Skip to main content
SpringerLink
Log in

The numéraire portfolio in semimartingale financial models

  • Published:
Finance and Stochastics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Algoet, P., Cover, T.M.: Asymptotic optimality and asymptotic equipartition property of log-optimal investment. Ann. Probab. 16, 876–898 (1988)

    MATH  Google Scholar 

  2. Aliprantis, C.D., Border, K.C.: Infinite-Dimensional Analysis: A Hitchhiker’s Guide, 2nd edn. Springer, Berlin (1999)

    MATH  Google Scholar 

  3. Ansel, J.-P., Stricker, C.: Couverture des actifs contingents et prix maximum. Ann. Inst. H. Poincaré 30, 303–315 (1994)

    MATH  Google Scholar 

  4. Becherer, D.: The numéraire portfolio for unbounded semimartingales. Finance Stoch. 5, 327–341 (2001)

    Article  MATH  Google Scholar 

  5. Bichteler, K.: Stochastic Integration with Jumps. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  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. 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. Christensen, M.M., Larsen, K.: No arbitrage and the growth optimal portfolio. Stoch. Anal. Appl. 25, 255–280 (2007)

    Article  MATH  Google Scholar 

  9. Dellacherie, C., Meyer, P.-A.: Probabilities and Potential B: Theory of Martingales. Elsevier, Amsterdam (1983)

    Google Scholar 

  10. Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994)

    Article  MATH  Google Scholar 

  11. Delbaen, F., Schachermayer, W.: The existence of absolutely continuous local martingale measures. Ann. Appl. Probab. 5, 926–945 (1995)

    MATH  Google Scholar 

  12. Delbaen, F., Schachermayer, W.: The no-arbitrage property under a change of numéraire. Stoch. Stoch. Rep. 53, 213–226 (1995)

    MATH  Google Scholar 

  13. Delbaen, F., Schachermayer, W.: Arbitrage possibilities in Bessel processes and their relations to local martingales. Probab. Theory Relat. Fields 102, 357–366 (1995)

    Article  MATH  Google Scholar 

  14. Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–260 (1998)

    Article  MATH  Google Scholar 

  15. Fernholz, R., Karatzas, I.: Relative arbitrage in volatility-stabilized markets. Ann. Finance 1, 149–177 (2005)

    Article  Google Scholar 

  16. Fernholz, R., Karatzas, I., Kardaras, C.: Diversity and relative arbitrage in equity markets. Finance Stoch. 9, 1–27 (2005)

    Article  MATH  Google Scholar 

  17. Föllmer, H., Kramkov, D.: Optional decompositions under constraints. Probab. Theory Relat. Fields 109, 1–25 (1997)

    Article  MATH  Google Scholar 

  18. Goll, T., Kallsen, J.: A complete explicit solution to the log-optimal portfolio problem. Ann. Appl. Probab. 13, 774–799 (2003)

    Article  MATH  Google Scholar 

  19. Goll, T., Rüschendorf, L.: Minimax and minimal distance martingale measures and their relationship to portfolio optimization. Finance Stoch. 5, 557–581 (2001)

    Article  MATH  Google Scholar 

  20. Jacod, J.: Calcul Stochastique et Problèmes de Martingales, Lecture Notes in Mathematics, vol. 714. Springer, Berlin (1979)

    MATH  Google Scholar 

  21. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003)

    MATH  Google Scholar 

  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. Kallsen, J.: σ-Localization and σ-martingales. Theory Probab. Appl. 48, 152–163 (2004)

    Article  Google Scholar 

  24. Karatzas, I., Kou, S.G.: On the pricing of contingent claims under constraints. Ann. Appl. Probab. 6, 321–369 (1996)

    Article  MATH  Google Scholar 

  25. Karatzas, I., Lehoczky, J.P., Shreve, S.E.: Equilibrium models with singular asset prices. Math. Finance 1(3), 11–29 (1991)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  27. Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Springer, Berlin (1998)

    MATH  Google Scholar 

  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. 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. Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  33. Long, J.B. Jr.: The numéraire portfolio. J. Financ. Econ. 26, 29–69 (1990)

    Article  Google Scholar 

  34. Mémin, J.: Espaces de semimartingales et changement de probabilité. Z. Wahrsch. Verwandte Geb. 52, 9–39 (1980)

    Article  MATH  Google Scholar 

  35. Platen, E.: A benchmark approach to finance. Math. Finance 16, 131–151 (2006)

    Article  MATH  Google Scholar 

  36. Schweizer, M.: Martingale densities for general asset prices. J. Math. Econ. 21, 363–378 (1992)

    Article  MATH  Google Scholar 

  37. Schweizer, M.: A minimality property of the minimal martingale measure. Stat. Probab. Lett. 42, 27–31 (1999)

    Article  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics and Statistics Departments, Columbia University, New York, NY, 10027, USA

    Ioannis Karatzas

  2. Mathematics and Statistics Department, Boston University, Boston, MA, 02215, USA

    Constantinos Kardaras

Authors
  1. Ioannis Karatzas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Constantinos Kardaras
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Constantinos Kardaras.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Karatzas, I., Kardaras, C. The numéraire portfolio in semimartingale financial models. Finance Stoch 11, 447–493 (2007). https://doi.org/10.1007/s00780-007-0047-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00780-007-0047-3

Keywords

Mathematics Subject Classification (2000)

JEL

Search

Navigation