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A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes


  • Lord, Roger
  • Fang, Fang
  • Bervoets, Frank
  • Oosterlee, Kees


A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the well-known risk-neutral valuation formula by recognising that it is a convolution. The resulting convolution is dealt with numerically by using the Fast Fourier Transform (FFT). This novel pricing method, which we dub the Convolution method, CONV for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within exponentially Lévy models, including the exponentially affine jump-diffusion models. For an M-times exercisable Bermudan option, the overall complexity is O(MN log(N)) with N grid points used to discretise the price of the underlying asset. It is shown how to price American options efficiently by applying Richardson extrapolation to the prices of Bermudan options.

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  • 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.
  • Handle: RePEc:pra:mprapa:1952

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    References listed on IDEAS

    1. Sam Howison, 2005. "A matched asymptotic expansions approach to continuity corrections for discretely sampled options. Part 2: Bermudan options," OFRC Working Papers Series 2005mf03, Oxford Financial Research Centre.
    2. Joanne Kennedy & Phil Hunt & Antoon Pelsser, 2000. "Markov-functional interest rate models," Finance and Stochastics, Springer, vol. 4(4), pages 391-408.
    3. Roger Lord & Christian Kahl, 2006. "Optimal Fourier Inversion in Semi-analytical Option Pricing," Tinbergen Institute Discussion Papers 06-066/2, Tinbergen Institute, revised 05 Jun 2007.
    4. 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.
    5. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    6. Geske, Robert & Johnson, Herb E, 1984. " The American Put Option Valued Analytically," Journal of Finance, American Finance Association, vol. 39(5), pages 1511-1524, December.
    7. 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.
    8. Sam Howison & Mario Steinberg, 2005. "A matched asymptotic expansions approach to continuity corrections for discretely sampled options. Part 1: barrier options," OFRC Working Papers Series 2005mf02, Oxford Financial Research Centre.
    9. Andricopoulos, Ari D. & Widdicks, Martin & Duck, Peter W. & Newton, David P., 2003. "Universal option valuation using quadrature methods," Journal of Financial Economics, Elsevier, vol. 67(3), pages 447-471, March.
    10. 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.
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    Cited by:

    1. Tat Lung Chan, 2017. "Singular Fourier-Pad\'e Series Expansion of European Option Prices," Papers 1706.06709,, revised Nov 2017.
    2. A S Hurn & Kenenth A Lindsay & Andrew McClelland, 2013. "On the Efficacy of Fourier Series Approximations for Pricing European and Digital Options," NCER Working Paper Series 90, National Centre for Econometric Research.
    3. Julien Hok & Tat Lung Chan, 2016. "Option pricing with Legendre polynomials," Papers 1610.03086,, revised Mar 2017.
    4. Tim Leung & Marco Santoli, 2014. "Accounting for earnings announcements in the pricing of equity options," Journal of Financial Engineering (JFE), World Scientific Publishing Co. Pte. Ltd., vol. 1(04), pages 1-46.
    5. Anastasia Borovykh & Cornelis W. Oosterlee & Andrea Pascucci, 2016. "Pricing Bermudan options under local L\'evy models with default," Papers 1604.08735,
    6. Chen, Ding & Härkönen, Hannu J. & Newton, David P., 2014. "Advancing the universality of quadrature methods to any underlying process for option pricing," Journal of Financial Economics, Elsevier, vol. 114(3), pages 600-612.
    7. Fang, Fang & Oosterlee, Kees, 2008. "A Novel Pricing Method For European Options Based On Fourier-Cosine Series Expansions," MPRA Paper 9319, University Library of Munich, Germany.
    8. Peter A. Forsyth & George Labahn, 2017. "$\epsilon$-Monotone Fourier Methods for Optimal Stochastic Control in Finance," Papers 1710.08450,, revised Apr 2018.
    9. Antonio Cosma & Stefano Galluccio & Paola Pederzoli & Olivier Scaillet, 2016. "Early exercise decision in American options with dividends, stochastic volatility and jumps," Papers 1612.03031,
    10. Andrey Itkin, 2014. "Splitting and Matrix Exponential approach for jump-diffusion models with Inverse Normal Gaussian, Hyperbolic and Meixner jumps," Papers 1405.6111,, revised May 2014.
    11. Imai Junichi, 2013. "Comparison of random number generators via Fourier transform," Monte Carlo Methods and Applications, De Gruyter, vol. 19(3), pages 237-259, October.
    12. Alexander Kushpel, 2014. "Pricing of basket options I," Papers 1401.1856,
    13. D. J. Manuge & P. T. Kim, 2014. "A fast Fourier transform method for Mellin-type option pricing," Papers 1403.3756,, revised Mar 2014.
    14. Adrien Genin & Peter Tankov, 2016. "Optimal importance sampling for L\'evy Processes," Papers 1608.04621,
    15. 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.
    16. Tim Leung & Haohua Wan, 2015. "ESO Valuation with Job Termination Risk and Jumps in Stock Price," Papers 1504.08073,
    17. Alexander Kushpel, 2015. "Pricing of high-dimensional options," Papers 1510.07221,
    18. 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.
    19. Sesana, Debora & Marazzina, Daniele & Fusai, Gianluca, 2014. "Pricing exotic derivatives exploiting structure," European Journal of Operational Research, Elsevier, vol. 236(1), pages 369-381.
    20. Stavros J. Sioutis, 2017. "Calibration and Filtering of Exponential L\'evy Option Pricing Models," Papers 1705.04780,
    21. Fabi'an Crocce & Juho Happola & Jonas Kiessling & Ra'ul Tempone, 2015. "Error analysis in Fourier methods for option pricing," Papers 1503.00019,, revised Nov 2015.
    22. repec:qut:auncer:2013_02 is not listed on IDEAS

    More about this item


    Option pricing; Bermudan options; American options; convolution; Lévy Processes; Fast Fourier Transform;

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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