Variable annuities with high water mark withdrawal benefit
We develop a continuous-time model for variable annuities that allow for periodic withdrawals proportional to the high water mark of the underlying account value as well as early surrender of the policy. We derive the Hamilton--Jacobi--Bellman equation characterizing the value of such a contract and the worst case policy holder behavior from an issuer's perspective. Based on these results, we construct a dynamic trading strategy which super-hedges the contract. To treat the problem numerically, we develop a semi-Lagrangian scheme based on a discretization of the underlying noise process.