On variable annuities with surrender charges

TL;DR

This paper provides a rigorous theoretical framework for valuing variable annuities with surrender charges, including a new methodology for handling non-monotonic optimal stopping boundaries.

Contribution

It introduces a novel approach to analyze and compute optimal surrender boundaries in variable annuities with complex payoff structures.

Findings

Derived a pricing formula for surrender options.

Proved the continuity and regularity of non-monotonic stopping boundaries.

Developed a new methodology for non-monotone optimal stopping problems.

Abstract

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.