Skip to main content
SpringerLink
Log in

Minimal q-entropy martingale measures for exponential time-changed Lévy processes

  • Published:
Finance and Stochastics Aims and scope Submit manuscript

Abstract

We consider structure preserving measure transforms for time-changed Lévy processes. Within this class of transforms preserving the time-changed Lévy structure, we derive equivalent martingale measures minimizing relative q-entropy. They combine the corresponding transform for the Lévy process with an Esscher transform on the time change. Structure preservation is found to be an inherent property of minimal q-entropy martingale measures under continuous time changes, whereas it imposes an additional restriction for discontinuous time changes.

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. Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  2. Barndorff-Nielsen, O.E., Shephard, N.: Modelling by Lévy processes for financial econometrics. In: Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S. (eds.) Lévy Processes—Theory and Applications, pp. 283–318. Birkhäuser, Boston (2001)

    Google Scholar 

  3. Barndorff-Nielsen, O.E., Shephard, N.: Normal modified stable processes. Theory Probab. Math. Stat. 65, 1–20 (2002)

    MathSciNet  Google Scholar 

  4. Carr, P., Geman, H., Madan, D.B., Yor, M.: Stochastic volatility for Lévy processes. Math. Finance 13, 345–382 (2003)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  6. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton (2004)

    MATH  Google Scholar 

  7. Esche, F., Schweizer, M.: Minimal entropy preserves the Lévy property: how and why. Stochastic Process. Appl. 115, 299–327 (2005)

    MATH  MathSciNet  Google Scholar 

  8. Föllmer, H., Schweizer, M.: Hedging of contingent claims under incomplete information. In: Davis, M.H.A., Elliott, R.J. (eds.) Applied Stochastic Analysis, pp. 389–414. Gordon and Breach, New York (1991)

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  10. Hubalek, F., Sgarra, C.: Esscher transforms and the minimal entropy martingale measure for exponential Lévy models. Quant. Finance 6, 125–145 (2006)

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  12. Jeanblanc, M., Klöppel, S., Miyahara, Y.: Minimal f q martingale measures for exponential Lévy processes. Ann. Appl. Probab. 17, 1615–1638 (2007)

    MATH  MathSciNet  Google Scholar 

  13. Kassberger, S., Liebmann, T.: q-optimal martingale measures for exponential Lévy processes. Working paper available at SSRN. http://ssrn.com/abstract=1098098 (2007)

  14. Luciano, E., Semeraro, P.: Multivariate time changes for Lévy asset models: characterization and calibration. J. Comput. Appl. Math. 233, 1937–1953 (2010)

    MATH  MathSciNet  Google Scholar 

  15. Madan, D.B., Seneta, E.: The variance gamma model for share market returns. J. Bus. 63, 511–524 (1990)

    Google Scholar 

  16. Nicolato, E., Venardos, E.: Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type. Math. Finance 13, 445–466 (2003)

    MATH  MathSciNet  Google Scholar 

  17. Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  18. Schoutens, W.: Lévy Processes in Finance. Wiley, Chichester (2003)

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  20. Schweizer, M.: A guided tour through quadratic hedging approaches. In: Jouini, S., Cvitanić, J., Musiela, M. (eds.) Option Pricing, Interest Rates and Risk Management, pp. 538–574. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  21. Selivanov, A.V.: On the martingale measures in exponential Lévy models. Theory Probab. Appl. 42, 261–274 (2005)

    MathSciNet  Google Scholar 

  22. Semeraro, P.: A multivariate variance gamma model for financial applications. Int. J. Theor. Appl. Finance 11, 1–18 (2008)

    MATH  MathSciNet  Google Scholar 

  23. Winkel, M.: The recovery problem for time-changed Lévy processes. Maphysto Research Report 2001-37. http://www.maphysto.dk/publications/MPS-RR/2001/37.pdf (2001)

Download references

Author information

Authors and Affiliations

  1. Centre for Practical Quantitative Finance, Frankfurt School of Finance & Management, Sonnemannstraße 9-11, 60314, Frankfurt am Main, Germany

    Stefan Kassberger

  2. Institute of Mathematical Finance, Ulm University, Helmholtzstr. 18, 89069, Ulm, Germany

    Thomas Liebmann

Authors
  1. Stefan Kassberger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Liebmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Liebmann.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kassberger, S., Liebmann, T. Minimal q-entropy martingale measures for exponential time-changed Lévy processes. Finance Stoch 15, 117–140 (2011). https://doi.org/10.1007/s00780-010-0133-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00780-010-0133-9

Keywords

Mathematics Subject Classification (2000)

JEL Classification

Search

Navigation