Mean-variance Hedging with Basis Risk

Basis risk arises in a number of financial and insurance risk management problems when the hedging assets do not perfectly match the underlying asset in a hedging program. Notable examples in insurance include the hedging for longevity risks, weather index based insurance products, variable annuities, etc. In the presence of basis risk, a perfect hedging is impossible, and in this paper, we adopt a mean-variance criterion to strike a balance between the expected hedging error and its variability. Under a time-dependent diffusion model setup, explicit optimal solutions are derived for hedging target being either a European option or a forward contract. The solutions are obtained by a delicate application of linear quadratic control theory, method of backward stochastic differential equation, and Malliavin calculus. A numerical example is presented to illustrate our theoretical results and their interesting implications