🤖 AI Summary
This paper addresses the pricing of variable annuities with a minimum guaranteed maturity benefit, modeling policyholder surrender behavior as an optimal stopping problem aimed at maximizing risk-neutral value—yielding an unbounded, time-varying, and discontinuous payoff structure. Using stochastic optimal control theory and variational inequality methods, complemented by PDE analysis and novel auxiliary value function constructions, we systematically characterize, for the first time, the nonemptiness and geometric structure of the surrender region—entirely determined by management fees and surrender charges. We derive three distinct representations of the value function, one of which is original to both actuarial science and American option literature. We establish a sufficient condition under which the optimal stopping time is necessarily delayed until maturity. Furthermore, we quantitatively elucidate the intrinsic mechanism through which fee structures govern the location of the surrender boundary and the intensity of early surrender incentives.
📝 Abstract
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.