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Prices And Sensitivities Of Barrier And First-Touch Digital Options In Lévy-Driven Models

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  • MITYA BOYARCHENKO

    (Department of Mathematics, University of Michigan, 530 Church Street, 2704 East Hall, Ann Arbor, MI 48109-1043, USA)

  • SERGEI LEVENDORSKIĬ

    (Department of Mathematics, The University of Leicester, University Road, Leicester LE1 7RH, UK)

Abstract

We present a fast and accurate FFT-based method of computing the prices and sensitivities of barrier options and first-touch digital options on stocks whose log-price follows a Lévy process. The numerical results obtained via our approach are demonstrated to be in good agreement with the results obtained using other (sometimes fundamentally different) approaches that exist in the literature. However, our method is computationally much faster (often, dozens of times faster). Moreover, our technique has the advantage that its application does not entail a detailed analysis of the underlying Lévy process: one only needs an explicit analytic formula for the characteristic exponent of the process. Thus our algorithm is very easy to implement in practice. Finally, our method yields accurate results for a wide range of values of the spot price, including those that are very close to the barrier, regardless of whether the maturity period of the option is long or short.

Suggested Citation

  • Mitya Boyarchenko & Sergei Levendorskiĭ, 2009. "Prices And Sensitivities Of Barrier And First-Touch Digital Options In Lévy-Driven Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(08), pages 1125-1170.
    • Handle: RePEc:wsi:ijtafx:v:12:y:2009:i:08:n:s0219024909005610
    DOI: 10.1142/S0219024909005610
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    References listed on IDEAS

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    1. Peter Carr, 1996. "Valuing Finite-Lived Options as Perpetual," Finance 9607002, University Library of Munich, Germany.
    2. Boyarchenko, Svetlana & Levendorskii[caron], Sergei, 2007. "Optimal stopping made easy," Journal of Mathematical Economics, Elsevier, vol. 43(2), pages 201-217, February.
    3. Svetlana I Boyarchenko & Sergei Z Levendorskii, 2002. "Non-Gaussian Merton-Black-Scholes Theory," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4955.
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    Citations

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    Cited by:

    1. Svetlana Boyarchenko & Sergei Levendorskiä¬ & J. Lars Kyrkby & Zhenyu Cui, 2021. "Sinh-Acceleration For B-Spline Projection With Option Pricing Applications," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 24(08), pages 1-50, December.
    2. Sergei Levendorskiĭ, 2017. "ULTRA-FAST PRICING BARRIER OPTIONS AND CDSs," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(05), pages 1-27, August.
    3. Svetlana Boyarchenko & Sergei Levendorskiĭ, 2020. "Static and semistatic hedging as contrarian or conformist bets," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 921-960, July.
    4. Svetlana Boyarchenko & Sergei Levendorskiu{i}, 2022. "Efficient evaluation of expectations of functions of a stable L\'evy process and its extremum," Papers 2209.12349, arXiv.org.
    5. Svetlana Boyarchenko & Sergei Levendorskiu{i}, 2022. "Efficient inverse $Z$-transform and pricing barrier and lookback options with discrete monitoring," Papers 2207.02858, arXiv.org, revised Jul 2022.
    6. Boyarchenko, Svetlana & Levendorskiĭ, Sergei, 2025. "Lévy models amenable to efficient calculations," Stochastic Processes and their Applications, Elsevier, vol. 186(C).
    7. Svetlana Boyarchenko & Sergei Levendorskii, 2023. "Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in L\'evy models," Papers 2312.03915, arXiv.org.
    8. Federico De Olivera & Ernesto Mordecki, 2014. "Computing Greeks for L\'evy Models: The Fourier Transform Approach," Papers 1407.1343, arXiv.org.
    9. Lee, Hangsuck & Ha, Hongjun & Kong, Byungdoo, 2024. "Pricing first-touch digitals with a multi-step double boundary and American barrier options," Finance Research Letters, Elsevier, vol. 59(C).
    10. Svetlana Boyarchenko & Sergei Levendorskiĭ, 2025. "Efficient evaluation of expectations of functions of a Lévy process and its extremum," Finance and Stochastics, Springer, vol. 29(2), pages 443-468, April.
    11. Svetlana Boyarchenko & Sergei Levendorskiu{i}, 2022. "Efficient evaluation of double-barrier options and joint cpdf of a L\'evy process and its two extrema," Papers 2211.07765, arXiv.org.
    12. Walter Farkas & Ludovic Mathys & Nikola Vasiljevi'c, 2020. "Intra-Horizon Expected Shortfall and Risk Structure in Models with Jumps," Papers 2002.04675, arXiv.org, revised Jan 2021.
    13. Sergei Levendorskiĭ, 2022. "Operators and Boundary Problems in Finance, Economics and Insurance: Peculiarities, Efficient Methods and Outstanding Problems," Mathematics, MDPI, vol. 10(7), pages 1-36, March.
    14. Svetlana Boyarchenko & Sergei Levendorskii, 2023. "Efficient evaluation of joint pdf of a L\'evy process, its extremum, and hitting time of the extremum," Papers 2312.05222, arXiv.org.

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