The minimum-variance hedge ratio

Question

You are long an asset S and want to hedge it by shorting a related but imperfect instrument F. Their return volatilities are \sigmaS = 20\% and \sigmaF = 25\%, with correlation \rho = 0.8. How many units of F do you short per unit of S to minimize risk, and what is your residual volatility?

Accepted Answer

Hold one unit of S and short h units of F. The portfolio return change is \Delta S - h\,\Delta F, with variance 	ext{Var}(h) = \sigmaS^2 - 2h\,ho\,\sigmaS\sigmaF + h^2\sigmaF^2. Minimize over h: \frac{d\,	ext{Var}}{dh} = -2ho\sigmaS\sigmaF + 2h\sigmaF^2 = 0 \implies \boxed{h^ = ho\,\frac{\sigmaS}{\sigmaF} = \frac{	ext{Cov}(S,F)}{	ext{Var}(F)}.} This is exactly the regression beta of S on F, the slope you'd get by regressing the asset's moves on the hedge instrument's moves. Plug in: h^ = 0.8 \cdot \frac{0.20}{0.25} = 0.8 \cdot 0.8 = 0.64. Short 0.64 units of F per unit of S, not 1.0…

The minimum-variance hedge ratio

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