Optimal Valuation of Variable Annuity Guaranteed LifetimeWithdrawal Benefits with Surrender Option

This paper extends the earlier works on the valuation of variable annuity guaranteed lifetime withdrawal benefits (VAGLWB) by introducing a surrender option to the contract. This new rider gives the policyholder an option to withdraw from the contract prior to his/her death subject to paying a penalty fee proportional to the current account value. The option is of American typed which can be exercised at any time during the duration of the contract. The valuation is formulated in terms of an optimal stopping problem of finding an exercise time of the option and the optimal account level at which the monetary value of the contract is maximized. The optimal solution to the stopping problem is derived under geometric Brownian motion dynamics of the equity price, the underlying investment vehicle of VAGLWB. The optimal value function is given explicitly in terms of the confluent hypergeometric function satisfying both continuous and smooth pasting conditions. Majorant and (super) harmonic properties of the value function are established to exhibit the optimality of the solution. In the absence of surrender option, the results reduce to that of Feng and Jing (2017). Since withdrawal from insurance contract is a natural feature/option in insurance, without considering the surrender option in the valuation would result in undervaluation of the insurance premium for VAGLWB.