Evaluating variable annuities with GMWB when exogenous factors influence the policy-holder withdrawals

We propose a model for evaluating variable annuities with guaranteed minimum withdrawal benefits in which a rational policy-holder, who would withdraw the optimal amounts maximizing the current policy value only with respect to the endogenous variables of the evaluation problem, acts in a more realistic context where her/his choices may be influenced by exogenous variables that may lead to withdraw sub-optimal amounts. The model is based on a trinomial approximation of the personal sub-account dynamics that, despite the presence of a downward jump due to the payed withdrawal at each anniversary of the contract, guarantees the reconnecting property. A backward induction scheme is used to compute the insurance fair fee paid for the guarantee.