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

Jose Da Fonseca, Patrick Wong

TL;DR

This paper tackles pricing of joint longevity products under dependent lifetimes by introducing a linear-rational mortality model built on the Wishart process and a Rogers-style state-price density to ensure positive mortality intensities. The approach yields closed-form expressions for the joint survival bond and, by extension, the joint survival annuity, with its distribution tractable through a one-dimensional integration of the Wishart MGF. It also delivers a closed-form GAO price that reduces to a one-dimensional integral and develops three fast, accurate approximations (Gaussian, spectral, gamma) based on cumulants, enhancing practical applicability. A key finding is that modeling dependence between annuitants materially affects option values, and the proposed framework provides efficient risk management tools (density, VaR, ES) and avenues for extensions to estimation, Greeks, and multi-population settings.

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.

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

TL;DR

This paper tackles pricing of joint longevity products under dependent lifetimes by introducing a linear-rational mortality model built on the Wishart process and a Rogers-style state-price density to ensure positive mortality intensities. The approach yields closed-form expressions for the joint survival bond and, by extension, the joint survival annuity, with its distribution tractable through a one-dimensional integration of the Wishart MGF. It also delivers a closed-form GAO price that reduces to a one-dimensional integral and develops three fast, accurate approximations (Gaussian, spectral, gamma) based on cumulants, enhancing practical applicability. A key finding is that modeling dependence between annuitants materially affects option values, and the proposed framework provides efficient risk management tools (density, VaR, ES) and avenues for extensions to estimation, Greeks, and multi-population settings.

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.
Paper Structure (11 sections, 16 theorems, 76 equations, 3 figures, 3 tables)

This paper contains 11 sections, 16 theorems, 76 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The modelling framework
  3. The linear-rational Wishart model
  4. Joint survival annuity valuation
  5. Guaranteed joint survival annuity option valuation
  6. Model implementation
  7. Sensitivity analyses
  8. Approximation methods
  9. Independence vs dependence
  10. Conclusion
  11. Figures and Tables

Key Result

Proposition 3.1

Suppose that $\omega$ in EQ:Wishart is such that $\omega=\beta \sigma^2$ with $\beta \in \mathbb{R}$ and $\beta \geq n+1$ then the moment generating function of $v_t$ for $\theta_1\in \mathbb{S}_n$ is given by with $\varsigma_t :=\int_0^te^{(t-s)m}\sigma^2 e^{(t-s)m^\top}ds$, which is given by $\textup{vec}(\varsigma_t)=\mathsf{A}^{-1}(e^{\mathsf{A}t}-I_{n^2})\textup{vec}(\sigma^2)$, $\vartheta_t

Figures (3)

  • Figure 1: Sensitivity analyses.
  • Figure 2: Comparison of approximation methods and errors.
  • Figure 3: Pricing difference for independence and dependence assumption.

Theorems & Definitions (30)

  • Proposition 3.1
  • Remark 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Remark 3.6
  • Proposition 4.1
  • Remark 4.2
  • Remark 4.3
  • Proposition 4.4
  • ...and 20 more