Cross Hedging – Sisi Tang

When our study group read John Hull’s Options, Futures, and Other Derivatives 10th Edition book section 3.4 Cross Hedging, the hedging ratio $h^*$ was given directly in (3.1). We filled in the derivation of it here.

Goal

We want to hedge an asset using a future contract whose underlying asset is different from the one being hedged. This usually happens for example in the case that the asset being hedged is not available in the future market. If we are going to sell an asset, then we take long position in a future contract; short otherwise.

Setting

$h$ : hedge ratio. The ratio of the size of the position taken in futures contracts to the size of the exposure.
$\Delta S$ : change of spot price of the asset to be hedged.
$\Delta F$ : change of price in the future contract.

Formulation

Mathematically, we want to minimize the variance of our portfolio value, i.e.,

$\begin{aligned}\mbox{Var}(\Delta S-h\Delta F) &= \mathbf{E}[(\Delta S-\overline{\Delta S}-h(\Delta F-\overline{\Delta F}))^2] \\&= \mathbf{E}[(\Delta S-\overline{\Delta S})^2] + h^2\mathbf{E}[(\Delta F-\overline{\Delta F})^2] \\ & \quad - 2h\mathbf{E}[(\Delta S-\overline{\Delta S})(\Delta F-\overline{\Delta F})] \\&= \sigma_{\Delta S}^2+ h^2\sigma_{\Delta F}^2 - 2h~\mbox{cov}(\Delta S,\Delta F) ,\end{aligned}$

where $\sigma_{\Delta S}, \sigma_{\Delta F}$ denote the standard deviation of $\Delta S$ , standard deviation of $\Delta F$ respectively. This is a quadratic fucntion in $h$ . The minimum of it is achieved at

$h=\frac{\mbox{cov}(\Delta S,\Delta F)}{\sigma_{\Delta F}^2}=\rho\frac{\sigma_{\Delta S}}{\sigma_{\Delta F}},$

where $\rho$ is the correlation of $\Delta S$ and $\Delta F$ . This formula for $h$ is exactly the slope of linear regression $\Delta S \sim \Delta F$ . Our derivation here also explains why linear regression is the minimum variance estimator.

Then, the variance

$\mbox{Var}(\Delta S-h\Delta F)=(1-\rho^2)\sigma_{\Delta S}^2.$