Annals of Finance

, 6:147

No arbitrage conditions for simple trading strategies

Research Article

Abstract

Strict local martingales may admit arbitrage opportunities with respect to the class of simple trading strategies. (Since there is no possibility of using doubling strategies in this framework, the losses are not assumed to be bounded from below.) We show that for a class of non-negative strict local martingales, the strong Markov property implies the no arbitrage property with respect to the class of simple trading strategies. This result can be seen as a generalization of a similar result on three dimensional Bessel process in Delbaen and Schachermayer (Math Finance 4:343–348, 1994). We also provide no arbitrage conditions for stochastic processes within the class of simple trading strategies with shortsale restriction.

Keywords

Simple trading strategies Arbitrage Sticky processes Shortsales restriction 

JEL Classification

G10 

References

  1. Bayraktar, E, Sayit, H.: Arbitrage free models in markets with transaction costs. Preprint. Available at http://arxiv.org/abs/0707.0336
  2. Delbaen F., Schachermayer W.: A general version of the fundamental theorem of asset pricing. Math Ann 300, 463–520 (1994a)CrossRefGoogle Scholar
  3. Delbaen F., Schachermayer W.: Arbitrage and free lunch with bounded risk for unbounded continuous processes. Math Finance 4, 343–348 (1994b)CrossRefGoogle Scholar
  4. Delbaen F., Schachermayer W.: Arbitrage possibilities in Bessel processes and their relations to local martingales. Prob Theory Relat Fields 102, 357–366 (1995)CrossRefGoogle Scholar
  5. Delbaen F., Shirakawa H.: No arbitrage condition for positive diffusion price processes. Financ Eng Jpn Markets 9(3–4), 159–168 (2002)Google Scholar
  6. Delbaen F., Schachermayer W.: The Mathematics of Arbitrage. Springer, Berlin (2006)Google Scholar
  7. Elworthy K.D., Li X.-M., Yor M.: The importance of strict local martingales; applications to radial Ornstein-Uhlenbeck processes. Probab. Theory Relat. Fields 115, 325–355 (1999)CrossRefGoogle Scholar
  8. Guasoni P.: No arbitrage with transaction costs, with fractional Brownian motion and Beyond. Math Finance 16(2), 469–588 (2006)Google Scholar
  9. Heath D, Schweizer M.: Martingales versus PDEs in finance: An equivalence result with examples. J Appl Prob 37, 947–957 (2000)CrossRefGoogle Scholar
  10. Jarrow, R., Protter, P., Sayit, H.: No Arbitrage Without Semimartingales. Ann Appl Prob (2009a) (to appear)Google Scholar
  11. Jarrow, R., Protter, P., Shimbo, K.: Asset price bubbles in incomplete markets. Math Finance (2009b) (to appear)Google Scholar
  12. Karatzas I., Shreve S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)Google Scholar
  13. Protter, P., Pal, S.: Analysis of strict local martingales via h-tranforms. Cornell University, Preprint (2008)Google Scholar
  14. Protter P.: Stochastic Integration and Differential Equations, 2nd edn, Version 2.1. Springer, Heidelberg (2005)Google Scholar
  15. Revuz D., Yor M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Heidelberg (1999)Google Scholar
  16. Sin C.A.: Complications with stochastic volatility models. Adv Appl Prob 30, 256–268 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsWorcester Polytechnic InstituteWorcesterUSA

Personalised recommendations