Finitely Additive Probabilities and the Fundamental Theorem of Asset Pricing

  • Constantinos KardarasEmail author


This work aims at a deeper understanding of the mathematical implications of the economically-sound condition of absence of arbitrages of the first kind in a financial market. In the spirit of the Fundamental Theorem of Asset Pricing (FTAP), it is shown here that the absence of arbitrages of the first kind in the market is equivalent to the existence of a finitely additive probability, weakly equivalent to the original and only locally countably additive, under which the discounted wealth processes become “local martingales”. The aforementioned result is then used to obtain an independent proof of the classical FTAP, as it appears in Delbaen and Schachermayer (Math. Ann. 300:463–520, 1994). Finally, an elementary and short treatment of the previous discussion is presented for the case of continuous-path semimartingale asset-price processes.


Asset Price Fundamental Theorem Local Probability Local Martingale Wealth Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aliprantis, C.D., Border, K.C.: Infinite-Dimensional Analysis – A Hitchhiker’s Guide, Second edn. Springer, Berlin (1999) zbMATHGoogle Scholar
  2. 2.
    Ansel, J.-P., Stricker, C.: Couverture des actifs contingents et prix maximum. Ann. Inst. Henri Poincaré Probab. Stat. 30, 303–315 (1994) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bhaskara Rao, K.P.S., Bhaskara Rao, M.: Theory of Charges: A Study of Finitely Additive Measures. Pure and Applied Mathematics, vol. 109. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, With a foreword by D.M. Stone (1983) zbMATHGoogle Scholar
  4. 4.
    Brannath, W., Schachermayer, W.: A bipolar theorem for L +0(Ω,F, P). In: Séminaire de Probabilités, XXXIII. Lecture Notes in Math., vol. 1709, pp. 349–354. Springer, Berlin (1999) CrossRefGoogle Scholar
  5. 5.
    Christensen, M.M., Larsen, K.: No arbitrage and the growth optimal portfolio. Stoch. Anal. Appl. 25, 255–280 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cvitanić, J., Schachermayer, W., Wang, H.: Utility maximization in incomplete markets with random endowment. Finance Stoch. 5, 259–272 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Delbaen, F., Schachermayer, W.: The no-arbitrage property under a change of numéraire. Stoch. Stoch. Rep. 53, 213–226 (1995) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Fernholz, E., Karatzas, I.: Stochastic Portfolio Theory: An Overview. Handbook of Numerical Analysis, vol. 15, pp. 89–167 (2009) Google Scholar
  11. 11.
    Föllmer, H., Kabanov, Y.M.: Optional decomposition and Lagrange multipliers. Finance Stoch. 2, 69–81 (1998) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Föllmer, H., Kramkov, D.: Optional decompositions under constraints. Probab. Theory Relat. Fields 109, 1–25 (1997) zbMATHCrossRefGoogle Scholar
  13. 13.
    Gilles, C., LeRoy, S.F.: Bubbles and charges. Int. Econ. Rev. 33, 323–339 (1992) zbMATHCrossRefGoogle Scholar
  14. 14.
    Ingersoll, J.E.: Theory of Financial Decision Making. Rowman and Littlefield Studies in Financial Economics. Rowman & Littlefield, Totowa (1987) Google Scholar
  15. 15.
    Kabanov, Y.M., Kramkov, D.O.: Large financial markets: asymptotic arbitrage and contiguity. Teor. Veroyatn. I Primenen. 39, 222–229 (1994) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Karatzas, I., Kardaras, C.: The numéraire portfolio in semimartingale financial models. Finance Stoch. 11, 447–493 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, vol. 113, Second edn. Springer, New York (1991) zbMATHCrossRefGoogle Scholar
  18. 18.
    Karatzas, I., Žitković, G.: Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31, 1821–1858 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Kardaras, C.: Market viability via absence of arbitrages of the first kind. Submitted for publication. Preprint available at (2009)
  20. 20.
    Kardaras, C., Platen, E.: On the semimartingale property of discounted asset-price processes. Submitted for publication. Preprint available at (2008)
  21. 21.
    Kreps, D.M.: Arbitrage and equilibrium in economies with infinitely many commodities. J. Math. Econ. 8, 15–35 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Loewenstein, M., Willard, G.A.: Local martingales, arbitrage, and viability. Free snacks and cheap thrills. Econom. Theory 16, 135–161 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Loewenstein, M., Willard, G.A.: Rational equilibrium asset-pricing bubbles in continuous trading models. J. Econ. Theory 91, 17–58 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Platen, E., Heath, D.: A Benchmark Approach to Quantitative Finance. Springer Finance. Springer, Berlin (2006) zbMATHCrossRefGoogle Scholar
  25. 25.
    Schweizer, M.: On the minimal martingale measure and the Föllmer-Schweizer decomposition. Stoch. Anal. Appl. 13, 573–599 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Yan, J.A.: Caractérisation d’une classe d’ensembles convexes de L 1 ou H 1. Seminar on Probability XIV. Lecture Notes in Math., vol. 784 pp. 220–222. Springer Berlin (1980) Google Scholar
  27. 27.
    Žitković, G.: A filtered version of the bipolar theorem of Brannath and Schachermayer. J. Theor. Probab. 15, 41–61 (2002) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Mathematics and Statistics DepartmentBoston UniversityBostonUSA

Personalised recommendations