Multifractality and sample size influence on Bitcoin volatility patterns

ArXiv ID: 2511.03314 “View on arXiv”

Authors: Tetsuya Takaishi

Abstract

The finite sample effect on the Hurst exponent (HE) of realized volatility time series is examined using Bitcoin data. This study finds that the HE decreases as the sampling period $Δ$ increases and a simple finite sample ansatz closely fits the HE data. We obtain values of the HE as $Δ\rightarrow 0$, which are smaller than 1/2, indicating rough volatility. The relative error is found to be $1%$ for the widely used five-minute realized volatility. Performing a multifractal analysis, we find the multifractality in the realized volatility time series, smaller than that of the price-return time series.

Keywords: Rough volatility, Hurst exponent, Realized volatility, Multifractal analysis, Bitcoin, Cryptocurrency

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced multifractal analysis with specialized estimators (MFDFA) and mathematical derivations for finite sample effects, but is heavily grounded in empirical Bitcoin data with detailed backtesting procedures for realized volatility.

  flowchart TD
    A["Research Goal: Examine finite sample effect on Hurst Exponent of Bitcoin realized volatility"]
    B["Data Input: Bitcoin High-Frequency Price Data"]
    C["Methodology: Compute Realized Volatility at various sampling intervals Δ"]
    D["Computational Process 1: Calculate Hurst Exponent for each Δ"]
    E["Computational Process 2: Perform Multifractal Analysis"]
    F["Outcome 1: HE decreases as Δ increases; HE→0 is < 0.5 (Rough Volatility)"]
    G["Outcome 2: Multifractality lower than price returns; 5-min RV relative error 1%"]
    
    A --> B
    B --> C
    C --> D
    C --> E
    D --> F
    E --> G