Mathematical Analysis and Numerical Methods for Pricing Pension Plans Allowing Early Retirement

Abstract

[Abstract] In this paper, we address the mathematical analysis and numerical solution ofa model for pricing a defined benefit pension plan. More precisely, the benefits received by themember of the plan depend on the average salary and early retirement is allowed. Thus, we formulatethe mathematical model in terms of an obstacle problem associated to a Kolmogorov equation inthe time region where the salary is being averaged. Previously to the initial averaging date, wepose a nonhomogeneous one factor Black–Scholes equation. After stating the model, we study theexistence and regularity of solutions. Moreover, we propose appropriate numerical methods based ona Lagrange–Galerkin discretization and an augmented Lagrangian active set method. Finally, somenumerical examples illustrate the performance of the numerical techniques and the properties of thesolution and the free boundary.