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

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.

💡 Research Summary

This paper tackles the valuation of variable annuity (VA) contracts that include an early surrender option, under a joint non‑Markovian framework that captures both rough stochastic volatility in the equity market and long‑range dependent mortality intensity. The equity index follows a rough Heston model, where the volatility process is driven by a fractional kernel with Hurst parameter (H_{\sigma}\in(0,1)). The force of mortality is modeled by a Volterra‑type stochastic integral equation (SVIE) with kernel (K_m(t)=t^{H_m-1/2}/\Gamma(H_m+1/2)), introducing a second Hurst parameter (H_m) that governs the memory of mortality. Both processes are calibrated to market and demographic data; the mortality calibration on U.S. male life‑table data yields (H_m\approx0.704), indicating a smooth, long‑range dependent mortality path.

The presence of an early surrender feature turns the valuation into an optimal stopping problem. Because the continuation value depends on the whole past trajectory of the two non‑Markovian processes, classical dynamic programming, PDE methods, or standard Least‑Squares Monte Carlo (LSMC) are infeasible. The authors therefore propose a “deep signature LSMC” algorithm. At each discretized exercise date they compute the truncated path signature of the joint equity‑mortality trajectory, thereby encoding the infinite‑dimensional history into a finite‑dimensional vector. This vector is fed into a deep neural network (a multilayer perceptron with ReLU activations and batch normalization) that learns to approximate the continuation value. The network is trained on a large set of Monte‑Carlo simulated paths; the regression target at each time step is the discounted future cash‑flows conditional on continuation. After training, the optimal surrender decision at each node is obtained by comparing the immediate surrender payoff with the network‑predicted continuation value. The fair fee is then determined by a bisection search that forces the contract’s net present value to zero.

A rigorous error analysis decomposes the total pricing error into four components: Monte‑Carlo sampling error (vanishing as (1/\sqrt{N_{\text{paths}}})), time‑discretization error (order (\Delta t)), signature truncation error (decaying polynomially with the truncation order), and neural‑network approximation error (which can be made arbitrarily small by increasing network capacity). The paper provides proofs that each term converges to zero as the corresponding discretization parameters are refined, establishing the theoretical stability of the method.

Numerical experiments explore the sensitivity of the fair fee to the Hurst parameters (H_{\sigma}) and (H_m). By varying each from 0.1 to 0.4 while keeping other parameters fixed, the authors find a monotone increase of the fair fee with both Hurst exponents. This reflects the intuition that rougher volatility (larger (H_{\sigma})) inflates the value of the early‑exercise feature, and that stronger long‑range dependence in mortality (larger (H_m)) raises the cost of the guaranteed minimum death benefit. The optimal surrender strategy is visualized in a three‑dimensional space of time, account value, and volatility; unlike the low‑dimensional free‑boundary in Markovian settings, the surrender region exhibits a complex surface that depends on the full path history. Sample paths are plotted with the learned surrender times, illustrating how the algorithm captures path‑dependent behavior.

The paper’s contributions are fourfold: (1) a novel valuation framework that integrates rough equity dynamics and Volterra mortality, (2) a deep‑learning‑enhanced LSMC algorithm that uses truncated signatures to overcome the curse of dimensionality, (3) a rigorous convergence analysis that quantifies all sources of numerical error, and (4) an extensive empirical study showing how roughness and long‑range dependence affect fair fees and surrender behavior. By providing both a practical computational tool and a solid theoretical foundation, the work advances the state of the art in actuarial pricing of variable annuities under realistic, memory‑rich risk factors.