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GMWB Riders in a Binomial Framework - Pricing, Hedging, and Diversification of Mortality Risk

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  • Cody B. Hyndman
  • Menachem Wenger

Abstract

We construct a binomial model for a guaranteed minimum withdrawal benefit (GMWB) rider to a variable annuity (VA) under optimal policyholder behaviour. The binomial model results in explicitly formulated perfect hedging strategies funded using only periodic fee income. We consider the separate perspectives of the insurer and policyholder and introduce a unifying relationship. Decompositions of the VA and GMWB contract into term-certain payments and options representing the guarantee and early surrender features are extended to the binomial framework. We incorporate an approximation algorithm for Asian options that significantly improves efficiency of the binomial model while retaining accuracy. Several numerical examples are provided which illustrate both the accuracy and the tractability of the binomial model. We extend the binomial model to include policy holder mortality and death benefits. Pricing, hedging, and the decompositions of the contract are extended to incorporate mortality risk. We prove limiting results for the hedging strategies and demonstrate mortality risk diversification. Numerical examples are provided which illustrate the effectiveness of hedging and the diversification of mortality risk under capacity constraints with finite pools.

Suggested Citation

  • Cody B. Hyndman & Menachem Wenger, 2014. "GMWB Riders in a Binomial Framework - Pricing, Hedging, and Diversification of Mortality Risk," Papers 1410.7453, arXiv.org, revised Jul 2016.
  • Handle: RePEc:arx:papers:1410.7453
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    References listed on IDEAS

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    1. Jingjiang Peng & Kwai Sun Leung & Yue Kuen Kwok, 2012. "Pricing guaranteed minimum withdrawal benefits under stochastic interest rates," Quantitative Finance, Taylor & Francis Journals, vol. 12(6), pages 933-941, October.
    2. Chen, Z. & Vetzal, K. & Forsyth, P.A., 2008. "The effect of modelling parameters on the value of GMWB guarantees," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 165-173, August.
    3. Massimo Costabile & Ivar Massabó & Emilio Russo, 2006. "An adjusted binomial model for pricing Asian options," Review of Quantitative Finance and Accounting, Springer, vol. 27(3), pages 285-296, November.
    4. Geske, Robert & Shastri, Kuldeep, 1985. "Valuation by Approximation: A Comparison of Alternative Option Valuation Techniques," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 20(01), pages 45-71, March.
    5. Bacinello, Anna Rita, 2005. "Endogenous model of surrender conditions in equity-linked life insurance," Insurance: Mathematics and Economics, Elsevier, vol. 37(2), pages 270-296, October.
    6. Milevsky, Moshe A. & Salisbury, Thomas S., 2006. "Financial valuation of guaranteed minimum withdrawal benefits," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 21-38, February.
    7. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    8. Bacinello, Anna Rita & Millossovich, Pietro & Olivieri, Annamaria & Pitacco, Ermanno, 2011. "Variable annuities: A unifying valuation approach," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 285-297.
    9. Hyndman, Cody B. & Wenger, Menachem, 2014. "Valuation perspectives and decompositions for variable annuities with GMWB riders," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 283-290.
    10. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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