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

TL;DR

This paper introduces a novel methodology using fractional Fourier transform to fit a seven-parameter Generalized Tempered Stable distribution to financial data, overcoming the challenge of unknown probability density functions.

Contribution

It provides a new approach for estimating GTS distribution parameters without the density function, improving fit accuracy for heavy-tailed and peaked financial data.

Findings

GTS distribution fits financial data significantly better than alternative models.

The methodology yields statistically significant parameter estimates for most parameters.

GTS outperforms other distributions like Kobol and Bilateral Gamma in goodness-of-fit tests.

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, $\mu$. 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.