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Pricing Variable Annuity Guarantees in a local volatility framework

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  • Deelstra, Griselda
  • Rayée, Grégory

Abstract

In this paper, we study the price of Variable Annuity Guarantees, particularly those of Guaranteed Annuity Options (GAO) and Guaranteed Minimum Income Benefit (GMIB), in the settings of a derivative pricing model where the underlying spot (the fund) is locally governed by a geometric Brownian motion with local volatility, while interest rates follow a Hull–White one-factor Gaussian model. Notwithstanding the fact that in this framework, the local volatility depends on a particularly complex expectation where no closed-form expression exists and it is neither directly related to European call prices or other liquid products, we present in this contribution a method based on Monte Carlo Simulations to calibrate the local volatility model. We further compare the Variable Annuity Guarantee prices obtained in three different settings, namely the local volatility, the stochastic volatility and the constant volatility models all combined with stochastic interest rates and show that an appropriate volatility modeling is important for these long-dated derivatives. More precisely, we compare the prices of GAO, GMIB Rider and barrier types GAO obtained by using the local volatility, stochastic volatility and constant volatility models.

Suggested Citation

  • Deelstra, Griselda & Rayée, Grégory, 2013. "Pricing Variable Annuity Guarantees in a local volatility framework," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 650-663.
    • Handle: RePEc:eee:insuma:v:53:y:2013:i:3:p:650-663
    DOI: 10.1016/j.insmatheco.2013.09.007
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    1. Griselda Deelstra & Gr�gory Ray�e, 2013. "Local Volatility Pricing Models for Long-Dated FX Derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 20(4), pages 380-402, September.
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    7. Ballotta, Laura & Haberman, Steven, 2003. "Valuation of guaranteed annuity conversion options," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 87-108, August.
    8. van Haastrecht, Alexander & Plat, Richard & Pelsser, Antoon, 2010. "Valuation of guaranteed annuity options using a stochastic volatility model for equity prices," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 266-277, December.
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    3. Gan, Guojun & Lin, X. Sheldon, 2015. "Valuation of large variable annuity portfolios under nested simulation: A functional data approach," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 138-150.
    4. Tang, Chun-Hua, 2018. "Subjective value of the guarantees embedded in public cash-balance pension plans," Journal of Pension Economics and Finance, Cambridge University Press, vol. 17(2), pages 231-250, April.
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    6. Hieber, Peter, 2017. "Cliquet-style return guarantees in a regime switching Lévy model," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 138-147.
    7. Delong, Łukasz, 2014. "Pricing and hedging of variable annuities with state-dependent fees," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 24-33.

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