Variable Annuities with High Water Mark Withdrawal Benefit

In this paper, we develop a continuous-time model for variable annuities allowing for periodic withdrawals proportional to the high water mark of the underlying account value as well as early surrender of the policy. We derive a HJB variational inequality characterizing the minimal superhedging price of such a contract and the worst-case policyholder behavior from an issuer's perspective. Based on these results, we construct a dynamic trading strategy which superreplicates the contract. In addition, we show how early surrender has to be penalized to disincentivize a worst-case policyholder from using this option. To treat the problem numerically, we develop a semi-Lagrangian scheme based on a discretization of the underlying noise process