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An operator splitting harmonic differential quadrature approach to solve Young’s model for life insurance risk

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  • Ballestra, Luca Vincenzo
  • Ottaviani, Massimiliano
  • Pacelli, Graziella

Abstract

This paper is concerned with the numerical approximation of a mathematical model for life insurance risk that has been presented quite recently by Young (2007, 2008). In particular, such a model, which consists of a system of several non-linear partial differential equations, is solved using a new numerical method that combines an operator splitting procedure with the differential quadrature (DQ) finite difference scheme. This approach allows one to reduce the partial differential problems to systems of linear equations of very small dimension, so that pricing portfolios of many life insurances becomes a relatively easily task. Numerical experiments are presented showing that the method proposed is very accurate and fast. In addition, the limit behavior of portfolios of life insurances as the number of contracts tends to infinity is investigated. This analysis reveals that the prices of portfolios comprising more than five thousand policies can be accurately approximated by solving a linear partial differential equation derived in Young (2007, 2008).

Suggested Citation

  • Ballestra, Luca Vincenzo & Ottaviani, Massimiliano & Pacelli, Graziella, 2012. "An operator splitting harmonic differential quadrature approach to solve Young’s model for life insurance risk," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 442-448.
    • Handle: RePEc:eee:insuma:v:51:y:2012:i:2:p:442-448
    DOI: 10.1016/j.insmatheco.2012.06.012
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    References listed on IDEAS

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    1. Bayraktar, Erhan & Young, Virginia R., 2007. "Hedging life insurance with pure endowments," Insurance: Mathematics and Economics, Elsevier, vol. 40(3), pages 435-444, May.
    2. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    3. Ludkovski, Michael & Young, Virginia R., 2008. "Indifference pricing of pure endowments and life annuities under stochastic hazard and interest rates," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 14-30, February.
    4. 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.
    5. Bayraktar, Erhan & Milevsky, Moshe A. & David Promislow, S. & Young, Virginia R., 2009. "Valuation of mortality risk via the instantaneous Sharpe ratio: Applications to life annuities," Journal of Economic Dynamics and Control, Elsevier, vol. 33(3), pages 676-691, March.
    6. Carl Chiarella & Boda Kang & Gunter H. Meyer & Andrew Ziogas, 2009. "The Evaluation Of American Option Prices Under Stochastic Volatility And Jump-Diffusion Dynamics Using The Method Of Lines," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(03), pages 393-425.
    7. Young, Virginia R., 2008. "Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 691-703, April.
    8. 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.
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    Cited by:

    1. Alessandro Andreoli & Luca Vincenzo Ballestra & Graziella Pacelli, 2018. "Pricing Credit Default Swaps Under Multifactor Reduced-Form Models: A Differential Quadrature Approach," Computational Economics, Springer;Society for Computational Economics, vol. 51(3), pages 379-406, March.

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