Hurst Exponent for Algorithmic Trading

This is the first post in a two-part series about the Hurst Exponent. Tom and I worked on this series together. I drew on some of his earlier work as well as other resources, including Quantstart.com.

UPDATE 03/01/16: The Python code below has been updated with a more accurate algorithm for calculating the Hurst Exponent.

Mean-reverting time series have long attracted quantitative traders. Some of the biggest names in quant trading are said to have built their fortunes by exploiting mean reversion in financial data—particularly in constructed spreads used in pairs trading. Naturally, identifying mean reversion is a key goal in algorithmic trading. However, this is harder than it seems, mainly due to the non-stationary nature of financial time series.

Ernie Chan’s Algorithmic Trading: Winning Strategies and Their Rationale is one of the better introductions to mean reversion. In his book, Ernie discusses several tools that help test whether a time series is mean-reverting. These include the Augmented Dickey-Fuller test and the Johansen test. We’ve explored both on Robot Wealth, using simple R code [here] and [here].

Another tool for testing mean reversion is the Hurst Exponent.

What is the Hurst Exponent?

The Hurst Exponent, denoted H, helps characterize the behavior of a time series. When H≈0.5H \approx 0.5H≈0.5, the series behaves like a random walk. Values of H>0.5H > 0.5H>0.5 indicate a trending series, where past movements are more likely to continue. In contrast, H<0.5H < 0.5H<0.5 suggests mean reversion, meaning the series tends to revert to a long-term average.

However, working with Hurst can be tricky. It often yields unexpected results, and outcomes vary across different implementations. We also found that while various methods agree when applied to synthetic data, they diverge when tested on real financial time series.

Why Should Algo Traders Care?

This post introduces Python code for calculating the Hurst Exponent. In Part 2, we’ll dig deeper into how it works and discuss how to use it in practice. Our goal is to demystify Hurst and show how to move beyond theory into something that can actually help in building trading strategies.

Python Code for Calculating the Hurst Exponent

Below is a simple Python script. It estimates the Hurst Exponent by plotting the relationship between lag and the standard deviation of lagged differences.

from numpy import *
from pylab import plot, show

# Create an arbitrary time series
ts = [0]
for i in range(1, 100000):
    ts.append(ts[i - 1] + random.randn())

# Compute standard deviations of differenced series for several lags
lags = range(2, 20)
tau = [sqrt(std(subtract(ts[lag:], ts[:-lag]))) for lag in lags]

# Plot on a log-log scale
plot(log(lags), log(tau))
show()

# Estimate Hurst Exponent as slope of the log-log plot
m = polyfit(log(lags), log(tau), 1)
hurst = m[0] * 2.0
print('hurst =', hurst)

In this code, we use lags from 2 to 20. These values are somewhat arbitrary but worked best when we tested them on synthetic data. Using larger lags tended to produce inaccurate results. Interestingly, this range matches the defaults in some standard implementations, such as MATLAB’s hurst function.

In Part 2, we’ll explore how the choice of lags affects the Hurst calculation and show how tuning this parameter can lead to practical use in developing trading systems.

If you’ve used the Hurst Exponent—or any of the other mean reversion tests we mentioned—feel free to share your experiences in the comments. Thanks for reading!