Optimal Reinsurance and Investment Strategies with Option Trading Under Loss-Dependent Premium Principle

This paper investigates the optimal delayed proportional reinsurance and investment strategies involving option trading for both an insurer and a reinsurer under a loss-dependent premium principle. The extrapolation bias method is employed to measure the dependency between future claims and historical claims, enabling dynamic premium adjustments. In addition to a risk-free asset, both parties invest in a stock modeled by the CEV (Constant Elasticity of Variance) process and a European call option written on the stock. The insurer and reinsurer engage in insurance businesses subject to common shock dependence, while jointly optimizing their interests under model uncertainty. The objective is to maximize the weighted sum of their terminal wealth’s expected exponential utility. Using stochastic control theory, an extended Hamilton-Jacobi-Bellman (HJB) equation is derived and solved, yielding closed-form solutions for the robust optimal reinsurance-investment strategies and the corresponding value functions. Finally, numerical analysis illustrates the impact of key model parameters on the optimal strategies, accompanied by economic interpretations.