Skip to main content
Springer Nature Link
Log in

An approximation scheme for Black-Scholes equations with delays

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

This paper addresses a finite difference approximation for an infinite dimensional Black-Scholes equation obtained by Chang and Youree (2007). The equation arises from a consideration of an European option pricing problem in a market in which stock prices and the riskless asset prices have hereditary structures. Under a general condition on the payoff function of the option, it is shown that the pricing function is the unique viscosity solution of the infinite dimensional Black-Scholes equation. In addition, a finite difference approximation of the viscosity solution is provided and the convergence results are proved.

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
€32.70 /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 includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Economics, 1973, 81: 637–659.

    Article  Google Scholar 

  2. J. M. Harrison and D. M. Kreps, Martingales and arbitrage in multi-period security markets, J. Economics Theory, 1979, 20: 381–408.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. M. Harrison and S. R. Pliska, Martingales and stochastic integrals in theory of continuous trading, Stochastic Processes Appl., 1981, 11: 215–260.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. C. Merton, Theory of rational option pricing, Bell J. Economics and Management Science, 1973, 4: 141–183.

    Article  MathSciNet  Google Scholar 

  5. R. C. Merton, Continuous-Time Finance, Basil Blackwell, Oxford, 1990.

    Google Scholar 

  6. A. N. Shiryaev, Y. M. Kabanov, D. O. Kramkov, and A. V. Melnikov, Toward the theory of pricing of options of both European and American types II: continuous time, Theory of Probability and Applications, 1994, 39: 61–102.

    Article  MathSciNet  Google Scholar 

  7. M. H. Chang and R. Youree, The European option with hereditary price structures: Basic theory, Applied Mathematics and Computation, 1999, 102: 279–296.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Chang and R. Youree, Infinite-Dimensional Black-Scholes with Hereditary Structure, Applied Mathematics and Optimization, 2007, 56: 395–424.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Arriojas, Y. Hu, S. E. Mohammed, and G. Pap, A delayed Black and Scholes formula, Stochastic Analysis and Applications, 2007, 25(2): 471–492.

    Article  MATH  MathSciNet  Google Scholar 

  10. Y. I. Kazmerchuk, A. V. Swishchuk, and J. H. Wu, The pricing of options for securities markets with delayed response, preprint, 2004a.

  11. Y. I. Kazmerchuk, A. V. Swishchuk, and J. H. Wu, Black-Scholes formula revisited: Securities markets with delayed response, preprint, 2004b.

  12. Y. I. Kazmerchuk, A. V. Swishchuk, and J. H. Wu, A continuous-time GARCH model for stochastic volatility with delay, preprint, 2004c.

  13. G. Barles and P. E. Souganidis, Convergence of approximative schemes for fully nonlinear second order equations, J. Asymptotic Analysis, 1991, 4: 557–579.

    MathSciNet  Google Scholar 

  14. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, Heidelberg, Berlin, 1977.

    MATH  Google Scholar 

  15. S. E. A. Mohammed, Stochastic Functional Differential Equations, Research Notes in Mathematics, Pitman Publishing, Boston-London-Melbourne, 1984, 99.

    MATH  Google Scholar 

  16. S. E. A. Mohammed, Stochastic differential equations with memory, theory, examples and applications, Stochastics Analysis and Related Topics 6, the Geido Workshop, 1996, Progress in Probability, Birkhauser, 1998.

  17. M. G. Crandall, H. Ishii, and P. L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 1992, 27: 1–67.

    Article  MATH  MathSciNet  Google Scholar 

  18. W. H. Fleming and H. M. Soner, Control Markov Processes and Viscosity Solutions, Springer Verlag, New York, 1993.

    Google Scholar 

  19. M. H. Chang, T. Pang, and M. Pemy, Finite difference approximations for stochastic control systems with delay, Stoch. Anal. Appl., 2008, 26: 451–470.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. H. Chang, T. Pang, and M. Pemy, Finite difference approximation for stochastic optimal stopping problems with delays, J. Ind. Manag. Optim., 2008, 4: 227–246.

    MATH  MathSciNet  Google Scholar 

  21. M. H. Chang, T. Pang, and M. Pemy, Optimal control of stochastic functional differential equations with a bounded memory, Stochastics, 2008, 80: 69–96.

    MATH  MathSciNet  Google Scholar 

  22. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd Edition, Springer-Verlag, Berlin, New York, 1991.

    MATH  Google Scholar 

  23. B. Larssen, Dynamic programming in stochastic control of systems with delay, Dr. Scient. thesis, University of Oslo, 2002.

  24. R. S. Liptser and A. N. Shiryaev, Statistics of Random Processes, Vols. 1 and 2, Springer-Verlag, Berlin, New York, 1977, 1978.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Division, US Army Research Office, P. O. Box 12211, RTP, NC, 27709, USA

    Mou-Hsiung Chang

  2. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Tao Pang

  3. Department of Mathematics, Towson University, Towson, MD, 21252, USA

    Moustapha Pemy

Authors
  1. Mou-Hsiung Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tao Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Moustapha Pemy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mou-Hsiung Chang.

Additional information

This research is partially supported by a grant W911NF-04-D-0003 from the US Army Research Office.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chang, MH., Pang, T. & Pemy, M. An approximation scheme for Black-Scholes equations with delays. J Syst Sci Complex 23, 438–455 (2010). https://doi.org/10.1007/s11424-010-0139-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-010-0139-6

Key words

Search

Navigation