pages 2006-2022, December.
    13. Tom Fischer, 2015. "No-Arbitrage Prices of Cash Flows and Forward Contracts as Choquet Representations," Papers 1506.01837, arXiv.org, revised Jun 2015.

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