Multifractality and Market Efficiency Across Asset Classes

The Fractal Market Hypothesis (FMH) is increasingly studied and applied by both finance academics and practitioners. We previously discussed the use of Detrended Fluctuation Analysis to estimate the Hurst exponent for major cryptocurrencies.

Continuing this line of research, Reference [1] applies Multifractal Detrended Fluctuation Analysis (MFDFA) to examine cryptocurrency, commodity, foreign exchange, and equity markets, specifically Bitcoin, Ethereum, crude oil, gold, EUR/USD, USD/JPY, the S&P 500, and the BIST 100 using data from January 1, 2018 to December 19, 2022.

In addition to the standard Hurst exponent used in prior studies, the paper also incorporates the q-th order fluctuation function to generalize the Hurst exponent. The scaling behavior of the resulting h(q) function serves as a measure of multifractality.

The authors pointed out,

The findings reveal that BTC, ETH, crude oil, and BIST 100 exhibit h (2) > 0.5, indicating long memory and persistent structure. These time series deviate from the mean and exhibit autocorrelations with trends and cycles, suggesting that increases or decreases in past periods are likely to continue in the future. In contrast, gold, EUR/USD, USD/JPY and S&P 500 have h (2) less than 0.5, implying short memory and mean-reverting behavior. These variables show negative autocorrelation, where increases or decreases in the past are often followed by opposite movements in the future. Importantly, none of the analyzed time series follow a random walk, indicating that past returns could be used to predict future returns…

The degree of multifractality varies significantly across the variables. EUR/USD has the lowest degree of multifractality, followed by USD/JPY, S&P 500, crude oil, ETH, gold, BTC, and BIST 100. Among the markets, the foreign exchange market demonstrates the lowest degree of multifractality, while the cryptocurrency and commodity markets exhibit higher degrees. Within the stock market, S&P 500 displays low multifractality, but BIST 100 stands out as the variable with the highest degree of multifractality among all analyzed. The findings further reveal that as the degree of multifractality increases, so does the associated risk.

In short, the paper finds that none of the analyzed assets follow a random walk, implying a predictable structure in returns. It shows that cryptocurrencies and commodities exhibit stronger persistence and higher multifractality than foreign exchange and developed equity markets, with BTC, ETH, crude oil, and BIST 100 displaying long memory, while gold, major FX pairs, and the S&P 500 exhibit mean-reverting behavior.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Temel, F., Tuğay, O. Testing the Fractal Market Hypothesis Using MFDFA Across Multiple Asset Classes. Comput Econ (2025).

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