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Fitting the seven-parameter Generalized Tempered Stable distribution to the financial data

Aubain Nzokem, Daniel Maposa

TL;DR

This work develops a FRFT-based maximum likelihood framework to fit the seven-parameter Generalized Tempered Stable distribution to financial returns, addressing the absence of closed-form densities. By modeling returns as a Lévy process with tempered stable components, the authors estimate seven parameters and assess fit against standard models using KS, AD, and Pearson tests. Empirical results on BTC, ETH, S&P 500, and SPY show that GTS can capture skewness and heavy tails, with ML estimates generally indicating negative location and significant tail/scale parameters; however, some stability indices may be insignificant and competitive alternative distributions (Kobol/CGMY) can perform similarly in certain cases. The study highlights both the methodological strength and computational demands of GTS fitting, and points toward future applications in simulating cumulative returns via OU-type processes.

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.

Fitting the seven-parameter Generalized Tempered Stable distribution to the financial data

TL;DR

This work develops a FRFT-based maximum likelihood framework to fit the seven-parameter Generalized Tempered Stable distribution to financial returns, addressing the absence of closed-form densities. By modeling returns as a Lévy process with tempered stable components, the authors estimate seven parameters and assess fit against standard models using KS, AD, and Pearson tests. Empirical results on BTC, ETH, S&P 500, and SPY show that GTS can capture skewness and heavy tails, with ML estimates generally indicating negative location and significant tail/scale parameters; however, some stability indices may be insignificant and competitive alternative distributions (Kobol/CGMY) can perform similarly in certain cases. The study highlights both the methodological strength and computational demands of GTS fitting, and points toward future applications in simulating cumulative returns via OU-type processes.

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, . 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.

Paper Structure

This paper contains 28 sections, 10 theorems, 72 equations, 10 figures, 23 tables.

Table of Contents

  1. Introduction
  2. Generalized Tempered Stable (GTS) Distribution
  3. GTS Distribution and Characteristic Exponent
  4. GTS Distribution and Lévy Process
  5. Multivariate Maximum Likelihood Method
  6. Maximum Likelihood Method: Numerical Approach
  7. Asymptotic Distribution of the Maximum Likelihood Estimator (MLE)
  8. Asymptotic Test and Confidence Interval
  9. Applications of the Log-likelihood Estimator to the Normal Distribution
  10. Fitting Tempered Stable Distribution to Cryptocurrencies: Bitcoin BTC and Ethereum
  11. Data Summaries
  12. Multidimensional Estimation Results for Cryptocurrencies
  13. Evaluation of the Method of Moments
  14. Fitting Tempered Stable Distribution to Traditional Indices: S&P 500 and SPY EFT
  15. Data Summaries
  16. ...and 13 more sections

Key Result

Theorem 1

Consider a variable $Y \sim GTS(\mu, \beta_{+}, \beta_{-}, \alpha_{+},\alpha_{-}, \lambda_{+}, \lambda_{-})$. The characteristic exponent can be written as:

Figures (10)

  • Figure 1: Daily Price
  • Figure 2: Daily Return
  • Figure 3: $\frac{\frac{df(x, V)}{dV_j}}{f(x, V)}$: Effect of parameters on the GTS probability density (Bitcoin Returns)
  • Figure 4: $\frac{\frac{df(x, V)}{dV_j}}{f(x, V)}$: Effect of parameters on the GTS probability density (Ethereum Returns)
  • Figure 5: Daily Price
  • ...and 5 more figures

Theorems & Definitions (15)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 4
  • proof : Proof:
  • Theorem 5
  • proof : Proof:
  • ...and 5 more