An efficient algorithm for Bermudan barrier option pricing


An efficient option pricing method based on Fourier-cosine expansions was presented by Fang and Oosterlee for European options in 2008, and later, this method was also used by them to price early-exercise options and barrier options respectively, in 2009. In this paper, this method is applied to price discretely American barrier options in which the monitored dates are many times more than the exercise dates. The corresponding algorithm is presented to practical option pricing. Numerical experiments show that this algorithm works very well and efficiently for different exponential Lévy asset models.

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  1. [1]

    A D Andricopoulos, M Widdicks, P D Duck, P D Newton. Universal option valuation using quadrature methods, J Financ Econ, 2003, 67: 447–471.

    Article  Google Scholar 

  2. [2]

    O E Barndorff-Nielsen. Normal inverse Gaussian distributions and stochastic volatility modelling, Scand J Statist, 1997, 24(1): 1–13.

    MathSciNet  MATH  Article  Google Scholar 

  3. [3]

    D S Bates. Jumps and stochastic volatility: exchange rate processes implicit in Deutsch mark options, Rev Financ Stud, 1996, 9: 69–107.

    Article  Google Scholar 

  4. [4]

    F Black, M Scholes. The pricing of options and corporate liabilities, J Polit Econ, 1973, 81(3): 637–654.

    Article  Google Scholar 

  5. [5]

    P Carr, H Geman, D B Madan, M Yor. The fine structure of asset returns: An empirical investigation, J Bus, 2002, 75(2): 305–332.

    Article  Google Scholar 

  6. [6]

    P Carr, D Madan. Option valuation using the fast Fourier transform, J Comput Financ, 1999, 2: 61–73.

    Google Scholar 

  7. [7]

    R Cont, P Tankov. Financial Modelling with Jump Processes, Chapman & Hall/CRC Press, 2003.

  8. [8]

    C Dean, J F Lawless, G E Willmot. A mixed Poisson-Inverse-Gaussian regression model, Canad J Statist, 1989, 17(2): 171–181.

    MathSciNet  MATH  Article  Google Scholar 

  9. [9]

    D Ding, S C U. Efficient option pricing methods based on Fourier series expansions, JMath Res Exposition, 2010, 31: 12–22.

    MathSciNet  Google Scholar 

  10. [10]

    F Fang, C W Oosterlee. A novel pricing method for European option based on Fourier-cosine series expansions, SIAM J Sci Comput, 2008, 31: 826–848.

    MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    F Fang, C W Oosterlee. Pricing early-exercise and discrete barrier options by Fourier-cosine series expansions, Numer Math, 2009, 114: 27–62.

    MathSciNet  MATH  Article  Google Scholar 

  12. [12]

    L Feng, V Linetsky. Pricing discretely monitored barrier options and defaultable bonds in Lévy process models: a fast Hilbert transform approach, Math Financ, 2008, 18: 337–384.

    MathSciNet  MATH  Article  Google Scholar 

  13. [13]

    B Gao, J Huang, M Subrahmanyam. The valuation of American barrier options using the decomposition technique, J Econ Dyn Control, 2000, 24: 1783–1827.

    MathSciNet  MATH  Article  Google Scholar 

  14. [14]

    S L Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev Financ Stud, 1993, 6: 327–343.

    Article  Google Scholar 

  15. [15]

    S G Kou. A jump-diffusion model for option pricing, Manage Sci, 2002, 48: 1086–1101.

    MATH  Article  Google Scholar 

  16. [16]

    R Lord, F Fang, F Bervoets, C W Oosterlee. A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes, SIAM J Sci Comput, 2008, 30: 1678–1705.

    MathSciNet  Article  Google Scholar 

  17. [17]

    D B Madan, P Carr, E Chang. The variance gamma process and option pricing, Eur Financ Rev, 1998, 2: 79–105.

    MATH  Article  Google Scholar 

  18. [18]

    C R Merton. Option pricing when underlying stock returns are discontinuous. J Financ Econ, 1976, 3: 125–144.

    MATH  Article  Google Scholar 

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Corresponding author

Correspondence to Ning-ying Huang.

Additional information

The work was partially supported by the research grants (UL020/08-Y4/MAT/JXQ01/FST and MYRG136(Y1-L2)-FST11-DD) from University of Macau.

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Ding, D., Huang, N. & Zhao, J. An efficient algorithm for Bermudan barrier option pricing. Appl. Math. J. Chin. Univ. 27, 49–58 (2012).

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MR Subject Classification

  • 42A10
  • 62P05
  • 65T40


  • American barrier option
  • Bermudan option
  • Fourier transform
  • Fourier-cosine expansion