Gamma variance model: Fractional Fourier Transform (FRFT)

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 prices. The Kolmogorov-Smirnov (KS) goodness-of-fit shows that the VG model fits better the cumulative distribution of the sample data than the CLM. The best VG model comes from the FRFT estimation.


Introduction
Several empirical studies have shown that asset returns are often characterized by peakedness, leptokurtosis and asymmetry. These facts provide evidence suggesting the assumptions of the Classical Lognormal Model (CLM) are not consistent with the empirical observations. A natural generalization of the CLM is the method of subordination [1,2], which has been used to reduce the theoretical-empirical gap. The subordinated process is obtained by substituting the physical time in the CLM by any independent and stationary increments random process, called the subordinator. If we consider the random process to be a Gamma process, we have a Variance Gamma (VG) model, which is the model the paper will be investigating. The Variance Gamma (VG) model was proposed by Madan [3]. In contrast to the CLM, the VG model does not have an explicit closed form of the probability density function and its derivatives. In the paper, the VG model has five parameters: parameters of location (µ), symmetric (δ), volatility (σ), and the Gamma parameters of shape (α) and scale (θ). The VG model density function is proven to be (1).
The integral (1) makes it difficult to utilize the density function and its derivatives, and to perform the Maximum likelihood method. However, in the literature, many studies have found a way to circumvent the lack of closed form by decreasing the number of parameters and using approximation function or analytical expression with modified Bessel function. In fact, [4] developed a procedure to approximate (1) by Chebyshev Polynomials expansion. [5] and [6] got (1) by analytical expression with modified Bessel function of second kind and third kind respectively. [7] got (1) through Gauss-Laguerre quadrature approximation with Laguerre  (1) and its derivatives. The paper is structured as follows; the next section presents the analytical framework. The third section presents the Variance Gamma (VG) model and the sample data, before performing the parameter estimations of the VG model and the Kolmogorov-Smirnov (KS) goodness-of-fit test.

Fast Fourier Transform (FFT)
The continuous Fourier transform (CFT) of function f (t) and its inverse are defined by: where i is the imaginary unit. The Fast Fourier Transform (FFT) is commonly used to evaluate the integrals (2) . The fundamentally inflexible nature [8] of FFT is the main weakness of the algorithm. The advantages of computing with the FRFT [8] can be found at three levels: (1)  The FRFT is set up on n-long sequence (x 1 , x 2 , . . . , x n ) and the Discrete Fourier Transform (DFT), G k (x, δ), was shown in [9] to be a composition of DF T −1 and DF T .
where DF T −1 is the inverse of the Discrete Fourier Transform (DFT). We assume that F [f ](t) is zero outside the interval [− a 2 , a 2 ], and β = a n is the step size of the n input values F [f ](t); we define t j = (j − n 2 )β for 0 ≤ j < n. We have also γ as the step size of the n output values of f (t) and x k = (k − n 2 )γ for 0 ≤ k < n. By choosing the step size β on the input side and the step size γ in the output side, we fix the FRFT parameter δ = βγ 2π . The density function f at x k can be written as (4). The proof is provided in [10].
In order to perform f (t) function from the Fourier Transform (FT), we assume a = 20, n = 2048, β = γ = a n . For more detail on FRFT, see [9].

Variance Gamma (VG) Distribution
The Variance Gamma distribution is an infinitely divisible distribution. The Fourier transform function has an explicit closed form in (6).
When δ = 0, we have Symmetric Variance Gamma (SVG) Model. It can be shown by Cumulantgenerating function [11] that  Parameter values: µ = −2, δ = 0, σ = 1, α = 1, θ = 1. As shown in Fig 2, the probability density is left asymmetric and right asymmetric when the parameter (δ) is negative and positive respectively. Forδ = 0, the density function is symmetric, as shown in (7). The shape parameter (α) impacts the peakedness and tails of the distribution, as illustrated in Fig 3 and (7); heavier is the tails, shorter is the peakedness. θ and σ have the same impact on the distribution. As shown in (7), both change only the variance.

Model for asset Price
The VG model was introduced by Madan [3]. The asset price is modeled on business time (k) as follows. µ, δ ∈ R, σ > 0, α > 0 and θ > 0 {T k } is the activity time process, a non-negative stationary independent increment, called the subordinator. µ is the drift of the physical time scale t, δ is the drift of the activity time process, and σ is the volatility. The density of Y j and its Fourier transform were provided in (6). See Appendix A.1 in [10] for proof of (6). Y k is the return variable of the stock or index price, we have (10) from (8) and (9).
The Classical Lognormal Model (CLM) in (11) is a special case of the Variance -Gamma Model. See Appendix A.1 in [10] for proof of (11).

SPY ETF data
The data comes from the SPY ETF, called SPDR S&P 500 ETF (SPY). The SPY is an Exchange-Traded Fund (ETF) managed by State Street Global Advisors that tracks the Standard & Poor's 500 index (S&P 500 ), which comprises 500 large and mid-cap US stocks. The SPY ETF is a well-diversified basket of assets, listed on the New York Stock Exchange (NYSE). Like other ETFs, SPY ETF provides the diversification of a mutual fund and the flexibility of a stock. The SPY ETF data was extracted from Yahoo finance. The daily data was adjusted for splits and dividends. The period spans from January 4, 2010 to December 30, 2020. 2768 daily SPY ETF prices were collected, around 252 observations per year, over 10 years. The dynamic of daily adjusted SPY ETF price is provided in Fig 4. Let the number of observations N = 2768, and the daily observed SPY ETF price S j on day t j with j = 1, . . . , N ; t 1 is the first observation date (January 4, 2010) and t N is the last observation date (December 30, 2020). The daily SPY ETF log return (y j ) is computed as in (12).
The results of the daily SPY ETF return are shown in Fig 5.

Variance Gamma (VG) Model Estimations
From a probability density function f (y, V ) with parameter V of size (p = 5) and the sample data y of size (M = 2755), we define the Likelihood function and its derivatives.
dV k dV j are computed with the FRFT on each y i with 1 ≤ i ≤ M . See Fig 4, Fig 5, Appendix B.2 and Appendix C.3 in [10], these figures display the shape of the quantities df (y i ,V ) dV j , d 2 f (y i ,V ) dV k dV j , which can be Odd or Even functions. The Newton Raphson Iteration process in (16) was implemented on the score function (I ′ (y, V )), and the Fisher information matrix (I ′′ (y, V )).
With initial value σ = α = θ = 1, δ = µ = 0, the maximization procedure convergences after 21 iterations for Asymmetric Variance-Gamma Model (AVG). The result of the iteration Process (16) are shown in Table 1.  The estimation of other models are summarized in Table 2. The method of moments provides the initial values for AVG1 and SVG1 maximization procedure. The results are labeled AVG1 for Asymmetric VG Model and SVG1 for Symmetric VG Model. Another initial value was chosen: σ = α = θ = 1, δ = µ = 0. The results are labeled AVG2 and SVG2 respectively for Asymmetric VG and Symmetric VG Models. The Maximum Likelihood estimations are summarized in Table 2.

Comparison of Variance Gamma (VG) Models
which VG model estimation fits the empirical distribution was also considered. The Kolmogorov-Smirnov (KS) test was performed under the null hypothesis (H0) that the sample {y 1 , y 2 . . . y n } comes from VG model. The Kolmogorov-Smirnov (K-S) estimator (D n ) is defined in (17).
F n (x) denotes the empirical cumulative distribution and n is the sample size. The VG cumulative distribution function (F ) was computed with FRFT from its Fourier. The cumulative distribution of D n [12] under the null hypothesis was computedand the density function was deduced. The computed density function is shown in Fig 6. Under the null hypothesis (H0), D n has a positively skewed distribution with means (µ = 0.0165) and standard deviation (σ = 5 * 10 −3 ).  [13,14], d n can be estimated as follows.
The statistics estimation for SVG2 model is shown in [10]. d − n = max((1)) = 0.023629, d + n = max((2)) = 0.021986 and d n = 0.023629. See Appendix E.5, Table E.5 in [10]. For each model, KS-Statistics (d n ) and P values were computed, and the results are provided in Table 3. maximum likelihood method, the SVG2 model has P values = 9.079%, which is high than the classical threshold 5%. Therefore, SVG2 model can not be rejected. See [10] for d n and P values computations. The daily SPY ETF return histogram was compared to the density function of two models (SVG2, CLM), as shown in Fig 8 and Fig 9. It results that the peakedness of the histogram explains the high level of the KS-Statistics in Table 3 and model rejection. For work related to Normal and exponential distributions, see [15,16,17,18]

Conclusion
The paper explores the use of FRFT based technique as a tool to compute and analyse the probability density function of the Variance-Gamma (VG) model; and also to perform the estimation of five parameters of the VG model. The results show that the VG model captures the peakedness and leptokurtosis properties of the daily SPY sample data. The estimations provide evidence that the VG model fits better than the CLM Model. The Kolmogorov-Smirnov (KS) goodness-of-fit test shows that the Maximum Likelihood method with FRFT produces a good estimation of the VG model, which fits the empirical distribution of the sample data.