On an Optimal Stopping Problem with a Discontinuous Reward

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 (VA) contract with guaranteed minimum maturity benefit, under the assumption that the policyholder's surrender behaviour maximizes the contract's risk-neutral value. 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 for understanding the link between fees and surrender functions and how they affect early surrender and the optimal surrender boundary. When the fee and surrender charge only depend on time, we develop three different representations of the value function; two are analogous to their American option counterpart, and one is new to the actuarial and American option pricing literature. Our results allow for the development of new algorithms for the valuation of variable annuity contracts. We provide three such algorithms, based on continuous-time Markov chain approximations. The efficiency of these three algorithms is studied numerically and compared to other commonly used approaches.