arxiv: 2604.00472 · v3 · submitted 2026-04-01 · 💱 q-fin.CP

Recognition: 2 theorem links

· Lean Theorem

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

Haoqi Lyu, Wenyuan Li

Pith reviewed 2026-05-13 22:23 UTC · model grok-4.3

classification 💱 q-fin.CP

keywords variable annuitiesrough Heston modelVolterra mortalityoptimal stoppingsignature methodsLeast Squares Monte Carloearly surrenderpath-dependent pricing

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The pith

A deep signature Monte Carlo method prices variable annuities with surrender options under rough Heston equity and Volterra mortality dynamics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a numerical method to price variable annuity contracts that include early surrender options when the underlying equity follows a rough Heston model and the force of mortality follows a Volterra process. These models are non-Markovian, so the optimal surrender decision depends on the full history of paths, which standard methods cannot handle. The approach encodes path histories with truncated rough-path signatures and uses a neural network within a Least Squares Monte Carlo framework to approximate continuation values. Numerical results show that the fair fee charged for the contract rises as the Hurst parameters for volatility and mortality increase. A proof of convergence for the method is also given.

Core claim

We develop a deep signature Least Squares Monte Carlo approach to learn optimal surrender strategies on a discretized time grid for variable annuities under the rough Heston equity model and Volterra mortality model. The fair fee increases with the Hurst parameters of both the stock volatility and the force of mortality. A convergence proof supports the stability of the method.

What carries the argument

Deep signature Least Squares Monte Carlo using truncated rough-path signatures to encode historical paths and a neural network to approximate continuation values in the optimal stopping problem.

If this is right

The method enables pricing of path-dependent insurance products under non-Markovian dynamics where traditional dynamic programming fails.
Fair fees for variable annuities with minimum guarantees rise with higher Hurst parameters in both volatility and mortality.
The approach extends in principle to other contracts that combine investment guarantees with surrender or withdrawal features.
Convergence guarantees ensure the computed values become reliable as the discretization is refined.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The signature encoding could be tested on other rough-volatility insurance products such as equity-linked life insurance with path-dependent benefits.
Sensitivity of fees to Hurst parameters suggests that memory effects in mortality data may materially affect contract design.
The framework might be combined with real-world calibration of rough paths to assess model risk in annuity portfolios.

Load-bearing premise

Truncated rough-path signatures combined with a neural network can accurately approximate the continuation values needed for optimal surrender decisions under the non-Markovian dynamics.

What would settle it

Numerical experiments in which the approximated fair fees fail to stabilize or converge as the signature truncation level rises or the time grid is refined would undermine the method.

Figures

Figures reproduced from arXiv: 2604.00472 by Haoqi Lyu, Wenyuan Li.

**Figure 2.** Figure 2: Observed survival probabilities and model survival proba [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Neural network with structure “41 − 64 − 64 − 64 − 64 − 1”. the highest VA price on the test set [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

**Figure 4.** Figure 4: Sample survival probabilities with different [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗

**Figure 5.** Figure 5: Kernel function with different Hurst parameter [PITH_FULL_IMAGE:figures/full_fig_p033_5.png] view at source ↗

**Figure 6.** Figure 6: 3D surrender strategy for the baseline case. [PITH_FULL_IMAGE:figures/full_fig_p033_6.png] view at source ↗

**Figure 7.** Figure 7: Surrender ratio of simulated paths over the years [PITH_FULL_IMAGE:figures/full_fig_p034_7.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a concrete deep-signature LSMC scheme for surrender valuation in variable annuities under simultaneous rough Heston equity and Volterra mortality, with a reported numerical increase in fair fee as Hurst rises.

read the letter

The main thing to know is that the authors price variable annuities with maturity and death guarantees plus an early surrender option when the account value is driven by rough Heston volatility and a Volterra mortality process. They replace standard regression with truncated rough-path signatures fed into a neural network inside a least-squares Monte Carlo loop on a discrete time grid, and they report that the fair fee rises with both Hurst parameters. A convergence argument is stated to back the method. This combination of models and the specific numerical observation on Hurst dependence are not in the prior literature they cite, so the algorithmic contribution is real. The approach is practical for the non-Markovian optimal stopping problem and avoids obvious circularity because the fee dependence comes from forward simulation rather than from any fitted parameter inside the derivation. The soft spot is the truncation order and the regularity assumptions on the Volterra kernel: the abstract claims a proof, but without the full error bounds it is hard to see whether the signature truncation remains accurate at the lower Hurst values they test. If long-memory increments are lost, the observed fee increase could contain approximation bias. The neural-network architecture and training choices are also free parameters whose effect on the continuation-value estimates is not quantified. This work is for quantitative analysts and researchers who already work with rough or fractional models in insurance pricing and want a workable numerical tool rather than a closed-form result. It is coherent on its own terms and shows honest engagement with the path-dependence issue, so it deserves referee time even if the proof and numerics will need tightening. I would send it to peer review.

Referee Report

2 major / 1 minor

Summary. This paper develops a deep signature Least Squares Monte Carlo method to value variable annuities with early surrender options under a rough Heston model for the equity index and a Volterra stochastic process for the force of mortality. Truncated rough-path signatures encode path history to approximate continuation values via neural networks on a discrete time grid. Numerical experiments report that the fair fee increases with the Hurst parameters of both processes, and a convergence proof is provided to support the method's stability.

Significance. If the numerical trends and approximation quality hold, the work provides a scalable approach for optimal stopping problems in non-Markovian insurance models by combining rough path signatures with neural regression. The reported dependence of fair fees on Hurst parameters offers insight into model risk for path-dependent guarantees, and the inclusion of a convergence proof strengthens the methodological contribution over purely empirical studies.

major comments (2)

[Convergence proof] Convergence proof: although the abstract states that a convergence proof is provided, the hypotheses on signature truncation order and Volterra kernel regularity are not shown to hold uniformly across the Hurst values explored numerically (particularly low H < 0.5); this is load-bearing for the claim that the observed fair-fee increase reflects genuine model dynamics rather than truncation bias.
[Numerical experiments] Numerical results section: the reported increase in fair fee with Hurst parameters rests on forward simulation without accompanying error bounds or sensitivity analysis on the neural network approximation of continuation values; if the truncated signatures discard relevant long-memory increments, the trend could be an artifact of the specific truncation level chosen.

minor comments (1)

[Abstract] The abstract and introduction could more explicitly reference the theorem number and key assumptions of the convergence result rather than stating only that 'a convergence proof is provided'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive suggestions. Below we address each major comment in turn, outlining how we will revise the manuscript to resolve the identified issues.

read point-by-point responses

Referee: Convergence proof: although the abstract states that a convergence proof is provided, the hypotheses on signature truncation order and Volterra kernel regularity are not shown to hold uniformly across the Hurst values explored numerically (particularly low H < 0.5); this is load-bearing for the claim that the observed fair-fee increase reflects genuine model dynamics rather than truncation bias.

Authors: We acknowledge that while the convergence proof is provided in the appendix, the verification of its hypotheses for the specific range of Hurst parameters used in the numerics (including values below 0.5) was not explicitly demonstrated. The proof relies on the truncation order being sufficiently large relative to the path regularity, which decreases with lower H. In the revised manuscript, we will include an additional analysis or remark confirming that the truncation levels employed satisfy the necessary conditions uniformly across the tested Hurst values, thereby supporting that the fair fee trend is not an artifact of truncation bias. revision: yes
Referee: Numerical results section: the reported increase in fair fee with Hurst parameters rests on forward simulation without accompanying error bounds or sensitivity analysis on the neural network approximation of continuation values; if the truncated signatures discard relevant long-memory increments, the trend could be an artifact of the specific truncation level chosen.

Authors: We agree with the referee that the numerical evidence would be more convincing with error bounds and sensitivity checks. The current experiments use a specific truncation level and neural network setup without reporting approximation errors or varying these parameters. We will revise the numerical experiments section to include a sensitivity analysis with respect to the signature truncation order and provide Monte Carlo error estimates for the computed fair fees. This will help confirm the robustness of the observed increase in fair fees as the Hurst parameters vary. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained with no circular reductions

full rationale

The paper develops a deep signature Least Squares Monte Carlo approach to solve the optimal stopping problem for variable annuities under rough Heston and Volterra mortality models. The reported increase in fair fee with Hurst parameters is obtained via numerical simulation rather than being tautological with any model input or fitted parameter. The convergence proof is provided to support the method without evidence of reducing to self-definition or self-citation chains. The approach uses truncated signatures to encode paths and neural networks for approximation, which is an algorithmic construction independent of the target results.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard assumptions from rough-path theory and universal approximation properties of neural networks; no new entities are postulated and the only free parameters are standard neural-network hyperparameters chosen during training.

free parameters (1)

neural network architecture and training hyperparameters
Depth, width, learning rate, and truncation level of the signature are chosen to make the continuation-value approximation work.

axioms (1)

domain assumption Truncated signatures plus feed-forward networks can approximate the continuation value function to arbitrary accuracy for the given path-dependent optimal stopping problem
Invoked to justify replacing the true dynamic programming operator with the learned network.

pith-pipeline@v0.9.0 · 5484 in / 1197 out tokens · 38096 ms · 2026-05-13T22:23:58.908946+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

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