Joint-Survival Annuity Derivative Valuation in the Linear-Rational Wishart Mortality Model

Abstract

This paper develops a linear-rational mortality model for couples based on a Wishart state process. The matrix-valued state variable permits a flexible dependence structure between the two mortality intensities while preserving positivity. Within this framework, we derive closed-form expressions for the joint-survival bond and tractable valuation formulas for the joint-survival annuity and the guaranteed joint-survival annuity option. We also characterize the mortality-intensity distribution and retain fast approximation methods for the guaranteed annuity option. To complement the theory, we implement the model on the Canadian dependent-lives portfolio. We benchmark the Wishart model against independence and a Frank copula, and we report the implications for joint-life pricing as well as guaranteed annuity option valuation. The estimation identifies a positive dependence structure consistent with the literature. Moreover, unlike static copula alternatives, the estimated Wishart model feeds directly into the closed-form guaranteed annuity option pricing and approximation formulas developed in the paper, thereby illustrating how the linear-rational Wishart framework can be taken from analytical derivations to a practical dependent-lives implementation.