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

TL;DR

This paper develops a novel deep learning-based method to value variable annuities with early surrender options under complex non-Markovian models involving rough Heston and Volterra processes.

Contribution

It introduces a deep signature Least Squares Monte Carlo approach using truncated signatures and neural networks to handle path-dependent valuation problems.

Findings

Fair fees increase with Hurst parameters of volatility and mortality.

The proposed method effectively approximates continuation values in non-Markovian models.

A convergence proof supports the method's stability.

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.