Fitting Tempered Stable Return Distributions with the Fractional Fourier Transform

Summary

This study examines the Generalized Tempered Stable distribution as a model for asset returns, comparing it with Normal and alpha-stable alternatives. It describes properties of the distribution and fits it to returns for the S&P 500, the SPY ETF, and Bitcoin. In the maximum-likelihood procedure, the Fractional Fourier Transform is used to evaluate the probability density function and its derivatives. The Kolmogorov-Smirnov test is used to assess goodness of fit.

The reported findings vary across assets and sides of their return distributions. The GTS distribution fits SPY ETF returns, while the Bitcoin right tail and S&P 500 left tail are described as fitting a Tempered Stable distribution. The Bitcoin left tail and S&P 500 right tail are instead modeled by a compound Poisson process. These are sample-specific conclusions; the brief account gives no sample period or further comparison metrics, so it does not establish that any one distribution is generally best for return modeling.

Key ideas

• The study evaluates Generalized Tempered Stable distributions as models of asset returns.
• The sample includes S&P 500, SPY ETF, and Bitcoin returns.
• Fractional Fourier Transform calculations support maximum-likelihood fitting of the density and its derivatives.
• The Kolmogorov-Smirnov test is used to assess fit.
• The preferred model differs between assets and between the left and right sides of return distributions.

Tags

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


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









The paper investigates the rich class of Generalized Tempered Stable distribution, an alternative to Normal distribution and the $α$-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

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.