This is the first post in a two-part series about the Hurst Exponent. Tom and I worked on this series together and I drew on some of his previously published work as well as other sources like the very useful Quantstart.com.
The idea behind the Hurst Exponent H is that it can supposedly help us determine whether a time series is a random walk (H ~ 0.5), trending (H > 0.5) or mean reverting (H < 0.5) for a specific period of time. However, if you’ve ever used Hurst, you know that it can be a bit bewildering: not only does it often give unexpected results, but it also returns different results depending on the implementation used in its calculation. Further, there are a few different methods for calculating Hurst; we found that these generally agree for a randomly generated time series, but disagree when we use real data.
How then can Hurst be of any value to algo traders?
The remainder of this post is devoted to presenting and discussing some Python code for calculating Hurst. In the next post, we are going to delve more deeply into the calculation and work out what’s going on. Our ultimate goal is to demystify the Hurst Exponent and show how to take it beyond some nice theory to something of practical value to algo traders.
Without further ado, here is the code for calculating the Hurst Exponent in Python. We determine Hurst by firstly calculating the standard deviation of the difference between a series and its lagged counterpart. We then repeat this calculation for a number of lags and plot the result as a function of the number of lags. If we plot this on a log-log scale, we end up with a straight line, the slope of which provides an estimate for the Hurst exponent. I found this article which describes this approach to calculating Hurst, as does this one.
from numpy import *
from pylab import plot, show
# first, create an arbitrary time series, ts
ts = 
for i in range(1,100000):
ts.append(ts[i-1]*1.0 + random.randn())
# calculate standard deviation of differenced series using various lags
lags = range(2, 20)
tau = [sqrt(std(subtract(ts[lag:], ts[:-lag]))) for lag in lags]
# plot on log-log scale
plot(log(lags), log(tau)); show()
# calculate Hurst as slope of log-log plot
m = polyfit(log(lags), log(tau), 1)
hurst = m*2.0
print 'hurst = ',hurst
You can see in the code that we used lags 2 through 20 for calculating H. These lags are somewhat arbitrary, but based on the best results obtained using synthetic data with known behaviour. Specifically, we found that if we set the maximum number of lags too high, the results became quite inaccurate. These values are the defaults used in some other implementations, such as the standard Hurst function in MATLAB.
In the next post, we will look at these lags in more detail and show how they are actually crucial for calculating Hurst in such a way that is useful and meaningful. We tweak this part of the calculation to uncover a practical application of Hurst in developing algo trading systems. Check out the follow on post Demystifying the Hurst Exponent – Part 2 here.
If you have used the Hurst Exponent, or indeed any of the other tests for mean reversion that we mentioned in this post, please share your experiences in the comments. Thanks!
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