Estimating Hurst Memory and Fitting Multifractal Market Structure

Summary

The article outlines a pipeline for characterizing multifractality in EURUSD five-minute returns. Starting from partition-function slopes calculated in an earlier installment, it extracts the scaling function tau(q), estimates the Hurst exponent using rescaled-range analysis, applies a three-part multifractality test, and derives the spectrum f(alpha) through a Legendre transform. It then compares the empirical spectrum with Lognormal, Binomial, Poisson, and Gamma distributions to identify a candidate multiplicative cascade for constructing a Multifractal Model of Asset Returns process.

It explains H as an indicator of scaling memory: values above one half imply persistence, below one half imply anti-persistence, and one half corresponds to random-walk scaling. The global exponent summarizes average behavior, whereas the spectrum represents variation in local scaling. These interpretations may motivate trend-following or mean-reversion hypotheses, but are not themselves evidence of profitable signals. The excerpt describes the analysis stages, but omits their detailed test results and fitted parameters; estimates also depend on data and method choices.

Key ideas

• The scaling function is obtained by subtracting one from partition-function regression slopes.
• A linear scaling function is associated with monofractal behavior, while curvature indicates multifractal scaling.
• The Hurst exponent summarizes average memory, while the multifractal spectrum describes variation in local scaling.
• The workflow uses a Legendre transform and compares four candidate distributions to fit the spectrum.
• Fractal measurements describe statistical structure but do not by themselves establish a profitable trading strategy.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.