Optimal Hedge Ratio

The Problem

A portfolio manager holds $N_S$ shares of stock $S$ and wants to hedge using $N_F$ futures contracts on a related asset $F$. Let ${\sigma }_S$ = standard deviation of $\Delta S$, ${\sigma }_F$ = standard deviation of $\Delta F$, and $\rho$ = correlation between $\Delta S$ and $\Delta F$.

1. Derive the minimum-variance hedge ratio $h^{\ast }=N_F/N_S$ that minimizes the variance of the hedged portfolio.
2. Compute $h^{\ast }$ when ${\sigma }_S=0.30,{\sigma }_F=0.25,\rho =0.85$.
3. Compute the hedge effectiveness — the percentage reduction in variance vs. the unhedged position.