Discussing the article: "Beyond GARCH (Part III): Building the MMAR and the Verdict"

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Check out the new article: Beyond GARCH (Part III): Building the MMAR and the Verdict.

With the multifractal parameters from Part 2 in hand, this article builds the full MMAR process. We construct the multiplicative cascade for trading time, generate Fractional Brownian Motion via Davies-Harte FFT, and combine both into X(t) = B_H[theta(t)]. A 100-path Monte Carlo simulation produces the volatility forecast, which we then pit against GARCH on the same EURUSD M5 data. Does Mandelbrot's fractal architecture outforecast Engle's conditional variance framework? Part 3 of a eight-part series leading to a native MQL5 library and Expert Advisor.

In Part 1 of this series, we covered the theoretical foundations of GARCH and the MMAR, loaded EURUSD 5-minute data, and confirmed via partition function analysis that the data exhibits multifractal scaling. In Part 2, we extracted the scaling function tau(q), estimated the Hurst exponent H, ran a rigorous multifractality decision test, and fitted the multifractal spectrum to four theoretical distributions. The result: our data is genuinely multifractal, and we now have the specific parameters, the Hurst exponent, the winning distribution type, and its fitted coefficients, needed to build the full MMAR process.

In this article, we put those parameters to work. We construct two MMAR components: a multiplicative cascade that creates multifractal trading time (the "deformed clock" that warps uniform time into market time) and Fractional Brownian Motion that provides long-memory structure. We combine them into the compound process X(t) = B_H[theta(t)]. Next, we run 1,000 Monte Carlo simulations to forecast volatility. We then fit a GARCH(1,1) model on the same training data. Finally, we compare both models head-to-head against realized volatility. No tricks, no cherry-picking, just a fair fight.

Author: Muhammad Minhas Qamar