A little about adaptive betas and pairs trading…

Why do they matter?

Beta captures elements of the risk return relationship between a pair of symbols. One method of calculating beta (β) is to find the slope of the line of best fit in a scatter chart of the prices of two symbols. This method uses all the observations in a given period and is non-adaptive or “fixed”. For example:

The beta equals 0.464. So long as the price ratio of Y divided by X (ie the beta ratio) remains stable it is a calculation method that works very well.

However, beta ratios can vary greatly. For these cases a beta calculation that adapts over time, whilst filtering out excessive noise, may improve returns. An example using the same data as the chart above:

A beta of 0.464 does not look all that marvellous an approximation considering this plot. The vertical axis is the price of Y divided by the price of X – the beta ratio – which fluctuates well away from 0.464; and the horizontal axis is the number of observations.

The chart also plots a time-varying beta derived from a method called the “Kalman Filter” to capture this variation whilst filtering out noise. Our next release has both this method and Robert Goodell Brown’s double exponential smoothing prediction (DESP) model.

Why is this important?

The beta determines the the ratio of money in symbol X to money in symbol Y in any potential pairs trade. If the beta is volatile the balance may be inaccurate using a fixed beta. If it is stable, no.

ArbMaker implements the beta ratio in two distinct ways:

1. Where the beta calculation is fixed the ratio is applied directly to the amounts per side in the back testing window. Under this method a pair is not money neutral (unless beta equals 1). For example, a beta of 2 means amount Y will be twice amount X.
2. Where the beta calculation is adaptive it is applied directly to the spread calculation so that spread Z = Yprice – βXprice. This means amounts Y and X are of equal size in the back tester.

A key implication of the second point is that the shape of the spread under time-adaptive betas can be ‘tuned’ for profit performance. We will follow this up in another post…