CalvoGarrido, Maria del Carmen and Pascucci, Andrea and Vázquez Cendón, Carlos (2012): Mathematical analysis and numerical methods for pricing pension plans allowing early retirement.

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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 BlackScholes equation is posed. After stating the model, existence and regularity of solutions are studied. Moreover, appropriate numerical methods based on a LagrangeGalerkin 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.
Item Type:  MPRA Paper 

Original Title:  Mathematical analysis and numerical methods for pricing pension plans allowing early retirement 
English Title:  Mathematical analysis and numerical methods for pricing pension plans allowing early retirement 
Language:  English 
Keywords:  retirement plans, options pricing, Kolmogorov equations, complementarity problem, numerical methods, augmented Lagrangian formulation 
Subjects:  G  Financial Economics > G2  Financial Institutions and Services > G23  Nonbank Financial Institutions; Financial Instruments; Institutional Investors G  Financial Economics > G1  General Financial Markets > G13  Contingent Pricing; Futures Pricing G  Financial Economics > G0  General > G00  General 
Item ID:  36494 
Depositing User:  Andrea Pascucci 
Date Deposited:  07. Feb 2012 15:04 
Last Modified:  12. Feb 2013 00:30 
References:  [1] A. Bermudez, M. R. Nogueiras and C. Vazquez, Numerical analysis of convectiondiusionreaction problems with higher order characteristics nite elements. Part I: Time discretization, SIAM J. Numer. Anal., 44 (2006), pp. 18291853. [2] A. Bermudez, M. R. Nogueiras and C. Vazquez, Numerical analysis of convectiondiusionreaction problems with higher order characteristics nite elements. Part II: Fully discretized scheme and quadrature formulas, SIAM J. Numer. Anal., 44 (2006), pp. 18541876. [3] A. Bermudez, M. R. Nogueiras and C. Vazquez, Numerical solution of variational inequalities for pricing Asian options by higher order LagrangeGalerkin methods, Appl. Numer. Math., 56 (2006), pp. 12561270. [4] A. Bermudez, M. R. Nogueiras and C. Vazquez, Comparison of two algorithms to solve a xedstrike Amerasian options pricing problem, in International Series in Numerical Mathematics, 154, 2006, pp. 95106. [5] Z. Bodie, Pensions as retirement income insurance, J. of Economic Literature, 18 (1990), pp. 2849. [6] N. L. Jr. Bowers, H. V. Gerber, J. C. Nickman and C. J. Nesbitt, Actuarial Mathematics, Society of Actuaries, Schaumburg, Illinois, 1997. [7] M.C. CalvoGarrido and C. Vazquez, Pricing pension plans based on average salary without early retirement: PDE modelling and numerical solution, to appear in J. of Computational Finance, (2012). [8] E. Chevalier, Optimal early retirement near the expiration of a pension plan, Finance Stoch., 10 (2006), pp. 204{ 221. [9] M. Di Francesco and A. Pascucci, On a class of degenerate parabolic equations of Kolmogorov type, AMRX Appl. Math. Res. Express, 3 (2005), pp. 77{116. [10] M. Di Francesco, A. Pascucci and S. Polidoro, The obstacle problem for a class of hypoelliptic ultraparabolic equations, Proc. R. Soc. London Ser. A: Math. Phys. Eng. Sci., 464 (2008), pp. 155176. [11] P. Faurre, Analyse numerique: Notes d'optimisation, Ecole Polytechnique, Palaiseau, 1988. [12] M. Frentz, K. Nystr�om, A. Pascucci, and S. Polidoro, Optimal regularity in the obstacle problem for Kol mogorov operators related to American Asian options, Math. Ann., 347 (2010), pp. 805{838. [13] A. Friedman and W. Shen, A variational inequality approach to nancial valuation of retirement benets based on salary, Finance and Stoch., 6 (2002), pp. 273{302. [14] T. K�arkk�ainen, K. Kunisch, and P. Tarvainen, Augmented Lagrangian active set methods for obstacle problems, J. Optim. Theory Appl., 119 (2003), pp. 499{533. [15] E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), pp. 29{63. [16] L. Monti, A. Pascucci, Obstacle problem for arithmetic Asian options, C.R. Acad. Sci. Paris, Ser. I, 347 (2009), pp. 1443{1446. [17] A. Pascucci, Free boundary and optimal stopping problems for American Asian options, Finance Stoch., 12 (2008), pp. 21{41. [18] O. Pironneau, On the transportdiusion algorithm and its application to the NavierStokes equations, Numer. Math., 38 (1982), pp. 309{332. [19] S. Polidoro, On a class of ultraparabolic operators of KolmogorovFokkerPlanck type, Matematiche (Catania), 49 (1994), pp. 53{105. [20] M. Sherris and W. Shen, Financial valuation of retirement benets based on salary, in Quantitative Methods in Finance Conference, Sidney, 1999. [21] R. S Varga, Matrix iterative analysis, PrenticeHall Inc., Englewood Clis, N.J., 1962. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/36494 