Pricing Bounds and Bang-Bang Analysis of the Polaris Variable Annuities

44 Pages Posted: 25 Oct 2017 Last revised: 15 Jul 2019

See all articles by Zhiyi Shen

Zhiyi Shen

University of Waterloo

Chengguo Weng

University of Waterloo

Date Written: August 6, 2018


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 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, preserving convexity and monotonicity of the continuation value, reducing the computational cost of the tuning parameter selection, and evading extrapolating 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.

Keywords: Variable Annuities; Bang-bang Solution; Least Squares Monte Carlo; Sieve Estimation

JEL Classification: G22

Suggested Citation

Shen, Zhiyi and Weng, Chengguo, Pricing Bounds and Bang-Bang Analysis of the Polaris Variable Annuities (August 6, 2018). Available at SSRN: or

Zhiyi Shen (Contact Author)

University of Waterloo ( email )

200 University Avenue West
Waterloo, Ontario N2L 3G1

Chengguo Weng

University of Waterloo ( email )

M3-200 Univ Ave W
Waterloo, Ontario N2L3G1
(1)888-4567 ext.31132 (Phone)

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