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Pricing first-touch digitals with a multi-step double boundary and American barrier options

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  • Lee, Hangsuck
  • Ha, Hongjun
  • Kong, Byungdoo

Abstract

A first-touch digital option delivers a payoff when an underlying variable first touches a given boundary, and most studies discuss its pricing by assuming a single flat boundary and a fixed payout amount. This paper derives explicit pricing formulas for first-touch digital options with a multi-step double boundary and customizable payoffs. To this end, we establish the probabilities of the first-hitting times of a Brownian motion for the multi-step double boundary. Furthermore, we explain how the first-touch digital option pricing formulas approximate an American knock-out multi-step barrier option.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:finlet:v:59:y:2024:i:c:s1544612323010711
    DOI: 10.1016/j.frl.2023.104699
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    References listed on IDEAS

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    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.
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    4. Lee, Hangsuck & Ha, Hongjun & Kong, Byungdoo & Lee, Minha, 2023. "Pricing multi-step double barrier options by the efficient non-crossing probability," Finance Research Letters, Elsevier, vol. 54(C).
    5. Tsvetelin S. Zaevski, 2023. "American strangle options with arbitrary strikes," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 43(7), pages 880-903, July.
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    8. 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.
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    14. Ju, Nengjiu, 1998. "Pricing an American Option by Approximating Its Early Exercise Boundary as a Multipiece Exponential Function," The Review of Financial Studies, Society for Financial Studies, vol. 11(3), pages 627-646.
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    Cited by:

    1. Lee, Hangsuck & Ha, Hongjun & Lee, Gaeun & Lee, Minha, 2024. "Valuing American options using multi-step rebate options," The North American Journal of Economics and Finance, Elsevier, vol. 74(C).

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    More about this item

    Keywords

    First-hitting time; Multi-step double boundary; Digital option; American barrier option;
    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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