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Robust barrier option pricing by frame projection under exponential Lévy dynamics

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  • J. Lars Kirkby

Abstract

We present an efficient method for robustly pricing discretely monitored barrier and occupation time derivatives under exponential Lévy models. This includes ordinary barrier options, as well as (resetting) Parisian options, delayed barrier options (also known as cumulative Parisian or Parasian options), fader options and step options (soft-barriers), all with single and double barriers, which have yet to be priced with more general Lévy processes, including KoBoL (CGMY), Merton’s jump diffusion and NIG. The method’s efficiency is derived in part from the use of frame-projected transition densities, which transform the problem into the Fourier domain and accelerate the convergence of intermediate expectations. Moreover, these expectations are approximated by Toeplitz matrix-vector multiplications, resulting in a fast implementation. We devise an augmentation approach that contributes to the method’s robustness, adding protection against mis-specifying a proper truncation support of the transition density. Theoretical convergence is verified by a series of numerical experiments which demonstrate the method’s efficiency and accuracy.

Suggested Citation

  • J. Lars Kirkby, 2017. "Robust barrier option pricing by frame projection under exponential Lévy dynamics," Applied Mathematical Finance, Taylor & Francis Journals, vol. 24(4), pages 337-386, July.
    • Handle: RePEc:taf:apmtfi:v:24:y:2017:i:4:p:337-386
    DOI: 10.1080/1350486X.2017.1384701
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    Citations

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    Cited by:

    1. Phelan, Carolyn E. & Marazzina, Daniele & Fusai, Gianluca & Germano, Guido, 2018. "Fluctuation identities with continuous monitoring and their application to the pricing of barrier options," European Journal of Operational Research, Elsevier, vol. 271(1), pages 210-223.
    2. Cui, Zhenyu & Kirkby, J. Lars & Nguyen, Duy, 2021. "A data-driven framework for consistent financial valuation and risk measurement," European Journal of Operational Research, Elsevier, vol. 289(1), pages 381-398.
    3. Kirkby, J. Lars, 2023. "Hybrid equity swap, cap, and floor pricing under stochastic interest by Markov chain approximation," European Journal of Operational Research, Elsevier, vol. 305(2), pages 961-978.
    4. 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.
    5. Jie Chen & Liaoyuan Fan & Lingfei Li & Gongqiu Zhang, 2022. "A multidimensional Hilbert transform approach for barrier option pricing and survival probability calculation," Review of Derivatives Research, Springer, vol. 25(2), pages 189-232, July.
    6. Cui, Zhenyu & Lars Kirkby, J. & Nguyen, Duy, 2019. "A general framework for time-changed Markov processes and applications," European Journal of Operational Research, Elsevier, vol. 273(2), pages 785-800.
    7. André Catalão & Rogério Rosenfeld, 2020. "Analytical Path-Integral Pricing Of Deterministic Moving-Barrier Options Under Non-Gaussian Distributions," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(01), pages 1-52, February.
    8. Orzechowski Arkadiusz, 2018. "Pricing Correlation Options: from the P. Carr And D. Madan Approach to the New Method Based on the Fourier Transform," Economics and Business Review, Sciendo, vol. 4(1), pages 16-28, April.

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