Income drawdown option with minimum guarantee

This paper deals with a constrained investment problem for a defined contribution (DC) pension fund where retirees are allowed to defer the purchase of the annuity at some future time after retirement. This problem has already been treated in the unconstrained case in a number of papers. The aim of this work is to deal with the more realistic case when constraints on the investment strategies and on the state variable are present. Due to the difficulty of the task, we consider, as a first step, the basic model of \cite{GerHabVig04}, where interim consumption and annuitization time are fixed. We extend their model by adding a no short-selling constraint on the control variable and a final capital requirement constraint on the state variable. This implies, in particular, no ruin. The mathematical problem is naturally formulated as a stochastic control problem with constraints on the control and the state variable, and is approached by the dynamic programming method. We write the non-linear Hamilton-Jacobi-Bellman equation for the problem and transform it into a dual one that is semi-linear, following a well-established duality procedure. In the special relevant case without running cost, we explicitly compute the value function for the problem and give the optimal strategy in feedback form. A numerical application ends the paper and shows the extent of applicability of the model to a DC pension fund in the decumulation phase.