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Consider I have two stocks $A$ and $B$, $A$ is trading at $\$40$ and $B$ at $\$30$. The standard deviation of its returns are $\sigma_A=25\%$ and $\sigma_B = 30\%$. Correlation between the returns is $\rho=0.95$. Suppose we have long position on one stock of $B$, how many stocks of $A$ should we short?

Now it is easy to calculate the hedge ration which is $h^{*}=\rho\frac{\sigma_B}{\sigma_A}=1.14$

Now it seems to be as if this is the answear, I should just short $1.14$ shares. However, I feel as if I should be scaling this with $\frac{\$30}{\$40}$ because the standard deviations are based on percentage changes and do not concern about the real value of the stocks. I feel as if the underlying assumption that when calculating percentage that we are always dividing by a constant when calulating the percentage between two time intervals when in fact the price is changing over time. Maybe the approximation that it is almost the same most of the time makes it almost right.

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There are two things:

First: You have one stock of $B$ (worth \$30) and the calculation tells you to short 1.14 stocks of $A$. Of course you can only short whole stocks. So you would have to decide wether to short 0,1 or 2 stocks. This is a question of contract size, or in this case just size.

Second: Usually we speak about hedging in portfolio context. In this case you have houndreds of stocks $B$ that make e.g. $x\%$ of your portfolio. Then you could short $1.14* x\%$ of stock $A$. This will be an amount in USD and again you have to calculate how many pieces of stock $A$ you will sell. Say $x = 50\%$ and your total portfolio is worht $10 000$ USD. Then you would want to short $50\%*1.14*10 000 = 5 700$. But $5700/40=142.5$ and you either short $142$ or $143$ pieces of stock $A$.

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