In this paper we provide a theoretical analysis of Variable Annuities with a
focus on the holder's right to an early termination of the contract. We obtain
a rigorous pricing formula and the optimal exercise boundary for the surrender
option. We also illustrate our theoretical results with extensive numerical
experiments. The pricing problem is formulated as an optimal stopping problem
with a time-dependent payoff which is discontinuous at the maturity of the
contract and non-smooth. This structure leads to non-monotonic optimal stopping
boundaries which we prove nevertheless to be continuous and regular in the
sense of diffusions for the stopping set. The lack of monotonicity of the
boundary makes it impossible to use classical methods from optimal stopping.
Also more recent results about Lipschitz continuous boundaries are not
applicable in our setup. Thus, we contribute a new methodology for non-monotone
stopping boundaries.