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Semi-analytical valuation for discrete barrier options under time-dependent Lévy processes

Author

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  • Lian, Guanghua
  • Zhu, Song-Ping
  • Elliott, Robert J.
  • Cui, Zhenyu

Abstract

Simple analytical solutions for the prices of discretely monitored barrier options do not yet exist in the literature. This paper presents a semi-analytical and fully explicit solution for pricing discretely monitored barrier options when the underlying asset is driven by a general Lévy process. The explicit formula only involves elementary functions, and the Greeks are also explicitly available with little additional computation. By performing a Z-transform, we reduce the valuation problem to an integral equation. This equation is solved analytically with the solution expressed in terms of a Fourier cosine series. We then manage to analytically carry out the Z-transform inversion, and obtain a semi-analytical formula for pricing discrete barrier options. We establish the theoretical error bound and analyze the convergence order of our method. Numerical implementation demonstrates that our numerical results are accurate and efficient, and match up with the results from the benchmark methods in the literature.

Suggested Citation

  • Lian, Guanghua & Zhu, Song-Ping & Elliott, Robert J. & Cui, Zhenyu, 2017. "Semi-analytical valuation for discrete barrier options under time-dependent Lévy processes," Journal of Banking & Finance, Elsevier, vol. 75(C), pages 167-183.
    • Handle: RePEc:eee:jbfina:v:75:y:2017:i:c:p:167-183
    DOI: 10.1016/j.jbankfin.2016.11.012
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    References listed on IDEAS

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    Cited by:

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    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. Xie, Fei & He, Zhijian & Wang, Xiaoqun, 2019. "An importance sampling-based smoothing approach for quasi-Monte Carlo simulation of discrete barrier options," European Journal of Operational Research, Elsevier, vol. 274(2), pages 759-772.
    4. Li, Zhe & Zhang, Wei-Guo & Liu, Yong-Jun & Zhang, Yue, 2019. "Pricing discrete barrier options under jump-diffusion model with liquidity risk," International Review of Economics & Finance, Elsevier, vol. 59(C), pages 347-368.
    5. Kontosakos, Vasileios E. & Mendonca, Keegan & Pantelous, Athanasios A. & Zuev, Konstantin M., 2021. "Pricing discretely-monitored double barrier options with small probabilities of execution," European Journal of Operational Research, Elsevier, vol. 290(1), pages 313-330.
    6. Sheng-Feng Luo & Hsin-Chieh Wong, 2023. "Continuity correction: on the pricing of discrete double barrier options," Review of Derivatives Research, Springer, vol. 26(1), pages 51-90, April.
    7. Castellano, Rosella & Corallo, Vincenzo & Morelli, Giacomo, 2022. "Structural estimation of counterparty credit risk under recovery risk," Journal of Banking & Finance, Elsevier, vol. 140(C).

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    More about this item

    Keywords

    Discrete barrier options; Lévy processes; Fourier-cosine series;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C3 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables

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