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Generalizing the reflection principle of Brownian motion, and closed-form pricing of barrier options and autocallable investments

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  • Lee, Hangsuck
  • Ahn, Soohan
  • Ko, Bangwon

Abstract

In this paper, we intend to generalize the well-known reflection principle, one of the most interesting properties of the Brownian motion. The essence of our generalization lies in its ability to stochastically eliminate arbitrary number of partial maximums (or minimums) in the joint events associated with the Brownian motion, thereby allowing us to express the joint probabilities in terms of the multivariate normal distribution functions. Due to the simplicity and versatility, our generalized reflection principle can be used to solve many probabilistic problems pertaining to the Brownian motion. To illustrate, we consider evaluating barrier options and autocallable structured product. Using the basic inclusion-exclusion principle, we obtain integrated pricing formulas for various barrier options under the Black-Scholes model, and derive an explicit pricing formula for the autocallable product, which is not known yet despite its popularity. These formulas are explored through numerical examples. The method of Esscher transform demonstrates its time-honored value during the derivation process.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:ecofin:v:50:y:2019:i:c:s1062940818306028
    DOI: 10.1016/j.najef.2019.101014
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    References listed on IDEAS

    as
    1. Tristan Guillaume, 2015. "Autocallable Structured Products," Post-Print hal-02979985, HAL.
    2. 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.
    3. Lee, Hangsuck, 2003. "Pricing equity-indexed annuities with path-dependent options," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 677-690, December.
    4. Serena Tiong, 2000. "Valuing Equity-Indexed Annuities," North American Actuarial Journal, Taylor & Francis Journals, vol. 4(4), pages 149-163.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. 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).
    2. 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).
    3. 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.
    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. Lee, Hangsuck & Kim, Eunchae & Ko, Bangwon, 2022. "Valuing lookback options with barrier," The North American Journal of Economics and Finance, Elsevier, vol. 60(C).

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    More about this item

    Keywords

    Autocallable structured product; Black-Scholes model; Esscher transform; 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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