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Gaussian Process Regression for Pricing Variable Annuities with Stochastic Volatility and Interest Rate

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  • Ludovic Gouden`ege
  • Andrea Molent
  • Antonino Zanette

Abstract

In this paper we investigate price and Greeks computation of a Guaranteed Minimum Withdrawal Benefit (GMWB) Variable Annuity (VA) when both stochastic volatility and stochastic interest rate are considered together in the Heston Hull-White model. We consider a numerical method the solves the dynamic control problem due to the computing of the optimal withdrawal. Moreover, in order to speed up the computation, we employ Gaussian Process Regression (GPR). Starting from observed prices previously computed for some known combinations of model parameters, it is possible to approximate the whole price function on a defined domain. The regression algorithm consists of algorithm training and evaluation. The first step is the most time demanding, but it needs to be performed only once, while the latter is very fast and it requires to be performed only when predicting the target function. The developed method, as well as for the calculation of prices and Greeks, can also be employed to compute the no-arbitrage fee, which is a common practice in the Variable Annuities sector. Numerical experiments show that the accuracy of the values estimated by GPR is high with very low computational cost. Finally, we stress out that the analysis is carried out for a GMWB annuity but it could be generalized to other insurance products.

Suggested Citation

  • Ludovic Gouden`ege & Andrea Molent & Antonino Zanette, 2019. "Gaussian Process Regression for Pricing Variable Annuities with Stochastic Volatility and Interest Rate," Papers 1903.00369, arXiv.org, revised Jul 2019.
  • Handle: RePEc:arx:papers:1903.00369
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    File URL: http://arxiv.org/pdf/1903.00369
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    References listed on IDEAS

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    1. M. Briani & L. Caramellino & A. Zanette, 2015. "A hybrid tree/finite-difference approach for Heston-Hull-White type models," Papers 1503.03705, arXiv.org, revised Dec 2017.
    2. Ryan Donnelly & Sebastian Jaimungal & Dmitri H. Rubisov, 2014. "Valuing guaranteed withdrawal benefits with stochastic interest rates and volatility," Quantitative Finance, Taylor & Francis Journals, vol. 14(2), pages 369-382, February.
    3. Maya Briani & Lucia Caramellino & Antonino Zanette, 2013. "A hybrid approach for the implementation of the Heston model," Papers 1307.7178, arXiv.org, revised Sep 2017.
    4. Dai, Tian-Shyr & Yang, Sharon S. & Liu, Liang-Chih, 2015. "Pricing guaranteed minimum/lifetime withdrawal benefits with various provisions under investment, interest rate and mortality risks," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 364-379.
    5. Xiaolin Luo & Pavel V. Shevchenko, 2014. "Valuation of Variable Annuities with Guaranteed Minimum Withdrawal and Death Benefits via Stochastic Control Optimization," Papers 1411.5453, arXiv.org, revised Apr 2015.
    6. Gan, Guojun, 2013. "Application of data clustering and machine learning in variable annuity valuation," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 795-801.
    7. Philippe Deprez & Pavel V. Shevchenko & Mario V. Wuthrich, 2017. "Machine Learning Techniques for Mortality Modeling," Papers 1705.03396, arXiv.org.
    8. Jan De Spiegeleer & Dilip B. Madan & Sofie Reyners & Wim Schoutens, 2018. "Machine learning for quantitative finance: fast derivative pricing, hedging and fitting," Quantitative Finance, Taylor & Francis Journals, vol. 18(10), pages 1635-1643, October.
    9. 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.
    10. Maya Briani & Lucia Caramellino & Antonino Zanette, 2017. "A hybrid approach for the implementation of the Heston model," Post-Print hal-00916440, HAL.
    11. 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.
    12. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    13. Ludovic Goudenege & Andrea Molent & Antonino Zanette, 2019. "Pricing and hedging GMWB in the Heston and in the BlackÔÇôScholes with stochastic interest rate models," Computational Management Science, Springer, vol. 16(1), pages 217-248, February.
    14. Bauer, Daniel & Kling, Alexander & Russ, Jochen, 2008. "A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities1," ASTIN Bulletin, Cambridge University Press, vol. 38(2), pages 621-651, November.
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    Cited by:

    1. Ludovic Gouden`ege & Andrea Molent & Antonino Zanette, 2019. "Variance Reduction Applied to Machine Learning for Pricing Bermudan/American Options in High Dimension," Papers 1903.11275, arXiv.org, revised Dec 2019.

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