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Modeling and Pricing Longevity Derivatives Using Stochastic Mortality Rates and the Esscher Transform

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  • Shuo-Li Chuang
  • Patrick Brockett

Abstract

The Lee-Carter mortality model provides a structure for stochastically modeling mortality rates incorporating both time (year) and age mortality dynamics. Their model is constructed by modeling the mortality rate as a function of both an age and a year effect. Recently the MBMM model (Mitchell et al. 2013) showed the Lee Carter model can be improved by fitting with the growth rates of mortality rates over time and age rather than the mortality rates themselves. The MBMM modification of the Lee-Carter model performs better than the original and many of the subsequent variants. In order to model the mortality rate under the martingale measure and to apply it for pricing the longevity derivatives, we adapt the MBMM structure and introduce a Lévy stochastic process with a normal inverse Gaussian (NIG) distribution in our model. The model has two advantages in addition to better fit: first, it can mimic the jumps in the mortality rates since the NIG distribution is fat-tailed with high kurtosis, and, second, this mortality model lends itself to pricing of longevity derivatives based on the assumed mortality model. Using the Esscher transformation we show how to find a related martingale measure, allowing martingale pricing for mortality/longevity risk–related derivatives. Finally, we apply our model to pricing a q-forward longevity derivative utilizing the structure proposed by Life and Longevity Markets Association.

Suggested Citation

  • Shuo-Li Chuang & Patrick Brockett, 2014. "Modeling and Pricing Longevity Derivatives Using Stochastic Mortality Rates and the Esscher Transform," North American Actuarial Journal, Taylor & Francis Journals, vol. 18(1), pages 22-37.
    • Handle: RePEc:taf:uaajxx:v:18:y:2014:i:1:p:22-37
    DOI: 10.1080/10920277.2013.873708
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    Citations

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    Cited by:

    1. Blake, David & Cairns, Andrew J.G., 2021. "Longevity risk and capital markets: The 2019-20 update," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 395-439.
    2. Blake, David & El Karoui, Nicole & Loisel, Stéphane & MacMinn, Richard, 2018. "Longevity risk and capital markets: The 2015–16 update," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 157-173.
    3. Liu, Yanxin & Li, Johnny Siu-Hang, 2018. "A strategy for hedging risks associated with period and cohort effects using q-forwards," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 267-285.
    4. Liu, Yanxin & Li, Johnny Siu-Hang, 2015. "The age pattern of transitory mortality jumps and its impact on the pricing of catastrophic mortality bonds," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 135-150.
    5. Yang Qiao & Chou-Wen Wang & Wenjun Zhu, 2024. "Machine learning in long-term mortality forecasting," The Geneva Papers on Risk and Insurance - Issues and Practice, Palgrave Macmillan;The Geneva Association, vol. 49(2), pages 340-362, April.
    6. Wang, Ling & Chiu, Mei Choi & Wong, Hoi Ying, 2021. "Volterra mortality model: Actuarial valuation and risk management with long-range dependence," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 1-14.
    7. Kung, Ko-Lun & Liu, I-Chien & Wang, Chou-Wen, 2021. "Modeling and pricing longevity derivatives using Skellam distribution," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 341-354.
    8. Ling Wang & Mei Choi Chiu & Hoi Ying Wong, 2020. "Volterra mortality model: Actuarial valuation and risk management with long-range dependence," Papers 2009.09572, arXiv.org.

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