Pricing bounds and bang-bang analysis of the Polaris variable annuities

This paper studies the no-arbitrage pricing of the ‘Polaris Income Plus Daily’ structured in the ‘Polaris Choice IV’ variable annuities recently issued by the American International Group. Distinct from the withdrawal benefits studied in the literature, Polaris allows the income base to ‘lock in’ the high water mark of the investment account over a certain monitoring period which is related to the timing of the policyholder's first withdrawal. By prudently introducing certain auxiliary state and decision variables, we manage to formulate the pricing model under a Markovian stochastic optimal control framework. By a slight modification of the fee structure, we show the existence of a bang-bang solution to the stochastic control problem: the optimal withdrawal strategy is among a few explicit choices. We consequently design a novel Least Squares Monte Carlo (LSMC) algorithm to approach the optimal solution. Convergence results are established for the algorithm by applying the theory of nonparametric sieve estimation. Compared with existing LSMCs, our algorithm possesses a number of advantages such as memory reduction, preservation of convexity and monotonicity of the continuation value, reducing computational cost of the tuning parameter selection, and evading extrapolation of the value function estimate. Finally, we prove that the obtained pricing result works as an upper bound of the no-arbitrage price of Polaris with the real fee structure. Numerical experiments show that this upper bound is fairly tight.