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Geometric stopping of a random walk and its applications to valuing equity-linked death benefits

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  • Gerber, Hans U.
  • Shiu, Elias S.W.
  • Yang, Hailiang

Abstract

We study discrete-time models in which death benefits can depend on a stock price index, the logarithm of which is modeled as a random walk. Examples of such benefit payments include put and call options, barrier options, and lookback options. Because the distribution of the curtate-future-lifetime can be approximated by a linear combination of geometric distributions, it suffices to consider curtate-future-lifetimes with a geometric distribution. In binomial and trinomial tree models, closed-form expressions for the expectations of the discounted benefit payment are obtained for a series of options. They are based on results concerning geometric stopping of a random walk, in particular also on a version of the Wiener–Hopf factorization.

Suggested Citation

  • Gerber, Hans U. & Shiu, Elias S.W. & Yang, Hailiang, 2015. "Geometric stopping of a random walk and its applications to valuing equity-linked death benefits," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 313-325.
    • Handle: RePEc:eee:insuma:v:64:y:2015:i:c:p:313-325
    DOI: 10.1016/j.insmatheco.2015.06.006
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    Cited by:

    1. Pirjol, Dan & Zhu, Lingjiong, 2016. "Discrete sums of geometric Brownian motions, annuities and Asian options," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 19-37.
    2. Deelstra, Griselda & Hieber, Peter, 2023. "Randomization and the valuation of guaranteed minimum death benefits," European Journal of Operational Research, Elsevier, vol. 309(3), pages 1218-1236.
    3. Kirkby, J. Lars & Nguyen, Duy, 2021. "Equity-linked Guaranteed Minimum Death Benefits with dollar cost averaging," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 408-428.
    4. Wenguang Yu & Yaodi Yong & Guofeng Guan & Yujuan Huang & Wen Su & Chaoran Cui, 2019. "Valuing Guaranteed Minimum Death Benefits by Cosine Series Expansion," Mathematics, MDPI, vol. 7(9), pages 1-15, September.
    5. Yichen Han & Dongchen Li & Kun Fan & Jiaxin Wan & Luyan Li, 2024. "Valuation of a Mixture of GMIB and GMDB Variable Annuity," Mathematics, MDPI, vol. 12(3), pages 1-22, January.
    6. Dan Pirjol & Lingjiong Zhu, 2016. "Discrete Sums of Geometric Brownian Motions, Annuities and Asian Options," Papers 1609.07558, arXiv.org.
    7. Søren Asmussen & Patrick J. Laub & Hailiang Yang, 2019. "Phase-Type Models in Life Insurance: Fitting and Valuation of Equity-Linked Benefits," Risks, MDPI, vol. 7(1), pages 1-22, February.

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    More about this item

    Keywords

    IM10; IE50; IM40; IB10; Equity-linked death benefits; Binomial and trinomial tree models; Random walk; Geometric stopping; Esscher transform;
    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
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics

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