On an Optimal Stopping Problem with a Discontinuous Reward

TL;DR

This paper analyzes an optimal stopping problem with a discontinuous, time-dependent reward, motivated by variable annuity pricing, providing conditions for optimal surrender timing and characterizing surrender regions.

Contribution

It introduces new analytical tools and representations for the value function in discontinuous reward optimal stopping problems, relevant to actuarial and financial contexts.

Findings

Optimal stopping occurs at maturity under certain conditions.

The shape of the surrender region is fully characterized by fee and charge functions.

Three representations of the value function are developed, including two analogous to American options.

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.