Generalized Tempered Stable Distribution

Comparing Bitcoin and Ethereum tail behavior via Q-Q analysis of cryptocurrency returns ArXiv ID: 2507.01983 “View on arXiv” 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. ...

Fitting the seven-parameter Generalized Tempered Stable distribution to the financial data ArXiv ID: 2410.19751 “View on arXiv” Authors: Unknown Abstract The paper proposes and implements a methodology to fit a seven-parameter Generalized Tempered Stable (GTS) distribution to financial data. The nonexistence of the mathematical expression of the GTS probability density function makes the maximum likelihood estimation (MLE) inadequate for providing parameter estimations. Based on the function characteristic and the fractional Fourier transform (FRFT), we provide a comprehensive approach to circumvent the problem and yield a good parameter estimation of the GTS probability. The methodology was applied to fit two heavily tailed data (Bitcoin and Ethereum returns) and two peaked data (S&P 500 and SPY ETF returns). For each index, the estimation results show that the six-parameter estimations are statistically significant except for the local parameter, $μ$. The goodness-of-fit was assessed through Kolmogorov-Smirnov, Anderson-Darling, and Pearson’s chi-squared statistics. While the two-parameter geometric Brownian motion (GBM) hypothesis is always rejected, the GTS distribution fits significantly with a very high p-value; and outperforms the Kobol, Carr-Geman-Madan-Yor, and Bilateral Gamma distributions. ...