In finance and insurance, we’ve long priced instruments like bonds and annuities based on the assumption that interest rates and mortality evolve fairly independently — and without memory. But COVID-19 shattered that illusion. Today, the joint dynamics of mortality and macroeconomics demand a rethink, and a recent paper by Zhou & Zhou offers exactly that.

Their proposal? A bivariate stochastic framework powered by mixed fractional Brownian motions (mfBm). This hybrid modeling approach brings together the short-term randomness of traditional Brownian motion with the long-memory characteristics of fractional processes — and applies it jointly to excess mortality and interest rates. It’s a bold move that challenges much of the status quo in actuarial finance.

Beyond Vasicek: Modeling the Memory of Crisis

At its core, the model extends the classic mean-reverting Vasicek process. But instead of assuming white noise, the authors use a mixture of Brownian and fractional Brownian motion:

$$ dr_t = (m_1 - \theta_1 r_t) dt + \sigma_1 (\alpha_1 dW_{1,t} + dB^{H_1}_{1,t}) $$
$$ d\mu_t = (m_2 - \theta_2 \mu_t) dt + \sigma_2 \left[ \alpha_2 (\rho dW_{1,t} + \sqrt{1 - \rho^2} dW_{2,t}) + dB^{H_2}_{2,t} \right] $$

Here, $r_t$ is the interest rate, $\mu_t$ is excess mortality, $W$ is Brownian motion, $B^H$ is fBm, and $\rho$ captures the instantaneous correlation between the short-term shocks to mortality and rates. Crucially, the Hurst parameters $H_1, H_2 > 0.5$ govern the long-range dependence — a persistent memory where past events echo far into the future.

Why Memory Matters: Implications for Pricing

Most actuarial models assume mortality or rates are either deterministic or Markovian. But empirical data suggests both show memory. For example:

Process Evidence of Memory Implication
Mortality Post-pandemic shifts persist Elevated tail risk in catastrophe bonds
Interest rates Structural regimes (e.g. low-rate era) Persistent deviation from mean forecasts

By embracing memory, the authors show that:

  • Zero-coupon bonds have higher fair prices when Hurst $H$ is large, due to more persistent rate paths.
  • Catastrophic mortality bonds (e.g. Swiss Re’s Vita VI) become more sensitive to tail mortality risk, especially when long-range dependence is strong in $\mu_t$.

This modeling shift isn’t cosmetic — it affects real-world pricing. For example, under their model calibrated to US weekly mortality (2015–2024) and 3-month T-bills, they derive a closed-form price for zero-coupon bonds and numerically price Vita VI, explaining its 3% coupon via calibrated mortality tail risk.

Calibrating the Unthinkable

The authors develop a two-stage calibration process:

  1. Under physical measure ($\mathbb{P}$): Estimate Hurst parameters, volatility, drift, and mean reversion using rescaled range analysis and least squares from observed mortality and rate data.
  2. Under risk-neutral measure ($\mathbb{Q}$): Calibrate market price of risk ($\gamma, \eta$) to match real-world catastrophe bond prices — including attachment and detachment point tuning.

This matters because post-pandemic, investors demand compensation not just for volatility, but for correlated persistence. The fair coupon increases notably as either the Hurst parameter or mortality volatility increases — as shown in their simulations.

Designing Bonds for a Long-Memory World

Sensitivity analysis in the paper shows:

  • Increasing mortality volatility dramatically raises expected loss and hence the fair coupon.
  • Raising the Hurst $H$ of $\mu_t$ also pushes the bond’s tail risk higher, requiring better pricing.
  • Longer-term bonds require more careful design of attachment (a) and detachment (b) points, as LRD amplifies cumulative effects.
Scenario Expected Loss Coupon Rate
Baseline 0.75% 5.41%
Double Mortality Volatility 6.47% 7.16%
No Long Memory (H = 0.5) 0.20% 5.14%

This means insurers and ILS investors should move away from static or short-memory models when designing mortality-linked securities. Incorporating mfBm not only improves realism but helps avoid underpricing the rare-but-extreme tail scenarios that pandemics make all too real.

Where Next?

This paper opens new directions:

  • Dynamic correlation structures: The current model uses constant $\rho$; future work could allow time-varying correlation responsive to systemic shocks.
  • Direct modeling of seasonal mortality: Instead of modeling excess mortality, integrating periodic components could improve sensitivity to flu or heat waves.
  • Index design: Exploring different mortality index structures (average vs max) can better match investor appetite.

In a world where pandemics and financial crises blur into one another, memory is not a bug — it’s a feature. The mixed fractional Brownian motion framework brings that memory to life, offering a path to more robust pricing, better risk transfer, and ultimately, stronger financial resilience.

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