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Valuation of piecewise linear barrier options

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  • Lee, Hangsuck
  • Ha, Hongjun
  • Lee, Minha

Abstract

This paper discusses the valuation of piecewise linear barrier options that generalize classical barrier options. We establish formulas for joint probabilities of the logarithmic returns of the underlying asset and its partial running maxima when the process has a piecewise constant drift. In particular, we show that our results embrace the famous reflection principle as a special case, and that our established proposition delivers useful scalability for computing desired probabilities related to various types of barriers. We derive the closed-form prices of piecewise linear barrier options under the Black–Scholes framework, which are obtainable with little effort by relying on the derived probabilities. In addition, we provide numerical examples and discuss how option prices respond to several types of piecewise linear barriers.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:ecofin:v:58:y:2021:i:c:s1062940821000929
    DOI: 10.1016/j.najef.2021.101470
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    References listed on IDEAS

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    1. Robert J. Elliott & Leunglung Chan & Tak Kuen Siu, 2005. "Option pricing and Esscher transform under regime switching," Annals of Finance, Springer, vol. 1(4), pages 423-432, October.
    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. Yong Yao, 2001. "State Price Density, Esscher Transforms, and Pricing Options on Stocks, Bonds, and Foreign Exchange Rates," North American Actuarial Journal, Taylor & Francis Journals, vol. 5(3), pages 104-117.
    4. 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..
    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. Bühlmann, Hans & Delbaen, Freddy & Embrechts, Paul & Shiryaev, Albert N., 1998. "On Esscher Transforms in Discrete Finance Models," ASTIN Bulletin, Cambridge University Press, vol. 28(2), pages 171-186, November.
    7. Naoto Kunitomo & Masayuki Ikeda, 1992. "Pricing Options With Curved Boundaries1," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 275-298, October.
    8. Badescu, Alex & Elliott, Robert J. & Siu, Tak Kuen, 2009. "Esscher transforms and consumption-based models," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 337-347, December.
    9. Cho H. Hui, 1997. "Time‐dependent barrier option values," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 17(6), pages 667-688, September.
    10. 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).
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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 & 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).
    3. 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.
    4. Battauz, Anna & De Donno, Marzia & Sbuelz, Alessandro, 2022. "On the exercise of American quanto options," The North American Journal of Economics and Finance, Elsevier, vol. 62(C).

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    More about this item

    Keywords

    Brownian motion; Piecewise linear barrier; Esscher transform; Refractive reflection principle; Barrier option;
    All these keywords.

    JEL classification:

    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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