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AN ACCURATE AND EFFICIENT NUMERICAL METHOD FOR BLACK-SCHOLES EQUATIONS

  • Published : 2009.10.31

Abstract

We present an efficient and accurate finite-difference method for computing Black-Scholes partial differential equations with multiunderlying assets. We directly solve Black-Scholes equations without transformations of variables. We provide computational results showing the performance of the method for two underlying asset option pricing problems.

Keywords

References

  1. F. Black and M. Sholes, The pricing of options and corporate liabilities, J. Political Economy 81 (1973), no. 3, 637–659 https://doi.org/10.1086/260062
  2. R. C. Y. Chin, T. A. Manteuffel, and J. de Pillis, ADI as a preconditioning for solving the convection-diffusion equation, SIAM J. Sci. Stat. Comput. 5 (1984), no. 2, 281-299. https://doi.org/10.1137/0905020
  3. D. J. Duffy, Finite Difference Methods in Financial Engineering : a partial differential equation approach, John Wiley and Sons, New York, 2006.
  4. W. Hackbusch, Iterative Solution of Large Linear Systems of Equations, Springer, New York, 1994
  5. H. Han and X.Wu, A fast numerical method for the Black-Scholes equation of American options, SIAM J. Numer. Anal. 41 (2003), 2081-2095. https://doi.org/10.1137/S0036142901390238
  6. E. G. Haug, The Complete Guide to Option Pricing Formulas, MaGraw-Hill, 2007.
  7. J. C. Hull, Options, Futures and Others, Prentice Hall, 2003.
  8. S. Ikonen and J. Toivanen, Operator splitting methods for American option pricing, Applied Mathematics Letters 17 (2004), 809-814. https://doi.org/10.1016/j.aml.2004.06.010
  9. Y. K. Kwok, Mathematical Models of Financial Derivatives, Springer, 1998.
  10. R. C. Merton, Theory of rational option pricing, Bell J. Econ. Manag. Sci. 4 (1973), no. 1, 141-183. https://doi.org/10.2307/3003143
  11. J. Persson and L. von Sydow, Pricing European multi-asset options using a space-time adaptive FD-method, Comput. Visual. Sci. 10 (2007), 173-183. https://doi.org/10.1007/s00791-007-0072-y
  12. C. Reisinger and G. Wittum, On multigrid for anisotropic equations and variational inequalities, Comput. Visual. Sci. 7 (2004), 189-197 https://doi.org/10.1007/s00791-004-0149-9
  13. Y. Saad and M. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7 (1986), 856-869. https://doi.org/10.1137/0907058
  14. Y. Saad and H. A. van der Vorst, Iterative solution of linear systems in the 20th century, J. Comput. Appl. Math. 123 (2000), 1-33. https://doi.org/10.1016/S0377-0427(00)00412-X
  15. R. Seydel, Tools for Computational Finance, Springer Verlag, Berlin, 2003.
  16. D. Tavella and C. Randall, Pricing Financial Instruments - The finite difference method, John Wiley and Sons, Inc., 2000.
  17. J. Topper, Financial Engineering with Finite Elements, John Wiley and Sons, New York, 2005.
  18. U. Trottenberg, C. Oosterlee, and A. Schuller, Multigrid, Academic press, 2001.
  19. H. A. Van Der Vorst, BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 13 (1992), no. 2, 631-644. https://doi.org/10.1137/0913035
  20. P. Wilmott, J. Dewynne, and S. Howison, Option Pricing : mathematical models and computation, Oxford Financia Press, Oxford, 1993.

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