An investor commits to sell 1 unit of asset on date T and would like to hedge it using futures position of size h at time T. Assume the spot prices today and at time T are S0 and ST , respectively. The futures prices today and at time T are F0 and FT , respectively.

(a) Prove that the variance of net amount received by the investor is σ 2 (∆S)+ h 2σ 2 (∆F)−2hcov(∆S, ∆F). σ 2 (∆S) is the variance of spot price change, σ 2 (∆F) is the variance of futures price change, and cov(∆S, ∆F) is the covariance between spot and futures price change.

(b) Prove that the futures position to minimum the variance of net amount received by the investor is h ∗ = cov(∆S,∆F) σ2(∆F) = ρ σ(∆S) σ(∆F) , where ρ is the correlation coefficient between ∆S and ∆F. Hint: to minimize an objective function, take the first order partial derivatives and let it equal to zero.