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A Unified Pricing of Variable Annuity Guarantees under the Optimal Stochastic Control Framework

Author

Listed:
  • Pavel V. Shevchenko

    () (The Commonwealth Scientific and Industrial Research Organisation, PO BOX 52, North Ryde, NSW 1670, Australia)

  • Xiaolin Luo

    () (The Commonwealth Scientific and Industrial Research Organisation, PO BOX 52, North Ryde, NSW 1670, Australia)

Abstract

In this paper, we review pricing of the variable annuity living and death guarantees offered to retail investors in many countries. Investors purchase these products to take advantage of market growth and protect savings. We present pricing of these products via an optimal stochastic control framework and review the existing numerical methods. We also discuss pricing under the complete/incomplete financial market models, stochastic mortality and optimal/sub-optimal policyholder behavior, and in the presence of taxes. For numerical valuation of these contracts in the case of simple risky asset process, we develop a direct integration method based on the Gauss-Hermite quadratures with a one-dimensional cubic spline for calculation of the expected contract value, and a bi-cubic spline interpolation for applying the jump conditions across the contract cashflow event times. This method is easier to implement and faster when compared to the partial differential equation methods if the transition density (or its moments) of the risky asset underlying the contract is known in closed form between the event times. We present accurate numerical results for pricing of a Guaranteed Minimum Accumulation Benefit (GMAB) guarantee available on the market that can serve as a numerical benchmark for practitioners and researchers developing pricing of variable annuity guarantees to assess the accuracy of their numerical implementation.

Suggested Citation

  • Pavel V. Shevchenko & Xiaolin Luo, 2016. "A Unified Pricing of Variable Annuity Guarantees under the Optimal Stochastic Control Framework," Risks, MDPI, Open Access Journal, vol. 4(3), pages 1-31, July.
  • Handle: RePEc:gam:jrisks:v:4:y:2016:i:3:p:22-:d:73342
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    References listed on IDEAS

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    1. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    2. Pavel V. Shevchenko & Xiaolin Luo, 2016. "Valuation of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Stochastic Interest Rate," Papers 1602.03238, arXiv.org, revised Jan 2017.
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    Citations

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    Cited by:

    1. Jin Sun & Pavel V. Shevchenko & Man Chung Fung, 2017. "A note on the impact of management fees on the pricing of variable annuity guarantees," Papers 1705.03787, arXiv.org, revised May 2017.
    2. Marcos Escobar & Mikhail Krayzler & Franz Ramsauer & David Saunders & Rudi Zagst, 2016. "Incorporation of Stochastic Policyholder Behavior in Analytical Pricing of GMABs and GMDBs," Risks, MDPI, Open Access Journal, vol. 4(4), pages 1-36, November.
    3. repec:eee:insuma:v:76:y:2017:i:c:p:104-117 is not listed on IDEAS
    4. repec:gam:jrisks:v:6:y:2018:i:3:p:103-:d:170856 is not listed on IDEAS

    More about this item

    Keywords

    variable annuity; guaranteed living and death benefits; guaranteed minimum accumulation benefit; optimal stochastic control; direct integration method;

    JEL classification:

    • C - Mathematical and Quantitative Methods
    • G0 - Financial Economics - - General
    • G1 - Financial Economics - - General Financial Markets
    • G2 - Financial Economics - - Financial Institutions and Services
    • G3 - Financial Economics - - Corporate Finance and Governance
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics
    • M4 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Accounting
    • K2 - Law and Economics - - Regulation and Business Law

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