<p>Market efficiency assumes that prices in financial markets are perfectly informative and, therefore, it is not possible to design trading strategies that outperform the market. The concept of efficiency has important implications for financial stability and, consequently, for financial policies. If asset returns exhibit persistent or anti-persistent behavior, then predictability based on past returns might be possible, which would be a clear violation of the weak form of efficiency. Many studies rely on the Hurst exponent to evaluate the level of memory of financial returns, and the purpose of this paper is to show that long memory or anti-persistence of financial returns is not incompatible with the random walk model or the efficient market hypothesis (EMH). The use of the Hurst exponent to demonstrate the inefficiency of financial markets using common estimators is troublesome, especially when applied to financial returns, since values of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11135_2025_2052_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$\hat{H} \ne 0.5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>H</mi> <mo stretchy="false">^</mo> </mover> <mo>≠</mo> <mn>0.5</mn> </mrow> </math></EquationSource> </InlineEquation> are not evidence against the random walk model or the EMH. Moreover, the high variability of Hurst exponent estimates and their dependence on the chosen algorithm should motivate careful use of this tool. This study proposes a simple theoretical explanation and an extensive simulation study to show that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11135_2025_2052_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$\hat{H} \ne 0.5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>H</mi> <mo stretchy="false">^</mo> </mover> <mo>≠</mo> <mn>0.5</mn> </mrow> </math></EquationSource> </InlineEquation> for financial returns is perfectly compatible with the random walk model. As a robustness check, both the traditional rescaled range and the wavelet lifting algorithms are used. Applications to real data are also discussed to show that the empirical values of the Hurst exponent are in the range suggested by the simulations, providing evidence that over-reliance on the Hurst exponent could lead to erroneous rejection of the random walk model. Specifically, the paper presents an application to the daily returns of stock market indices (DJIA and S&P 500) over a period of more than 30 years and cryptocurrencies (Bitcoin and Ethereum) over a period of more than 5 years.</p>

Random walks, Hurst exponent, and market efficiency

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2025年1月24日

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Giuseppe Pernagallo

摘要

Market efficiency assumes that prices in financial markets are perfectly informative and, therefore, it is not possible to design trading strategies that outperform the market. The concept of efficiency has important implications for financial stability and, consequently, for financial policies. If asset returns exhibit persistent or anti-persistent behavior, then predictability based on past returns might be possible, which would be a clear violation of the weak form of efficiency. Many studies rely on the Hurst exponent to evaluate the level of memory of financial returns, and the purpose of this paper is to show that long memory or anti-persistence of financial returns is not incompatible with the random walk model or the efficient market hypothesis (EMH). The use of the Hurst exponent to demonstrate the inefficiency of financial markets using common estimators is troublesome, especially when applied to financial returns, since values of $\hat{H} \ne 0.5$ $\hat{H} \neq 0.5$ are not evidence against the random walk model or the EMH. Moreover, the high variability of Hurst exponent estimates and their dependence on the chosen algorithm should motivate careful use of this tool. This study proposes a simple theoretical explanation and an extensive simulation study to show that $\hat{H} \ne 0.5$ $\hat{H} \neq 0.5$ for financial returns is perfectly compatible with the random walk model. As a robustness check, both the traditional rescaled range and the wavelet lifting algorithms are used. Applications to real data are also discussed to show that the empirical values of the Hurst exponent are in the range suggested by the simulations, providing evidence that over-reliance on the Hurst exponent could lead to erroneous rejection of the random walk model. Specifically, the paper presents an application to the daily returns of stock market indices (DJIA and S&P 500) over a period of more than 30 years and cryptocurrencies (Bitcoin and Ethereum) over a period of more than 5 years.