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

Abstract

While the majority of earlier studies used autocorrelation-based methodologies to explore the dependency structure for Bitcoin, this paper follows Benoit Mandelbrot in taking a fractal point of view. It shows that both Bitcoin and S&P 500 returns exhibit fractal-like behavior. Further evidence suggests that the infinite-variance-hypothesis cannot be rejected for both assets supporting Mandelbrot’s (1963) early study on cotton price changes. This result holds across non-overlapping subsamples. Following Mandelbrot (2008), Hurst exponents are estimated using rescaled/range analysis. The key findings are that (i) Bitcoin returns exhibit a higher level of persistence than S&P 500 returns across various subsamples, (ii) the level of persistence in Bitcoin returns has not changed across time, (iii) the S&P 500 moved from efficiency in the first subsample to inefficiency in the ex-post June 17, 2018 period, (iv) even if it was assumed that the variance of S&P 500 returns is finite, the kurtosis remains statistically undefined. The study concludes that correlation-based methods used to explore the S&P 500 universe result in misleading answers.