Discussing the article: "Beyond GARCH (Part II): Measuring the Fractal Dimension of Markets"
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Check out the new article: Beyond GARCH (Part II): Measuring the Fractal Dimension of Markets.
Building on the partition function analysis from Part 1, this article deepens the theoretical foundation before completing the analytical pipeline. We first give a full treatment of the Hurst exponent: what it measures, what it implies about market memory, and why it matters for the MMAR. This is followed by an intuitive exploration of multifractal spectra and what f(α) reveals about volatility heterogeneity. We then move to implementation: extracting the scaling function τ(q), estimating H via R/S analysis, and fitting the multifractal spectrum across four candidate distributions. By the end, we have the complete parameter set needed to construct the MMAR process in Part 3. Part 2 of an eight-part series.
The Hurst exponent controls the roughness of a path. Consider three Fractional Brownian Motion paths, each 1,000 steps long but with different H values. At H = 0.3 (anti-persistent), the path is extremely jagged. Every upward move is likely followed by a downward move, creating a rapidly oscillating, rough trajectory that constantly reverses direction. At H = 0.5 (the standard random walk), the path has moderate roughness, the familiar irregular wandering of a coin-flip process. At H = 0.7 (persistent), the path is noticeably smoother. Upward moves tend to continue upward, producing longer sweeps and gentler undulations. The path still wanders randomly, but it does so in broader, more coherent strokes.
Mathematically, this roughness is captured by the Hölder regularity of the path. An FBM path with Hurst exponent H is almost surely Hölder continuous with exponent H, meaning that the increments over a time step delta_t scale as delta_t^H. Larger H means increments grow faster with the time step, which means the path covers more ground in smooth sweeps. Smaller H means increments grow more slowly, which means the path changes direction frequently and the total variation is higher. This geometric interpretation matters for trading: a persistent market (H > 0.5) rewards trend-following strategies, while an anti-persistent market (H < 0.5) rewards mean-reversion strategies. A random walk (H = 0.5) rewards neither, and any apparent pattern is noise.
Author: Muhammad Minhas Qamar