  The paper examines the Fractional Fourier Transform (FRFT) based technique as
a tool for obtaining probability density function and its derivatives, and
mainly for fitting stochastic model with the fundamental probabilistic
relationships of infinite divisibility. The probability density functions are
computed and the distributional proprieties such as leptokurtosis, peakedness,
and asymmetry are reviewed for Variance-Gamma (VG) model and Compound Poisson
with Normal Compounding model. The first and second derivatives of probability
density function of the VG model are also computed in order to build the Fisher
information matrix for the Maximum likelihood method. The VG model has been
increasingly used as an alternative to the Classical Lognormal Model (CLM) in
modelling asset price. The VG model with fives parameters was estimated by the
FRFT. The data comes from the daily SPY ETF price data. The Kolmogorov-Smirnov
(KS) goodness-of-fit shows that the VG model fits better the empirical
cumulative distribution than the CLM. The best VG model comes from the FRFT
estimation.
