Cubic B-Spline Collocation Method for Pricing Path Dependent Options

  • Geraldine Tour
  • Désiré Yannick Tangman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8584)


We consider an efficient pricing method based on the cubic B-spline collocation method in which the approximation to the exact solution can be expressed as a linear combination of cubic B-spline basis functions. An earlier work has proposed to use such technique to solve the generalised Black–Scholes PDE to price European options only. In this work, we extend the application of the B-spline collocation method to price path-dependent options such as American, Barrier and Asian options under the Black–Scholes model. Our numerical results show that the new scheme is a method of choice for option pricing. Indeed the scheme has the same computational complexity as the standard second order finite difference scheme since they both require the solution of tridiagonal linear systems. However our numerical experiments reveal that the B-spline collocation method is more accurate in the infinity norm when solving convectively dominated financial problems and yields second order convergent prices and hedging parameters for both continuous and discretely sampled path-dependent options.


collocation method cubic B-splines basis functions path-dependent option pricing 


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  1. 1.
    Wu, L., Kwok, Y.K.: A front-fixing difference method for the valuation of American options. Journal of Financial Engineering 6, 83–97 (1997)Google Scholar
  2. 2.
    Ikonen, S., Toivanen, J.: Operator splitting methods for American option pricing. Applied Mathematics Letters 17, 809–814 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Forsyth, P.A., Vetzal, K.R.: Quadratic convergence of a penalty method for valuing American options. SIAM J. Sci. Comput. 23, 2096–2123 (2002)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Kou, S.G.: Discrete barrier and lookback options. Handbooks in Operations Research and Management Science 15, 343–373 (2008)CrossRefGoogle Scholar
  5. 5.
    Turnbull, S., Wakerman, L.: A quick algorithm for pricing European average options. Journal of Financial and Quantitative Analysis 26, 377–389 (1992)CrossRefGoogle Scholar
  6. 6.
    Linetsky, V.: Spectral expansions for Asian (average price) options. Operations Research 52, 856–867 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Věceř, J.: A new PDE approach for pricing arithmetic average Asian options. The Journal of Computational Finance 4, 105–113 (2001)Google Scholar
  8. 8.
    Prenter, P.M.: Splines and Variational Methods. Wiley (1975)Google Scholar
  9. 9.
    de Boor, C.: A Practical guide to splines. Springer (1972)Google Scholar
  10. 10.
    Mittal, R.C., Arora, G.: Numerical solution of the coupled viscous Burger’s equation. Communications in Nonlinear Science and Numerical Simulation 16, 1304–1313 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Mittal, R.C., Jain, R.K.: Redefined cubic B-splines collocation method for solving convection-diffusion equations. Applied Mathematical Modelling 36, 5555–5573 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Mittal, R.C., Jain, R.K.: Numerical solutions of nonlinear Fisher’s reaction-diffusion equation with modified cubic B-spline collocation method. Mathematical Sciences (2013)Google Scholar
  13. 13.
    Kadalbajoo, M.K., Tripathi, L.P., Kumar, A.: A cubic B-spline collocation method for a numerical solution of the generalized Black–Scholes equation. Mathematical and Computer Modelling 55, 1483–1505 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Black, F., Scholes, M.: The pricing of options and other corporate liabilities. Journal of Political Economy 81, 637–654 (1973)CrossRefGoogle Scholar
  15. 15.
    Zhu, Y.-L., Wu, X., Chern, I.-L., Sun, Z.-Z.: Derivatives Securities and Different Methods. Springer, New York (2004)CrossRefGoogle Scholar
  16. 16.
    Tangman, D.Y., Peer, A.A.I., Rambeerich, N., Bhuruth, M.: Fast simplified approaches to Asian option pricing. The Journal of Computational Finance 14, 3–36 (2011)Google Scholar
  17. 17.
    Strang, G.: On the construction and comparison of difference schemes. SIAM Journal on Numerical Analysis 29, 209–228 (1968)MathSciNetGoogle Scholar
  18. 18.
    Fang, F., Oosterlee, C.W.: Pricing Early-Exercise and Discrete Barrier Options by Fourier-Cosine Series Expansions. Numerische Mathematik 114, 27–62 (2009)Google Scholar
  19. 19.
    Zhang, B., Oosterlee, C.W.: Efficient Pricing of Asian Options under Lévy Processes Based on Fourier Cosine Expansions, Report submitted to TU Delft University of Technology (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Geraldine Tour
    • 1
  • Désiré Yannick Tangman
    • 1
  1. 1.Department of MathematicsUniversity of MauritiusRéduitMauritius

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