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Multi-step Reflection Principle and Barrier Options

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  • Hangsuck Lee
  • Gaeun Lee
  • Seongjoo Song

Abstract

This paper examines a class of barrier options-multi-step barrier options, which can have any finite number of barriers of any level. We obtain a general, explicit expression of option prices of this type under the Black-Scholes model. Multi-step barrier options are not only useful in that they can handle barriers of different levels and time steps, but can also approximate options with arbitrary barriers. Moreover, they can be embedded in financial products such as deposit insurances based on jump models with simple barriers. Along the way, we derive multi-step reflection principle, which generalizes the reflection principle of Brownian motion.

Suggested Citation

  • Hangsuck Lee & Gaeun Lee & Seongjoo Song, 2021. "Multi-step Reflection Principle and Barrier Options," Papers 2105.15008, arXiv.org.
    • Handle: RePEc:arx:papers:2105.15008
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    References listed on IDEAS

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    1. Lee, Hangsuck, 2003. "Pricing equity-indexed annuities with path-dependent options," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 677-690, December.
    2. Lee, Hangsuck & Ko, Bangwon & Song, Seongjoo, 2019. "Valuing step barrier options and their icicled variations," The North American Journal of Economics and Finance, Elsevier, vol. 49(C), pages 396-411.
    3. Serena Tiong, 2000. "Valuing Equity-Indexed Annuities," North American Actuarial Journal, Taylor & Francis Journals, vol. 4(4), pages 149-163.
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    5. Ng, Andrew Cheuk-Yin & Li, Johnny Siu-Hang, 2011. "Valuing variable annuity guarantees with the multivariate Esscher transform," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 393-400.
    6. Gerber, Hans U. & Shiu, Elias S.W. & Yang, Hailiang, 2012. "Valuing equity-linked death benefits and other contingent options: A discounted density approach," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 73-92.
    7. Tristan Guillaume, 2010. "Step double barrier options," Post-Print hal-00924266, HAL.
    8. Naoto Kunitomo & Masayuki Ikeda, 1992. "Pricing Options With Curved Boundaries1," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 275-298, October.
    9. Emmanuel Gobet, 2009. "Advanced Monte Carlo methods for barrier and related exotic options," Post-Print hal-00319947, HAL.
    10. Ya-Wen Hwang & Shih-Chieh Chang & Yang-Che Wu, 2015. "Capital Forbearance, Ex Ante Life Insurance Guaranty Schemes, and Interest Rate Uncertainty," North American Actuarial Journal, Taylor & Francis Journals, vol. 19(2), pages 94-115, April.
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    Cited by:

    1. Lu, Yu-Ming & Lyuu, Yuh-Dauh, 2023. "Very fast algorithms for implied barriers and moving-barrier options pricing," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 251-271.

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