Skip to main content
SpringerLink
Log in

Black-Scholes Formula in Subdiffusive Regime

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

Abstract

In the classical approach the price of an asset is described by the celebrated Black-Scholes model. In this paper we consider a generalization of this model, which captures the subdiffusive characteristics of financial markets. We introduce a subdiffusive geometric Brownian motion as a model of asset prices exhibiting subdiffusive dynamics. We find the corresponding fractional Fokker-Planck equation governing the dynamics of the probability density function of the introduced process. We prove that the considered model is arbitrage-free and incomplete. We find the corresponding subdiffusive Black-Scholes formula for the fair prices of European options and show how these prices can be evaluated using Monte-Carlo methods. We compare the obtained results with the classical ones.

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. Bachelier, L.: Théorie de la spéculation. Ann. Ec. Norm. Supér. 17, 21–86 (1900)

    MathSciNet  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  ADS  MathSciNet  Google Scholar 

  3. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–659 (1973)

    Article  Google Scholar 

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

    Google Scholar 

  5. Eliazar, I., Klafter, J.: Spatial gliding, temporal trapping, and anomalous transport. Physica D 187, 30–50 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Hilfer, R.: Analytical representations for relaxation functions of glasses. J. Non-Cryst. Solids 305, 122–126 (2002)

    Article  ADS  Google Scholar 

  7. Hurst, S.R., Platen, E., Rachev, S.T.: Subordinated market index models: a comparison. Financ. Eng. Jpn. Mark. 4, 97–124 (1995)

    Article  Google Scholar 

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

    Google Scholar 

  9. Janicki, A., Weron, A.: Simulation and Chaotic Behaviour of α-Stable Stochastic Processes. Dekker, New York (1994)

    Google Scholar 

  10. Magdziarz, M.: Stochastic representation of subdiffusion processes with time-dependent drift. Stoch. Proc. Appl. (2009). doi:10.1016/j.spa.2009.05.006

    Google Scholar 

  11. Magdziarz, M.: Path properties of subdiffusion—a martingale approach (2009, submitted)

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

    Article  ADS  Google Scholar 

  13. 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 

  14. 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 

  15. Mantegna, R.N., Stanley, H.E.: An Introduction to Econophysics—Correlation and Complexity in Finance. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  16. Meerschaert, M.M., Benson, D.A., Scheffler, H.P., Baeumer, B.: Stochastic solution of space-time fractional diffusion equations. Phys. Rev. E 65, 041103 (2002)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Metzler, R., Barkai, E., Klafter, J.: Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82, 3563–3567 (1999)

    Article  ADS  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. Musiela, M., Rutkowski, M.: Martingale Methods in Financial Modeling. Springer, Berlin (1997)

    Google Scholar 

  21. Piryatinska, A., Saichev, A.I., Woyczynski, W.A.: Models of anomalous diffusion: the subdiffusive case. Physica A 349, 375–420 (2005)

    Article  ADS  Google Scholar 

  22. Protter, P.: Stochastic Integration and Differential Equations. A New Approach. Springer, Berlin (1990)

    MATH  Google Scholar 

  23. Samko, S.G., Kilbas, A.A., Maritchev, D.I.: Integrals and Derivatives of the Fractional Order and Some of Their Applications. Gordon and Breach, Amsterdam (1993)

    Google Scholar 

  24. Samuelson, P.A.: Rational theory of warrant pricing. Ind. Manag. Rev. 6, 13–31 (1965)

    Google Scholar 

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

    MATH  Google Scholar 

  26. Sokolov, I.M.: Lévy flights from a continuous-time process. Phys. Rev. E 63, 011104 (2000)

    Article  ADS  Google Scholar 

  27. Sokolov, I.M.: Solutions of a class of non-Markovian Fokker-Planck equations. Phys. Rev. E 66, 041101 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  28. Stanislavsky, A.A.: Black-Scholes model under subordination. Physica A 318, 469–474 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. Stanislavsky, A.A., Weron, K., Weron, A.: Diffusion and relaxation controlled by tempered-stable processes. Phys. Rev. E 78, 051106 (2008)

    Article  ADS  Google Scholar 

  30. Weron, R.: On the Chambers-Mallows-Stuck method for simulating skewed stable random variables. Stat. Probab. Lett. 28, 165–171 (1996)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Hugo Steinhaus Center, Institute of Mathematics and Computer Science, Wroclaw University of Technology, 50-370, Wroclaw, Poland

    Marcin Magdziarz

Authors
  1. Marcin Magdziarz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcin Magdziarz.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Magdziarz, M. Black-Scholes Formula in Subdiffusive Regime. J Stat Phys 136, 553–564 (2009). https://doi.org/10.1007/s10955-009-9791-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-009-9791-4

Keywords

Search

Navigation