Mathematical Analysis: The Efficiency of Bitcoin

Abstract

Do financial markets have memory: is every price movement independent from previous prices? This paper investigates the extent to which Bitcoin's market efficiency has transitioned towards the random walk model (H = 0.5), in comparison to the S&P 500, measured using a sliding window Hurst Exponent analysis. Bitcoin, a new asset with relatively minimal institutional surveillance, was expected to hold significant memory, while the S&P 500 serves as a mature, efficient benchmark. Daily prices from January 5, 2017 to March 27, 2026 were processed into log returns and analyzed using Rescaled Range analysis across a 252-day sliding window, producing 2,069 H values per asset. Markov chains and steady-state vectors were then applied to model transitions between Memory, Random Walk, and Mean-Reverting states, and their long-run destinations. Results show Bitcoin's full-period H (0.649) exceeded the S&P 500's (0.607) via our manual method, though Bitcoin's average H declined across most sliding-window eras (0.643 → 0.597 → 0.601 → 0.569), falling below the S&P 500's average (0.581) in the most recent recovery era. However, steady-state vectors reveal Bitcoin still spends 80.4% of its long-run time in the Memory state, an 18.4 percentage-point gap above the S&P 500's 62.0%. The evidence supports that Bitcoin's transition toward efficiency is significant but structurally incomplete, favoring the Fractal Market Hypothesis-that Bitcoin has transitioned toward, and temporarily beaten, real-world mature market efficiency-rather than full convergence under the Efficient Market Hypothesis.