  Variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) entitle
the policy holder to periodic withdrawals together with a terminal payoff
linked to the performance of an equity fund. In this paper, we consider the
valuation of a general class of GMWB annuities, allowing for step-up, bonus and
surrender features, taking also into account mortality risk and death benefits.
When dynamic withdrawals are allowed, the valuation of GMWB annuities leads to
a stochastic optimal control problem, which we address here by dynamic
programming techniques. Adopting a Hull-White interest rate model, correlated
with the equity fund, we propose an efficient tree-based algorithm. We perform
a thorough analysis of the determinants of the market value of GMWB annuities
and of the optimal withdrawal strategies. In particular, we study the impact of
a low/negative interest rate environment. Our findings indicate that
low/negative rates profoundly affect the optimal withdrawal behaviour and, in
combination with step-up and bonus features, increase significantly the fair
values of GMWB annuities, which can only be compensated by large management
fees.

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