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Asymptotic arbitrage in large financial markets

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Author Info

  • Y.M. Kabanov

    (Central Economics and Mathematics Institute of the Russian Academy of Sciences, Moscow)

  • D.O. Kramkov

    (Steklov Mathematical Institute of the Russian Academy of Sciences, Gubkina str., 8, 117966 Moscow, Russia)

Abstract

A large financial market is described by a sequence of standard general models of continuous trading. It turns out that the absence of asymptotic arbitrage of the first kind is equivalent to the contiguity of sequence of objective probabilities with respect to the sequence of upper envelopes of equivalent martingale measures, while absence of asymptotic arbitrage of the second kind is equivalent to the contiguity of the sequence of lower envelopes of equivalent martingale measures with respect to the sequence of objective probabilities. We express criteria of contiguity in terms of the Hellinger processes. As examples, we study a large market with asset prices given by linear stochastic equations which may have random volatilities, the Ross Arbitrage Pricing Model, and a discrete-time model with two assets and infinite horizon. The suggested theory can be considered as a natural extension of Arbirage Pricing Theory covering the continuous as well as the discrete time case.

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Bibliographic Info

Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 2 (1998)
Issue (Month): 2 ()
Pages: 143-172

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Handle: RePEc:spr:finsto:v:2:y:1998:i:2:p:143-172

Note: received: January 1996; final version received: October 1996
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Web page: http://www.springerlink.com/content/101164/

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Related research

Keywords: Large financial market; continuous trading; asymptotic arbitrage; APM; APT; semimartingale; optional decomposition; contiguity; Hellinger process;

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Citations

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Cited by:
  1. Lépinette-Denis, Emmanuel & Kabanov, Yuri, 2012. "Consistent Price Systems and Arbitrage Opportunities of the Second Kind in Models with Transaction Costs," Economics Papers from University Paris Dauphine 123456789/4652, Paris Dauphine University.
  2. Miklós Rásonyi, 2004. "Arbitrage pricing theory and risk-neutral measures," Decisions in Economics and Finance, Springer, vol. 27(2), pages 109-123, December.
  3. Morten Christensen & Eckhard Platen, 2004. "A General Benchmark Model for Stochastic Jump Sizes," Research Paper Series 139, Quantitative Finance Research Centre, University of Technology, Sydney.
  4. Dmitry B. Rokhlin, 2007. "Asymptotic arbitrage and num\'eraire portfolios in large financial markets," Papers math/0702849, arXiv.org.
  5. Paolo Guasoni & Constantinos Kardaras & Scott Robertson & Hao Xing, 2014. "Abstract, classic, and explicit turnpikes," Finance and Stochastics, Springer, vol. 18(1), pages 75-114, January.
  6. Fatma Haba & Antoine Jacquier, 2013. "Asymptotic arbitrage in the Heston model," Papers 1302.6491, arXiv.org, revised Apr 2014.
  7. Bas Peeters & Cees L. Dert & Andr� Lucas, 2003. "Black Scholes for Portfolios of Options in Discrete Time: the Price is Right, the Hedge is wrong," Tinbergen Institute Discussion Papers 03-090/2, Tinbergen Institute.
  8. Dokuchaev, N. G. & Savkin, Andrey V., 2004. "Universal strategies for diffusion markets and possibility of asymptotic arbitrage," Insurance: Mathematics and Economics, Elsevier, vol. 34(3), pages 409-419, June.
  9. Igor Evstigneev & Dhruv Kapoor, . "Arbitrage in Stationary Markets," Swiss Finance Institute Research Paper Series 07-32, Swiss Finance Institute.
  10. Jouini, Elyes & Napp, Clotilde & Schachermayer, Walter, 2005. "Arbitrage and state price deflators in a general intertemporal framework," Journal of Mathematical Economics, Elsevier, vol. 41(6), pages 722-734, September.
  11. Björk, Tomas & Näslund, Bertil, 1996. "Diversified Portfolios in Continuous Time," Working Paper Series in Economics and Finance 122, Stockholm School of Economics.
  12. Bas Peeters & Cees L. Dert & Andr� Lucas, 2003. "Black Scholes for Portfolios of Options in Discrete Time: the Price is Right, the Hedge is wrong," Tinbergen Institute Discussion Papers 03-090/2, Tinbergen Institute.
  13. Irene Klein & Thorsten Schmidt & Josef Teichmann, 2013. "When roll-overs do not qualify as num\'eraire: bond markets beyond short rate paradigms," Papers 1310.0032, arXiv.org.
  14. Miklós Rásonyi, 2008. "A note on arbitrage in term structure," Decisions in Economics and Finance, Springer, vol. 31(1), pages 73-79, May.
  15. Irene Klein & Emmanuel Lepinette & Lavinia Ostafe, 2012. "Large Financial Markets and Asymptotic Arbitrage with Small Transaction Costs," Papers 1211.0443, arXiv.org.
  16. Winslow Strong, 2011. "Fundamental theorems of asset pricing for piecewise semimartingales of stochastic dimension," Papers 1112.5340, arXiv.org.
  17. Kraft, Holger & Steffensen, Mogens, 2008. "How to invest optimally in corporate bonds: A reduced-form approach," Journal of Economic Dynamics and Control, Elsevier, vol. 32(2), pages 348-385, February.
  18. Klaas Schulze, 2008. "Asymptotic Maturity Behavior of the Term Structure," Bonn Econ Discussion Papers bgse11_2008, University of Bonn, Germany.
  19. Fischer, Tom, 2007. "A law of large numbers approach to valuation in life insurance," Insurance: Mathematics and Economics, Elsevier, vol. 40(1), pages 35-57, January.

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