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Valuing guaranteed withdrawal benefits with stochastic interest rates and volatility

Author

Listed:
  • Ryan Donnelly
  • Sebastian Jaimungal
  • Dmitri H. Rubisov

Abstract

Guaranteed withdrawal benefits are long term contracts which provide investors with equity participation while guaranteeing them a secured income stream. Due to the long investment horizons involved, stochastic volatility and stochastic interest rates are important factors to include in their valuation. Moreover, investors are typically allowed to participate in a mixed fund composed of both equity and fixed-income securities. Here, we develop an efficient method for valuing these path-dependent products through re-writing the problem in the form of an Asian styled claim and a dimensionally reduced partial differential equation (PDE). The PDE is then solved using an Alternating Direction Implicit method. Furthermore, we derive an analytical closed form approximation and compare this approximation with the PDE results and find excellent agreement. We illustrate the various effects of the parameters on the valuation through numerical experiments and discuss their financial implications.

Suggested Citation

  • Ryan Donnelly & Sebastian Jaimungal & Dmitri H. Rubisov, 2014. "Valuing guaranteed withdrawal benefits with stochastic interest rates and volatility," Quantitative Finance, Taylor & Francis Journals, vol. 14(2), pages 369-382, February.
    • Handle: RePEc:taf:quantf:v:14:y:2014:i:2:p:369-382
    DOI: 10.1080/14697688.2013.837580
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    Citations

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    Cited by:

    1. Ludovic Goudenège & Andrea Molent & Antonino Zanette, 2021. "Gaussian process regression for pricing variable annuities with stochastic volatility and interest rate," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 44(1), pages 57-72, June.
    2. Ludovic Goudenege & Andrea Molent & Antonino Zanette, 2019. "Pricing and hedging GMWB in the Heston and in the Black–Scholes with stochastic interest rate models," Computational Management Science, Springer, vol. 16(1), pages 217-248, February.
    3. Yanhong Zhong & Guohe Deng, 2019. "Geometric Asian Options Pricing under the Double Heston Stochastic Volatility Model with Stochastic Interest Rate," Complexity, Hindawi, vol. 2019, pages 1-13, January.
    4. Claudio Fontana & Francesco Rotondi, 2022. "Valuation of general GMWB annuities in a low interest rate environment," Papers 2208.10183, arXiv.org, revised Aug 2023.
    5. Yaowen Lu & Duy-Minh Dang, 2023. "A semi-Lagrangian $\epsilon$-monotone Fourier method for continuous withdrawal GMWBs under jump-diffusion with stochastic interest rate," Papers 2310.00606, arXiv.org.
    6. Man Chung Fung & Katja Ignatieva & Michael Sherris, 2019. "Managing Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives," Risks, MDPI, vol. 7(1), pages 1-25, January.
    7. Kang, Boda & Ziveyi, Jonathan, 2018. "Optimal surrender of guaranteed minimum maturity benefits under stochastic volatility and interest rates," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 43-56.
    8. Ludovic Gouden`ege & Andrea Molent & Antonino Zanette, 2019. "Gaussian Process Regression for Pricing Variable Annuities with Stochastic Volatility and Interest Rate," Papers 1903.00369, arXiv.org, revised Jul 2019.
    9. Maciej Augustyniak & Mathieu Boudreault, 2017. "Mitigating Interest Rate Risk in Variable Annuities: An Analysis of Hedging Effectiveness under Model Risk," North American Actuarial Journal, Taylor & Francis Journals, vol. 21(4), pages 502-525, October.
    10. 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.
    11. Fontana, Claudio & Rotondi, Francesco, 2023. "Valuation of general GMWB annuities in a low interest rate environment," Insurance: Mathematics and Economics, Elsevier, vol. 112(C), pages 142-167.
    12. Moenig, Thorsten, 2021. "Variable annuities: Market incompleteness and policyholder behavior," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 63-78.
    13. Andrea Molent, 2019. "Taxation of a GMWB Variable Annuity in a Stochastic Interest Rate Model," Papers 1901.11296, arXiv.org, revised May 2020.

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