A Stackelberg reinsurance-investment game under $\alpha$-maxmin mean-variance criterion and stochastic volatility

TL;DR

This paper models a Stackelberg game between an insurer and reinsurer under an $oldsymbol{ extalpha}$-maxmin mean-variance criterion with stochastic volatility, deriving equilibrium strategies through advanced mathematical equations and analyzing effects of ambiguity and risk attitudes.

Contribution

It introduces a novel Stackelberg reinsurance-investment model under $oldsymbol{ extalpha}$-maxmin mean-variance preferences with stochastic volatility, providing explicit equilibrium strategies and numerical insights.

Findings

Excess-of-loss reinsurance is optimal for the insurer.

Equilibrium strategies are characterized by Riccati differential equations.

Ambiguity and risk attitudes significantly influence strategies and premiums.

Abstract

This paper investigates a Stackelberg game between an insurer and a reinsurer under the $\alpha$-maxmin mean-variance criterion. The insurer can purchase per-loss reinsurance from the reinsurer. With the insurer's feedback reinsurance strategy, the reinsurer optimizes the reinsurance premium in the Stackelberg game. The financial market consists of cash and stock with Heston's stochastic volatility. Both the insurer and reinsurer maximize their respective $\alpha$-maxmin mean-variance preferences in the market. The criterion is time-inconsistent and we derive the equilibrium strategies by the extended Hamilton-Jacobi-Bellman equations. Similar to the non-robust case in Li and Young (2022), excess-of-loss reinsurance is the optimal form of reinsurance strategy for the insurer. The equilibrium investment strategy is determined by a system of Riccati differential equations. Besides, the equations determining the equilibrium reinsurance strategy and reinsurance premium rate are given semi-explicitly, which is simplified to an algebraic equation in a specific example. Numerical examples illustrate that the game between the insurer and reinsurer makes the insurance more radical when the agents become more ambiguity aversion or risk aversion. Furthermore, the level of ambiguity, ambiguity attitude, and risk attitude of the insurer (reinsurer) have similar effects on the equilibrium reinsurance strategy, reinsurance premium, and investment strategy.