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Optimal surrender strategies and valuations of path-dependent guarantees in variable annuities

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  • Jeon, Junkee
  • Kwak, Minsuk

Abstract

In this paper, we consider variable annuity contracts with guaranteed minimum maturity benefits in which the embedded guarantees can be surrendered any time before the maturity date has been reached. We derive the optimal surrender strategies and the values of three different types of guarantees embedded in VA contracts: (1) constant guarantees, (2) constant guarantees with an up-and-out barrier, and (3) path-dependent guarantees with a lookback feature. The valuation of the embedded guarantees with a surrender option can be formulated as a free boundary problem, and we can obtain a corresponding non-homogeneous partial differential equation (PDE). We apply the Mellin transform to the non-homogeneous PDE in order to obtain an integral equation that produces the value of the embedded guarantee as well as the optimal surrender boundary in a variable annuity contract. We present the properties of the fair fee rate and the optimal surrender strategy with numerical examples.

Suggested Citation

  • Jeon, Junkee & Kwak, Minsuk, 2018. "Optimal surrender strategies and valuations of path-dependent guarantees in variable annuities," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 93-109.
    • Handle: RePEc:eee:insuma:v:83:y:2018:i:c:p:93-109
    DOI: 10.1016/j.insmatheco.2018.09.006
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    References listed on IDEAS

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    1. Jing-Zhi Huang & Marti G. Subrahmanyam & G. George Yu, 1999. "Pricing And Hedging American Options: A Recursive Integration Method," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar, chapter 8, pages 219-239, World Scientific Publishing Co. Pte. Ltd..
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    3. Shen, Yang & Sherris, Michael & Ziveyi, Jonathan, 2016. "Valuation of guaranteed minimum maturity benefits in variable annuities with surrender options," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 127-137.
    4. Bernard, Carole & MacKay, Anne & Muehlbeyer, Max, 2014. "Optimal surrender policy for variable annuity guarantees," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 116-128.
    5. Bernard, Carole & Kwak, Minsuk, 2016. "Semi-static hedging of variable annuities," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 173-186.
    6. T. F. Coleman & Y. Kim & Y. Li & M. Patron, 2007. "Robustly Hedging Variable Annuities With Guarantees Under Jump and Volatility Risks," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 74(2), pages 347-376, June.
    7. Coleman, Thomas F. & Li, Yuying & Patron, Maria-Cristina, 2006. "Hedging guarantees in variable annuities under both equity and interest rate risks," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 215-228, April.
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    Cited by:

    1. 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.
    2. Jeon, Junkee & Koo, Hyeng Keun & Park, Kyunghyun, 2021. "Finite horizon portfolio selection with durable goods," Mathematical Social Sciences, Elsevier, vol. 111(C), pages 55-67.
    3. Anne Mackay & Marie-Claude Vachon, 2023. "On an Optimal Stopping Problem with a Discontinuous Reward," Papers 2311.03538, arXiv.org, revised Nov 2023.

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    More about this item

    Keywords

    Variable annuities; Path-dependent guarantee; Optimal surrender strategy; Free boundary problem; Mellin transform;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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