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Fourier based methods for the management of complex life insurance products

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  • Ballotta, Laura
  • Eberlein, Ernst
  • Schmidt, Thorsten
  • Zeineddine, Raghid

Abstract

This paper proposes a framework for the valuation and the management of complex life insurance contracts, whose design can be described by a portfolio of embedded options, which are activated according to one or more triggering events. These events are in general monitored discretely over the life of the policy, due to the contract terms. Similar designs can also be found in other contexts, such as counterparty credit risk for example. The framework is based on Fourier transform methods as they allow to derive convenient closed analytical formulas for a broad spectrum of underlying dynamics. Multidimensionality issues generated by the discrete monitoring of the triggering events are dealt with efficiently designed Monte Carlo integration strategies. We illustrate the tractability of the proposed approach by means of a detailed study of ratchet variable annuities, which can be considered a prototypical example of these complex structured products.

Suggested Citation

  • Ballotta, Laura & Eberlein, Ernst & Schmidt, Thorsten & Zeineddine, Raghid, 2021. "Fourier based methods for the management of complex life insurance products," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 320-341.
    • Handle: RePEc:eee:insuma:v:101:y:2021:i:pb:p:320-341
    DOI: 10.1016/j.insmatheco.2021.08.009
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    1. Dhaene, Jan & Kukush, Alexander & Luciano, Elisa & Schoutens, Wim & Stassen, Ben, 2013. "On the (in-)dependence between financial and actuarial risks," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 522-531.
    2. Ernst Eberlein & Sebastian Raible, 1999. "Term Structure Models Driven by General Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 9(1), pages 31-53, January.
    3. Kijima, Masaaki & Wong, Tony, 2007. "Pricing of Ratchet equity-indexed annuities under stochastic interest rates," Insurance: Mathematics and Economics, Elsevier, vol. 41(3), pages 317-338, November.
    4. Laura Ballotta & Ernst Eberlein & Thorsten Schmidt & Raghid Zeineddine, 2020. "Variable annuities in a Lévy-based hybrid model with surrender risk," Quantitative Finance, Taylor & Francis Journals, vol. 20(5), pages 867-886, May.
    5. X. Sheldon Lin & Ken Seng Tan, 2003. "Valuation of Equity-Indexed Annuities Under Stochastic Interest Rates," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 72-91.
    6. Fusai, Gianluca & Germano, Guido & Marazzina, Daniele, 2016. "Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options," European Journal of Operational Research, Elsevier, vol. 251(1), pages 124-134.
    7. 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.
    8. Xiaoqun Wang & Ken Seng Tan, 2013. "Pricing and Hedging with Discontinuous Functions: Quasi-Monte Carlo Methods and Dimension Reduction," Management Science, INFORMS, vol. 59(2), pages 376-389, July.
    9. Abdou Kelani & François Quittard-Pinon, 2017. "Pricing and Hedging Variable Annuities in a Lévy Market : A Risk Management Perspective," Post-Print hal-02313300, HAL.
    10. Ning Cai & Yingda Song & Steven Kou, 2015. "A General Framework for Pricing Asian Options Under Markov Processes," Operations Research, INFORMS, vol. 63(3), pages 540-554, June.
    11. Dahl, Mikkel & Moller, Thomas, 2006. "Valuation and hedging of life insurance liabilities with systematic mortality risk," Insurance: Mathematics and Economics, Elsevier, vol. 39(2), pages 193-217, October.
    12. Ernst Eberlein & Marcus Rudmann, 2018. "Hybrid Lévy Models: Design and Computational Aspects," Applied Mathematical Finance, Taylor & Francis Journals, vol. 25(5-6), pages 533-556, November.
    13. Ernst Eberlein & Kathrin Glau & Antonis Papapantoleon, 2010. "Analysis of Fourier Transform Valuation Formulas and Applications," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(3), pages 211-240.
    14. Ernst Eberlein & Jean Jacod & Sebastian Raible, 2005. "Lévy term structure models: No-arbitrage and completeness," Finance and Stochastics, Springer, vol. 9(1), pages 67-88, January.
    15. Loisel, Stéphane & Milhaud, Xavier, 2011. "From deterministic to stochastic surrender risk models: Impact of correlation crises on economic capital," European Journal of Operational Research, Elsevier, vol. 214(2), pages 348-357, October.
    16. Tanskanen, Antti Juho & Lukkarinen, Jani, 2003. "Fair valuation of path-dependent participating life insurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 595-609, December.
    17. Kolkiewicz, A. W. & Tan, K. S., 2006. "Unit-Linked Life Insurance Contracts with Lapse Rates Dependent on Economic Factors," Annals of Actuarial Science, Cambridge University Press, vol. 1(1), pages 49-78, March.
    18. Dahl, Mikkel, 2004. "Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 113-136, August.
    19. Xiaoqun Wang, 2006. "On the Effects of Dimension Reduction Techniques on Some High-Dimensional Problems in Finance," Operations Research, INFORMS, vol. 54(6), pages 1063-1078, December.
    20. Nolte, Sven & Schneider, Judith C., 2017. "Don’t lapse into temptation: a behavioral explanation for policy surrender," Journal of Banking & Finance, Elsevier, vol. 79(C), pages 12-27.
    21. Anna Rita Bacinello & Enrico Biffis & Pietro Millossovich, 2010. "Regression-based algorithms for life insurance contracts with surrender guarantees," Quantitative Finance, Taylor & Francis Journals, vol. 10(9), pages 1077-1090.
    22. Ning Cai & Yingda Song & Nan Chen, 2017. "Exact Simulation of the SABR Model," Operations Research, INFORMS, vol. 65(4), pages 931-951, August.
    23. Laura Ballotta, 2010. "Efficient Pricing of Ratchet Equity-Indexed Annuities in a Variance-Gamma Economy," North American Actuarial Journal, Taylor & Francis Journals, vol. 14(3), pages 355-368.
    24. Di Giacinto, Marina & Federico, Salvatore & Gozzi, Fausto & Vigna, Elena, 2014. "Income drawdown option with minimum guarantee," European Journal of Operational Research, Elsevier, vol. 234(3), pages 610-624.
    25. Nikolay Gudkov & Katja Ignatieva & Jonathan Ziveyi, 2019. "Pricing of guaranteed minimum withdrawal benefits in variable annuities under stochastic volatility, stochastic interest rates and stochastic mortality via the componentwise splitting method," Quantitative Finance, Taylor & Francis Journals, vol. 19(3), pages 501-518, March.
    26. Mark Broadie & Özgür Kaya, 2006. "Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes," Operations Research, INFORMS, vol. 54(2), pages 217-231, April.
    27. Biffis, Enrico, 2005. "Affine processes for dynamic mortality and actuarial valuations," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 443-468, December.
    28. Stéphane Loisel, 2011. "Surrender risk and correlation crises," Post-Print hal-00671919, HAL.
    29. 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, vol. 4(4), pages 1-36, November.
    30. Sesana, Debora & Marazzina, Daniele & Fusai, Gianluca, 2014. "Pricing exotic derivatives exploiting structure," European Journal of Operational Research, Elsevier, vol. 236(1), pages 369-381.
    31. Cui, Zhenyu & Lars Kirkby, J. & Nguyen, Duy, 2017. "A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps," European Journal of Operational Research, Elsevier, vol. 262(1), pages 381-400.
    32. Xavier Milhaud & Stéphane Loisel & Véronique Maume-Deschamps, 2011. "Surrender triggers in life insurance: what main features affect the surrender behavior in a classical economic context?," Post-Print hal-00450003, HAL.
    33. Ballotta, Laura, 2005. "A Lévy process-based framework for the fair valuation of participating life insurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 37(2), pages 173-196, October.
    34. Abdou Kélani & François Quittard-Pinon, 2017. "Pricing and Hedging Variable Annuities in a Lévy Market: A Risk Management Perspective," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 84(1), pages 209-238, March.
    35. Brennan, Michael J. & Schwartz, Eduardo S., 1976. "The pricing of equity-linked life insurance policies with an asset value guarantee," Journal of Financial Economics, Elsevier, vol. 3(3), pages 195-213, June.
    36. Russo, Vincenzo & Giacometti, Rosella & Fabozzi, Frank J., 2017. "Intensity-based framework for surrender modeling in life insurance," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 189-196.
    37. Christian Knoller & Gunther Kraut & Pascal Schoenmaekers, 2016. "On the Propensity to Surrender a Variable Annuity Contract: An Empirical Analysis of Dynamic Policyholder Behavior," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 83(4), pages 979-1006, December.
    38. Ning Cai & Steven Kou, 2012. "Pricing Asian Options Under a Hyper-Exponential Jump Diffusion Model," Operations Research, INFORMS, vol. 60(1), pages 64-77, February.
    39. 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.
    40. Bacinello, Anna Rita & Ortu, Fulvio, 1996. "Fixed income linked life insurance policies with minimum guarantees: Pricing models and numerical results," European Journal of Operational Research, Elsevier, vol. 91(2), pages 235-249, June.
    41. Jennifer Alonso-García & Oliver Wood & Jonathan Ziveyi, 2018. "Pricing and hedging guaranteed minimum withdrawal benefits under a general Lévy framework using the COS method," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 1049-1075, June.
    42. Laura Ballotta, 2009. "Pricing and capital requirements for with profit contracts: modelling considerations," Quantitative Finance, Taylor & Francis Journals, vol. 9(7), pages 803-817.
    43. Ballotta, Laura & Fusai, Gianluca & Marazzina, Daniele, 2019. "Integrated structural approach to Credit Value Adjustment," European Journal of Operational Research, Elsevier, vol. 272(3), pages 1143-1157.
    44. Milevsky, Moshe A. & David Promislow, S., 2001. "Mortality derivatives and the option to annuitise," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 299-318, December.
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    More about this item

    Keywords

    Fourier transforms; Hybrid models; Lévy processes; Structured products; Variable annuities;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • 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
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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