Risk & Reward

Optimal Hedge Ratio

You are long one share of A and want to short B to cancel its risk. How many shares of B minimize the variance of the hedged position?

solvedeasy1 min

You hold one share of stock A and want to hedge it by shorting hh shares of stock B. Stock A's return has variance σA2\sigma_A^2, stock B's has variance σB2\sigma_B^2, and their returns have correlation ρ\rho. How many shares of B should you short to minimize the variance of the hedged return rA−h rBr_A - h\,r_B?

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#The variance is a parabola in the hedge

Expanding the variance of the hedged return,

Var⁡(rA−h rB)=σA2−2ρ σAσB h+σB2 h2,(1)\Var(r_A - h\,r_B) = \sigma_A^2 - 2\rho\,\sigma_A \sigma_B\,h + \sigma_B^2\,h^2, \tag{1}

an upward parabola in hh since the coefficient σB2\sigma_B^2 is positive.

#Set the derivative to zero

Differentiating in hh and solving,

ddh Var⁡(rA−h rB)=−2ρ σAσB+2σB2 h=0⟹h∗=ρ σAσB.(2)\frac{d}{dh}\,\Var(r_A - h\,r_B) = -2\rho\,\sigma_A \sigma_B + 2\sigma_B^2\,h = 0 \quad\Longrightarrow\quad h^{*} = \rho\,\frac{\sigma_A}{\sigma_B}. \tag{2}

The second derivative 2σB2>02\sigma_B^2 > 0 confirms this is the minimum.

h*hedge ratio hvariance
The hedged variance is a parabola in the number of shares shorted. Its vertex, where the derivative vanishes, sits at h equal to rho times the ratio of the two volatilities.

#Read it off

Shorting h∗=ρ σA/σBh^{*} = \rho\,\sigma_A/\sigma_B shares leaves a residual variance of

σA2−(ρ σAσB)2σB2=σA2 (1−ρ2).(3)\sigma_A^2 - \frac{(\rho\,\sigma_A \sigma_B)^2}{\sigma_B^2} = \sigma_A^2\,(1 - \rho^2). \tag{3}

The hedge strips away the fraction ρ2\rho^2 of A's variance, exactly the squared correlation, so a tightly correlated B hedges well and an uncorrelated B does nothing at all.