Fitting Generalized Tempered Stable distribution: Fractional Fourier Transform (FRFT) Approach

TL;DR

This paper introduces a novel approach using Fractional Fourier Transform to fit the Generalized Tempered Stable distribution, effectively modeling asset returns like SPY ETF and Bitcoin with improved accuracy.

Contribution

It develops a new FRFT-based method for fitting GTS distributions and demonstrates its effectiveness on real financial data, outperforming traditional models.

Findings

GTS distribution fits SPY ETF return distribution well

Bitcoin BTC's right tail and S&P 500's left tail are modeled by GTS

Certain return segments are better modeled by compound Poisson processes

Abstract

The paper investigates the rich class of Generalized Tempered Stable distribution, an alternative to Normal distribution and the $\alpha$-Stable distribution for modelling asset return and many physical and economic systems. Firstly, we explore some important properties of the Generalized Tempered Stable (GTS) distribution. The theoretical tools developed are used to perform empirical analysis. The GTS distribution is fitted using S&P 500, SPY ETF and Bitcoin BTC. The Fractional Fourier Transform (FRFT) technique evaluates the probability density function and its derivatives in the maximum likelihood procedure. Based on the results from the statistical inference and the Kolmogorov-Smirnov (K-S) goodness-of-fit, the GTS distribution fits the underlying distribution of the SPY ETF return. The right side of the Bitcoin BTC return, and the left side of the S&P 500 return underlying distributions fit the Tempered Stable distribution; while the left side of the Bitcoin BTC return and the right side of the S&P 500 return underlying distributions are modelled by the compound Poisson process