Option Pricing for Time-Change Exponential Lévy Model Under Memm

Abstract

The purpose of this article is to study the rational evaluation of European options price when the underlying price process is described by a time-change Lévy process. European option pricing formula is obtained under the minimal entropy martingale measure (MEMM) and applied to several examples of particular time-change Lévy processes. It can be seen that the framework in this paper encompasses the Black-Scholes model and almost all of the models proposed in the subordinated market.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Applebaum, D. Lévy processes and stochastic calculus. Cambridge, Cambridge University Press, 2004

  2. 2.

    Carr, P., Wu, L. Time-changed Lévy processes and option pricing. Journal of Financial Economics, 71:113–141 (2004)

    Article  Google Scholar 

  3. 3.

    Chan, T. Pricing contingent claims on stocks driven by Lévy processes. Ann. Appl. Probab., 9:504–528 (1999)

    Article  MATH  Google Scholar 

  4. 4.

    Delbaen, F., Schachermayer, W. The variance-optimal martingale measure for continuous processes. Bernoulli, 2:81–106 (1996)

    Article  MATH  Google Scholar 

  5. 5.

    Frittelli, M. The minimal entropy martingale measures and the valuation problem in incomplete markets. Mathematical Finance, 10:39–52 (2000)

    Article  MATH  Google Scholar 

  6. 6.

    Fujiwara, T., Miyahara, Y. The minimal entropy martingale measures for geometric Lévy processes. Finance and Stochastics, 7:509–531 (2003)

    Article  MATH  Google Scholar 

  7. 7.

    Hurst, S.R., Platen, E., Rachev, S.T. Option pricing for a logstable asset price model. Mathematical and Computer Modelling, 29:105–119 (1999)

    Article  MATH  Google Scholar 

  8. 8.

    Miyahara, Y. Geometric Lévy processes & MEMM:Pricing model and related estimation problems. Asia- Pacific Financial Market, 8:45–60 (2001)

    Article  MATH  Google Scholar 

  9. 9.

    Sato, K. Lévy processes and infinitely divisible distributions. Cambridge, Cambridge University Press, 1999

  10. 10.

    Schweizer, M. On the minimal martingale measure and the Föllmer-Schweizer decomposition. Stochast. Annal. Appl., 13:573–599 (1995)

    Article  MATH  Google Scholar 

  11. 11.

    Vollert, A. Margrabe’s option to exchange in a paretian-stable subordinated market. Mathematical and Computer Modelling, 34:1185–1197 (2001)

    Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xu Chen.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chen, X., Wan, Jp. Option Pricing for Time-Change Exponential Lévy Model Under Memm. Acta Mathematicae Applicatae Sinica, English Series 23, 651–664 (2007). https://doi.org/10.1007/s10255-007-0403

Download citation

Keywords

  • Option pricing
  • Lévy processes
  • time-change
  • subordination
  • MEMM

2000 MR Subject Classification

  • 60G51
  • 91B28
  • 91B70