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Mathematical analysis and numerical methods for pricing pension plans allowing early retirement

Author

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  • Calvo-Garrido, Maria del Carmen
  • Pascucci, Andrea
  • Vázquez Cendón, Carlos

Abstract

In this paper, we address the mathematical analysis and numerical solution of a model for pricing a defined benefit pension plan. More precisely, the benefits received by the member of the plan depend on the average salary and early retirement is allowed. Thus, the mathematical model is posed as an obstacle problem associated to a Kolmogorov equation in the time region where the salary is being averaged. Previously to the initial averaging date, a nonhomogeneous one factor Black-Scholes equation is posed. After stating the model, existence and regularity of solutions are studied. Moreover, appropriate numerical methods based on a Lagrange-Galerkin discretization and an augmented Lagrangian active set method are proposed. Finally, some numerical examples illustrate the performance of the numerical techniques and the properties of the solution and the free boundary.

Suggested Citation

  • Calvo-Garrido, Maria del Carmen & Pascucci, Andrea & Vázquez Cendón, Carlos, 2012. "Mathematical analysis and numerical methods for pricing pension plans allowing early retirement," MPRA Paper 36494, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:36494
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    File URL: https://mpra.ub.uni-muenchen.de/36494/1/MPRA_paper_36494.pdf
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    References listed on IDEAS

    as
    1. Andrea Pascucci, 2008. "Free boundary and optimal stopping problems for American Asian options," Finance and Stochastics, Springer, vol. 12(1), pages 21-41, January.
    2. Bodie, Zvi, 1990. "Pensions as Retirement Income Insurance," Journal of Economic Literature, American Economic Association, vol. 28(1), pages 28-49, March.
    3. Avner Friedman & Weixi Shen, 2002. "A variational inequality approach to financial valuation of retirement benefits based on salary," Finance and Stochastics, Springer, vol. 6(3), pages 273-302.
    4. E. Chevalier, 2006. "Optimal Early Retirement Near the Expiration of a Pension Plan," Finance and Stochastics, Springer, vol. 10(2), pages 204-221, April.
    5. E. Chevalier, 2006. "Optimal Early Retirement Near the Expiration of a Pension Plan," Finance and Stochastics, Springer, vol. 10(2), pages 204-221, April.
    6. Laura Monti & Andrea Pascucci, 2009. "Obstacle problem for Arithmetic Asian options," Papers 0910.4257, arXiv.org.
    7. T. Kärkkäinen & K. Kunisch & P. Tarvainen, 2003. "Augmented Lagrangian Active Set Methods for Obstacle Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 499-533, December.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    retirement plans; options pricing; Kolmogorov equations; complementarity problem; numerical methods; augmented Lagrangian formulation;
    All these keywords.

    JEL classification:

    • G23 - Financial Economics - - Financial Institutions and Services - - - Non-bank Financial Institutions; Financial Instruments; Institutional Investors
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G00 - Financial Economics - - General - - - General

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