# 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

• Not sure if this answers directly to your question, but it might be helpful quant.stackexchange.com/questions/16481/… Apr 13 at 21:26
• This does shed some more color, thanks for the link. I think (and pls correct me if I'm wrong) is that the discussion there is basically stationarity vs non, and measuring returns whether is just change over some horizon or percent change if it's log, puts assets to compare on the same plane. Once they are on the same plane (as mentioned in the link, like vs like), then we can regress them to produce a coefficient that would minimize the change between the assets and therefore we can get our hedge ratio. Apr 13 at 22:36

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.

• Thank you for your answer, and it intuitively makes sense to measure two assets (however different they are) via change vs. change, or returns. Do we simply just discard the constant from the model as a way of saying that's errors or unexplained part and just use the beta coefficient? Also, as mentioned above sort of a related question. If we were to take asset x and use regression to imply a fair value for asset y, then would using price be ok? Apr 13 at 22:29
• If the residuals are homoscedastic, then it would be okay. If they are heteroskedastic, there are methods meant for that to treat the errors, such as the nonlinear OLS/Kalman filtration methodology for option pricing model calibration. On the price regression, I am honestly not sure, because I haven't worked much with them. Apr 14 at 8:37
• Hi, if you feel my response has helped you, you can give an upvote (if you haven't done so) or accept it as the solution (if it answered your question). :) Apr 21 at 13:07

# 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