Hurst exponent

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In fractal geometry, the generalized Hurst exponent, (named after Harold Edwin Hurst (1880-1978))

Hq = H(q),

for a time series

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

is defined by the scaling properties of its structure functions Sq(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. 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

Hq = ½,

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

Hq = 0.

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

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

[edit] Reference

  • 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.

[edit] External links

  • [1] Scientio's ChaosKit product calculates hurst exponents amongst other Chaotic measures. Access is provided free online via a web service.
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