  We study an optimal stopping problem with an unbounded, time-dependent and
discontinuous reward function.This problem is motivated by the pricing of a
variable annuity contract with guaranteed minimum maturity benefit, under the
assumption that the policyholder's surrender behaviour maximizes the
risk-neutral value of the contract. We consider a general fee and surrender
charge function, and give a condition under which optimal stopping always
occurs at maturity. Using an alternative representation for the value function
of the optimization problem, we study its analytical properties and the
resulting surrender (or exercise) region. In particular, we show that the
non-emptiness and the shape of the surrender region are fully characterized by
the fee and the surrender charge functions, which provides a powerful tool to
understand their interrelation and how it affects early surrenders and the
optimal surrender boundary. Under certain conditions on these two functions, we
develop three representations for the value function; two are analogous to
their American option counterpart, and one is new to the actuarial and American
option pricing literature.
