Optimal hedging with basis risk under mean–variance criterion

Basis risk occurs naturally in a number of financial and insurance risk management problems. A notable example is in the context of hedging a derivative where the underlying security is either non-tradable or not sufficiently liquid. Other examples include hedging longevity risk using index-based longevity instrument and hedging crop yields using weather derivatives. These applications give rise to basis risk and it is imperative that such a risk needs to be taken into consideration for the adopted hedging strategy. In this paper, we consider the problem of hedging a European option using another correlated and liquidly traded asset and we investigate an optimal construction of hedging portfolio involving such an asset. The mean–variance criterion is adopted to evaluate the hedging performance, and a subgame Nash equilibrium is used to define the optimal solution. The problem is solved by resorting to a dynamic programming procedure and a change-of-measure technique. A closed-form optimal control process is obtained under a diffusion model setup. The solution we obtain is highly tractable and to the best of our knowledge, this is the first time the analytical solution exists for dynamic hedging of general European options with basis risk under the mean–variance criterion. Examples on hedging European call options are presented to foster the feasibility and importance of our optimal hedging strategy in the presence of basis risk.