Scaling a Return-Based Stock Hedge to Dollar Notional
Summary
The document asks how many shares of one stock to short against a long position in another when their return volatilities and correlation are known. It computes a minimum-variance hedge ratio from the correlation and the ratio of return volatilities, then questions whether that ratio should be adjusted for the stocks’ different prices.
The answer distinguishes the hedge ratio in return terms from the practical conversion into shares. For a portfolio-sized position, it applies the ratio to the dollar exposure being hedged, then divides the desired short dollar amount by the hedge stock’s price to estimate shares. This makes clear why a return-based ratio is not automatically a share count when prices differ. Actual orders must also account for integer share sizes, and the example does not discuss estimation error, changing correlations, transaction costs, or ongoing rebalancing.
Key ideas
- A return-based hedge ratio uses correlation and relative return volatility.
- The ratio alone does not specify a practical number of shares when the two stocks have different prices.
- Scale the hedge ratio by the dollar exposure being hedged, then convert the target dollar position into shares.
- Whole-share constraints can make the implemented hedge differ from its theoretical target.
- The example does not address changing estimates or rebalancing costs.
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Full text
# Hedging using relative values
# Hedging using relative values
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.
## Answer by Richi Wa (score 1)
https://quant.stackexchange.com/a/14763
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 $A$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.