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.

Keywords: Generalized Tempered Stable distribution, fractional Fourier transform, maximum likelihood estimation, heavy-tailed distributions, financial returns, Equities & Cryptocurrencies

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail
  • Why: The paper is highly theoretical with dense mathematics, featuring complex Lévy measure expressions, characteristic exponents, and cumulative functions derived via fractional Fourier transforms, but it also includes concrete backtests on financial data (Bitcoin, S&P 500) and standard goodness-of-fit metrics, making it mathematically advanced yet empirically grounded.
  flowchart TD
    A["Research Goal: Fit GTS distribution<br>to financial data"] --> B["Data Collection<br>Bitcoin, Ethereum, S&P 500, SPY Returns"]
    B --> C["Methodology: MLE & Fractional Fourier Transform"]
    C --> D{"Parameter Estimation"}
    D -->|Success| E["Statistical Testing<br>K-S, A-D, Chi-squared"]
    D -->|Failure| F["Refine Parameters"]
    F --> C
    E --> G["Key Findings"]
    G --> H["GBM Hypothesis Rejected"]
    G --> I["GTS Outperforms<br>Kobol, CGMY, Bilateral Gamma"]