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

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