# Mean-variance hedging of unit linked life insurance contracts in a jump-diffusion model

## TL;DR

This paper develops a mean-variance hedging framework for unit linked life insurance in a jump-diffusion setting, deriving explicit strategies through a system of PIDEs and demonstrating robustness of payoffs via simulations.

## Contribution

It introduces a novel equilibrium approach using extended HJB equations for optimal hedging in jump-diffusion models with mortality risk, providing explicit solutions and robustness analysis.

## Key findings

- Explicit closed-form optimal strategies derived.
- Strategies are robust to jump size distribution.
- Variance is unaffected by insurance horizon or longevity asset maturity.

## Abstract

We consider a time-consistent mean-variance portfolio selection problem of an insurer and allow for the incorporation of basis (mortality) risk. The optimal solution is identified with a Nash subgame perfect equilibrium. We characterize an optimal strategy as solution of a system of partial integro-differential equations (PIDEs), a so called extended Hamilton-Jacobi-Bellman (HJB) system. We prove that the equilibrium is necessarily a solution of the extended HJB system. Under certain conditions we obtain an explicit solution to the extended HJB system and provide the optimal trading strategies in closed-form. A simulation shows that the previously found strategies yield payoffs whose expectations and variances are robust regarding the distribution of jump sizes of the stock. The same phenomenon is observed when the variance is correctly estimated, but erroneously ascribed to the diffusion components solely. Further, we show that differences in the insurance horizon and the time to maturity of a longevity asset do not add to the variance of the terminal wealth.