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

TL;DR

This paper introduces a new linear-rational joint survival mortality model based on the Wishart process, providing explicit formulas and efficient approximations for valuing joint survival annuities and options, enhancing modeling flexibility and computational efficiency.

Contribution

It develops a novel linear-rational Wishart mortality model with explicit distribution formulas and polynomial approximations, improving joint mortality risk modeling and valuation methods.

Findings

Closed-form expression for joint survival annuity

Explicit distribution of mortality intensities and dependencies

Fast polynomial approximation for guaranteed joint survival options

Abstract

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.