Hurst exponent

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In fractal geometry, the generalized Hurst exponent, named H in honor of both Harold Edwin Hurst (1880-1978) and Ludwig Otto Hölder (1859-1937) by Benoît Mandelbrot, is referred to as the "index of dependence," and is the relative tendency of a time series to either strongly regress to the mean or 'cluster' in a direction. [1]

H was originally developed in hydrology for the practical matter of determining optimum dam sizing for the Nile river's volatile rain and drought conditions that had been observed over a long period of time. The Hurst exponent is non-deterministic in that it expresses what is actually observed in nature; it is not calculated so much as it is estimated. The accuracy and fidelity of H are also limited in the extreme for small measurements of roughness by the Heisenberg Uncertainty Principle, a boundary for all measurements.

There are a variety of techniques that exist for estimating H, however assessing the accuracy of the estimation can be a complicated issue. Mathematically, in one technique, the Hurst exponent can be estimated such that:

H_q = H(q),

for a time series

g(t) (t = 1, 2,...)

may be defined by the scaling properties of its structure functions S_q(τ):

$S_q = \langle |g(t + \tau) - g(t)|^q \rangle_T \sim \tau^{qH(q)}, \,$

where q > 0, τ is the time lag and averaging is over the time window

$T \gg \tau,\,$

usually the largest time scale of the system.

Practically, in nature, there is no limit to time, and thus H is non-deterministic as it may only be estimated based on the observed data; e.g., the most dramatic daily move upwards ever seen in a stock market index can always be exceeded during some subsequent day.

H is directly related to fractal dimension, D, such that D = 2 - H. The values of the Hurst exponent vary between 0 and 1, with higher values indicating a smoother trend, less volatility, and less roughness.

In the above mathematical estimation technique, the function H(q) contains information about averaged generalized volatilities at scale τ (only q = 1, 2 are used to define the volatility). In particular, the H₁ exponent indicates persistent (H₁ > ½) or antipersistent (H₁ < ½) behavior of the trend.

For the BRW (brown noise, 1/f²) one gets

H_q = ½,

while for the pink noise (1/f) and white noise we have

H_q = 0.

For the popular Levy stable processes and truncated Levy processes with parameter α it has been found that

H_q = q/α for q < α and H_q = 1 for q ≥ α.

Note added Oct. 2007: in the above definition two separate requirements are mixed together as if they would be one. Here are the two independent requirements: (i) stationarity of the increments, x(t+T)-x(t)=x(T) in distribution. this is the condition that yields long time autocorrelations. (ii) Self-similarity of the stochastic then yields variance scaling, but is not needed for long time memory. E.g., both Markov processes (i.e., memory-free processes) and fractional Brownian motion scale at the level of 1-point densities (simple averages), but neither scales at the level of pair correlations or, correspondingly, the 2-point probability density.

An efficient market requires a martingale condition, and unless the variance is linear in the time this produces nonstationary increments, x(t+T)-x(t)≠x(T). Martingales are Markovian at the level of pair correlations, meaning that pair correlations cannot be used to beat a martingale market. Stationary increments with nonlinear variance, on the other hand, induce the long time pair memory of fBm that would make the market beatable at the level of pair correlations. Such a market would necessarily be far from "efficient".

[edit] References

Mandelbrot, Benoît B., The (Mis)Behavior of Markets, A Fractal View of Risk, Ruin and Reward (Basic Books, 2004), pp. 186-195
A.Z. Gorski et al., Financial multifractality and its subtleties: an example of DAX. Published in Physica, vol. 316 (2002), pp. 496 - 510
T. Di Matteo, "Multi-scaling in Finance", Quantitative Finance, Vol. 7, No. 1 (2007) 21-36.
T. Di Matteo, T. Aste and M. M. Dacorogna, "Long term memories of developed and emerging markets: using the scaling analysis to characterize their stage of development", Journal of Banking & Finance 29/4 (2005) 827-851.
T. Di Matteo, T. Aste and M. M. Dacorogna, "Scaling behaviors in differently developed markets", Physica A 324 (2003) 183-188.

References for note added Oct., 2007: Joseph L. McCauley, Kevin E. Bassler, and Gemunu H. Gunaratne , Martingales, Detrending Data, and the Efficient Market Hypothesis, Physica A37, 202, 2008.