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Pricing Early-Exercise and Discrete Barrier Options by Fourier-Cosine Series Expansions


  • Fang, Fang
  • Oosterlee, Kees


We present a pricing method based on Fourier-cosine expansions for early-exercise and discretely-monitored barrier options. The method works well for exponential Levy asset price models. The error convergence is exponential for processes characterized by very smooth transitional probability density functions. The computational complexity is $O((M-1) N \log{N})$ with $N$ a (small) number of terms from the series expansion, and $M$, the number of early-exercise/monitoring dates.

Suggested Citation

  • Fang, Fang & Oosterlee, Kees, 2008. "Pricing Early-Exercise and Discrete Barrier Options by Fourier-Cosine Series Expansions," MPRA Paper 9248, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:9248

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    References listed on IDEAS

    1. Lord, Roger & Fang, Fang & Bervoets, Frank & Oosterlee, Kees, 2007. "A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes," MPRA Paper 1952, University Library of Munich, Germany.
    2. Kenneth J. Singleton & Len Umantsev, 2002. "Pricing Coupon-Bond Options And Swaptions In Affine Term Structure Models," Mathematical Finance, Wiley Blackwell, vol. 12(4), pages 427-446.
    3. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    4. Ariel Almendral & Cornelis W. Oosterlee, 2007. "On American Options Under the Variance Gamma Process," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(2), pages 131-152.
    5. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    6. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
    7. Mark Broadie & Yusaku Yamamoto, 2003. "Application of the Fast Gauss Transform to Option Pricing," Management Science, INFORMS, vol. 49(8), pages 1071-1088, August.
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    Cited by:

    1. A. Golbabai & L. Ballestra & D. Ahmadian, 2014. "A Highly Accurate Finite Element Method to Price Discrete Double Barrier Options," Computational Economics, Springer;Society for Computational Economics, vol. 44(2), pages 153-173, August.
    2. Tat Lung Chan, 2017. "Singular Fourier-Pad\'e Series Expansion of European Option Prices," Papers 1706.06709,, revised Nov 2017.
    3. Jun Cheng & Jin Zhang, 2012. "Analytical pricing of American options," Review of Derivatives Research, Springer, vol. 15(2), pages 157-192, July.
    4. Ki Wai Chau & Cornelis W. Oosterlee, 2016. "On the wavelets-based SWIFT method for backward stochastic differential equations," Papers 1611.06098,
    5. Carolyn E. Phelan & Daniele Marazzina & Gianluca Fusai & Guido Germano, 2017. "Hilbert transform, spectral filtering and option pricing," Papers 1706.09755,
    6. Carl Chiarella & Boda Kang & Gunter H. Meyer, 2010. "The Evaluation Of Barrier Option Prices Under Stochastic Volatility," Research Paper Series 266, Quantitative Finance Research Centre, University of Technology, Sydney.
    7. Amirhossein Sobhani & Mariyan Milev, 2017. "A Numerical Method for Pricing Discrete Double Barrier Option by Legendre Multiwavelet," Papers 1703.09129,, revised Mar 2017.
    8. Adrien Genin & Peter Tankov, 2016. "Optimal importance sampling for L\'evy Processes," Papers 1608.04621,
    9. Laura Ballota & Griselda Deelstra & Grégory Rayée, 2015. "Quanto Implied Correlation in a Multi-Lévy Framework," Working Papers ECARES ECARES 2015-36, ULB -- Universite Libre de Bruxelles.
    10. Marjon Ruijter & Kees Oosterlee, 2012. "Two-dimensional Fourier cosine series expansion method for pricing financial options," CPB Discussion Paper 225, CPB Netherlands Bureau for Economic Policy Analysis.
    11. Alev{s} v{C}ern'y, 2016. "Discrete-Time Quadratic Hedging of Barrier Options in Exponential L\'{e}vy Model," Papers 1603.03747,
    12. Helin Zhu & Fan Ye & Enlu Zhou, 2013. "Fast Estimation of True Bounds on Bermudan Option Prices under Jump-diffusion Processes," Papers 1305.4321,
    13. Fei Cong & Cornelis W. Oosterlee, 2017. "Accurate and Robust Numerical Methods for the Dynamic Portfolio Management Problem," Computational Economics, Springer;Society for Computational Economics, vol. 49(3), pages 433-458, March.
    14. Rob Aalbers & Marjon Ruijter & Kees Oosterlee, 2014. "The social discount rate under a stochastic A2 scenario," CPB Discussion Paper 296, CPB Netherlands Bureau for Economic Policy Analysis.
    15. Chunfa Wang, 2017. "Pricing European Options by Stable Fourier-Cosine Series Expansions," Papers 1701.00886,, revised Jan 2017.

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    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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