Finance and Stochastics

, Volume 11, Issue 4, pp 447–493 | Cite as

The numéraire portfolio in semimartingale financial models

Article
First Online:
Received:
Accepted:

Abstract

We study the existence of the numéraire portfolio under predictable convex constraints in a general semimartingale model of a financial market. The numéraire portfolio generates a wealth process, with respect to which the relative wealth processes of all other portfolios are supermartingales. Necessary and sufficient conditions for the existence of the numéraire portfolio are obtained in terms of the triplet of predictable characteristics of the asset price process. This characterization is then used to obtain further necessary and sufficient conditions, in terms of a no-free-lunch-type notion. In particular, the full strength of the “No Free Lunch with Vanishing Risk” (NFLVR) condition is not needed, only the weaker “No Unbounded Profit with Bounded Risk” (NUPBR) condition that involves the boundedness in probability of the terminal values of wealth processes. We show that this notion is the minimal a-priori assumption required in order to proceed with utility optimization. The fact that it is expressed entirely in terms of predictable characteristics makes it easy to check, something that the stronger NFLVR condition lacks.

Keywords

Numéraire portfolio Semimartingale Predictable characteristics Free lunch Supermartingale deflator Log-utility 

Mathematics Subject Classification (2000)

60H05 60H30 91B28 

JEL

G11 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Algoet, P., Cover, T.M.: Asymptotic optimality and asymptotic equipartition property of log-optimal investment. Ann. Probab. 16, 876–898 (1988) MATHGoogle Scholar
  2. 2.
    Aliprantis, C.D., Border, K.C.: Infinite-Dimensional Analysis: A Hitchhiker’s Guide, 2nd edn. Springer, Berlin (1999) MATHGoogle Scholar
  3. 3.
    Ansel, J.-P., Stricker, C.: Couverture des actifs contingents et prix maximum. Ann. Inst. H. Poincaré 30, 303–315 (1994) MATHGoogle Scholar
  4. 4.
    Becherer, D.: The numéraire portfolio for unbounded semimartingales. Finance Stoch. 5, 327–341 (2001) MATHCrossRefGoogle Scholar
  5. 5.
    Bichteler, K.: Stochastic Integration with Jumps. Cambridge University Press, Cambridge (2002) MATHGoogle Scholar
  6. 6.
    Cherny, A.S., Shiryaev, A.N.: Vector stochastic integrals and the fundamental theorems of asset pricing. Proc. Steklov Math. Inst. 237, 12–56 (2002) Google Scholar
  7. 7.
    Cherny, A.S., Shiryaev, A.N.: On stochastic integrals up to infinity and predictable criteria for integrability. In: Séminaire de Probabilités XXXVIII. Lecture Notes in Mathematics, vol. 1857, pp. 165–185. Springer, New York (2004) Google Scholar
  8. 8.
    Christensen, M.M., Larsen, K.: No arbitrage and the growth optimal portfolio. Stoch. Anal. Appl. 25, 255–280 (2007) MATHCrossRefGoogle Scholar
  9. 9.
    Dellacherie, C., Meyer, P.-A.: Probabilities and Potential B: Theory of Martingales. Elsevier, Amsterdam (1983) Google Scholar
  10. 10.
    Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994) MATHCrossRefGoogle Scholar
  11. 11.
    Delbaen, F., Schachermayer, W.: The existence of absolutely continuous local martingale measures. Ann. Appl. Probab. 5, 926–945 (1995) MATHGoogle Scholar
  12. 12.
    Delbaen, F., Schachermayer, W.: The no-arbitrage property under a change of numéraire. Stoch. Stoch. Rep. 53, 213–226 (1995) MATHGoogle Scholar
  13. 13.
    Delbaen, F., Schachermayer, W.: Arbitrage possibilities in Bessel processes and their relations to local martingales. Probab. Theory Relat. Fields 102, 357–366 (1995) MATHCrossRefGoogle Scholar
  14. 14.
    Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–260 (1998) MATHCrossRefGoogle Scholar
  15. 15.
    Fernholz, R., Karatzas, I.: Relative arbitrage in volatility-stabilized markets. Ann. Finance 1, 149–177 (2005) CrossRefGoogle Scholar
  16. 16.
    Fernholz, R., Karatzas, I., Kardaras, C.: Diversity and relative arbitrage in equity markets. Finance Stoch. 9, 1–27 (2005) MATHCrossRefGoogle Scholar
  17. 17.
    Föllmer, H., Kramkov, D.: Optional decompositions under constraints. Probab. Theory Relat. Fields 109, 1–25 (1997) MATHCrossRefGoogle Scholar
  18. 18.
    Goll, T., Kallsen, J.: A complete explicit solution to the log-optimal portfolio problem. Ann. Appl. Probab. 13, 774–799 (2003) MATHCrossRefGoogle Scholar
  19. 19.
    Goll, T., Rüschendorf, L.: Minimax and minimal distance martingale measures and their relationship to portfolio optimization. Finance Stoch. 5, 557–581 (2001) MATHCrossRefGoogle Scholar
  20. 20.
    Jacod, J.: Calcul Stochastique et Problèmes de Martingales, Lecture Notes in Mathematics, vol. 714. Springer, Berlin (1979) MATHGoogle Scholar
  21. 21.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003) MATHGoogle Scholar
  22. 22.
    Kabanov, Y.M.: On the FTAP of Kreps–Delbaen–Schachermayer. In: Kabanov, Y.M., Rozovskii, B.L., Shiryaev, A.N. (eds.) Statistics and Control of Random Processes. The Liptser Festschrift. Proceedings of Steklov Mathematical Institute Seminar, pp. 191–203. World Scientific, Singapore (1997) Google Scholar
  23. 23.
    Kallsen, J.: σ-Localization and σ-martingales. Theory Probab. Appl. 48, 152–163 (2004) CrossRefGoogle Scholar
  24. 24.
    Karatzas, I., Kou, S.G.: On the pricing of contingent claims under constraints. Ann. Appl. Probab. 6, 321–369 (1996) MATHCrossRefGoogle Scholar
  25. 25.
    Karatzas, I., Lehoczky, J.P., Shreve, S.E.: Equilibrium models with singular asset prices. Math. Finance 1(3), 11–29 (1991) MATHCrossRefGoogle Scholar
  26. 26.
    Karatzas, I., Lehoczky, J.P., Shreve, S.E., Xu, G.-L.: Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control Optim. 29, 707–730 (1991) CrossRefGoogle Scholar
  27. 27.
    Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Springer, Berlin (1998) MATHGoogle Scholar
  28. 28.
    Kardaras, C.: The numéraire portfolio and arbitrage in semimartingale models of financial markets. Ph.D. dissertation, Columbia University (2006) http://people.bu.edu/kardaras/kardaras_thesis.pdf
  29. 29.
    Kardaras, C.: No-free-lunch equivalences for exponential Lévy models under convex constraints on investment. Preprint (electronic access): http://people.bu.edu/kardaras/nfl_levy.pdf; Math. Finance (2006, to appear)
  30. 30.
    Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999) MATHCrossRefGoogle Scholar
  31. 31.
    Kramkov, D., Schachermayer, W.: Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 13, 1504–1516 (2003) MATHCrossRefGoogle Scholar
  32. 32.
    Levental, S., Skorohod, A.V.: A necessary and sufficient condition for absence of arbitrage with tame portfolios. Ann. Appl. Probab. 5, 906–925 (1995) MATHGoogle Scholar
  33. 33.
    Long, J.B. Jr.: The numéraire portfolio. J. Financ. Econ. 26, 29–69 (1990) CrossRefGoogle Scholar
  34. 34.
    Mémin, J.: Espaces de semimartingales et changement de probabilité. Z. Wahrsch. Verwandte Geb. 52, 9–39 (1980) MATHCrossRefGoogle Scholar
  35. 35.
    Platen, E.: A benchmark approach to finance. Math. Finance 16, 131–151 (2006) MATHCrossRefGoogle Scholar
  36. 36.
    Schweizer, M.: Martingale densities for general asset prices. J. Math. Econ. 21, 363–378 (1992) MATHCrossRefGoogle Scholar
  37. 37.
    Schweizer, M.: A minimality property of the minimal martingale measure. Stat. Probab. Lett. 42, 27–31 (1999) MATHCrossRefGoogle Scholar
  38. 38.
    Stricker, C., Yan, J.-A.: Some remarks on the optional decomposition theorem. In: Séminaire de Probabilités XXXII. Lecture Notes in Mathematics, vol. 1686, pp. 56–66. Springer, New York (1998) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Mathematics and Statistics DepartmentsColumbia UniversityNew YorkUSA
  2. 2.Mathematics and Statistics DepartmentBoston UniversityBostonUSA

Personalised recommendations