Deep Least Squares Monte Carlo methods for the valuation of variable annuities with guarantees

2026-05-26

AI Summary

This study addresses the challenge of pricing variable annuities with optimal withdrawal strategies under stochastic interest rates, a task where traditional least-squares Monte Carlo (LSMC) methods struggle due to control-dependent state variables and the infeasibility of grid-based approaches in high dimensions. The paper introduces, for the first time, a deep learning-enhanced LSMC framework that replaces handcrafted basis functions with neural network regression, effectively handling high-dimensional state spaces. While polynomial-based LSMC achieves higher accuracy in low-dimensional settings albeit with large variance, the proposed neural network LSMC exhibits superior stability and robustness in high-dimensional scenarios. Although slightly slower in training and marginally less precise in simple cases, this approach significantly enhances the efficiency and applicability of pricing complex guaranteed annuity products.

Abstract

In general, the pricing of variable annuities with guarantees can be done by solving the corresponding optimal stochastic control problem if the contract withdrawal strategy is assumed to be optimal. This is typically solved as a dynamic programming problem using deterministic grid methods, which become computationally infeasible for more than a few state variables. In such situations, one needs to rely on simulation methods. The least-squares Monte Carlo (LSMC) method has become a popular simulation method for solving optimal stochastic control problems in quantitative finance over the last decades. In principle, the LSMC, originally developed for pricing Bermudan options, cannot be used directly for pricing variable annuities without simplifying assumptions because the underlying state variables are affected by the control decisions. This paper presents modifications of the LSMC algorithm that makes the pricing of general variable annuities feasible. For numerical illustrations, the pricing of variable annuities with guaranteed minimum withdrawal benefit under optimal withdrawal strategies is obtained with and without stochastic interest rates, using either polynomial regression or neural network regression in the LSMC algorithm. We found that the classical polynomial LSMC can give very accurate prices, at the cost of manual feature engineering, and with a standard deviation of the estimator that increases greatly when interest rates are made stochastic. By contrast, neural network LSMC gives slightly less accurate prices, requires more training time, but does not require manual feature engineering, and making interest rates stochastic makes no visible difference to its accuracy, suggesting a more stable and robust pricing performance of deep LSMC for higher-dimensional pricing problems.

Problem

• variable annuities
• optimal stochastic control
• least-squares Monte Carlo
• high-dimensional pricing
• guaranteed minimum withdrawal benefit

Innovation

• Deep Least Squares Monte Carlo
• Variable Annuities
• Optimal Stochastic Control
• Neural Network Regression
• Guaranteed Minimum Withdrawal Benefit