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

We propose an evaluation framework for variable annuities with guaranteed minimum withdrawal benefits aimed at considering a more realistic context where the policy-holder takes the decisions. In particular, in a rational context, where the policy-holder withdraws the optimal amounts maximizing the current policy value only with respect to the endogenous variables of the evaluation problem, we take into account the effect of exogenous factors that may lead the holder to withdraw sub-optimal amounts. For the sake of completeness, we propose an evaluation model based on a lattice approximation due to its flexibility and ease of implementation that is useful also for practitioners. We discretize the personal sub-account dynamics by a trinomial tree that, despite the presence of a downward jump due to the paid 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.