Advertisement

Positivity

, Volume 6, Issue 3, pp 359–368 | Cite as

No Arbitrage: On the Work of David Kreps

  • W. Schachermayer
Article

Abstract

Since the seminal papers by Black, Scholes and Merton on the pricing of options (Nobel Prize for Economics, 1997), the theory of No Arbitrage plays a central role in Mathematical Finance. Pioneering work on the relation between no arbitrage arguments and martingale theory has been done in the late seventies by M. Harrison, D. Kreps and S. Pliska. In the present note we give a brief survey on the relation of the theory of No-Arbitrage to coherent pricing of derivative securities. We focus on a seminal paper published by D. Kreps in 1981, and give a solution to an open problem posed in this paper.

No Arbitrage No Free Lunch Coherent Pricing of Derivative Securities 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ansel, J. P. and Stricker, C.: Couverture des actifs contingents et prix maximum, Ann. Inst. Henri Poincaré 30 (1994), 303-315.Google Scholar
  2. 2.
    Artzner, P. and Heath, D.: Approximate completeness with multiple martingale measures, Math. Finance 5 (1995), 1-11.Google Scholar
  3. 3.
    Bewley, T.: Existence of equilibria in economics with infinitely many commodities,Journal of Economic Theory 4 (1972), 514-540.Google Scholar
  4. 4.
    Black, F., and Scholes, M.: The pricing of options and corporate liabilities, Journal of Political Economy 81 (1973), 637-659.Google Scholar
  5. 5.
    Clark, S. A.: The valuation problem in arbitrage price theory, Journal of Mathematical Economics 22 (1993), 463-478.Google Scholar
  6. 6.
    Clark, S. A.: Arbitrage approximation theory, Journal of Mathematical Economics 33 (2000), 167-181.Google Scholar
  7. 7.
    Dalang, R. C., Morton, A. and Willinger,W.: Equivalent Martingale measures and no-arbitrage in stochastic, Stochastics and Stochastics Reports 29 (1990), 185-201.Google Scholar
  8. 8.
    Delbaen, F.: Representing martingale measures when asset prices are continuous and bounded, Mathematical Finance 2 (1992), 107-130.Google Scholar
  9. 9.
    Delbaen, F. and Schachermayer, W.: A general version of the fundamental theorem of asset pricing, Math. Annalen 300 (1994), 463-520.Google Scholar
  10. 10.
    Delbaen, F. and Schachermayer, W.: The No-arbitrage property under a change of numéraire, Stochastics and Stochastic Reports 53 (1995), 213-226.Google Scholar
  11. 11.
    Delbaen, F. and Schachermayer, W.: The Fundamental theorem of asset pricing for unbounded stochastic processes, Mathematische Annalen 312 (1998), 215-250.Google Scholar
  12. 12.
    Duffie, D. and Huang, C.-F.: Multiperiod security markets with differential information; martingales and resolution times, J. Math. Economics 15 (1986), 283-303.Google Scholar
  13. 13.
    Dybvig, P. and Ross, S.: Arbitrage, The new Palgrave dictionary of economics 1 (1987), 100-106.Google Scholar
  14. 14.
    El Karoui, N. and Quenez, M.-C.: Dynamic programming and pricing of contingent claims in an incomplete market, SIAM J. Control Optimization 33 (1995), 29-66.Google Scholar
  15. 15.
    Halmos, P. R. and Savage, L. J.: Application of the Radon-Nikodym theorem to the theory of sufficient statistics, Ann. Math. Statistics 20 (1949), 225-241.Google Scholar
  16. 16.
    Harrison, J. M. and Kreps, D. M.: Martingales and arbitrage in multiperiod securities markets, J. Econ. Theory 20 (1979), 381-408.Google Scholar
  17. 17.
    Harrison, J. M. and Pliska, S. R.: Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and Applications 11 (1981), 215-260.Google Scholar
  18. 18.
    Hart, O.: Monopolistic competition in a large economy with differentiated commodities, Review of Economic Studies 46 (1979), 1-30.Google Scholar
  19. 19.
    Jacka, S. D.: A martingale representation result and an application to incomplete financial markets, Mathematical Finance 2 (1992), 239-250.Google Scholar
  20. 20.
    Klein, I., and Schachermayer, W.: A quantitative and a dual version of the Halmos-Savage theorem with applications to mathematical finance, The Annals of Probability 24 (1996), 867-881.Google Scholar
  21. 21.
    Kreps, D. M.: Arbitrage and equilibrium in economies with infinitely many commodities, Journal of Mathematical Economics 8 (1981), 15-35.Google Scholar
  22. 22.
    Mas-Colell, A.: A model of equilibrium with differentiated commodities, Journal of Mathematical Economics 2 (1975), 263-296.Google Scholar
  23. 23.
    Merton, R. C.: Theory of rational option pricing, Bell J. Econom. Manag. Sci. 4 (1973), 141-183.Google Scholar
  24. 24.
    Schachermayer, W.: A Counter-example to several problems in the theory of asset pricing, Math. Finance 3 (1993), 217-229.Google Scholar
  25. 25.
    Schachermayer, W.: Martingale measures for discrete time processes with infinite horizon, Math. Finance 4 (1994), 25-55.Google Scholar
  26. 26.
    Schachermayer, W.: Introduction to the Mathematics of Financial Markets, Preprint, to appear in Springer Lecture Notes on the St. Flour Summer School, 2000.Google Scholar
  27. 27.
    Schäfer, H. H.: Topological Vector Spaces, Graduate Texts in Mathematics., Springer, Berlin and New York, 1966.Google Scholar
  28. 28.
    Stricker, C.: Arbitrage et lois de martingale, Ann. Inst. H. Poincarè Probab. Statist. 26 (1990), 451-460.Google Scholar
  29. 29.
    Yan, J. A.: 1980, Caracterisation d' 'une classe d'ensembles convexes de L 1 ou H 1, Lecture Notes in Mathematics, Vol. 784, Springer, Berlin and New York, 1980, pp. 220-222.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • W. Schachermayer
    • 1
  1. 1.Department of Financial and Actuarial MathematicsVienna University of TechnologyViennaAustria

Personalised recommendations