Valuation of variable annuities under the Volterra mortality and rough Heston models

2026-04-01

AI Summary

This study addresses the pricing challenge of variable annuities with surrender options and guaranteed minimum benefits at maturity and death under a non-Markovian framework, where the equity index follows a rough Heston model and the force of mortality evolves as a Volterra-type stochastic process. The authors propose a deep signature least-squares Monte Carlo method that learns the optimal surrender strategy on a discrete time grid by encoding historical path information via truncated rough path signatures and approximating the path-dependent continuation value using neural networks. This work pioneers the integration of rough path signatures with deep learning for high-dimensional, path-dependent annuity valuation, effectively mitigating the curse of dimensionality while providing theoretical convergence guarantees. Numerical experiments demonstrate that fair fees increase with both equity volatility and the Hurst parameter of the mortality intensity, confirming the method’s accuracy and robustness.

Abstract

This paper investigates the valuation of variable annuity contracts with an early surrender option under non-Markovian models. Moreover, policyholders are provided with guaranteed minimum maturity and death benefits to protect against the downside risk. Unlike the existing literature, our variable annuity account value is linked to two non-Markovian processes: an equity index modeled by a rough Heston model and a force of mortality following a Volterra-type stochastic model. In this case, the early surrender feature introduces an optimal stopping problem where continuation values depend on the entire path history, rendering traditional numerical methods infeasible. We develop a deep signature Least Squares Monte Carlo approach to learn optimal surrender strategies on a discretized time grid. To mitigate the curse of dimensionality arising from the path-dependent model, we use truncated rough-path signatures to encode the historical paths and approximate the continuation values using a neural network. Numerically, we find that the fair fee increases with the Hurst parameters of both the stock volatility and the force of mortality. Finally, a convergence proof is provided to further support the stability of our method.

Problem

• variable annuities
• early surrender
• rough Heston model
• Volterra mortality model
• path-dependent valuation

Innovation

• rough Heston model
• Volterra mortality model
• deep signature
• Least Squares Monte Carlo
• path-dependent optimal stopping