Cointegration Trading Alongside Log Prices Vs. Prices

In my recent book, I highlighted a departure betwixt cointegration (pair) trading of cost spreads too log cost spreads. Suppose the cost spread hA*yA-hB*yB of ii stocks H5N1 too B is stationary. We should simply maintain the number of shares of stocks H5N1 too B fixed, inward the ratio hA:hB, too curt this spread when it is much higher than average, too long this spread when it is much lower. On the other hand, for a stationary log cost spread hA*log(yA)-hB*log(yB), nosotros ask to maintain the market values of stocks H5N1 too B fixed, inward the ratio hA:hB, which way that at the halt of every bar, nosotros ask to rebalance the shares of H5N1 too B due to cost changes.

For virtually cointegrating pairs that I accept studied, both the cost spreads too the log cost spreads are stationary, then it doesn't thing which 1 nosotros purpose for our trading strategy. However, for an odd duad where its log cost spread cointegrates but cost spread does non (Hat tip: Adam G. for drawing my attending to 1 such example), the implication is quite significant. H5N1 stationary cost spread way that prices differences are mean-reverting, a stationary log cost spread way that returns differences are mean-reverting. For example, if stock H5N1 typically grows 2 times equally fast equally B, but has been growing 2.5 times equally fast recently, nosotros tin await the growth charge per unit of measurement differential to decrease going forward. We would nonetheless curt H5N1 too long B, but nosotros would locomote out this set when the growth rates of H5N1 vs B provide to a 2:1 ratio, too non when the cost spread of H5N1 vs B returns to a historical mean. In fact, the cost spread of H5N1 vs B should proceed to growth over the long term.

This much is slow to understand. But thank yous to a reader Ferenc F. who referred me to a newspaper past times Fernholz too Maguire, I realize at that spot is a uncomplicated mathematical human relationship betwixt stock H5N1 too B inward companionship for their log prices to cointegrate.

Let us commencement amongst a formula derived past times these authors for the alter inward log marketplace value P of a portfolio of 2 stocks: d(logP) = hA*d(log(yA))+hB*d(log(yB))+gamma*dt.

The gamma inward this equation is

gamma=1/2*(hA*varA + hB*varB), where varA is the variance of stock H5N1 minus the variance of the portfolio marketplace value, too ditto for varB.

Note that this formula holds for a portfolio of whatever ii stocks, non simply when they are cointegrating. But if they are inward fact cointegrating, too if hA too hB are the weights which practise the stationary portfolio P, nosotros know that d(logP) cannot accept a non-zero long term drift term represented past times gamma*dt. So gamma must live on zero. Now inward companionship for gamma to live on zero, the covariance of the ii stocks must live on positive (no surprise here) too equal to the average of the variances of the ii stocks. I invite the reader to verify this decision past times expressing the variance of the portfolio marketplace value inward terms of the variances of the private stocks too their covariance, too too to extend it to a portfolio amongst north stocks. This cointegration examine for log prices is sure enough simpler than the green CADF or Johansen tests! (The cost to pay for this simplicity? We must assume normal distributions of returns.)

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