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Valuing step barrier options and their icicled variations

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  • Lee, Hangsuck
  • Ko, Bangwon
  • Song, Seongjoo

Abstract

This paper intends to investigate an interesting class of barrier options, called step barrier options, whose barrier levels are a piecewise constant function of time. These options, while having transparent, simple, and flexible payoff structures, allow for explicit pricing formulas under the Black-Scholes model, and thus can be easily embedded into equity-linked products to enhance the yield or reduce the downside risk. Moreover, the class can be further generalized by attaching vertical branches of barriers to the horizontal one as in Lee and Ko (2018). Using the actuarial method of Esscher transform and the factorization formula, we derive the option pricing formulas under a more general framework with vertical branches attached to horizontal barriers. We explore the formulas through numerical examples, demonstrating their applicability to equity-linked investment with the step barrier option embedded.

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  • 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.
    • Handle: RePEc:eee:ecofin:v:49:y:2019:i:c:p:396-411
    DOI: 10.1016/j.najef.2018.09.001
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    1. Gerber, Hans U. & Shiu, Elias S. W., 1996. "Actuarial bridges to dynamic hedging and option pricing," Insurance: Mathematics and Economics, Elsevier, vol. 18(3), pages 183-218, November.
    2. Lee, Hangsuck, 2003. "Pricing equity-indexed annuities with path-dependent options," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 677-690, December.
    3. Serena Tiong, 2000. "Valuing Equity-Indexed Annuities," North American Actuarial Journal, Taylor & Francis Journals, vol. 4(4), pages 149-163.
    4. Jang, Ji-Wook & Krvavych, Yuriy, 2004. "Arbitrage-free premium calculation for extreme losses using the shot noise process and the Esscher transform," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 97-111, August.
    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.
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    Citations

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    Cited by:

    1. Lee, Hangsuck & Ha, Hongjun & Lee, Minha, 2021. "Valuation of piecewise linear barrier options," The North American Journal of Economics and Finance, Elsevier, vol. 58(C).
    2. Hangsuck Lee & Gaeun Lee & Seongjoo Song, 2022. "Multi‐step reflection principle and barrier options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(4), pages 692-721, April.
    3. Hangsuck Lee & Seongjoo Song & Gaeun Lee, 2023. "Insurance guaranty premiums and exchange options," Mathematics and Financial Economics, Springer, volume 17, number 3, February.
    4. Hangsuck Lee & Hongjun Ha & Minha Lee, 2022. "Piecewise linear double barrier options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(1), pages 125-151, January.
    5. 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).
    6. Lee, Hangsuck & Lee, Minha & Ko, Bangwon, 2022. "A semi-analytic valuation of two-asset barrier options and autocallable products using Brownian bridge," The North American Journal of Economics and Finance, Elsevier, vol. 61(C).
    7. Lee, Hangsuck & Jeong, Himchan & Lee, Gaeun, 2023. "Valuing rebate options and equity-linked products," The North American Journal of Economics and Finance, Elsevier, vol. 68(C).
    8. Hangsuck Lee & Hongjun Ha & Minha Lee, 2022. "Piecewise linear boundary crossing probabilities, barrier options, and variable annuities," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(12), pages 2248-2272, December.
    9. Lee, Hangsuck & Lee, Gaeun & Song, Seongjoo, 2023. "Min–max multi-step barrier options and their variants," The North American Journal of Economics and Finance, Elsevier, vol. 67(C).
    10. Lee, Hangsuck & Choi, Yang Ho & Lee, Gaeun, 2022. "Multi-step barrier products and static hedging," The North American Journal of Economics and Finance, Elsevier, vol. 61(C).
    11. Lee, Hangsuck & Kim, Eunchae & Ko, Bangwon, 2022. "Valuing lookback options with barrier," The North American Journal of Economics and Finance, Elsevier, vol. 60(C).
    12. Hangsuck Lee & Gaeun Lee & Seongjoo Song, 2021. "Multi-step Reflection Principle and Barrier Options," Papers 2105.15008, arXiv.org.

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    More about this item

    Keywords

    Black-Scholes model; Esscher transform; Step barrier option; Icicled barrier option; Reflection principle;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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