Omega ratio optimization with actuarial and financial applications

The omega ratio is an interesting performance measure because it focuses on both downside losses and upside gains, and actuarial/financial instruments are reflecting more and more asymmetry and heavy tails. This paper focuses on the omega ratio optimization in general Banach spaces, which applies for both infinite-dimensional approaches and more classical ones. New Fritz John-like and Karush Kuhn Tucker-like optimality conditions and duality results will be provided, despite the fact that omega is neither differentiable nor convex. Then, the focus is on both portfolio selection and optimal reinsurance, classic problems in Financial Mathematics and Actuarial Mathematics, respectively. The new duality results apply in order to study the potential ill-posedness of the financial problem. It will be provided further evidence about the relationship between the ill-posedness of problems involving omega and the ill-posedness of alternative problems avoiding omega and only involving coherent risk measures. The new optimality conditions apply in order to characterize and solve the actuarial problem. The solution is often a “bang-bang reinsurance contract”, i.e., a contract saturating the constraints of the ceded risk sensitivity with respect to the global (ceded plus retained) risk. In this sense, omega may be “essentially similar” to other deviation/downside risk measures previously studied, though the use of omega will also provoke some modifications in the final optimal results.