方差-Gamma（VG）模型：分数傅立叶变换（FRFT）
Variance-Gamma (VG) model: Fractional Fourier Transform (FRFT)
摘要：
本文研究了分数阶傅里叶变换（FRFT）技术作为一种获得概率密度函数及其导数的工具，主要用于拟合具有无限可除性的基本概率关系的随机模型。计算了方差-伽马（VG）模型的概率密度函数，并讨论了VG模型的分布特性。VG模型越来越多地被用作经典对数正态模型（CLM）的替代品，用于模拟资产价格。VG模型由FRFT估计。数据来自SPY ETF历史数据。Kolmogorov-Smirnov（KS）拟合优度表明VG模型比CLM模型更好地拟合样本数据的累积分布。最好的VG模型来自FRFT估计。
英文摘要：
The paper examines the Fractional Fourier Transform (FRFT) based technique as a tool for obtaining the 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 are reviewed for Variance-Gamma (VG) model. The VG model has been increasingly used as an alternative to the Classical Lognormal Model (CLM) in modelling asset prices. The VG model was estimated by the FRFT. The data comes from the SPY ETF historical data. The Kolmogorov-Smirnov (KS) goodness-of-fit shows that the VG model fits the cumulative distribution of the sample data better than the CLM. The best VG model comes from the FRFT estimation.
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