Abstract: In this talk, I will first discuss the no-arbitrage pricing of Polaris variable annuities (VAs), which were issued by the American International Group in recent years. Variable annuities are prevailing equity-linked insurance products that provide the policyholder with the flexibility of dynamic withdrawals, mortality protection, and guaranteed income payments against a market decline. The Polaris allows a shadow account to lock in the high watermark of the investment account over a monitoring period that depends on the policyholder’s choice of his/her first withdrawal time. This feature makes the insurer’s payouts depend on policyholder’s withdrawal behaviours and significantly complicates the pricing problem. By prudently introducing certain auxiliary state variables, we manage to formulate the pricing problem into solving a convoluted stochastic optimal control framework and developing a computationally efficient algorithm to approach the solution. Driven by the challenges from the pricing Polaris VAs, in the second part of the talk, I will introduce a regression-based Monte Carlo algorithm, which we propose to solve a class of general stochastic optimal control problems numerically. The algorithm has three pillars: a construction of auxiliary stochastic control model, an artificial simulation of the post-action value of state process, and a shape-preserving sieve estimation method. The algorithm enjoys many merits, including obviating forward simulation and control randomization, eliminating in-sample bias, evading extrapolating the value function, and alleviating the computational burden of the tuning parameter selection. This talk is based on two joint works with Chengguo Weng from the University of Waterloo.