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Probability of multiple crossings and pricing of double barrier options

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  • Choe, Geon Ho
  • Koo, Ki Hwan

Abstract

This paper derives pricing formulas of standard double barrier option, generalized window double barrier option and chained option. Our method is based on probabilitic approach. We derive the probability of multiple crossings of curved barriers for Brownian motion with drift, by repeatedly applying the Girsanov theorem and the reflection principle. The price of a standard double barrier option is presented as an infinite sum that converges very rapidly. Although the price formula of standard double barrier option is the same with Kunitomo and Ikeda (1992), our method gives an intuitive interpretation for each term in the infinite series. From the intuitive interpretation we present the way how to approximate the infinite sum in the pricing formula and an error bound for the given approximation. Guillaume (2003) and Jun and Ku (2013) assumed that barriers are constant to price barrier options. We extend constant barriers of window double barrier option and chained option to curved barriers. By employing multiple crossing probabilities and previous skills we derive closed formula for prices of 16 types of the generalized chained option. Based on our analytic formulas we compute Greeks of chained options directly.

Suggested Citation

  • Choe, Geon Ho & Koo, Ki Hwan, 2014. "Probability of multiple crossings and pricing of double barrier options," The North American Journal of Economics and Finance, Elsevier, vol. 29(C), pages 156-184.
    • Handle: RePEc:eee:ecofin:v:29:y:2014:i:c:p:156-184
    DOI: 10.1016/j.najef.2014.05.007
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    References listed on IDEAS

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    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
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    3. Tristan Guillaume, 2003. "Window double barrier options," Post-Print hal-00924247, HAL.
    4. 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.
    5. Tristan Guillaume, 2010. "Step double barrier options," Post-Print hal-00924266, HAL.
    6. Naoto Kunitomo & Masayuki Ikeda, 1992. "Pricing Options With Curved Boundaries1," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 275-298, October.
    7. R. C. Heynen & H. M. Kat, 1995. "Lookback options with discrete and partial monitoring of the underlying price," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(4), pages 273-284.
    8. Antoon Pelsser, 2000. "Pricing double barrier options using Laplace transforms," Finance and Stochastics, Springer, vol. 4(1), pages 95-104.
    9. Naoto Kunitomo & Masayuki Ikeda, 2000. "Correction: Pricing Options with Curved Boundaries (Mathematical Finance 1992, 2, 275–297)," Mathematical Finance, Wiley Blackwell, vol. 10(4), pages 459-459, October.
    10. Hélyette Geman & Marc Yor, 1996. "Pricing And Hedging Double‐Barrier Options: A Probabilistic Approach," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 365-378, October.
    11. Grant Armstrong, 2001. "Valuation formulae for window barrier options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 8(4), pages 197-208.
    12. Paolo Baldi & Lucia Caramellino & Maria Gabriella Iovino, 1999. "Pricing General Barrier Options: A Numerical Approach Using Sharp Large Deviations," Mathematical Finance, Wiley Blackwell, vol. 9(4), pages 293-321, October.
    13. Mahmoud Hamada & Michael Sherris, 2003. "Contingent claim pricing using probability distortion operators: methods from insurance risk pricing and their relationship to financial theory," Applied Mathematical Finance, Taylor & Francis Journals, vol. 10(1), pages 19-47.
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    Cited by:

    1. 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.
    2. Lee, Hangsuck & Jeong, Himchan & Lee, Minha, 2022. "Multi-step double barrier options," Finance Research Letters, Elsevier, vol. 47(PA).
    3. 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

    Multiple crossing; Double barrier; Exotic option; Window option; Chained option;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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