  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
