Optimal hedge ratio

We will call hedge ratio, the proportion of the position taken in futures to the size of the exposure, and we will denote it by $h$. This is the ratio that will minimize the risk of the position, the risk being measured as the volatility, in the spirit of the CAPM. Therefore, if an investor is long one unit of the asset, she has to short $h$ units of the futures -a short hedge. The initial position is worth

$$\Pi \left( t\right) =S\left( t\right) -hF\left( t,T\right)$$   ,

where the final position will be worth

$$\Pi \left( \tau \right) =S\left( \tau \right) -hF\left( \tau ,T\right)$$   .

Remember that both $S\left( \tau \right)$ and $F\left( \tau ,T\right)$ are unknown at time $t$, which makes $\Pi \left( \tau \right)$ a random variable. The change in the value of the position will therefore be

$$\Delta \Pi =\Delta S-h\Delta F$$.

Alternatively, for the long hedge the change in the value of the position will be

$$\Delta \Pi =h\Delta F-\Delta S$$.

In both cases the variance of $\Delta \Pi$ is equal to

$$\sigma _{\Delta \Pi }^{2}=\sigma _{\Delta S}^{2}+h^{2}\sigma _{\D... ...{2}-2h\rho _{\Delta S,\Delta F}\sigma _{\Delta S}\sigma _{\Delta F}\text{.%% }$$

For the variance to be minimized, the f.o.c. is

$$\frac{\partial }{\partial h}\sigma _{\Delta \Pi }^{2}=0$$,

which gives as a result that

$$h=\rho _{\Delta S,\Delta F}\frac{\sigma _{\Delta S}}{\sigma _{\Delta F}}%%$$   .

One can recall the similarities of the above procedure with the Markovitz type mean-variance portfolio analysis. Observe that the hedge ratio can be rewritten as $h=\frac{cov\left( \Delta S,\Delta F\right) }{var\left( \Delta F\right) }$ to illustrate the similarities.