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

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

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.

• Item Type: MPRA Paper
• 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 - Non-bank 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

References

[1] A. Bermudez, M. R. Nogueiras and C. Vazquez, Numerical analysis of convection-diusion-reaction pro-blems with higher order characteristics nite elements. Part I: Time discretization, SIAM J. Numer. Anal., 44 (2006), pp. 1829-1853.

[2] A. Bermudez, M. R. Nogueiras and C. Vazquez, Numerical analysis of convection-diusion-reaction problems with higher order characteristics nite elements. Part II: Fully discretized scheme and quadrature formulas, SIAM J. Numer. Anal., 44 (2006), pp. 1854-1876.

[3] A. Bermudez, M. R. Nogueiras and C. Vazquez, Numerical solution of variational inequalities for pricing Asian options by higher order Lagrange-Galerkin methods, Appl. Numer. Math., 56 (2006), pp. 1256-1270.

[4] A. Bermudez, M. R. Nogueiras and C. Vazquez, Comparison of two algorithms to solve a xed-strike Amerasian options pricing problem, in International Series in Numerical Mathematics, 154, 2006, pp. 95-106.

[5] Z. Bodie, Pensions as retirement income insurance, J. of Economic Literature, 18 (1990), pp. 28-49.

[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. Calvo-Garrido 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. 155-176.

[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 transport-diusion algorithm and its application to the Navier-Stokes equations, Numer. Math., 38 (1982), pp. 309{332.

[19] S. Polidoro, On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck 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, Prentice-Hall Inc., Englewood Clis, N.J., 1962.