Paper: arXiv 2507.01983

Authors: A. H. Nzokem

Abstract

The cryptocurrency market presents both significant investment opportunities and higher risks relative to traditional financial assets. This study examines the tail behavior of daily returns for two leading cryptocurrencies, Bitcoin and Ethereum, using seven-parameter estimates from prior research, which applied the Generalized Tempered Stable (GTS) distribution. Quantile-quantile (Q-Q) plots against the Normal distribution reveal that both assets exhibit heavy-tailed return distributions. However, Ethereum consistently shows a greater frequency of extreme values than would be expected under its Bitcoin-modeled counterpart, indicating more pronounced tail risk.

Complexity vs Empirical Score

  • Math Complexity: 7.0/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced stochastic calculus with the 7-parameter Generalized Tempered Stable distribution, including Lévy measures, characteristic exponents, and quantile estimation theorems, indicating high mathematical density. Empirical rigor is strong due to detailed parameter estimation with standard errors, goodness-of-fit tests, and specific data sourcing, though it lacks full backtest code or implementation details.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Compare Bitcoin & Ethereum Tail Risk"] --> B["Method: Q-Q Analysis<br>vs. Normal Distribution"]
  B --> C["Data Input:<br>Daily Returns"]
  C --> D["Computational Process:<br>Fit & Compare with<br>GTS-Distribution Parameters"]
  D --> E["Outcome 1:<br>Heavy Tails for Both"]
  D --> F["Outcome 2:<br>Ethereum Shows<br>More Extreme Values"]

Cited References (127)

  1. The generalised hyperbolic distribution and its subclass in the analysis of a new era of cryptocurrencies: Ethereum and its financial risk (2019) — link
  2. Ethereum: The OG Smart Contract Blockchain (2025) — link
  3. The basics of bitcoins and blockchains: an introduction to cryptocurrencies and the technology that powers them (2018)
  4. Bitcoin: A peer-to-peer electronic cash system (2008)
  5. Tail weight measures for distributions. A new tail weight coefficient (2017)
  6. Heavy-tailed distributions: properties and tests (1974)
  7. Testing For Normality (2002) — link
  8. Market uncertainty and correlation between Bitcoin and Ether (2022) — link
and 119 more
  1. Modelling the volatility of Bitcoin returns using GARCH models (2019) — link
  2. Measuring heavy-tailedness of distributions (2017) — link
  3. Robust measures of tail weight (2006) — link
  4. Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio Selection, and Option Pricing Ed. 1 (2005)
  5. THE VARIATION OF CERTAIN SPECULATIVE PRICES (1963)
  6. On the meaning and use of kurtosis. (1997)
  7. A Measure for Peakedness (1983) — link
  8. A note on finding peakedness in bivariate normal distribution using Mathematica (2007)
  9. A new index for measuring the peakedness of a probability distribution (2024)
  10. Cryptocurrency Market Statistics (2025) — link
  11. A New Measure of the Tail-Heaviness of a Probability Distribution (2024)
  12. Simple tests for peakedness, fat tails and leptokurtosis based on quantiles (2003)
  13. On measuring skewness and kurtosis (2009)
  14. Kurtosis and Spread (1990)
  15. Kurtosis: a critical review (1988)
  16. Kurtosis as Peakedness, 1905–2014. R.I.P. (2014) — link
  17. What Is Kurtosis?: An Influence Function Approach (1987)
  18. Inference for quantile measures of kurtosis, peakedness, and tail weight (2017) — link
  19. Measuring skewness and kurtosis (1984)
  20. On more robust estimation of skewness and kurtosis (2004)
  21. A comparison of some new measures of skewness (2003)
  22. Measuring skewness with respect to the mode (1995)
  23. Measuring skewness: a forgotten statistic? (2011)
  24. Five-parameter Variance-Gamma Process: Lévy versus probability density (2024) — link
  25. Q-Q Plots with Confidence Bands for Comparing Several Populations (1982)
  26. Positions and QQ Plots (2004)
  27. Variations of Q–Q Plots: The Power of Our Eyes! (2016) — link
  28. Using the normal quantile plot to explore meta-analytic data sets. (1998)
  29. Tempering stable processes (2007) — link
  30. Foundations of modern probability (1997)
  31. Series representations of L 'e vy processes from the perspective of point processes (2001)
  32. On simulation of tempered stable random variates (2011)
  33. Representing the CGMY and Meixner L 'e vy processes as time changed Brownian motions (2008)
  34. Tempered stable distributions and processes in finance: numerical analysis (2010)
  35. Monte Carlo Option Pricing for Tempered Stable (CGMY) Processes (2006) — link
  36. Q-Q Plot (Quantile to Quantile Plot) (2008) — link
  37. Probability plotting methods for the analysis for the analysis of data (1968) — link
  38. A simple method for generating gamma variables (2000)
  39. Stochastic simulation (2009)
  40. Sampling exponentially tilted stable distributions (2011-12) — link
  41. Efficiently sampling nested Archimedean copulas (2011) — link
  42. A note on Bayesian nonparametric priors derived from exponentially tilted Poisson–Kingman models (2007) — link
  43. Generalized gamma measures and shot-noise Cox processes (1999)
  44. Generating Random Numbers and Random Variables (2003) — link
  45. Random variate generation for exponential and gamma tilted stable distributions (2021)
  46. Exact simulation for a class of tempered stable distributions (2018)
  47. Fitting Generalized Tempered Stable distribution: Fractional Fourier Transform (FRFT) Approach (2022) — link
  48. Enhanced the Fast Fractional Fourier Transform (FRFT) scheme using the closed Newton-Cotes rules (2023) — link
  49. The Ornstein–Uhlenbeck Process and Variance Gamma Process: Parameter Estimation and Simulations (2023) — link
  50. Self-Decomposable Laws Associated with General Tempered Stable (GTS) Distribution and their Simulation Applications (2024) — link
  51. Bitcoin versus S & P 500 Index: Return and Risk Analysis (2024) — link
  52. European Option Pricing Under Generalized Tempered Stable Process: Empirical Analysis (2023) — link
  53. Numerical solution of a Gamma - integral equation using a higher order composite Newton-Cotes formulas (2021) — link
  54. Financial volatility, L 'e vy processes and power variation
  55. Random variate generation for exponentially and polynomially tilted stable distributions (2009)
  56. Series Representations of L 'e vy Processes from the Perspective of Point Processes (2001) — link
  57. Tempered stable models in finance: theory and applications (2009)
  58. Tempering stable processes (2007)
  59. Financial modeling with L 'e vy processes: Lecture notes (2010)
  60. An introduction to L' e vy processes with applications in finance (2008) — link
  61. Fluctuations of L 'e vy processes with applications: Introductory Lectures (2014)
  62. An introduction to probability theory and its applications, Volume 2 (1971)
  63. Chance and Stability: Stable Distributions and their Applications (2011) — link
  64. On the representation of stable laws by integrals (1964)
  65. Accurate and efficient numerical calculation of stable densities via optimized quadrature and asymptotics (2018)
  66. A note on the generation of random normal deviates (1958)
  67. Numerical calculation of stable densities and distribution functions (1997)
  68. A Method for Simulating Stable Random Variables (1976)
  69. Stable non-Gaussian random processes: stochastic models with infinite variance (2017) — link
  70. Univariate Stable Distributions: Models for Heavy Tailed Data (2020) — link
  71. Stable Distributions (2005) — [link](https://doi.org/10.1007/3-540-27395-6\textunderscore 1)
  72. One-dimensional stable distributions (1986)
  73. Precision and Accuracy (2007) — link
  74. On the Combinatorics of Cumulants (2000) — link
  75. Fourier transform methods in finance (2010)
  76. Fourier-Based Valuation Methods in Mathematical Finance (2014) — link
  77. Analysis of Fourier transform valuation formulas and applications (2010)
  78. Likelihood ratio tests for model selection and non-nested hypotheses (1989)
  79. Relations between cumulants and central moments and their applications (2021) — link
  80. Parametric Estimation of Tempered Stable Laws (2024) — link
  81. Simulating stock prices using geometric Brownian motion model under normal and convoluted distributional assumptions (2023) — link
  82. L 'e vy processes and infinitely divisible distributions (1999)
  83. A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa (1995) — link
  84. Bilateral gamma distributions and processes in financial mathematics (2008) — link
  85. A new tempered stable distribution and its application to finance (2009)
  86. Stable and Tempered Stable Distributions (2011) — link
  87. Stochastic volatility for L 'e vy processes (2003)
  88. Non-Gaussian Merton-Black-Scholes Theory (2002)
  89. Gamma variance model: Fractional Fourier Transform ( FRFT ) (2021) — link
  90. Modelling tail risk with tempered stable distributions: an overview (2021)
  91. Quantile-based inference for tempered stable distributions (2019)
  92. Handbook of Heavy-Tailed Distributions in Asset Management and Risk Management (2019) — link
  93. Basic Results on L 'e vy Processes (2001) — link
  94. Elements of large-sample theory (1999)
  95. A Course in the Large Sample Theory of Statistical Inference (2023)
  96. Statistical theory and inference (2014)
  97. Statistical Inference (2024) — link
  98. Anderson–Darling Test (2008) — link
  99. Anderson–Darling Tests of Goodness-of-Fit (2011) — link
  100. Sulla determinazione empirica di una legge didistribuzione (1933)
  101. Evaluating the anderson-darling distribution (2004)
  102. EDF statistics for goodness of fit and some comparisons (1974)
  103. Distribution of the Anderson-Darling statistic (1961)
  104. Pricing European Options under Stochastic Volatility Models: Case of Five-Parameter Variance-Gamma Process (2023) — link
  105. L 'e vy distributions (1997) — link
  106. Tempered stable distributions and processes (2013)
  107. Do financial returns have finite or infinite variance? A paradox and an explanation (2010)
  108. The advanced theory of statistics (1945)
  109. The variance gamma process and option pricing (1998)
  110. L 'e vy processes in finance: pricing financial derivatives (2003)
  111. The Kolmogorov-Smirnov test for goodness of fit (1951)
  112. Evaluating Kolmogorov’s distribution (2003)
  113. Computing the Kolmogorov-Smirnov distribution when the underlying CDF is purely discrete, mixed, or continuous (2020)
  114. Rachunek prawdopodobie 'n stwa i statystyka matematyczna w zadaniach (1999)
  115. Applied statistical inference (2014)
  116. Fitting Infinitely divisible distribution: Case of Gamma-Variance Model (2021) — link
  117. Wiley Series in Computational Statistics (2013)
  118. Stochastic and Renewal Methods Applied to Epidemic Models (2020) — link
  119. The Application of Statistical Analysis to the Cryptocurrency Market (2022)

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