Advertisement

A New Howard–Crandall–Douglas Algorithm for the American Option Problem in Computational Finance

  • Nawdha Thakoor
  • Dhiren Kumar Behera
  • Désiré Yannick Tangman
  • Muddun BhuruthEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 711)

Abstract

The unavailability of a closed-form formula for the American option price means that the price needs to be approximated by numerical techniques. The valuation problem can be formulated either as a linear complementarity problem or a free-boundary value problem. Both approaches require a discretisation of the associated partial differential equation, and it is common to employ standard second-order finite difference approximations. This work develops a new procedure for the linear complementarity formulation. Howard’s algorithm is used to solve the discrete problem obtained through a higher-order Crandall–Douglas discretisation. Speed and error comparisons indicate that this approach is more efficient than the procedures for solving the free-boundary value problem.

Keywords

Computational finance American option Policy iteration Howard’s algorithm 

References

  1. 1.
    McCartin, B.J., Labadie, S.M.: Accurate and efficient pricing of vanilla stock options via the Crandall-Douglas scheme. Appl. Math. Comput. 143, 39–60 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Crandall, S.H.: An optimum implicit recurrence formula for the heat conduction equation. Q. Appl. Math. 13, 318–320 (1955)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Douglas Jr., J.: The solution of the diffusion equation by a high order correct difference equation. J. Math. Phys. 35, 145–151 (1956)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Elliott, C.M., Ockendon, J.R.: Weak and variational methods for moving boundary problems. Research notes in Mathematics, Pitman, Boston, Mass. 59 (1982)Google Scholar
  5. 5.
    Howard, R.A.: Dynamic Programming and Markov Processes. The MIT Press, Cambridge, MA (1960)Google Scholar
  6. 6.
    Reisinger, C., Witte, J.H.: On the use of policy iteration as an easy way of pricing American options. SIAM J. Financial Math. 3, 459–478 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Seydel, U.R.: Tools for Computational Finance. Springer-Verlag, Heidelberg (2006)Google Scholar
  9. 9.
    Han, H., Wu, X.: A fast numerical method for the Black-Scholes equation of American options. SIAM J. Numer. Anal. 41, 2081–2095 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Saib, A.A.E.F., Tangman, D.Y., Thakoor, N., Bhuruth, M.: On some finite difference algorithms for pricing American options and their implementation in Mathematica. In: Proceedings of the 11th International Conference on Computational and Mathematical Methods in Science and Engineering, pp. 1029–1040. Alicante, Spain (2011)Google Scholar
  11. 11.
    Muthuraman, K.: A moving boundary approach to American option pricing. J. Econ. Dyn. Control 32, 3520–3537 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Brennan, M.J., Schwartz, E.S.: The valuation of American put options. J. Finance 32, 449–462 (1977)CrossRefGoogle Scholar
  13. 13.
    Bokanowski, O., Maroso, S., Zidani, H.: Some convergence results for Howard’s algorithm. SIAM J. Numer. Anal. 47, 3001–3026 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Nawdha Thakoor
    • 1
  • Dhiren Kumar Behera
    • 2
  • Désiré Yannick Tangman
    • 1
  • Muddun Bhuruth
    • 1
    Email author
  1. 1.Department of MathematicsUniversity of MauritiusReduitMauritius
  2. 2.Mechanical Engineering DepartmentIndira Gandhi Institute of Technology, SarangDhenkanalIndia

Personalised recommendations