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

2026-02-06

AI Summary

This study addresses the efficient pricing of joint-life annuities and associated guaranteed options by proposing a linear rational joint mortality model based on the Wishart process. It is the first to incorporate the Wishart process into multi-life mortality modeling, leveraging its matrix-affine structure and the positive definiteness of state variables to explicitly capture the positive correlation among mortality intensities. The framework yields closed-form expressions for both annuity and option prices. By further integrating polynomial expansion techniques, the model achieves highly accurate approximations with low computational complexity. This work not only provides an explicit joint distribution of mortality intensities but also substantially enhances the pricing efficiency and practical applicability of complex longevity risk products.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Life-Long and Continual Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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.

Problem

• joint survival annuity
• mortality risk
• Wishart process
• derivative valuation
• joint mortality

Innovation

• Wishart process
• linear-rational framework
• joint survival annuity
• mortality intensity dependency
• polynomial expansion