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Fast Numerical Method for Pricing of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Optimal Withdrawal Strategy

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  • Xiaolin Luo
  • Pavel Shevchenko

Abstract

A variable annuity contract with Guaranteed Minimum Withdrawal Benefit (GMWB) promises to return the entire initial investment through cash withdrawals during the policy life plus the remaining account balance at maturity, regardless of the portfolio performance. Under the optimal withdrawal strategy of a policyholder, the pricing of variable annuities with GMWB becomes an optimal stochastic control problem. So far in the literature these contracts have only been evaluated by solving partial differential equations (PDE) using the finite difference method. The well-known Least-Squares or similar Monte Carlo methods cannot be applied to pricing these contracts because the paths of the underlying wealth process are affected by optimal cash withdrawals (control variables) and thus cannot be simulated forward in time. In this paper we present a very efficient new algorithm for pricing these contracts in the case when transition density of the underlying asset between withdrawal dates or its moments are known. This algorithm relies on computing the expected contract value through a high order Gauss-Hermite quadrature applied on a cubic spline interpolation. Numerical results from the new algorithm for a series of GMWB contract are then presented, in comparison with results using the finite difference method solving corresponding PDE. The comparison demonstrates that the new algorithm produces results in very close agreement with those of the finite difference method, but at the same time it is significantly faster; virtually instant results on a standard desktop PC.

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  • Xiaolin Luo & Pavel Shevchenko, 2014. "Fast Numerical Method for Pricing of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Optimal Withdrawal Strategy," Papers 1410.8609, arXiv.org.
    • Handle: RePEc:arx:papers:1410.8609
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    References listed on IDEAS

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    1. Bacinello, Anna Rita & Millossovich, Pietro & Olivieri, Annamaria & Pitacco, Ermanno, 2011. "Variable annuities: A unifying valuation approach," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 285-297.
    2. Milevsky, Moshe A. & Salisbury, Thomas S., 2006. "Financial valuation of guaranteed minimum withdrawal benefits," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 21-38, February.
    3. P. Forsyth & K. Vetzal & R. Zvan, 2002. "Convergence of numerical methods for valuing path-dependent options using interpolation," Review of Derivatives Research, Springer, vol. 5(3), pages 273-314, October.
    4. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    5. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    6. Xiaolin Luo & Pavel V. Shevchenko, 2014. "Fast and Simple Method for Pricing Exotic Options using Gauss-Hermite Quadrature on a Cubic Spline Interpolation," Papers 1408.6938, arXiv.org, revised Dec 2014.
    7. Bauer, Daniel & Kling, Alexander & Russ, Jochen, 2008. "A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities 1," ASTIN Bulletin, Cambridge University Press, vol. 38(2), pages 621-651, November.
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    Cited by:

    1. Paolo Angelis & Roberto Marchis & Antonio L. Martire & Emilio Russo, 2022. "A flexible lattice framework for valuing options on assets paying discrete dividends and variable annuities embedding GMWB riders," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 45(1), pages 415-446, June.
    2. Luo, Xiaolin & Shevchenko, Pavel V., 2015. "Valuation of variable annuities with guaranteed minimum withdrawal and death benefits via stochastic control optimization," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 5-15.
    3. Huansang Xu & Ruyi Liu & Marek Rutkowski, 2023. "Equity Protection Swaps: A New Type of Investment Insurance for Holders of Superannuation Accounts," Papers 2305.09472, arXiv.org, revised Apr 2024.

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