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Minimum Variance Hedge Ratio

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This note briefly explains what's the minimum variance hedge ratio and how to derive it in a cross hedge, where the asset to be hedged is not the same as underlying asset.

The Hedge Ratio hh

The hedge ratio hh is the ratio of the size of the hedging position to the exposure of the asset to be hedged:

h=NFNA=size of hedging positionsize of exposureh=\frac{N_F}{N_A}=\frac{\text{size of hedging position}}{\text{size of exposure}}

Apparently, if we vary hh, the variance (risk) of the combined hedged position will also change.

The (Optimal) Minimum-Variance Hedge Ratio hh^*

Our objective in hedging is to manage the variance (risk) of our position, making it as low as possible by setting the hedge ratio hh to be the optimal hedge ratio hh^* that minimises the variance of the combined hedged position.

Hedge where A=AA'=A

It's relatively easy when the underlying asset of the futures (AA') is the same as the asset to be hedged (AA), as they have a perfect correlation and the same variance. Thus, as long as the hedge ratio h=1h=1, where the size of hedging position equals the exposure of the asset held, the perfect correlation and same variance ensure the value changes in the hedging position offset the changes in the value of asset to be hedged, so that the variance of the hedged position is minimum at zero (ignoring other basis risks). This means, the optimal minimum-variance hedge ratio h=1h^*=1.

Cross Hedge where AAA' \neq A

When the underlying asset of the futures (AA') differ from asset to be hedged (AA), the optimal hedge ratio hh^* that minimises the portfolio variance is not necessarily 1 anymore.

Let's now derive hh^*.

Let's consider a short hedge, where we long StS_t and short h×Fth\times F_t, hence:

The optimal hedge ratio hh^* is the hedge ratio that minimises the variance of ΔC\Delta C.

h=argminhVar(ΔC)=argminhVar(ΔSth×ΔFt)h^* =\underset{h}{\operatorname{argmin}} \text{Var}(\Delta C) =\underset{h}{\operatorname{argmin}} \text{Var}(\Delta S_t-h\times \Delta F_t)

We also know that

Var(ΔSth×ΔFt)=σS2+h2σF22h(ρσSσF)\text{Var}(\Delta S_t-h\times \Delta F_t) = \sigma^2_S + h^2\sigma^2_F - 2h(\rho \sigma_S \sigma_F)

To minimise the variance, the first-order condition (FOC) is that

Var(ΔC)h=2hσF22(ρσSσF)=0\frac{\partial \text{Var}(\Delta C)}{\partial h}=2h\sigma^2_F-2(\rho \sigma_S \sigma_F)=0

The optimal hedge ratio hh^* is the hh that solves the FOC above. Therefore,

h=ρσSσFh^* = \rho \frac{\sigma_S}{\sigma_F}

Intuition

The optimal hedge ratio hh^* describes the optimal NF/NAN_F/N_A, so that the optimal size of the hedging position:

NF=h×NAN_F^* = h^* \times N_A

If ρ=1\rho=1 and σF=σS\sigma_F=\sigma_S, then h=1h^*=1:

If ρ=1\rho=1 and σF=2σS\sigma_F=2\sigma_S, then h=0.5h^*=0.5:

If ρ<1\rho<1, then hh^* depends on ρ\rho and σS/σF{\sigma_S}/{\sigma_F}: