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Exact arbitrage, well-diversified portfolios and asset pricing in large markets

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  • Khan, M. Ali
  • Sun, Yeneng

Abstract

In a model of a nancial market with an atomless continuum of assets, we give a precise and rigorous meaning to the intuitive idea of a \well-diversi ed' portfolio and to a notion of \exact arbitrage'. We show this notion to be necessary and su cient for an APT pricing formula to hold, to be strictly weaker than the more conventional notion of \asymptotic arbitrage', and to have novel implications for the continuity of the cost functional as well as for various versions of APT asset pricing. We further justify the idealized measure-theoretic setting in terms of a pricing formula based on \essential' risk, one of the three components of a tri-variate decomposition of an asset's rate of return, and based on a speci c index portfolio constructed from endogenously extracted factors and factor loadings. Our choice of factors is also shown to satisfy an optimality property that the rst m factors always provide the best approximation. We illustrate how the concepts and results translate to markets with a large but nite number of assets, and relate to previous work.
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  • Khan, M. Ali & Sun, Yeneng, 2003. "Exact arbitrage, well-diversified portfolios and asset pricing in large markets," Journal of Economic Theory, Elsevier, vol. 110(2), pages 337-373, June.
  • Handle: RePEc:eee:jetheo:v:110:y:2003:i:2:p:337-373
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    References listed on IDEAS

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    1. Judd, Kenneth L., 1985. "The law of large numbers with a continuum of IID random variables," Journal of Economic Theory, Elsevier, vol. 35(1), pages 19-25, February.
    2. Sun, Yeneng, 1998. "A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN1," Journal of Mathematical Economics, Elsevier, vol. 29(4), pages 419-503, May.
    3. Ross, Stephen A., 1976. "The arbitrage theory of capital asset pricing," Journal of Economic Theory, Elsevier, vol. 13(3), pages 341-360, December.
    4. Mordecai Kurz & Mukul Majumdar, 1972. "Efficiency Prices in Infinite Dimensional Spaces: a Synthesis," Review of Economic Studies, Oxford University Press, vol. 39(2), pages 147-158.
    5. Werner, Jan, 1997. "Diversification and Equilibrium in Securities Markets," Journal of Economic Theory, Elsevier, vol. 75(1), pages 89-103, July.
    6. Reisman, Haim, 1988. "A General Approach to the Arbitrage Pricing Theory (APT)," Econometrica, Econometric Society, vol. 56(2), pages 473-476, March.
    7. Al-Najjar, Nabil I., 1998. "Factor Analysis and Arbitrage Pricing in Large Asset Economies," Journal of Economic Theory, Elsevier, vol. 78(2), pages 231-262, February.
    8. Khan, M. Ali & Sun, Yeneng, 2001. "Asymptotic Arbitrage and the APT with or without Measure-Theoretic Structures," Journal of Economic Theory, Elsevier, vol. 101(1), pages 222-251, November.
    9. Anderson, Robert M., 1991. "Non-standard analysis with applications to economics," Handbook of Mathematical Economics,in: W. Hildenbrand & H. Sonnenschein (ed.), Handbook of Mathematical Economics, edition 1, volume 4, chapter 39, pages 2145-2208 Elsevier.
    10. Huberman, Gur, 1982. "A simple approach to arbitrage pricing theory," Journal of Economic Theory, Elsevier, vol. 28(1), pages 183-191, October.
    11. M. Ali Khan & Yeneng Sun, 1999. "Weak measurability and characterizations of risk," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 13(3), pages 541-560.
    12. M. Ali Khan & Yeneng Sun, 1996. "Hyperfinite Asset Pricing Theory," Cowles Foundation Discussion Papers 1139, Cowles Foundation for Research in Economics, Yale University.
    13. Chamberlain, Gary, 1983. "Funds, Factors, and Diversification in Arbitrage Pricing Models," Econometrica, Econometric Society, vol. 51(5), pages 1305-1323, September.
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    Cited by:

    1. Birge, John R. & Yang, Song, 2007. "A model for tax advantages of portfolios with many assets," Journal of Banking & Finance, Elsevier, vol. 31(11), pages 3269-3290, November.

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