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Pricing multi-step double barrier options by the efficient non-crossing probability

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

Abstract

This paper considers pricing multi-step double barrier options. The non-crossing probability for a multi-step double boundary is vital in valuing the options. We extend an explicit formula for the non-crossing probability using the solutions to the relevant Fokker–Planck equations. The derived formula not only provides a new look at the non-crossing probability but also outperforms the existing probability formula relying on multivariate normal distribution functions in terms of efficiency by avoiding multi-dimensional numerical integrals. We demonstrate the merits of the new expression of the non-crossing probability via numerical experiments and apply it to valuing multi-step double barrier options.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:finlet:v:54:y:2023:i:c:s1544612323001459
    DOI: 10.1016/j.frl.2023.103772
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    References listed on IDEAS

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    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. 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.
    3. Lee, Hangsuck & Jeong, Himchan & Lee, Minha, 2022. "Multi-step double barrier options," Finance Research Letters, Elsevier, vol. 47(PA).
    4. David Raab & Edward Green, 1961. "A cosine approximation to the normal distribution," Psychometrika, Springer;The Psychometric Society, vol. 26(4), pages 447-450, December.
    5. Peter Buchen & Otto Konstandatos, 2009. "A New Approach to Pricing Double-Barrier Options with Arbitrary Payoffs and Exponential Boundaries," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(6), pages 497-515.
    6. Tetsuhisa Miwa & A. J. Hayter & Satoshi Kuriki, 2003. "The evaluation of general non‐centred orthant probabilities," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(1), pages 223-234, February.
    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.
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    10. Cho H. Hui, 1997. "Time‐dependent barrier option values," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 17(6), pages 667-688, September.
    11. Sheldon Lin, X., 1998. "Double barrier hitting time distributions with applications to exotic options," Insurance: Mathematics and Economics, Elsevier, vol. 23(1), pages 45-58, October.
    12. Lee, Hangsuck & Ahn, Soohan & Ko, Bangwon, 2019. "Generalizing the reflection principle of Brownian motion, and closed-form pricing of barrier options and autocallable investments," The North American Journal of Economics and Finance, Elsevier, vol. 50(C).
    13. Otto Konstandatos, 2018. "Methods for Analytical Barrier Option Pricing with Multiple Exponential Time-Varying Boundaries," Research Paper Series 396, Quantitative Finance Research Centre, University of Technology, Sydney.
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    Cited by:

    1. 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).
    2. Lee, Hangsuck & Ko, Bangwon & Lee, Minha, 2023. "The pricing and static hedging of multi-step double barrier options," Finance Research Letters, Elsevier, vol. 55(PA).

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    More about this item

    Keywords

    Non-crossing probability; Fokker–Planck equation; Multi-step double barrier options;
    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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