Choosing Return or Log-Price Regressions for Hedge Ratios

# Producing hedge ratios via regression via returns and not price


# Producing hedge ratios via regression via returns and not price












I'm a quant student and I need someone to clearly and plainly explain to me better than my professor did about this topic. Please be patient if my question seems very basic.

to find hedge ratios or betas, we typically use linear regression / OLS. However, to get the data, say SPY vs QQQQ, why is it that we want to sample changes over x horizon or returns versus just simply regress price? For example, one data point would be (if sampling period is every 5min), 5 ticks changed over the last 5min versus just taking the price at every 5min?

If we want to say, use QQQQ and regression to imply what SPY price is, would then we use price be ok?

last, why is the fact that we only use the beta and disregard the constant coefficient when constructing the hedge?

Thank you

## Answer by KaiSqDist (score 1)

https://quant.stackexchange.com/a/78977

If you regressed returns of two assets against each other and produced a linear model, the model is basically saying that for a certain return observed for asset X, on average, you would observe a certain return for asset Y. The beta or "hedge ratio" you describe would be a way to "equalize" the returns of both assets.

For example, if asset X is much less volatile than asset Y, a possible linear model you could obtain from regressing returns of Y on X would be Ret_Y = 1.5*Ret_X + C. Therefore, to hedge asset Y, one would need to purchase more of asset X to offset the returns from asset Y, which makes sense.

If you used prices in the regression, you could get a linkage between the prices, but I don't quite see how you could use it to effectively implement a hedge.

## Answer by mark leeds (score 0)

https://quant.stackexchange.com/a/79383

## This is not an answer but I needed space.

This hedge ratio question is intimately connected to pairs trading. I wouldn't do it justice ( it's quite a large topic with quite a few technical details ) so I'll leave it to this undergrad honors thesis at the link below. It gives a very nice explanation of the statistical issues when one is trying to compute hedge ratios. Method 3 starts in Section 3.4 and uses logs of prices. This model is closest to the model you described but obviously not the same.

Note that using returns ( rather than log prices ) in the regression model is not a good idea because returns tend to be I(0). In the Engle Granger cointegration setting, the regression variables need to be I(1) and logs of prices meet that criteria.

https://ses.library.usyd.edu.au/bitstream/handle/2123/4072/Thesis_Schmidt.pdf?sequence=1

### ADDENDUM: 06-14-2024

I just wanted to add that, in above, one might argue that prices are I(1) so we can use them also. I thought about this recently and it's not a good idea ( atleast as far as I can tell. I welcome corrections ) because if you regress $y$ prices against $x$ prices, then the ratio being estimated is $\hat{\beta_{1}} =\frac{p_{y}}{p_{x}}$. So, $\hat{\beta_{1}}$ represents how many cents y is expected to change for every say 100 cent increase in the price of $x$. But a hedge ratio that is a function of cents ( or any units for that matter ) is not helpful because this in turn implies that the magnitudes of each of the prices matter. One definitely seeks a hedge ratio that is independent of the magnitudes of the prices of the things being regressed and log prices achieve this. This is because when log prices are regressed, the beta estimate becomes $ \hat{\beta_{2}} = \frac{log_{y}}{log_{x}}$ and represents an estimate of: "how much log(y) of $y$ is obtained for every unit percent increase in the log(x) of $x$. But notice that "log(y) of $y$" is the return of $y$ and "$log(x)$ of $x$" is the return of $x$. This is the advantage of using log price rather than price. By using log prices, the $\beta$ relation is a return relation rather than a price relation.