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 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.

PDE models for the valuation of a non callable defaultable coupon bond under an extended JDCEV model

“…In order to obtain a numerical approach of the value of a non callable defaultable coupon bond we need to solve the Cauchy problem (7) for u = u 1 and u = u 2 with maturity T 1 = t i for i = 1, ..., M , that is each coupon payment date for both cases. Once these problems are solved, the value of the bond is given by expression (6) in which an integral term appears. This integral term will be approximated by means of the classical composite trapezoidal rule.…”

“…The convergence properties of this Lagrange-Galerkin method have been mathematically analyzed in [3,4] for time and space discretization. More recently, it has been applied to the valuation of pension plans without and with early retirement in [5] and [6], respectively. In order to apply this set of numerical techniques, first a localization procedure is used to cope with the initial formulation in an unbounded domain.…”

Mathematical analysis of obstacle problems for pricing fixed-rate mortgages with prepayment and default options

“…For example, in [6] the authors prove the existence of a strong solution for an obstacle problem linked to a non-uniformly parabolic operator of Kolmogorov type. This kind of Kolmogorov operators also appear in the complementarity problems that arise in the pricing of other financial products, such as American Asian options [10], pension plans with early retirement [2] or stock loans [12]. This paper is organized as follows.…”

A new numerical method for pricing fixed-rate mortgages with prepayment and default options

“…In this paper, we numerically solve the original equation by proposing the PDE discretization with the techniques developed in [4] for Asian options and more recently applied to pension plans in [5] and [6]. More precisely, we use a characteristics method to discretize first order terms and a Crank-Nicolson scheme that evaluates the functions at the previous time step in the basis of the characteristics, which consists on a different approach from the one proposed in [17].…”