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Scaling properties of extreme price fluctuations in Bitcoin markets

Stjepan Begušić, Zvonko Kostanjčar, H. Eugene Stanley, Boris Podobnik

arXiv:1803.08405v1q-fin.ST

TL;DR

Scaling behavior in cryptocurrency return tails had not been reported, motivating a study of Bitcoin across exchanges and time intervals. Using trade data and tail-estimation methods, the paper finds power-law tails with 2 < α < 2.5 and a finite second moment.

  • Problem

    Although other statistical features of cryptocurrency returns had been analyzed, evidence on scaling properties in their distribution tails had not been reported.

    Source: §1 · p. 2

  • Method

    The paper analyzes Bitcoin returns across exchanges and time intervals, estimating tail exponents with two methods and statistically testing the power-law hypothesis using resampling.

    Source: §3 · p. 4, §5 · p. 8

  • Results

    2 < α < 2.5 characterizes the reported Bitcoin return tails, with power-law decay found across multiple exchanges and intervals and a finite second moment.

    Source: §1 · p. 2, §4 · p. 7, §5 · p. 8

  • Takeaways & Limitations

    The finite second moment is a foundation for risk estimation methods and portfolio optimization scenarios that include Bitcoin.

    Source: §1 · p. 2, §4 · p. 7

Abstract

from arXiv · show

Detection of power-law behavior and studies of scaling exponents uncover the characteristics of complexity in many real world phenomena. The complexity of financial markets has always presented challenging issues and provided interesting findings, such as the inverse cubic law in the tails of stock price fluctuation distributions. Motivated by the rise of novel digital assets based on blockchain technology, we study the distributions of cryptocurrency price fluctuations. We consider Bitcoin returns over various time intervals and from multiple digital exchanges, in order to investigate the existence of universal scaling behavior in the tails, and ascertain whether the scaling exponent supports the presence of a finite second moment. We provide empirical evidence on slowly decaying tails in the distributions of returns over multiple time intervals and different exchanges, corresponding to a power-law. We estimate the scaling exponent and find an asymptotic power-law behavior with 2 < {\alpha} < 2.5 suggesting that Bitcoin returns, in addition to being more volatile, also exhibit heavier tails than stocks, which are known to be around 3. Our results also imply the existence of a finite second moment, thus providing a fundamental basis for the usage of standard financial theories and covariance-based techniques in risk management and portfolio optimization scenarios.

1. Introduction

Stock returns provide a prominent power-law benchmark, but scaling properties in cryptocurrency return tails had not been reported. This study examines Bitcoin across exchanges and time intervals, finding heavier-than-stock power-law tails with a finite second moment.

1. Introduction

  • Stock price-fluctuation tails follow an inverse cubic law, with cumulative distributions decaying at α ≃ 3 across high-frequency to daily intervals.The finite second moment associated with this pattern has direct implications for risk management and portfolio optimization using covariance-based methods.Source: §1 · p. 1
  • Scaling properties in cryptocurrency return tails had not been reported, despite analyses of other statistical features.The paper frames this evidence gap as relevant to understanding cryptocurrency price fluctuations and investment approaches.Source: §1 · p. 2
  • 2 < α < 2.5 describes Bitcoin’s estimated tail exponent, with power-law behavior reported across multiple exchanges and time intervals.The exponent is below the stock inverse-cubic benchmark but outside the Levy-stable region, implying a finite second moment.Source: §1 · p. 2

2. Data

The analysis uses trade-level BTC/USD data from five exchanges, spanning different operating periods and substantial changes in trading activity. Returns are constructed across time intervals and standardized to make distributions comparable across exchanges and intervals.

2. Data

  • The study analyzes trade-level BTC/USD data from five large Bitcoin exchanges—Mt. Gox, BTC-e, Bitstamp, Bitfinex, and Kraken—with each data point recording a trade price in U.S. dollars and traded quantity.Data availability varies across time periods.Source: §2 · p. 2
  • Figure 1 plots daily trade counts and 30-day moving averages; gaps indicate missing dataset periods.Counts are shown on a logarithmic scale, with transparent lines for daily counts and solid lines for moving averages.Source: Figure 1 · p. 3
  • Trading activity shifted from the 2013 boom and subsequent stagnation to expansion in activity and exchange count in 2017.The considered period spans seven years, and exchanges differ in their operating periods and user bases.Source: §2 · p. 3
  • Returns are calculated from prices sampled at fixed intervals ranging from 1 minute to 1 day, using over 80 million trades.Each interval’s price is the last traded price during the preceding interval; the text introduces log returns as changes in log price.Source: §2 · p. 3
  • Returns for each exchange and interval are standardized by subtracting the mean and dividing by the standard deviation.This standardization allows empirical distributions to be compared across different intervals and exchanges.Source: §2 · p. 3

3. Estimation of tail exponents in distributions of price fluctuations

The section outlines methods for estimating Bitcoin return-tail exponents and testing whether the observed tails plausibly follow a power law. It describes a cutoff-selection procedure and a resampling-based hypothesis test.

3. Estimation of tail exponents in distributions of price fluctuations

  • α ≃3 characterizes the inverse cubic law in cumulative financial-return tails, a scaling behavior reported across time scales from 1 minute to 1 day.The corresponding probability-density tails decay with α + 1 ≃4.Source: §3 · p. 4
  • Tail-exponent estimation requires selecting a lower bound xmin, after which log-log regression or the Hill maximum-likelihood estimator can be applied.The Hill estimator assumes the tail data follow a power law.Source: §3 · p. 4
  • The Hill estimator uses n tail observations and is asymptotically normal and consistent as n grows, with standard error ˆα/√n.Source: §3 · p. 4
  • The cutoff xmin is selected by minimizing the Kolmogorov–Smirnov distance between empirical and fitted distributions over x ≥ xmin.Source: §3 · p. 4
  • The resampling test compares the empirical fit's KS statistic with statistics from 1,000 synthetic power-law datasets to assess the null hypothesis of power-law data.A small p-value leads to rejection; the analyses use a 95% confidence level.Source: §3 · p. 4

4. Results

Bitcoin return tails show power-law behavior across exchanges and time intervals, with estimated intraday exponents mostly between 2 and 2.5. Tests find no significant evidence that exponents change over time or with liquidity, while estimates at longer intervals fluctuate amid scarce data.

4. Results

  • Estimated exponents drift upward over time, but regression tests cannot reject a zero time trend; power-law tests reject none of 30 hypotheses after Bonferroni correction.Before multiple-test correction, 28 of 30 tests could not reject the power-law hypothesis at 95% confidence.Source: §4 · p. 5, Table 2 · p. 5
  • Liquidity correlations are positive but not statistically significant, providing insufficient evidence that trading volume affects scaling exponents.The correlations are 0.22 for positive and 0.19 for negative tails, with p-values of 0.13 and 0.19, respectively.Source: §4 · p. 5
  • At 1-, 5-, and 10-minute intervals, return tails across exchanges decay more slowly than Gaussian tails and show similar behavior.Figure 2 compares the exchange tails with fitted power laws and a standard-normal reference.Source: §4 · p. 5, Figure 2 · p. 6
  • Intraday Bitcoin return exponents generally fall between 2 and 2.5 across exchanges, and most tests do not reject power-law tails.The estimates show no obvious positive-negative tail asymmetry; Table 3 and Table 4 report the high-frequency exponent estimates.Source: §4 · p. 6, Table 3 · p. 6, Table 4 · p. 7
  • Exponents outside the Levy-stable range imply a finite second moment for Bitcoin return distributions.This matters for risk modeling scenarios.Source: §4 · p. 7
  • At longer intervals, most exponent estimates remain between 2 and 2.5, but estimates fluctuate strongly as data become scarce.The paper states that decade-scale investigations of these returns are not yet possible for this developing asset class.Source: §4 · p. 7, Figure 3 · p. 7

5. Conclusion

Across five Bitcoin exchanges and return intervals from 1 minute to 1 day, the authors find power-law tails with exponents 2 < α < 2.5. They report heavier tails than stock returns, alongside a finite second moment.

5. Conclusion

  • 2 < α < 2.5, and the authors report power-law tails in Bitcoin return distributions across multiple exchanges and intervals from 1 minute to 1 day.The analysis uses over 80 million trades from five exchanges over 2010–2017.Source: §5 · p. 8
  • Bitcoin returns have heavier tails than stock returns, for which the paper cites the inverse cubic law.The authors state that extreme price fluctuations are considerably more frequent for Bitcoin.Source: §5 · p. 8
  • The estimated exponents imply a finite second moment for Bitcoin price fluctuations, with the authors also noting a central-limit-theorem implication and convergence to a Gaussian limit.Source: §5 · p. 8
  • The findings provide a basis for using standard financial theories and covariance-based techniques in risk management and portfolio optimization scenarios.Source: §5 · p. 8
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