On variable annuities with surrender charges

Abstract

In this paper we provide a theoretical analysis of Variable Annuities (VAs) with a focus on the holder's right to an early termination of the contract. Motivated by risk-management considerations, and in line with existing literature, we assume that the surrender option is optimally exercised from a purely financial perspective (i.e., we consider the worst-case scenario for the insurer). In this context, we rigorously derive the pricing formula for the VA and characterise the optimal surrender time. 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.

Version accepted for publication: 49 pages, 6 figures. Expanded Introduction and numerical analysis. Added new Section 8