Fitting Infinitely divisible distribution: Case of Gamma-Variance Model

Abstract: 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.