Skip to main content
Springer Nature Link
Log in

RETRACTED ARTICLE: The distribution of the maximum of a variance gamma process and path-dependent option pricing

  • Published:
Finance and Stochastics Aims and scope Submit manuscript

This article was retracted on 29 March 2016

Abstract

Although numerical procedures often supply a required accuracy, closed-form expressions allow one to escape any accumulation of errors. In this paper, we discuss the possibility of obtaining explicit results for a variance gamma process. We derive the exact distribution of the maximum of the variance gamma process over a finite interval of time and establish the prices of path-dependent options including digital barrier, fixed-strike lookback, and lookback options. The obtained formulas are based on values of hypergeometric functions.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Almendral, A., Oosterlee, C.W.: On American options under the variance gamma process. Appl. Math. Finance 14, 131–152 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bateman, H., Erdélyi, A.: Higher Transcendental Functions, vol. I. McGraw-Hill, New York (1953)

    MATH  Google Scholar 

  3. Bianchi, M.L., Rachev, S.T., Kim, Y.S., Fabozzi, F.J.: Tempered stable distributions and processes in finance: numerical analysis. In: Corazzo, M., Pizzi, C. (eds.) Mathematical and Statistical Methods for Actuarial Sciences and Finance, pp. 33–42. Springer, Milan (2010)

    Chapter  Google Scholar 

  4. Boyarchenko, M., Levendorskiĭ, S.: Ghost calibration and pricing barrier options and CDS in spectrally one-sided Lévy models: the parabolic Laplace inversion method (2014). Preprint. Available at: http://ssrn.com/abstract=2445318

  5. Boyarchenko, S., Levendorskiĭ, S.: Efficient pricing barrier options and CDS in Lévy models with stochastic interest rate (2015). Preprint. Available at: http://ssrn.com/abstract=2544271

  6. Carr, P., Lee, R., Wu, L.: Variance swaps on time-changed Lévy processes. Finance Stoch. 16, 335–355 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carr, P., Wu, L.: Time-changed Lévy processes and option pricing. J. Financ. Econ. 71, 113–141 (2004)

    Article  Google Scholar 

  8. Carr, P., Madan, D.: Option valuation using the fast Fourier transform. J. Comput. Finance 2, 61–73 (1999)

    Google Scholar 

  9. Eberlein, E.: Fourier based valuation methods in mathematical finance. In: Benth, F., Kholodnyi, V., Laurence, P. (eds.) Quantitative Energy Finance, pp. 85–114. Springer, New York (2014)

    Chapter  Google Scholar 

  10. Eberlein, E., Madan, D.: On correlating Lévy processes. J. Risk 13, 3–16 (2010)

    Google Scholar 

  11. Eberlein, E., Papapantoleon, A., Shiryaev, A.N.: On the duality principle in option pricing: semimartingale setting. Finance Stoch. 12, 265–292 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Finlay, R., Seneta, E.: Stationary-increment student and variance-gamma processes. J. Appl. Probab. 43, 441–453 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products. Academic Press, New York (1980)

    MATH  Google Scholar 

  14. Guillaume, F.: The \(\alpha\mbox{VG}\) model for multivariate asset pricing: calibration and extension. Rev. Deriv. Res. 16, 25–52 (2013)

    Article  MATH  Google Scholar 

  15. Hirsa, A., Madan, D.: Pricing American options under variance gamma. J. Comput. Finance 7, 63–80 (2004)

    Google Scholar 

  16. Ivanov, R.V., Ano, K.: On predicting the ultimate maximum for exponential Lévy processes. Electron. Commun. Probab. 17, 745–753 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Ivanov, R.V.: Closed form pricing of European options for a family of normal-inverse Gaussian processes. Stoch. Models 29, 435–450 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ivanov, R.V.: On the exact distribution of the maximum of the exponential of the generalized normal-inverse Gaussian process with respect to a martingale measure. Commun. Stoch. Anal. 7, 511–521 (2013)

    MathSciNet  Google Scholar 

  19. Ivanov, R.V.: The analytical formula for the distribution function of the variance gamma process and its application to option pricing. Stoch. Anal. Appl. (2012, submitted). Available at: http://ssrn.com/abstract=2617237

  20. Korn, K., Korn, E., Kroisandt, G.: Monte Carlo Methods and Models in Finance and Insurance. Chapman and Hall, Boca Raton (2010)

    Book  MATH  Google Scholar 

  21. Madan, D., Carr, P., Chang, E.: The variance gamma process and option pricing. Eur. Finance Rev. 2, 79–105 (1998)

    Article  MATH  Google Scholar 

  22. Madan, D., Milne, F.: Option pricing with VG martingale components. Math. Finance 1, 39–55 (1991)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  24. Musiela, M., Rutkowski, M.: Martingale Methods in Financial Modelling. Springer, Berlin (2005)

    MATH  Google Scholar 

  25. Rosiński, J.: Tempering stable processes. Stoch. Process. Appl. 117, 677–707 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schoutens, W., Van Damme, G.: The beta-variance gamma process. Rev. Deriv. Res. 14, 263–282 (2011)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Seneta, E.: The early years of the variance-gamma process. In: Fu, M., et al. (eds.) Advances in Mathematical Finance, pp. 3–19. Birkhäuser, Boston (2000)

    Google Scholar 

  29. Shiryaev, A.N.: Essentials of Stochastic Finance. Facts, Models, Theory. World Scientific, River Edge (1999)

    Google Scholar 

  30. Whittaker, E.T., Watson, G.N.: A Course in Modern Analysis. Cambridge University Press, New York (1990)

    MATH  Google Scholar 

  31. Yor, M.: Some remarkable properties of gamma processes. In: Advances in Mathematical Finance. Applied and Numerical Harmonic Analysis, pp. 37–47 (2007)

    Chapter  Google Scholar 

Download references

Acknowledgements

I am grateful to Professor A.N. Shiryaev for important discussions. I should like to thank the anonymous referees for important corrections, which have made the work essentially better.

Author information

Authors and Affiliations

  1. Laboratory of Control under Incomplete Information, Trapeznikov Institute of Control Sciences of RAS, Profsoyuznaya 65, 117997, Moscow, Russian Federation

    Roman V. Ivanov

Authors
  1. Roman V. Ivanov
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Roman V. Ivanov.

Additional information

The article "The distribution of the maximum of a variance gamma process and path-dependent option pricing" published in Volume 19/4, pages 979-993, DOI 10.1007/s00780-015-0277-8 has been retracted by agreement between the author Roman V. Ivanov and the journal's Editors Martin Schweizer and Chris Rogers. The retraction has been agreed because the paper contains a fundamental error which invalidates the results of the paper. The paper studies a variance gamma process as a time change of Brownian motion, but in Section 5 the proof of Theorem 2.1 incorrectly assumes that the pathwise maximum of the time-changed Brownian motion is the time-change of the maximum of the Brownian motion. We are grateful to Alexey Kuznetsov for detecting this error and drawing it to our attention.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ivanov, R.V. RETRACTED ARTICLE: The distribution of the maximum of a variance gamma process and path-dependent option pricing. Finance Stoch 19, 979–993 (2015). https://doi.org/10.1007/s00780-015-0277-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00780-015-0277-8

Keywords

Mathematics Subject Classification

JEL Classification