Pricing Asian Option by Solving Black–Scholes PDE Using Gauss–Seidel Method

  • W. S. KohEmail author
  • R. R. Ahmad
  • S. H. Jaaman
  • J. Sulaiman
Conference paper


The main purpose of this paper is to study the pricing of the Asian option by using Gauss–Seidel iterative method via the finite difference approximation equation. Actually, Asian option is an option that is taking the average price of the underlying asset over the lifetime of the option. To solve the proposed problem numerically, a two-dimensional Black–Scholes Partial Differential Equation (PDE) governing the Asian option is discretized by using the second-order Crank–Nicolson discretization scheme. Then, the linear system generated from the discretization process is solved by using Gauss–Seidel iterative method. The results of the numerical computation were shown and discussed.


Asian option Black–Scholes partial differential equation Crank–Nicolson approximation scheme Gauss–Seidel iterative method 


  1. 1.
    Kemna, A.G., Vorst, A.C.: A pricing method for options based on average asset values. J. Bank. Financ. 14(1), 113–129 (1990). Scholar
  2. 2.
    Boyle, P., Broadie, M., Glasserman, P.: Monte Carlo methods for security pricing. J. Econ. Dyn. Control. 21(8), 1267–1321 (1997). Scholar
  3. 3.
    Rogers, L.C.G., Shi, Z.: The value of an Asian option. J. App. Probab. 32(4), 1077–1088 (1995). Scholar
  4. 4.
    Vecer, J.: A new PDE approach for pricing arithmetic average Asian options. J. Comput. Financ. 4(4), 105–113 (2001). Scholar
  5. 5.
    Elshegmani, Z.A., Ahmad, R.R., Jaaman, S.H.: On the modified arithmetic Asian option equation and its analytical solution. App. Math. Sci. 5(25–28), 1217–1227 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Avramidis, A.N., L’Ecuyer, P.: Efficient Monte Carlo and quasi–Monte Carlo option pricing under the variance gamma model. Manag. Sci 52(12), 1930–1944 (2006). Scholar
  7. 7.
    Levy, E.: Pricing European average rate currency options. J. Int. Money Financ. 11(5), 474–491 (1992). Scholar
  8. 8.
    Fusai, G., Meucci, A.: Pricing discretely monitored Asian options under Lévy processes. J. Bank. Financ. 32(10), 2076–2088 (2008). Scholar
  9. 9.
    Lee, T.Y., Chin, S.T.: A simple Crank-Nicolson scheme for Asian option. In: Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA 2010). UTAR, Kuala Lumpur, pp. 381–294 (2010)Google Scholar
  10. 10.
    Dubois, F., Lelièvre, T.: Efficient pricing of Asian options by the PDE approach. J. Comput. Financ. 8(2), 55–64 (2004). Scholar
  11. 11.
    Geman, H., Yor, M.: Bessel processes, asian options and perpetuities. Math. Financ. 3(4), 349–375 (1993). Scholar
  12. 12.
    Elshegmani, Z.A., Ahmad, R.R.: Solving an Asian option PDE via the Laplace transform. ScienceAsia 39(SUPPL. 1), 67–69 (2013). Scholar
  13. 13.
    Barucci, E., Polidoro, S., Vespri, V.: Some results on partial differential equations and Asian options. Math. Model. Methods Appl. Sci. 11(03), 475–497 (2001). Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • W. S. Koh
    • 1
    • 2
    Email author
  • R. R. Ahmad
    • 2
  • S. H. Jaaman
    • 2
  • J. Sulaiman
    • 3
  1. 1.INTI International UniversityNilaiMalaysia
  2. 2.Universiti KebangsaanBangiMalaysia
  3. 3.Universiti Malaysia SabahKota KinabaluMalaysia

Personalised recommendations