A Fractal and Comparative View of the Memory of Bitcoin and S&P 500 Returns

The majority of previous studies used autocorrelation-based methodologies to explore the dependency structure for Bitcoin, but this paper follows Benoit Mandelbrot in taking a fractal point of view. This perspective showed that Bitcoin and S&P 500 returns exhibit fractal-like behavior. Additional evidence suggested that the infinite variance hypothesis cannot be rejected for either asset supporting Mandelbrot’s (1963) early study on cotton price changes. This result held across non-overlapping subsamples. Following Mandelbrot (2008), Hurst exponents were estimated using rescaled/range analysis. The key findings are that (a) Bitcoin returns exhibit a higher level of persistence than S&P 500 returns across various subsamples, (b) the level of persistence in Bitcoin returns did not change over time, (c) the S&P 500 moved from efficiency in the first subsample to inefficiency in the ex-post June 17, 2018, period, (d) even if it was assumed that the variance of S&P 500 returns was finite, the kurtosis remained statistically undefined. The study concluded that the correlation-based methods used to explore the S&P 500 universe result in misleading answers.