Optimal Initiation of Guaranteed Lifelong Withdrawal Benefit with Dynamic Withdrawals

We consider pricing a Guaranteed Lifelong Withdrawal Benefit (GLWB) that consists of the early phase of accumulation of benefit base and the later income phase of annuities. The most recent form of the GLWB provides flexibility in allowing additional purchases in the accumulation phase, dynamic withdrawals in the income phase, dynamic initiation into the income phase, and complete surrender right throughout the life of the contract. The policyholder chooses the initiation of the income phase optimally based on a combination of factors, like the age-dependent scheduled withdrawal rates, penalty charge rate, and bonus and ratchet provisions. Using the bang-bang control analysis, we show that the strategy space of the optimal policies is limited to four choices: maximum allowable purchase, zero withdrawal, withdrawal at the contractual amount, or complete surrender. We construct the Fourier transform algorithm for effective pricing of GLWB products with policy fund value under the general two-dimensional Markov process of the fund value and its variance. Our pricing model includes complex path dependent features arising from the ratchet and bonus provisions, dynamic control of withdrawals and additional purchases, together with optimality in the time of initiation of the income phase. We also analyze various pricing properties of the GLWB based on the effective and accurate Fourier transform algorithms. In particular, we examine the impact of various contractual specifications of the GLWB on the optimal decision of initiation of the income phase and optimal withdrawal strategies.