Joint survival annuity derivative valuation in the linear-rational Wishart mortality model

This study proposes a linear-rational joint survival mortality model based on the Wishart process. The Wishart process, which is a stochastic continuous matrix affine process, allows for a general dependency between the mortality intensities that are constructed to be positive. Using the linear-rational framework along with the Wishart process as state variable, we derive a closed-form expression for the joint survival annuity, as well as the guaranteed joint survival annuity option. Exploiting our parameterisation of the Wishart process, we explicit the distribution of the mortality intensities and their dependency. We provide the distribution (density and cumulative distribution) of the joint survival annuity. We also develop some polynomial expansions for the underlying state variable that lead to fast and accurate approximations for the guaranteed joint survival annuity option. These polynomial expansions also significantly simplify the implementation of the model. Overall, the linear-rational Wishart mortality model provides a flexible and unified framework for modelling and managing joint mortality risk.

💡 Research Summary

The paper introduces a novel joint‑mortality framework that combines the linear‑rational (LR) pricing approach with a Wishart matrix‑valued stochastic process as the state variable. Traditional multi‑life mortality models rely on exponential‑affine (EA) specifications, such as the Duffie‑Kan vector‑affine process, which suffer from two major drawbacks: (i) the diffusion matrix must satisfy restrictive constraints that limit the admissible correlation structures between mortality intensities, and (ii) the joint survival bond, being an exponential function of the state, leads to a sum of exponentials when pricing a guaranteed joint survival annuity option (GAO). The distribution of this sum is unknown, preventing a closed‑form option price.

To overcome these issues, the authors adopt the Wishart process, a continuous‑time affine process taking values in the cone of positive‑definite matrices. The Wishart process naturally captures a rich dependence structure among several mortality intensities because each intensity is expressed as a linear functional of the matrix. Moreover, the Wishart moment‑generating function is available in closed form via a matrix Riccati differential equation that can be solved analytically, making the process as tractable as the scalar CIR model.

Within the LR framework, the authors model the discounted joint survival zero‑coupon bond as a linear‑rational function of the Wishart state:

 SB(t,T)=P(t,T)·E_Q