Skip to main content
SpringerLink
Log in

Geometric Brownian Motion with Tempered Stable Waiting Times

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

One of the earliest system that was used to asset prices description is Black-Scholes model. It is based on geometric Brownian motion and was used as a tool for pricing various financial instruments. However, when it comes to data description, geometric Brownian motion is not capable to capture many properties of present financial markets. One can name here for instance periods of constant values. Therefore we propose an alternative approach based on subordinated tempered stable geometric Brownian motion which is a combination of the popular geometric Brownian motion and inverse tempered stable subordinator. In this paper we introduce the mentioned process and present its main properties. We propose also the estimation procedure and calibrate the analyzed system to real data.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Barkai, E., Silbey, R.: Distribution of single-molecule line widths. Chem. Phys. Lett. 310, 287–295 (1999)

    Article  ADS  Google Scholar 

  2. Barkai, E., Metzler, R., Klafter, J.: From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61, 132–138 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  3. Baeumer, B., Meerschaert, M.M.: Tempered stable lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233, 2438–2448 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)

    Article  Google Scholar 

  5. Gajda, J., Magdziarz, M.: Fractional Fokker-Planck equation with tempered alpha-stable waiting times. Langevin picture and computer simulation. Phys. Rev. E 82, 011117 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  6. Gajda, J., Magdziarz, M.: Kramer’s escape problem for fractional Klein-Kramers equation with tempered α-stable waiting times. Phys. Rev. E 84, 021137 (2011)

    Article  ADS  Google Scholar 

  7. Galloway, M.L., Nolder, C.A.: Subordination, self-similarity, and option pricing. J. Appl. Math. Decis. Sci. (2008), Article ID 397028

  8. Hougaard, P.: A class of multivariate failure time distributions. Biometrika 73, 671–678 (1986)

    MathSciNet  MATH  Google Scholar 

  9. Janczura, J., Wyłomańska, A.: Subdynamics of financial data from fractional Fokker-Planck equation. Acta Phys. Pol. B 40(5), 1341–1351 (2009)

    Google Scholar 

  10. Janczura, J., Orzeł, S., Wyłomańska, A.: Subordinated alpha-stable Ornstein-Uhlenbeck process as a tool of financial data description. Physica A 390, 4379–4387 (2011)

    Article  ADS  Google Scholar 

  11. Jeon, J.-H., Tejedor, V., Burov, S., Barkai, E., Selhuber-Unkel, Ch., Berg-Sørensen, K., Odderhede, L., Metzler, R.: In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett. 106, 048103 (2011)

    Article  ADS  Google Scholar 

  12. Jha, R., Kaw, P.K., Kulkarni, D.R., Parikh, J.C., et al.: Evidence of lévy stable process in tokamak edge turbulence. Phys. Plasmas 10, 699–704 (2003)

    Article  ADS  Google Scholar 

  13. Kim, Y.S., Rachev, S.T., Bianchi, M.L., Fabozzi, F.J.: A new tempered stable distribution and its application to finance. In: Bol, G., Rachev, S.T., Wuerth, R. (eds.) Risk Assessment: Decisions in Banking and Finance. Physica-Verlag/Springer, Heidelberg (2007)

    Google Scholar 

  14. Kim, Y.S., Chung, D.M., Rachev, S.T., Bianchi, M.L.: The modified tempered stable distribution, GARCH models and option pricing. Probab. Math. Stat. 29(1), 91–117 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Klafter, J., Sokolov, I.M.: First Steps in Random Walks: From Tools to Applications. Oxford University Press, Oxford (2011)

    MATH  Google Scholar 

  16. Magdziarz, M., Weron, A., Weron, K.: Fractional Fokker-Planck dynamics: stochastic representation and computer simulation. Phys. Rev. E 75, 016708 (2007)

    Article  ADS  Google Scholar 

  17. Magdziarz, M., Weron, A., Klafter, J.: Equivalence of the fractional Fokker-Planck and subordinated Langevin equations: the case of a time-dependent force. Phys. Rev. Lett. 101, 210601 (2008)

    Article  ADS  Google Scholar 

  18. Magdziarz, M.: Langevin picture of subdiffusion with infinitely divisible waiting times. J. Stat. Phys. 135, 763–772 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Magdziarz, M., Gajda, J.: Anomalous dynamics of Black-Scholes model time-changed by inverse subordinators. Acta Phys. Pol. B 43(5), 1093–1110 (2012)

    Article  Google Scholar 

  20. Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, 141–183 (1973)

    Article  MathSciNet  Google Scholar 

  21. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Montroll, E.W., Scher, H.: Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries. J. Stat. Phys. 9, 101–135 (1973)

    Article  ADS  Google Scholar 

  23. Orzeł, S., Wyłomańska, A.: Calibration of the subdiffusive arithmetic Brownian motion with tempered stable waiting-times. J. Stat. Phys. 143(3), 447–454 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Weigel, A.V., Simon, B., Tamkun, M.M., Krapf, D.: Ergodic and nonergodic processes coexist in the plasma membrane as observed by single molecule tracking. Proc. Natl. Acad. Sci. USA 108, 6438–6443 (2011)

    Article  ADS  Google Scholar 

  26. Wong, I.Y., Gardel, M.L., Reichman, D.R., Weeks, E., Valentine, M.T., Bausch, A.R., Weitz, D.A.: Anomalous diffusion probes microstructure dynamics of entangled F-actin networks. Phys. Rev. Lett. 92, 178101 (2004)

    Article  ADS  Google Scholar 

  27. Wyłomańska, A.: Arithmetic brownian motion subordinated by tempered stable and inverse tempered stable processes. Physica A (2012). doi:10.1016/j.physa.2012.050.72

    Google Scholar 

Download references

Acknowledgements

The research of Janusz Gajda has been partially supported by the European Union within the European Social Fund.

Author information

Authors and Affiliations

  1. Institute of Mathematics and Computer Science, Hugo Steinhaus Center, Wrocław University of Technology, Wroclaw, Poland

    Janusz Gajda & Agnieszka Wyłomańska

Authors
  1. Janusz Gajda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Agnieszka Wyłomańska
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Janusz Gajda.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gajda, J., Wyłomańska, A. Geometric Brownian Motion with Tempered Stable Waiting Times. J Stat Phys 148, 296–305 (2012). https://doi.org/10.1007/s10955-012-0537-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-012-0537-3

Keywords

Search

Navigation