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Optimal surrender of guaranteed minimum maturity benefits under stochastic volatility and interest rates

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  • Kang, Boda
  • Ziveyi, Jonathan

Abstract

In this paper we analyse how the policyholders’surrender behaviour is influenced by changes in various sources of risk impacting a variable annuity (VA) contract embedded with a guaranteed minimum maturity benefit rider that can be surrendered anytime prior to maturity. We model the underlying mutual fund dynamics by combining a Heston (1993) stochastic volatility model together with a Hull and White (1990) stochastic interest rate process. The model is able to capture the smile/skew often observed on equity option markets (Grzelak and Oosterlee 2011) as well as the influence of the interest rates on the early surrender decisions as noted from our analysis. The annuity provider charges management fees which are proportional to the level of the mutual fund as a way of funding the VA contract. To determine the optimal surrender decisions, we present the problem as a 4-dimensional free-boundary partial differential equation (PDE) which is then solved efficiently by the method of lines (MOL) approach. The MOL algorithm facilitates simultaneous computation of the prices, fair management fees, optimal surrender boundaries and hedge ratios of the variable annuity contract as part of the solution at no additional computational cost. A comprehensive analysis on the impact of various risk factors in influencing the policyholder’s surrender behaviour is carried out, highlighting the significance of both stochastic volatility and interest rate parameters in influencing the policyholder’s surrender behaviour. With the aid of the hedge ratios obtained from the MOL, we construct an effective dynamic hedging strategy to mitigate the provider’s risk and compare different hedging performances when the policyholders’ surrender behaviour is either optimal or sub-optimal.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:insuma:v:79:y:2018:i:c:p:43-56
    DOI: 10.1016/j.insmatheco.2017.12.012
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    References listed on IDEAS

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    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. 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.
    6. Anne Mackay & Marie-Claude Vachon, 2023. "On an Optimal Stopping Problem with a Discontinuous Reward," Papers 2311.03538, arXiv.org, revised Nov 2023.
    7. Belssing Taruvinga, 2019. "Solving Selected Problems on American Option Pricing with the Method of Lines," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 4-2019, January-A.
    8. Antonio L. Martire & Emilio Russo & Alessandro Staino, 2023. "Surrender and path-dependent guarantees in variable annuities: integral equation solutions and benchmark methods," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 46(1), pages 177-220, June.
    9. 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.
    10. Boda Kang & Christina Nikitopoulos Sklibosios & Erik Schlogl & Blessing Taruvinga, 2019. "The Impact of Jumps on American Option Pricing: The S&P 100 Options Case," Research Paper Series 397, Quantitative Finance Research Centre, University of Technology, Sydney.
    11. Jennifer Alonso Garcia & Michael Sherris & Samuel Thirurajah & Jonathan Ziveyi, 2020. "Taxation and policyholder behavior: the case of guaranteed minimum accumulation benefits," ULB Institutional Repository 2013/307889, ULB -- Universite Libre de Bruxelles.

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