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Infinitely many securities and the fundamental theorem of asset pricing

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Author Info
Alejandro Balbas ()
Anna Downarowicz ()

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Abstract

Several authors have pointed out the possible absence of martingale measures for static arbitrage-free markets with an infinite number of available securities. This paper addresses this caveat by drawing on projective systems of probability measures. Firstly, it is shown that there are two distinct sorts of models whose treatment is necessarily different. Secondly, and more important, we analyze those situations for which one can provide a projective system of ó .additive measures whose projective limit may be interpreted as a risk-neutral probability. Hence, the Fundamental Theorem of Asset Pricing is extended so that it can apply for models with infinitely many assets.

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Paper provided by Universidad Carlos III, Departamento de Economía de la Empresa in its series Business Economics Working Papers with number wb043513.

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Date of creation: Aug 2004
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Handle: RePEc:cte:wbrepe:wb043513

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  1. Back, Kerry & Pliska, Stanley R., 1991. "On the fundamental theorem of asset pricing with an infinite state space," Journal of Mathematical Economics, Elsevier, vol. 20(1), pages 1-18. [Downloadable!] (restricted)
  2. Chateauneuf, A. & Kast, R. & Lapied, A., 1992. "Choquet Pricing for Financial Markets with Frictions," G.R.E.Q.A.M. 92a11, Universite Aix-Marseille III.
  3. Jouini, Elyes & Kallal, Hedi, 2001. "Efficient Trading Strategies in the Presence of Market Frictions," Review of Financial Studies, Oxford University Press for Society for Financial Studies, vol. 14(2), pages 343-69.
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  4. Schachermayer, W., 1992. "A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time," Insurance: Mathematics and Economics, Elsevier, vol. 11(4), pages 249-257, December. [Downloadable!] (restricted)
  5. J. Jacod & A.N. Shiryaev, 1998. "Local martingales and the fundamental asset pricing theorems in the discrete-time case," Finance and Stochastics, Springer, vol. 2(3), pages 259-273. [Downloadable!] (restricted)
  6. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July. [Downloadable!] (restricted)
  7. Balbas, Alejandro & Miras, Miguel Angel & Munoz-Bouzo, Maria Jose, 2002. "Projective system approach to the martingale characterization of the absence of arbitrage," Journal of Mathematical Economics, Elsevier, vol. 37(4), pages 311-323, July. [Downloadable!] (restricted)
  8. Clark, Stephen A., 1993. "The valuation problem in arbitrage price theory," Journal of Mathematical Economics, Elsevier, vol. 22(5), pages 463-478. [Downloadable!] (restricted)
  9. Walter Schachermayer, 2004. "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time," Mathematical Finance, Blackwell Publishing, vol. 14(1), pages 19-48. [Downloadable!] (restricted)
  10. 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. [Downloadable!] (restricted)
  11. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447. [Downloadable!] (restricted)
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