[Paper Review] Volterra mortality model: Actuarial valuation and risk management with long-range dependence

Ling Wang, Mei Choi Chiu+1 more|arXiv (Cornell University)|Sep 20, 2020

Insurance, Mortality, Demography, Risk Management30 references4 citations

TL;DR

This paper proposes a novel class of Volterra mortality models that incorporate long-range dependence (LRD) into actuarial valuation while preserving analytical tractability. By leveraging affine Volterra processes, the model enables closed-form solutions for survival probabilities and mortality-linked products, and derives optimal mean-variance hedging strategies under LRD, offering the first tractable framework for LRD-affected longevity risk management.

ABSTRACT

While abundant empirical studies support the long-range dependence (LRD) of mortality rates, the corresponding impact on mortality securities are largely unknown due to the lack of appropriate tractable models for valuation and risk management purposes. We propose a novel class of Volterra mortality models that incorporate LRD into the actuarial valuation, retain tractability, and are consistent with the existing continuous-time affine mortality models. We derive the survival probability in closed-form solution by taking into account of the historical health records. The flexibility and tractability of the models make them useful in valuing mortality-related products such as death benefits, annuities, longevity bonds, and many others, as well as offering optimal mean-variance mortality hedging rules. Numerical studies are conducted to examine the effect of incorporating LRD into mortality rates on various insurance products and hedging efficiency.

Motivation & Objective

• To address the lack of analytically tractable stochastic mortality models that incorporate long-range dependence (LRD) in actuarial valuation and risk management.
• To develop a dynamic mortality model that preserves the affine structure for tractable pricing while capturing LRD in mortality rates.
• To derive closed-form solutions for survival probabilities and mortality-linked products such as death benefits, annuities, and longevity bonds under LRD.
• To formulate and solve a mean-variance hedging problem for longevity risk using a non-Markovian, non-semimartingale Volterra process.
• To empirically assess the impact of LRD on product valuations and hedging efficiency through numerical studies.

Proposed method

• The model is built on affine Volterra processes, allowing it to inherit the analytical tractability of affine models while incorporating LRD through long-memory kernels.
• The survival probability is derived in closed form using the conditional expectation of the integrated intensity process under the filtration generated by historical health records.
• The model extends the doubly stochastic mortality framework by embedding a Volterra process with a long-memory kernel, ensuring the mortality rate exhibits LRD.
• Mean-variance hedging is formulated using a backward stochastic differential equation (BSDE) framework, enabling optimal hedging strategies despite the non-Markovian and non-semimartingale nature of the process.
• The optimal hedging rule is derived via linear-quadratic control, with the solution expressed in terms of Riccati-type equations and stochastic integrals involving the volatility and correlation structure.
• Numerical validation is conducted by comparing the Volterra model with Markovian alternatives in pricing and hedging performance under varying LRD parameters.

Experimental results

Research questions

• RQ1 Can a tractable stochastic mortality model incorporate long-range dependence (LRD) while maintaining analytical solvability for actuarial valuation?
• RQ2 How does the inclusion of LRD affect the pricing of mortality-linked financial products such as longevity bonds and death benefits?
• RQ3 What is the optimal mean-variance hedging strategy for longevity risk when mortality rates exhibit LRD?
• RQ4 How does the performance of LRD-affected hedging strategies compare to those based on Markovian models in terms of variance reduction and risk exposure?
• RQ5 What is the impact of LRD on life expectancy forecasts and funding adequacy in pension schemes?

Key findings

• The proposed Volterra mortality model yields a closed-form solution for the survival probability by taking into account historical health records, enabling exact actuarial valuation under LRD.
• The model provides the first set of analytical formulas for pricing death benefits, survival benefits, and longevity bonds under long-range dependent mortality rates.
• The mean-variance hedging strategy for longevity risk is derived explicitly under the Volterra framework, with the optimal hedge depending on the long-memory kernel and volatility structure.
• Numerical studies show that models ignoring LRD systematically underestimate life expectancy, leading to material mispricing in pension and insurance products.
• The optimal hedging rule derived under the Volterra model demonstrates superior risk reduction compared to Markovian benchmarks, especially in long-horizon and high-LRD regimes.
• The model’s tractability allows for efficient calibration and simulation, making it suitable for real-world implementation in risk management and product pricing.