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Pricing guaranteed minimum withdrawal benefits under stochastic interest rates


  • Jingjiang Peng
  • Kwai Sun Leung
  • Yue Kuen Kwok


We consider the pricing of variable annuities with the Guaranteed Minimum Withdrawal Benefit (GMWB) under the Vasicek stochastic interest rate framework. The holder of the variable annuity contract pays an initial purchase payment to the insurance company, which is then invested in a portfolio of risky assets. Under the GMWB, the holder can withdraw a specified amount periodically over the term of the contract such that the return of the entire initial investment is guaranteed, regardless of the market performance of the underlying asset portfolio. The investors have the equity participation in the reference investment portfolio with protection on the downside risk. The guarantee is financed by paying annual proportional fees. Under the assumption of deterministic withdrawal rates, we develop the pricing formulation of the value function of a variable annuity with the GMWB. In particular, we derive the analytic approximation solutions to the fair value of the GMWB under both equity and interest rate risks, obtaining both the lower and upper bounds on the price functions. The pricing behavior of the embedded GMWB under various model parameter values is also examined.

Suggested Citation

  • 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.
  • Handle: RePEc:taf:quantf:v:12:y:2012:i:6:p:933-941
    DOI: 10.1080/14697680903436606

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    Cited by:

    1. Feng, Runhuan & Jing, Xiaochen, 2017. "Analytical valuation and hedging of variable annuity guaranteed lifetime withdrawal benefits," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 36-48.
    2. Dai, Tian-Shyr & Yang, Sharon S. & Liu, Liang-Chih, 2015. "Pricing guaranteed minimum/lifetime withdrawal benefits with various provisions under investment, interest rate and mortality risks," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 364-379.
    3. Pavel V. Shevchenko & Xiaolin Luo, 2016. "Valuation of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Stochastic Interest Rate," Papers 1602.03238,, revised Jan 2017.
    4. repec:eee:insuma:v:76:y:2017:i:c:p:104-117 is not listed on IDEAS
    5. Cody B. Hyndman & Menachem Wenger, 2014. "GMWB Riders in a Binomial Framework - Pricing, Hedging, and Diversification of Mortality Risk," Papers 1410.7453,, revised Jul 2016.
    6. Jan Baldeaux & Fung & Katja Ignatieva & Eckhard Platen, 2015. "A Hybrid Model for Pricing and Hedging of Long-dated Bonds," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(4), pages 366-398, September.
    7. 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.
    8. Forsyth, Peter & Vetzal, Kenneth, 2014. "An optimal stochastic control framework for determining the cost of hedging of variable annuities," Journal of Economic Dynamics and Control, Elsevier, vol. 44(C), pages 29-53.
    9. Huang, Yao Tung & Kwok, Yue Kuen, 2014. "Analysis of optimal dynamic withdrawal policies in withdrawal guarantee products," Journal of Economic Dynamics and Control, Elsevier, vol. 45(C), pages 19-43.

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