An Introduction to Hurst Exponent

As an example of an application of Mathematics to finance, we will look at Hurst exponents, omitting the technical details.

Given a time series, the Hurst exponent (H) of the mathematical object is a single number between 0 and 1. What can a single number tell us about the series?
It can be interpreted in many ways, one of them is that it measures the jaggedness or smoothness of the series.
It helps us classify time series. For example, one of the basic questions, is whether a time series is purely random (a random walk or Brownian movement) or not.
Many people have suggested that financial data such as stock prices are random, Hurst exponent helps explain that it is not.

H = 1/2 is a random walk with no memory of past states, H between 1/2 and 1 is a persistent time series, where the series has long term memory, and H between 0 and 1/2 is an anti-persistent time series (the persistence works in a negative way). A mean reverting series for example is anti-persistent, but the converse is not always true. In common parlance, "what goes up must come down, and vice versa" applies to reverting series.

Geometrically, anti-persistent series are more jagged than a random walk. Persistent series are smoother than a random walk.

In the extreme case of H very close to 0, the series is almost like a space filling curve.
If H is close to 1, the series is a single line (can be broken and curved).

From this, we can understand that H is also measuring the fractal dimension D. The relation can be shown to be

D = 2 - H

H is also related to the color of noise, white noise has H=1/2, brown noise H=0, and pink noise has H between 0 and 1/2.

H was first introduced by H.E. Hurst, a hydrologist who studied data of river overflows of the river Nile. He wanted to know whether the overflows occurs randomly or not. In so doing he had a wonderful idea, defining H by rescaling (or normalization).

Let the original series be x1, x2............ xn We divide the series into p buckets, each of size m, with n = m.p For each bucket, we calculate the mean and the standard deviation S.

For each bucket, define a new series y1, y2, ...yj.....ym, where yj is the cumulative sum of all (x - mean) from index 1 to j in the bucket.

Define R for the bucket as the largest minus the smallest yj. R will always be positive.
Then divide R by S: R/S. R/S is defined for each bucket 1,2, ...i,....p. Take the average of R/S for all buckets, called this the R/S for the bucket-size.

Now vary the bucket-size and calculate the corresponding R/S.

Hypothesize that (R/S) obeys a power law in the bucket size b, so that

(R/S)_b = c. b^H
with c a constant.

The Hurst exponent is the power in the power law relation above. When H = 1/2, we have Einstein's formula for Brownian movement.

To calculate or estimate H, take the log of the power law equation, and H becomes the slope of the line. The precise algorithm in C++ for estimating H can be found in here.

Some of the attraction of the Hurst exponent, is that it is related to many fields of Mathematics, fractals, chaos, wavelets, spectral analysis, statistics, etc. Accordingly there are many ways of estimating H, unfortunately they don't always give the same answer.

Hurst used the definition to investigate time series data from river discharges, tree rings, rainfalls, and others. He found they have H between 0.6 and 0.8 indicating persistent series with long term memory.

Edgar Peters introduced H into the financial world with his Fractal Market Hypothesis, see e.g. his book Fractal Market Analysis, applying chaos theory to investments and economics.
Peters applied H to stock and bond returns, forex exchange, and their volatilities.

Here is an example of a 5-day return of GBPUSD exchange rates:

I estimated H using Karagianis et.al. software called SELFIS

The result is H = 0.645:

The data uses 5-day returns, as it happens, longer day returns have more persistence than short term returns.

Despite its application to finance, Hurst exponent is not, perhaps to the disappointment of many, a forecasting tool, it shows theoretical predictability, but it is not a prediction method.