# Fitting Infinitely divisible distribution:
Case of Gamma-Variance Model

A.H.Nzokem1

###### Abstract.

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 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.

###### Key words and phrases:

Infinitely divisible distributions, Variance Gamma (VG) Model, Classical
Lognormal Model (CLM), Fractional Fourier Transform (FRFT), Maximum
likelihood, Function Characteristic, SPDR $S\&P500$ ETF (SPY), Kolmogorov-
Smirnov (K-S)

1<EMAIL_ADDRESS>
##  1\. Introduction

Empirical studies have shown that asset returns are often characterized by
peakedness, leptokurtosis and asymmetry. Most of these empirical properties
were reviewed in [1] as a set of stylized empirical facts that emerge from
price variations in various types of financial markets. These facts provide
evidence suggesting the assumptions of the Classical Lognormal Model (CLM) are
not consistent with the empirical observations. In order to reduce the
theoretical-empirical gap, it has been used in literature two sophisticated
stochastic processes with the fundamental probabilistic relationships of
infinite divisibility. A natural generalization of the Brownian motion, which
is the Lévy process. As a stochastic process, the Lévy process has independent
and stationary increments; and as a function, the process is right continuous
with left limit. However, the Lévy process model is usually incomplete, as the
associated risk-neutral equivalent martingale measure is usually not unique;
and this situation leads to many different possible prices for European
options[2, 3].
Another way to generalize the Classical Lognormal Model (CLM) is the method of
subordination [4, 5, 3]. 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
[6] as an alternative to the Classical Lognormal Model (CLM). In contrast to
the CLM, the VG model does not have an explicit closed-form for probability
density function and its derivatives. However, the fundamental probabilistic
relationship of infinite indivisibility allows the characteristic function of
the VG model and the Fourier of probability density function to have a closed-
form function. In the study, the VG model has five parameters: parameters of
location ($\mu$), symmetric ($\delta$), volatility ($\sigma$), and the Gamma
parameters of shape ($\alpha$) and scale ($\theta$). The VG model density
function is proven to be (1.1).

$\displaystyle f(y)$
$\displaystyle=\frac{1}{\sigma\Gamma(\alpha)\theta^{\alpha}}\int_{0}^{+\infty}\frac{1}{\sqrt{2\pi
v}}e^{-\frac{(y-\mu-\delta
v)^{2}}{2v\sigma^{2}}}v^{\alpha-1}e^{-\frac{v}{\theta}}\,dv $ (1.1)

The integral in (1.1) 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 making use of approximation
function or analytical expression with modified second kind Bessel function.
In fact, [7] developed a procedure to approximate (1.1) by Chebyshev
Polynomials expansion. [8] and [9] got (1.1) by analytical expression with
modified Bessel function of second kind and third kind respectively. [3] got
(1.1) through Gauss-Laguerre quadrature approximation with Laguerre polynomial
of degree $10$.[5] used the Fast Fourier Transform (FFT).
The paper will implement the Fractional Fourier Transform (FRFT)-based
technique on the Fourier Transform of the VG model function in order to obtain
probability density function and its derivatives. Contrary to the Fast Fourier
Transform (FFT), the advantage of using the FRFT mentioned in [10] can be
found at three levels:(1) both the input function values and the output
transform function values are equally spaced; (2) a large fraction of the
density function is either zero or smaller than the computer machine epsilon;
and (3) only a limited range of the density function are required.
The paper is structured as follows, the next section presents the analytical
framework, with an overview of the FRFT computing formulas, their applications
on three infinitely divisible distributions, and a review of the Maximum
likelihood method. The third section will present 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.

##  2\. Analytical Framework

The continuous Fourier transform (CFT) of function $f(t)$ and its inverse are
defined by:

$\displaystyle
\mathscr{F}[f](x)=\int_{-\infty}^{+\infty}\\!f(y)e^{-ixy}\,\mathrm{d}y$ (2.1)
$\displaystyle
f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\\!\mathscr{F}[f](y)e^{ixy}\,\mathrm{d}y$
(2.2)

where i is the imaginary unit.
The Fast Fourier Transform (FFT) is commonly used to evaluate the integrals
(2.1) and (2.2) . The fundamentally inflexible nature [10] of FFT is the main
weakness of the algorithm. The advantages of computing with the FRFT mentioned
in [10] can be found at three levels: (1) both the input function values
$f(x_{k})$ and the output transform values $\mathscr{F}[f](x_{k})$ are equally
spaced; (2) a large fraction of $f(x_{k})$ are either zero or smaller than the
computer machine epsilon; and (3) only a limited range of $f(x_{k})$ are
required.
The FRFT is set up on $n$-long sequence ($x_{1}$, $x_{2}$, …, $x_{n}$) by:

$ G_{k}(x,\delta)=\sum_{j=0}^{n-1}\\!x_{i}e^{-2\pi ijk\delta}\hskip 14.22636pt
\hbox{$0\leq k<M$} $ (2.3)

For $\delta=\frac{1}{n}$, we have the Discrete Fourier Transform (DFT). In the
general scheme, the parameter $\delta$ is not limited to fractions; it can
take complex number. It is shown in [11] that $ G_{k}(x,\delta)$ is a
composition of ${DFT}^{-1}$ and ${DFT}$.

$ G_{k}(x,\delta)=e^{-\pi
ik^{2}\delta}{DFT}_{k}^{-1}[{{DFT}_{j}(y){DFT}_{j}(z)}] $ (2.4)

where $y$ and $z$ are two 2n-long sequences defined by:

$\displaystyle y_{j}$ $\displaystyle=e^{-2\pi ij^{2}\delta} $
$\displaystyle\quad z_{j}$ $\displaystyle=e^{-2\pi ij^{2}\delta} $
$\displaystyle\quad \hskip 14.22636pt \hbox{$0\leq k<n$} $ (2.5)
$\displaystyle y_{j}$ $\displaystyle=0 $ $\displaystyle\quad z_{j}$
$\displaystyle=e^{-2\pi i(j-2m)^{2}\delta} $ $\displaystyle\quad \hskip
14.22636pt \hbox{$n\leq k<2n$} $ (2.6)

As shown in $(\ref{eq: l3})$, the FRFT is efficiently computed through the FFT
on 2n-point. We assume that $\mathscr{F}[f](t)$ is zero outside the interval
$[-\frac{a}{2},\frac{a}{2}]$, and $\beta=\frac{a}{n}$ is the step size of the
$n$ input values $\mathscr{F}[f](t)$; we define $t_{j}=(j-\frac{n}{2})\beta$
for $0\leq j<n$. We have also $\gamma$ as the step size of the $n$ output
values of $f(t)$ and $x_{k}=(k-\frac{n}{2})\gamma$ for $0\leq k<n$. By
choosing the step size $\beta$ on the input side and the step size $\gamma$ in
the output side, we fix the FRFT parameter $\delta=\frac{\beta\gamma}{2\pi}$.
The density function $f$ at $x_{k}$ can be written as (2.7). The proof is
provided in Appendix A.1.

$\displaystyle \hat{f}(x_{k})=\frac{\gamma}{2\pi}e^{-\pi
i(k-\frac{n}{2})n\delta}G_{k}(\mathscr{F}[f](y_{j})e^{-\pi
ijn\delta}),-\delta) $ (2.7)

In order to perform $f(t)$ function from the Fourier Transform (FT), we assume
$a=20$, $n=2048$, $\beta=\gamma=\frac{a}{n}$. For more detail on FRFT, See
[10], [11].

###  2.1. Infinitely Divisible Distributions

The FRFT probability density functions are computed for each of the three
infinitely divisible distributions: Normal distribution with two parameters,
Variance Gamma (VG) Distribution with five parameters, and Compound Poisson
with Normal compounding with three parameters. The distributional properties
are reviewed and the results also show that the Kurtosis statistics tell very
little about the peakedness[12].
For detailed proof of infinite divisibility for each distribution. See
[13],[14],[15]

####  2.1.1. Normal distribution

$\displaystyle X\sim\mu+N(0,\sigma^{2})$

The normal distribution is an infinitely divisible distribution. It was shown
in [13] that the normal distribution $N(\mu,\sigma)$ is the $n^{th}$
convolution power of a probability measure
$N(\frac{\mu}{n},\frac{\sigma}{n})$, for every n. The density and the Fourier
transform functions have an explicit closed form as shown below.

$\displaystyle\mathscr{F}[f](x)$
$\displaystyle=e^{-uix-\frac{1}{2}\sigma^{2}x^{2}}$ (2.8) $\displaystyle f(x)$
$\displaystyle=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(y-\mu)^{2}}{2\sigma^{2}}}$
(2.9)

For mean ($\mu=-2$) and standard deviation ($\sigma=1$), Fig 1(a) displays the
FRFT estimation of the Normal density function and Fig 1(b) shows the absolute
value between the estimations and the exact values of the normal density
function.

(a) $\hat{f}(t)$ with $\mu=-2$, $\sigma=1$

(b) $\sup_{t}{|f(t)-\hat{f}(t)|}=1.8738e-15$

Figure 1. FRFT estimation of the Normal density function

The highest computation error is $1.8738e-15$, which is negligible. The
probability density function is symmetric with the Kurtosis statistic $3$ for
the normal distribution.

####  2.1.2. Variance Gamma (VG) Distribution

$\displaystyle X$ $\displaystyle=\mu+\gamma V+\sigma\sqrt{V}Z$ $\displaystyle
Z$ $\displaystyle\sim N(0,1)\hskip 14.22636pt \hbox{and }\hskip 14.22636pt
V\sim\Gamma(\alpha,\theta) $

The Variance Gamma distribution is an infinitely divisible distribution. The
density function is (2.11) and does not have a closed form. However, the
Fourier transform function has explicit closed form (2.10). See Appendix A.1
for proof

$\displaystyle \mathscr{F}[f](x)$ $\displaystyle=\frac{e^{-i\mu
x}}{\left(1+\frac{1}{2}\theta\sigma^{2}x^{2}+i\gamma\theta x\right)^{\alpha}}
$ (2.10) $\displaystyle f(y)$
$\displaystyle=\frac{1}{\sigma\Gamma(\alpha)\theta^{\alpha}}\int_{0}^{+\infty}\frac{1}{\sqrt{2\pi
v}}e^{-\frac{(y-\mu-\delta
v)^{2}}{2v\sigma^{2}}}v^{\alpha-1}e^{-\frac{v}{\theta}}\,dv $ (2.11)

When $\delta=0$, we have Symmetric Variance Gamma (SVG) Model, which is a
special case of the Variance Gamma (VG) Model. It can be shown through
Cumulant-generating function[16] that

$\displaystyle E(X)$ $\displaystyle=\mu$ $\displaystyle Var(X)$
$\displaystyle=\alpha\theta\sigma^{2}$ $\displaystyle Skew(X)$
$\displaystyle=0$ $\displaystyle Kurt(X)$ $\displaystyle=3(1+\frac{1}{\alpha})
$ (2.12)

For Parameter values: $\mu=-2$, $\delta=0$, $\sigma=1$, $\alpha=1$,
$\theta=1$. Fig 2 displays the FRFT estimation of the probability density
functions. As shown in Fig 2(b), the probability density is left asymmetric
and right asymmetric when the parameter ($\delta$) is negative and positive
respectively. For $\delta=0$, the density function is symmetric, as shown in
$(\ref{eq:l011})$. The shape parameter ($\alpha$) impacts the peakedness and
tails of the distribution, as illustrated in Fig 2(c) and $(\ref{eq:l011})$;
heavier is the tails, shorter is the peakedness. $\theta$ and $\sigma$ have
the same impact on the distribution. As shown in $(\ref{eq:l011})$, both
change only the variance.

(a) $\hat{f}$:$\mu=-2$,$\delta=0$,$\sigma=1$,$\alpha=1$,$\theta=1$

(b) $\hat{f}(t)$ and symmetric parameter ($\delta$)

(c) $\hat{f}(t)$ and shape parameter ($\alpha$)

(d) $\hat{f}(t)$ and scale parameter ($\theta$)

Figure 2. FRFT estimation of the Variance Gamma density function

####  2.1.3. Compound Poisson with Normal Compounding

$\displaystyle X$ $\displaystyle=\sum_{i=1}^{N(\lambda)}X_{j}$ $\displaystyle
N(\lambda)\sim Poisson(\lambda)\hskip 14.22636pt $ $\displaystyle\hbox{and
}\hskip 14.22636pt X_{j}\sim\mu+N(0,\sigma^{2})$

The density function was shown in [17] to be (2.14). However, the Fourier
Transform density has explicit closed form $(\ref{eq:l12})$. See Appendix A.1
for proof.

$\displaystyle \mathscr{F}[f](x)$
$\displaystyle=e^{\lambda(e^{-uix-\frac{1}{2}\sigma^{2}x^{2}}-1)} $ (2.13)
$\displaystyle f(x)$
$\displaystyle=\sum_{n=1}^{\infty}\frac{e^{-\lambda}\lambda^{n}}{n!}\frac{e^{-\frac{1}{2}\frac{(x-n\mu)^{2}}{n\sigma^{2}}}}{\sigma\sqrt{2n\pi}}
$ (2.14)

It can be shown through Cumulant-generating function[16] that

$\displaystyle E(X)$ $\displaystyle=\lambda\mu$ $\displaystyle Var(X)$
$\displaystyle=\lambda(\mu^{2}+\sigma^{2})$ $\displaystyle Skew(X)$
$\displaystyle=\frac{\mu(\mu^{2}+3\sigma^{2})}{\sqrt{\lambda}(\mu^{2}+\sigma^{2})^{\frac{3}{2}}}$
$\displaystyle Kurt(X)$
$\displaystyle=3+\frac{\mu^{4}+6\sigma^{2}\mu^{2}+3\sigma^{4}}{\lambda(\mu^{2}+\sigma^{2})^{2}}
$ (2.15)

For Parameter values: $\mu=-0.1$, $\sigma=0.25$, $\lambda=16$. The FRFT
produces the density functions in Fig 3. As shown in Fig 3(b), the probability
density is asymmetric, shifts to the left and to the right when the parameter
($\mu$) is negative and positive respectively. For $\mu=0$, we have a
symmetric density function. $\mu$ and $\sigma$ impact the symmetry, peakedness
and tails of the distribution, as illustrated in Fig 3(c) and
$(\ref{eq:l013})$; heavier is the tails, shorter is the peakedness. The
Poisson parameter ($\lambda$) impacts all the statistics in $(\ref{eq:l013})$.
$\lambda$ affects the asymmetry, peakedness and tails of the distribution
simultaneously.

(a) $\hat{f}(t)$, $\mu=-0.25$, $\sigma=0.25$, $\lambda=8$

(b) $\hat{f}(t)$ and drift parameter ($\mu$)

(c) $\hat{f}(t)$ and Volatility parameter ($\sigma$)

(d) $\hat{f}(t)$ and Poisson parameter ($\lambda$)

Figure 3. FRFT estimation of the Compound Poisson with Normal Compounding
density function

###  2.2. Review of the Maximum Likelihood Method

From a probability density function $f(x,V)$ with parameter $V$ of size $p$
and the sample data $x$ of size $m$, we definite the Likelihood Function as
shown in (2.16).

$\displaystyle L(x,V)$ $\displaystyle=\prod_{j=1}^{m}f(x_{j},V) $ (2.16)
$\displaystyle l(x,V)$ $\displaystyle=\sum_{j=1}^{m}log(f(x_{j},V)) $ (2.17)

$\displaystyle \frac{dl(x,V)}{dV_{j}}$
$\displaystyle=\sum_{i=1}^{m}\frac{\frac{df(x_{i},V)}{dV_{j}}}{f(x_{i},V)}\hskip
14.22636pt \hbox{$1\leq j\leq p$}$ (2.18) $\displaystyle
\frac{d^{2}l(x,V)}{dV_{k}dV_{j}}$
$\displaystyle=\sum_{i=1}^{m}\left(\frac{\frac{d^{2}f(x_{i},V)}{dV_{k}dV_{j}}}{f(x_{i},V)}-\frac{\frac{df(x_{i},V)}{dV_{k}}}{f(x_{i},V)}\frac{\frac{df(x_{i},V)}{dV_{j}}}{f(x_{i},V)}\right)\hskip
14.22636pt \hbox{$1\leq k\leq p\ \ and\ \ 1\leq j\leq p$} $ (2.19)

In order to perform the Maximum of the likelihood function (2.16), the
quantities $\frac{dl(x,V)}{dV_{j}}$ and $\frac{d^{2}l(x,V)}{dV_{k}dV_{j}}$
must be computed. We need to compute $\frac{df(x,V)}{dV_{j}}$ and
$\frac{d^{2}f(x,V)}{dV_{k}dV_{j}}$, which are the first and second order
derivative of the density function respectively. Variance Gamma (VG)
distribution has five parameters ($\mu$, $\delta$, $\sigma$, $\alpha$,
$\theta$), using the closed form of Fourier transform in (2.10) and the FRFT
algorithm, we perform the first derivative functions:$\frac{df}{d\mu}$,
$\frac{df}{d\delta}$, $\frac{df}{d\sigma}$, $\frac{df}{d\alpha}$ and
$\frac{df}{d\theta}$. The results of the computation are shown in Fig 4.

(a) $\frac{d\hat{f}}{du}(t)$

(b) $\frac{d\hat{f}}{d\delta}(t)$

(c) $\frac{d\hat{f}}{d\alpha}(t)$

(d) $\frac{d\hat{f}}{d\theta}(t)$

Figure 4.  First order derivative of VG density function with $\mu=-2$,
$\delta=0$, $\sigma=1$, $\alpha=1$, $\theta=1$

As shown in Fig 4, we have two Odd functions in Fig 4(a) and 4(b) generated by
the drift parameters $\mu$ and $\delta$; and the remainings are Even
functions. The functions in Fig 4(c), and Fig 4(d) are symmetric along the
vertical line $x=-2$, whereas $\frac{df(x,V)}{d\mu}$ and
$\frac{df(x,V)}{d\delta}$ have rotational symmetry at the point ($-2$,$0$). In
addition, each graph has positive area values and negative area values. Having
such properties is a necessary condition to find
$V(\mu,\delta,\sigma,\alpha,\theta)$ such as
$(\frac{dl(x,v)}{d\mu},\frac{dl(x,v)}{d\delta},\frac{dl(x,v)}{d\sigma},\frac{dl(x,v)}{d\alpha},\frac{dl(x,v)}{d\theta})$
equals zero.
The quantities $\frac{d^{2}l(x,V)}{dV_{k}dV_{j}}$ in (2.19) are critical in
computing the Hessian Matrix and the Fisher Information Matrix. For VG
distribution, We need fifteen second derivative functions. The results of four
functions are shown in Fig 5. The remaining functions are reported in Appendix
B.2 and Appendix C.3.

(a) $\frac{d^{2}\hat{f}}{d^{2}\mu}(t)$

(b) $\frac{d^{2}\hat{f}}{d^{2}\delta}(t)$

(c) $\frac{d^{2}\hat{f}}{d^{2}\alpha}(t)$

(d) $\frac{d^{2}\hat{f}}{d^{2}\theta}(t)$

Figure 5. Second order derivative of VG density function with $\mu=-2$,
$\delta=0$, $\sigma=1$, $\alpha=1$, $\theta=1$

As shown in Fig 5, all four functions are symmetric along the vertical line
$x=-2$. We have the same characteristics for $\frac{d^{2}\hat{f}}{d\delta
d\mu}$, $\frac{d^{2}\hat{f}}{d^{2}\sigma}$, $\frac{d^{2}\hat{f}}{d\alpha
d\sigma}$, $\frac{d^{2}\hat{f}}{d\theta d\sigma}$,
$\frac{d^{2}\hat{f}}{d\theta d\alpha}$ in Appendix B.2 and Appendix C.3. The
remaining derivatives have rotational symmetry at the point ($-2$,$0$). These
features can be checked analytically through the Inverse Fourier Transform in
(2.2) and $\mathscr{F}[f]$ in (2.10).
Given the parameters $V(\mu,\delta,\sigma,\alpha,\theta)$ and the sample data
set $x$, from the previous development and computations, we have the
quantities:

$\displaystyle
I^{\prime}(x,V)=\left(\frac{dl(x,V)}{d\mu},\frac{dl(x,V)}{d\delta},\frac{dl(x,V)}{d\sigma},\frac{dl(x,V)}{d\alpha},\frac{dl(x,V)}{d\theta}\right)
$ (2.20) $\displaystyle I^{\prime\prime}(x,V)$
$\displaystyle=\begin{bmatrix}\frac{d^{2}l}{d^{2}\mu}&\frac{d^{2}l}{d\delta
d\mu}&\frac{d^{2}l}{d\sigma d\mu}&\frac{d^{2}l}{d\alpha
d\mu}&\frac{d^{2}l}{d\theta d\mu}\\\ \frac{d^{2}l}{d\delta
d\mu}&\frac{d^{2}l}{d^{2}\delta}&\frac{d^{2}l}{d\sigma d\delta}
&\frac{d^{2}l}{d\delta d\alpha}&\frac{d^{2}l}{d\theta d\delta}\\\
\frac{d^{2}l}{d\sigma d\mu}& \frac{d^{2}l}{d\sigma
d\delta}&\frac{d^{2}l}{d^{2}\sigma}&\frac{d^{2}l}{d\alpha
d\sigma}&\frac{d^{2}l}{d\theta d\sigma}\\\ \frac{d^{2}l}{d\alpha
d\mu}&\frac{d^{2}l}{d\alpha d\delta}&\frac{d^{2}l}{d\alpha
d\sigma}&\frac{d^{2}l}{d^{2}\alpha}&\frac{d^{2}l}{d\theta d\alpha}\\\
\frac{d^{2}l}{d\theta d\mu}&\frac{d^{2}l}{d\theta
d\delta}&\frac{d^{2}l}{d\theta d\sigma}&\frac{d^{2}l}{d\alpha
d\theta}&\frac{d^{2}l}{d^{2}\theta}\\\ \end{bmatrix} $ (2.21)

$I^{\prime}(x,V)$ is the score function with the property [18] that
$E(I^{\prime}(x,V))=0$. $-I^{\prime\prime}(x,V)$ is the observed Fisher
information and $I(V)=E(I^{\prime\prime}(x,V))$ is the Fisher information
matrix.
Maximizing L(x,V) in (2.16) with respect to $V$ is equivalent to solving the
equation system (2.22).

$\displaystyle I^{\prime}(x,V)=0 $ (2.22)

The solutions of (2.22) are provided by the Newton–Raphson Iteration Algorithm
(2.23). In fact, the $Taylor$’s expansion can be applied to the score function
to derive the Newton–Raphson Iteration Algorithm.

$\displaystyle
V^{n+1}=V^{n}+{\left(I^{\prime\prime}(x,V^{n})\right)^{-1}}I^{\prime}(x,V^{n})
$ (2.23)

For more detail on Maximum likelihood and Newton–Raphson Iteration procedure,
see [18],[19],[20],[21],[22].

##  3\. Fitting Variance Gamma (VG) Model

###  3.1. VG Model and Asset/Index price

The VG model was introduced by Madan et al[6, 9]. The idea [23] was to model
the asset price on bussiness time rather than on calender time. The asset
price is modelled on bussiness time $(k)$ as follows:

$\displaystyle Y_{k}=\mu+\delta V_{k}+\sigma\sqrt{V_{k}}Z\quad\quad Z\sim
N(0,1)\quad\quad V_{k}\sim\Gamma(\alpha,\theta) $ (3.1) $\displaystyle
S_{K}=S_{k-1}e^{\sum_{j=k}^{K}Y_{j}}\quad\quad T_{K}=\sum_{k=1}^{K}V_{k} $
(3.2)

Where $\mu,\delta\in R$ , $\sigma>0$, $\alpha>0$ and $\theta>0$. $\mu$ is the
drift of the physical time scale, $\delta$ is the drift of the activity time
process, and $\sigma$ is the volatility. $\\{T_{k}\\}$ is the activity time
process[24] or intrinsic time process[5] and is supposed to be non-negative
stationary independent increments. This process is associated with either the
flow of new price-sensitive information or the cumulative trading volume
process. $Y_{k}$ is the return variable of the stock or index price, we have
(3.3) from (3.1) and (3.2).

$\displaystyle Y_{k}=log(\frac{S_{k}}{S_{k-1}})\quad\quad E(Y_{k}|V_{k})\sim
N(\mu+\delta V_{k},V_{k}\sigma^{2})\quad\quad V_{k}\sim\Gamma(\alpha,\theta) $
(3.3)

The density of $Y_{j}$ in (3.1) and (3.3) has the following expression

$\displaystyle f(y)$
$\displaystyle=\frac{1}{\sigma\Gamma(\alpha)\theta^{\alpha}}\int_{0}^{+\infty}\frac{1}{\sqrt{2\pi
v}}e^{-\frac{(y-\mu-\delta
v)^{2}}{2v\sigma^{2}}}v^{\alpha-1}e^{-\frac{v}{\theta}}\,dv $ (3.4)

The Fourier transform of (3.4) can be shown to be .

$\displaystyle F[f](x)=E[e^{-ixY_{j}}]=\frac{e^{-i\mu
x}}{\left(1+\frac{1}{2}\theta\sigma^{2}x^{2}+i\delta\theta x\right)^{\alpha}}
$ (3.5)

See A.1 for proof (3.5).
For $\alpha=\frac{1}{\theta}$ and $0<\theta<<1$, $Y_{k}$ in (3.1) becomes

$\displaystyle Y_{k}\sim N(\mu+\delta,\sigma^{2}) $ (3.6)

See A.1 for proof (3.6)

The relationship between means $(E(Y_{j}))$, variance $(Var(Y_{j}))$, Skewness
$(Skew(Y_{j}))$ and Kurtosis $(Kurt(Y_{j}))$ indicators and the VG model with
five parameters can be shown by Cumulant-generating function[16].

$\displaystyle E(Y_{j})$ $\displaystyle=\mu+\alpha\theta\delta$ (3.7)
$\displaystyle Var(Y_{j})$
$\displaystyle=\alpha(\delta^{2}\theta^{2}+\theta\sigma^{2})$ (3.8)
$\displaystyle Skew(Y_{j})$
$\displaystyle=\frac{\delta(2\delta^{2}+3\frac{\sigma^{2}}{\theta})}{\alpha^{\frac{1}{2}}(\delta^{2}+\frac{\sigma^{2}}{\theta})^{\frac{3}{2}}}$
(3.9) $\displaystyle Kurt(Y_{j})$
$\displaystyle=3\left(1+\frac{2\delta^{4}+\frac{\sigma^{4}}{\theta^{2}}+\frac{4\sigma^{2}\delta^{2}}{\theta}}{\alpha(\delta^{2}+\frac{\sigma^{2}}{\theta})^{2}}\right)
$ (3.10)

By applying the method of Moments, the system of equations has more parameters
than equations. The issue can be overcome by imposing[3] $E(V)=1$ or
$\sigma=1$. $E(V)=1$ was chosen in [9, 25] and the implication was that the
gamma shape and scale parameters should be equal $(\alpha=\theta)$. In the
next section, we will assume $\sigma=1$ in order to solve the system of
equation (3.7)…(3.10). However, the constraints will be relaxed in the Maximum
Likelihood method where all five VG parameters will be estimated.

###  3.2. 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 is a well-diversified basket of assets, listed on the
New York Stock Exchange (NYSE). Like other ETFs, SPY provides the
diversification of a mutual fund and the flexibility of a stock.
The SPY 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 price was collected, around 252 observations per
year, over 10 years. The dynamic of daily adjusted SPY price is provided in
Fig 6.
As a whole, the SPY price has an increasing trend during the period 2010-2020.
The increase is temporally disrupted in the first quarter of 2020 by the
coronavirus pandemic.

Figure 6. Daily SPY ETF Price

Let the number of observations $N=2768$, and the daily observed SPY price
$S_{j}$ on day $t_{j}$ with $j=1,\dots,N$; $t_{1}$ 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 (3.11).

$\displaystyle y_{j}=\log(S_{j}/S_{j-1})\hskip 28.45274pt \hbox{
$j=2,\dots,N$} $ (3.11)

One observation will be lost in the computation process. The results of the
daily SPY ETF return are shown in Fig 7.

(a) Daily SPY return and COVID-19 pandemic

(b) Daily SPY return sample without outliers

Figure 7. Daily SPY ETF Return ($\%$)

The volatility of the SPY ETF is higher in the First quarter of $2020$ as
displayed in Fig 7(a). The SPY ETF is an inexpensive way for investors to gain
diversified exposure to the U.S. stock market; like others stocks and
securities, it was unusually volatile in $2020$ amid the coronavirus pandemic
and massive disruptions in the global economy.
$13$ daily return observations was identified as outliers and removed from the
data set in order to avoid negative impact on the statistics. See Table 1
below. By assuming the independent, identically distribution in the SPY ETF
return sample, the method of moments was used to compute statistics in Table 1
for both samples with outliers and without outliers. The outlier effects are
greater on Kurtosis, Skewness, and Variance. The sample Skewness (0.4687) is
between -0.5 and 0.5, which is the skewness interval for symmetric [5, 26].

Table 1. Impact of outliers on sample Statistics Statistics | All | Without outliers
---|---|---
$N$ | $2768$ | $2755$
$\hat{E(Y)}$ | $0.0509$ | $0.0541$
$\hat{\sigma^{2}(Y)}$ | $1.1905$ | $0.9487$
$\hat{Skew(Y)}$ | $-0.8633$ | $-0.4687$
$\hat{Kurt(Y)}$ | $17.3572$ | $6.6853$

The shape of the daily SPY ETF return distribution are estimated from the
sample using two non parametric density estimators: Histogram in Fig 8(a) and
Kernel density function in Fig8(b). Both graphs show the Leptokurtosis[25]
with sharp peak.

(a) Daily SPY ETF return Histogram

(b) Daily SPY ETF return Kernel density

Figure 8. Non-parametric density estimation

###  3.3. VG Model Estimations

Based on the VG model assumption that $\sigma=1$, the equations (3.7)-(3.10)
was solved given the statistics provided in Table 1. The parameter values
$\mu$, $\delta$, $\alpha$, and $\theta$ are shown in Table 2. We have two
models, the asymmetric Variance-Gamma model (AVG) with chronometer drift
negative ($\delta=-0.1399$); and the symmetric Variance-Gamma model (SVG)
without drift.
As shown Table 2, when $\delta=0$, the drift of the physical time scale $\mu$
decreases, but keeps the positive sign. The magnitude of $\alpha$ and $\theta$
remains the same. The Gamma shape parameter ($\alpha$) is less than one. In
fact, The larger the sample kurtosis, the smaller the Gamma shape parameter
($\alpha$) as shown in the relationship ($Kurtosis=3(1+\frac{1}{\alpha})$) for
SVG model.

Table 2. Method of Moments VG Parameters Estimations  Model | Parameters | Statistics
---|---|---
| $\hat{\mu}=0.1841$ | $Skew(Y)=-0.4689$
| $\hat{\delta}=-0.1399$
AVG | $\hat{\alpha}=0.8479$ | $Kurt(Y)=6.6853$
| $\hat{\theta}=1.0954$
| $\hat{\mu}=0.0541$ | $Skew(Y)=0$
SVG | $\hat{\alpha}=0.8140$
| $\hat{\theta}=1.1654$ | $\hat{Kurt(Y)}=6.6853$

Regarding the estimation by Maximum likelihood method, the previous assumption
on the volatility ($\sigma=1$) doest not hold and was dropped. The method has
a local property that makes the result depending on the initial value. The
estimations by Moments method were used as initial values in the maximization
procedure.

Table 3. Maximum Likelihood VG Parameters Estimations  Model | Parameters | Statistics
---|---|---
$\hat{\mu}=0.1683$ | $\hat{Skew(Y)}=-0.382$
$\hat{\delta}=-0.1089$ | $\hat{Kurt(Y)}=6.329$
$\hat{\sigma}=0.8987$ | Log(ML)=-3562
AVG1 | $\hat{\alpha}=0.9284$ | $P\\_{value}=0$
$\hat{\theta}=1.0546$
$\hat{\mu}=0.0510$ | $\hat{Skew(Y)}=0$
$\hat{\sigma}=0.9378$ | $\hat{Kurt(Y)}=6.534$
SVG1 | $\hat{\alpha}=0.8490$ | $Log(ML)=-3559$
$\hat{\theta}=1.0929$ | $P\\_{value}=0$
$\hat{\mu}=0.0848$ | $\hat{Skew(Y)}=-0.173$
$\hat{\delta}=-0.0577$ | $\hat{Kurt(Y)}=6.412$
AVG2 | $\hat{\sigma}=1.0295$ | $Log(ML)=-3548.65$
$\hat{\alpha}=0.8845$ | $P\\_{value}=0$
$\hat{\theta}=0.9378$
$\hat{\mu}=0.0652$ | $\hat{Skew(Y)}=0$
$\hat{\sigma}=0.9939$ | $\hat{Kurt(Y)}=6.421$
SVG2 | $\hat{\alpha}=0.8770$ | $Log(ML)=-3552.29$
$\hat{\theta}=0.9937$ | $P\\_{value}=0$
CLM | $\hat{\mu}=0.0541$ | $\hat{Kurt(Y)}=3$
$\hat{\sigma}=0.9740$ | Log(ML)=-3836.16

In table 3, the results are labeled AVG1 for the Asymmetric Variance-Gamma
Model and SVG1 for the Symmetric Variance-Gamma Model. Another initial value
was chosen with $\sigma=\alpha=\theta=1$, and $\delta=\mu=0$. The results are
labeled AVG2 and SVG2 respectively for Asymmetric Variance-Gamma Model and
Symmetric Variance-Gamma Model. As shown in Appendix D.4, the maximization
procedure convergences after 21 iterations For AVG2, as shown in Table 5. SVG2
also converges, after 15 iterations. See Table 6 in Appendix D.4.
It appears that the drift of the physical time scale $\mu$ is positive, and
the drift of the activity time process $\delta$ is negative. The estimation of
the Maximum likelihood underestimates the value of the kurtosis.

###  3.4. Comparison of VG Models

For the Classical Lognormal Model (CLM), the Method of Moments and the Maximum
likelihood yield the same estimation of the mean ($\hat{\mu}$) and standard
deviation ($\hat{\sigma}$), given a high sample size. The results are
displayed in Table 3.
It is shown in (3.3) that the VG model is a special case of the CLM. It is
also known that the parameter values in Table 3 maximized the likelihood
function. Therefore, it becomes appropriate to perform the Likelihood Ratio
test between the normal distribution and the four other VG models. The
Likelihood Ratio estimator (W) is defined as follows.

$\displaystyle W$ $\displaystyle=2(L_{H_{1}}-L_{H_{0}}) $ $\displaystyle W$
$\displaystyle\sim\chi^{2}(r) $ $\displaystyle P\\_{value}$
$\displaystyle=prob(\chi^{2}>\hat{W}|H_{0}) $ (3.12)

where $L_{H_{0}}$ is the Maximum of the Log-likelihood under $H_{0}$. Under
$H_{0}$: $Y_{k}\sim N(\mu,\sigma^{2})$.
$L_{H_{1}}$ is the Maximum of the Log-likelihood under $H_{1}$. Under $H_{1}$:
$Y_{k}\sim VG(\mu,\delta,\sigma,\alpha,\theta)$. $P\\_values$ is the
probability that the real values are even more extreme (more in the tail) than
our test statistic ($\hat{W}$). For Likelihood Ratio test review, see section
$5.5$ in [22], and [20], [27]
The value of the Log-likelihood ($Log(ML)$) is displayed in Table 3 for each
model. There is a huge discrepancy between the Log-likelihood of the CLM and
VG model, which produces $P\\_{value}$ almost zero. Therefore, the null
hypothesis ($H_{0}$) of normal distribution, is rejected based on the sample
data. The results provide evidence that the Variance-Gamma model is an
improvement compare to the classical Normal model.

The natural question is to know which VG model fits the empirical distribution
of the sample of daily SPY ETF return $\\{y_{1},y_{2}\dots y_{n}\\}$. In order
to compare the empirical distribution to the VG model theoretical
distribution, the Kolmogorov-Smirnov (KS) test was performed under the null
hypothesis (H0) that the sample $\\{y_{1},y_{2}\dots y_{n}\\}$ comes from VG
model.
The cumulative distribution function of the theoretical distribution needs to
be computed. The density function ($f$) does not have closed-form, the same
for the cumulative function($F$) in (3.14). However, we know the closed form
of the Fourier of the density function ($\mathscr{F}[f]$) and the relationship
in (3.15) provides the Fourier of the cumulative distribution function
($\mathscr{F}[F]$).

$\displaystyle Y$ $\displaystyle\sim VG(\mu,\delta,\sigma,\alpha,\theta)$
(3.13) $\displaystyle F(x)$
$\displaystyle=\int_{-\infty}^{x}f(t)\mathrm{d}t\hskip 14.22636pt \hbox{$f$ is
the density function of $Y$}$ (3.14) $\displaystyle \mathscr{F}[F](x)$
$\displaystyle=\frac{\mathscr{F}[f](x)}{ix}+\pi\mathscr{F}[f](0)\delta(x) $
(3.15) $\displaystyle \delta(x)$ $\displaystyle= \begin{cases} 0
&\quad\text{if }x\neq 0\\\ \infty &\quad\text{if }x=0\end{cases} $ (3.16)

See Appendix A.1 for (3.15) proof.
The cumulative distribution function ($F$) was computed from the inverse of
the Fourier of the cumulative distribution ($\mathscr{F}[F]$). The results for
SVG2 model is shown in Appendix E.5, Fourth column. The same methodology was
followed for the remaining VG models. The Kolmogorov-Smirnov (K-S) estimator
($D_{n}$) is defined as follows.

$\displaystyle D_{n}$ $\displaystyle=\sup_{x}{|F(x)-F_{n}(x)|} $
$\displaystyle P_{value}$ $\displaystyle=prob(D_{n}>d_{n}|H_{0}) $ (3.17)

where $n$ is the sample size, $F_{n}(x)$ denotes the empirical cumulative
distribution of $\\{y_{1},y_{2}\dots y_{n}\\}$ . The distribution of the
Kolmogorov’s goodness-of-fit measure $D_{n}$ has been studied extensively in
the litterature [28],[29],[30]. It was shown that $D_{n}$ distribution is
independent of the theoretical distribution ($F$) under the null hypothesis
(H0). More recently in [31], it was determined numerically the exact
distribution of $D_{n}$. [31] also provides a package KSgeneral[32] in the R
software for computing the cumulative distribution of $D_{n}$ when F(x) is
continuous and $n$ is a natural number.
The cumulative distribution of $D_{n}$ under the null hypothesis was computed
and the density function was deduced. The computed density function is shown
in Fig 9. Under the null hypothesis (H0), $D_{n}$ has a positively skewed
distribution with means ($\mu=0.0165$) and standard deviation
($\sigma=5*10^{-3}$).

Figure 9. Kolmogorov-Smirnov Estimator ($\hat{D_{n}}$) probability density
($n=2755$) under the null hypothesis $H{0}$

$d_{n}$ is the value of the KS estimator ($D_{n}$) computed from the sample
$\\{y_{1},y_{2}\dots y_{n}\\}$. Based on [33, 34], $d_{n}$ can be estimated as
follows.

$\displaystyle d^{+}_{n}$ $\displaystyle=\sup_{0\leq j\leq
P}{|F(x_{j})-F_{n}(x_{j})|} $ $\displaystyle d^{-}_{n}$
$\displaystyle=\sup_{1\leq j\leq P}{|F(x_{j})-F_{n}(x_{j-1})|}$ $\displaystyle
d_{n}$ $\displaystyle=Max(d^{+}_{n},d^{-}_{n}) $ (3.18)

The results for SVG2 model is shown in Appendix E.5, Table E.5, Fifth and
Sixth columns. From the Appendix E.5: $d^{-}_{n}=max((1))=0.023629$,
$d^{+}_{n}=max((2))=0.021986$ and $d_{n}=0.023629$.
For each estimation method and VG model, KS-Statistics ($d_{n}$) and P_values
were computed and the results are provided in Table 4.

Table 4. Kolmogorov-Smirnov (KS) test Method | Model | KS-Statistics ($d_{n}$) | P_values
---|---|---|---
Moments | SVG | 0.058145 | 0.00000154%
| AVG | 0.027211 | 3.3191%
| AVG1 | 0.054290 | 0.00001691%
Likelihood | SVG1 | 0.036763 | 0.1136%
| AVG2 | 0.028182 | 2.4668%
| SVG2 | 0.023629 | 9.0788%
| CLM | 0.095791 | 0%

As shown in Table 4, the VG models from the method of moments do not fit the
sample data distribution. The $P\\_values$ is less than $5\%$ and the null
hypothesis $H_{0}$ can not be accepted at some extent. The CLM does not fit
the sample data distribution at $5\%$. Regarding the maximum likelihood
method, the SVG2 model has $P\\_values=9.079\%$, which is high than the
classical threshold $5\%$. Therefore, we can not reject the null hypothesis
that SVG2 model fits the empirical distribution.

(a) Daily SPY ETF return histogram

(b) $\hat{\mu}=0.0541$,$\hat{\sigma}=0.9740$

(c) Moments method ($\hat{f}$): $\hat{\mu}=0.1841$, $\hat{\delta}=-0.1399$,
$\sigma=1$, $\hat{\alpha}=0.8479$, $\hat{\theta}=1.0954$.

(d) Maximum likelihood ($\hat{f}$): $\hat{\mu}=0.0652$, $\delta=0$,
$\hat{\sigma}=0.9939$, $\hat{\alpha}=0.8770$, $\hat{\theta}=0.9937$.

Figure 10. Probability density functions for the daily SPY return

The histogram of the daily SPY ETF return was compared to some models in Table
4 as shown in Fig 10. Compare to the CLM in Fig 10(b), the peakedness of
histogram explains the high level of the KS-Statistics ($d_{n}=0.095791$). But
to a lesser extent, the same explanation holds for some AVG models like SVG,
AVG1, SVG1 in table 4.
For work related to Normal and exponential distributions , see [35, 36, 37,
38]

##  4\. Conclusion

The paper explores the use of FRFT based technique as a tool to obtain the
probability density function and its derivatives for infinitely divisible
distribution. 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 the probability density
function of the VG model are also computed in order to build the Fisher
information matrix for the Maximum likelihood method.
We perform the maximum likelihood to estimate the Variance Gamma (VG) model
with five parameters. The sample data comes from the daily SPY ETF Price data
from January 04, 2010 to December 30, 2020. The method of moments and the
maximum likelihood were performed on the sample data. Independently of the
methods of estimation, the result shows that the VG model provides a better
fit than the Classical Lognormal Model (CLM). The Kolmogorov-Smirnov (KS)
goodness-of-fit test shows that the maximum likelihood method with FRFT
produces a good estimation for the VG model, which fits the empirical
distribution of the sample data.
It will be interesting to derive the European option price from the risk-
neutral VG model and compare the results with those from the Black-Scholes
model.

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##  Appendix A.1 Some Proofs

Proof (2.7):

$\displaystyle
f(x_{k})=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\mathscr{F}[f](y)e^{ix_{k}y}\mathrm{d}y\approx\frac{1}{2\pi}\int_{-a/2}^{a/2}\mathscr{F}[f](y)e^{ix_{k}y}\mathrm{d}y$
$\displaystyle=\frac{\gamma}{2\pi}\sum_{j=0}^{n-1}\\!\mathscr{F}[f](y_{j})e^{2\pi
i(k-\frac{n}{2})(j-\frac{n}{2})\delta}$
$\displaystyle=\frac{\gamma}{2\pi}e^{-\pi
i(k-\frac{n}{2})n\delta}G_{k}(\mathscr{F}[f](y_{j})e^{-\pi
ijn\delta}),-\delta)$

Proof (2.10) and (3.5)

$\displaystyle\mathscr{F}[f](x)$
$\displaystyle=E[e^{-ixY_{j}}]=E(E[e^{-ixY_{j}}|V_{j}])\hskip
14.22636ptY_{j}|_{V_{j}=v}\sim N(\mu+\delta v,v\sigma^{2})\hskip
14.22636pt\hbox{and }\hskip 14.22636ptV_{j}\sim\Gamma(\alpha,\theta)$
$\displaystyle=E\left(e^{-ix(\mu+\delta
V_{j})-\frac{1}{2}\sigma^{2}x^{2}V_{j}}\right)=\frac{1}{\Gamma(\alpha)\theta^{\alpha}}\int_{0}^{+\infty}v^{\alpha-1}e^{-i(\mu+\delta
v)x}e^{-(\frac{1}{2}\sigma^{2}x^{2}+\frac{1}{\theta})v}\mathrm{d}v$
$\displaystyle=\frac{e^{-i\mu
x}}{(\frac{1}{2}\theta\sigma^{2}x^{2}+1)^{\alpha}}E[e^{-i\delta xW}]\hskip
14.22636pt\hbox{and }\hskip
14.22636ptW\sim\Gamma(\alpha,\frac{1}{2}\sigma^{2}x^{2}+\frac{1}{\theta})$
$\displaystyle=\frac{e^{-i\mu
x}}{(1+\frac{1}{2}\theta\sigma^{2}x^{2}+i\delta\theta x)^{\alpha}}$

Proof (2.13)

$\displaystyle\mathscr{F}[f](x)$
$\displaystyle=E[e^{-ix\sum_{i=1}^{N(\lambda)}X_{j}}]=E\left(E[e^{-ix\sum_{i=1}^{N(\lambda)}X_{j}}|N(\lambda)]\right)\hskip
14.22636pt\hskip 14.22636ptN(\lambda)\sim Poisson(\lambda)\hskip
14.22636pt\hbox{and }\hskip 14.22636ptX_{j}\sim N(\mu,\sigma^{2})$
$\displaystyle=E\left(E[e^{-ix\sum_{i=1}^{k}X_{j}}|N(\lambda)=k]\right)=E\left(E[e^{-ixX_{j}}]^{N(\lambda)}\right)=e^{\lambda(E[e^{-ixX_{j}}]-1)}=e^{\lambda(e^{-uix-\frac{1}{2}\sigma^{2}x^{2}}-1)}$

Proof (3.6):

$\displaystyle\mathscr{F}[f](x)$ $\displaystyle=\lim_{\theta\to
0}\frac{e^{-i\mu x}}{(1+\frac{1}{2}\theta\sigma^{2}x^{2}+i\delta\theta
x)^{\frac{1}{\theta}}}$ $\displaystyle=e^{-i\mu x}\lim_{\theta\to
0}e^{-\frac{1}{\theta}log(1+\frac{1}{2}\theta\sigma^{2}x^{2}+i\delta\theta
x)}$ $\displaystyle=e^{-i(\mu+\delta)x-\frac{1}{2}\sigma^{2}x^{2}}$

Proof (3.15):
Fourier of the step function

$\displaystyle u(x)$ $\displaystyle=\begin{cases}1&\quad\text{if }0\leq x\\\
0&\quad\text{if }0>x\end{cases}$ $\displaystyle\mathscr{F}[u](y)$
$\displaystyle=\frac{1}{iy}+\pi\delta(y)$

The integral is transformed into convolution of $f$ and $u$

$\displaystyle F(x)$
$\displaystyle=\int_{-\infty}^{x}f(t)\mathrm{d}t=\int_{-\infty}^{+\infty}f(\tau)u(x-\tau)\mathrm{d}\tau=f(x)*u(x)$

Using the convolution theorem

$\displaystyle\mathscr{F}[F](x)=\mathscr{F}[f](x)\mathscr{F}[u](x)\hskip
28.45274pt\hbox{and}\hskip
28.45274pt\mathscr{F}[F](x)=\frac{\mathscr{F}[f](x)}{ix}+\pi\mathscr{F}[f](0)\delta(x)$

##  Appendix B.2 second order derivative of the VG density function

(a) $\frac{d^{2}\hat{f}}{d\delta d\mu}(t)$

(b) $\frac{d^{2}\hat{f}}{d\sigma d\mu}(t)$

(c) $\frac{d^{2}\hat{f}}{d\alpha d\mu}(t)$

(d) $\frac{d^{2}\hat{f}}{d\theta d\mu}(t)$

(e) $\frac{d^{2}\hat{f}}{d\sigma d\delta}(t)$

(f) $\frac{d^{2}\hat{f}}{d\alpha d\delta}(t)$

Figure 11. Second order derivative of the VG density function with $\mu=-2$,
$\delta=0$, $\sigma=1$, $\alpha=1$, $\theta=1$

##  Appendix C.3 second order derivative of the VG density function

(a) $\frac{d^{2}\hat{f}}{d\theta d\delta}(t)$

(b) $\frac{d^{2}\hat{f}}{d^{2}\sigma}(t)$

(c) $\frac{d^{2}\hat{f}}{d\alpha d\sigma}(t)$

(d) $\frac{d^{2}\hat{f}}{d\theta d\sigma}(t)$

(e) $\frac{d^{2}\hat{f}}{d\theta d\alpha}(t)$

Figure 12. Second order derivative of the VG density function with $\mu=-2$,
$\delta=0$, $\sigma=1$, $\alpha=1$, $\theta=1$

##  Appendix D.4 AVG Model parameter estimated
by Newton – Raphson Iteration Algorithm (2.23)

Table 5. Results of AVG2 Model Parameters Estimations

Iterations | $\mu$ | $\delta$ | $\sigma$ | $\alpha$ | $\theta$ | $Log(ML)$ | $||\frac{d(Log(ML))}{d(\theta)}||$
---|---|---|---|---|---|---|---
1 | 0 | 0 | 1 | 1 | 1 | -3582.8388 | 598.743231
2 | 0.05905599 | -0.0009445 | 1.03195903 | 0.9130208 | 1.03208412 | -3561.5099 | 833.530396
3 | 0.06949925 | 0.00400035 | 1.04101444 | 0.88478895 | 1.05131996 | -3559.5656 | 447.807305
4 | 0.07514039 | 0.00055771 | 1.17577397 | 0.67326429 | 1.17778666 | -3569.6221 | 211.365781
5 | 0.08928373 | -0.0263716 | 1.03756321 | 0.83842661 | 0.94304967 | -3554.4434 | 498.289445
6 | 0.08676498 | -0.0521887 | 1.03337015 | 0.85591875 | 0.95066351 | -3550.6419 | 204.467192
7 | 0.086995 | -0.0608517 | 1.02788937 | 0.87382621 | 0.95054954 | -3549.8465 | 66.8039738
8 | 0.08542912 | -0.058547 | 1.02705241 | 0.88258411 | 0.94321299 | -3549.7023 | 15.3209117
9 | 0.08478622 | -0.0576654 | 1.02995166 | 0.88447791 | 0.93670036 | -3549.6921 | 1.14764198
10 | 0.08477798 | -0.0577736 | 1.02922308 | 0.88449072 | 0.93831041 | -3549.692 | 0.17287708
11 | 0.08476475 | -0.0577271 | 1.02960343 | 0.88450434 | 0.93755549 | -3549.692 | 0.07850459
12 | 0.08477094 | -0.0577488 | 1.02942608 | 0.8844984 | 0.93790784 | -3549.692 | 0.03723941
13 | 0.08476804 | -0.0577386 | 1.02950937 | 0.88450117 | 0.93774266 | -3549.692 | 0.01732146
14 | 0.0847694 | -0.0577434 | 1.02947043 | 0.88449987 | 0.93781995 | -3549.692 | 0.00813465
15 | 0.08476876 | -0.0577411 | 1.02948868 | 0.88450048 | 0.93778375 | -3549.692 | 0.00380345
16 | 0.08476906 | -0.0577422 | 1.02948014 | 0.88450019 | 0.9378007 | -3549.692 | 0.00178206
17 | 0.08476892 | -0.0577417 | 1.02948414 | 0.88450033 | 0.93779276 | -3549.692 | 0.00083415
18 | 0.08476898 | -0.0577419 | 1.02948226 | 0.88450026 | 0.93779648 | -3549.692 | 0.00039063
19 | 0.08476895 | -0.0577418 | 1.02948314 | 0.88450029 | 0.93779474 | -3549.692 | 0.00018289
20 | 0.08476897 | -0.0577419 | 1.02948273 | 0.88450028 | 0.93779555 | -3549.692 | 8.56E-05
21 | 0.08476896 | -0.0577418 | 1.02948292 | 0.88450029 | 0.93779517 | -3549.692 | 4.01E-05

Table 6. Results of SVG2 Model Parameters Estimations

Iterations | $\mu$ | $\sigma$ | $\alpha$ | $\theta$ | $Log(ML)$ | $||\frac{d(Log(ML))}{d(\theta)}||$
---|---|---|---|---|---|---
1 | 0 | 1 | 1 | 1 | -3582.8388 | 582.057918
2 | 0.05883419 | 1.03172736 | 0.91367846 | 1.03172736 | -3561.9657 | 1096.4626
3 | 0.06789625 | 1.03903655 | 0.89847415 | 1.03903655 | -3560.2306 | 648.976501
4 | 0.07426127 | 1.06554771 | 0.8507772 | 1.06554771 | -3559.7139 | 270.990173
5 | 0.07141101 | 0.86066094 | 1.14468838 | 0.86066094 | -3567.8236 | 267.821258
6 | 0.06686856 | 0.94781927 | 0.92720936 | 0.94781927 | -3559.7962 | 1338.53688
7 | 0.06739235 | 0.95934348 | 0.90760199 | 0.95934348 | -3558.2696 | 773.42228
8 | 0.0665979 | 0.97263343 | 0.88806105 | 0.97263343 | -3556.6619 | 428.519657
9 | 0.06662118 | 0.98488196 | 0.87499518 | 0.98488196 | -3555.0979 | 203.689715
10 | 0.06614857 | 0.99253289 | 0.87413657 | 0.99253289 | -3554.2092 | 47.3050935
11 | 0.06523245 | 0.99394889 | 0.87669649 | 0.99394889 | -3554.1354 | 1.62304319
12 | 0.06515753 | 0.99386332 | 0.87702703 | 0.99386332 | -3554.1352 | 0.00262982
13 | 0.06515746 | 0.99386267 | 0.87702846 | 0.99386267 | -3554.1352 | 2.69E-08
14 | 0.06515746 | 0.99392282 | 0.87702846 | 0.99374235 | -3554.1352 | 7.64E-05
15 | 0.06515746 | 0.99392282 | 0.87702846 | 0.99374236 | -3554.1352 | 2.27E-12

##  Appendix E.5 SVG2 Cumulative Function ($\hat{F}$) versus Empirical
Function
($\hat{F_{n}}$),
$(1)=|\hat{F(x_{j})}-\hat{F_{n}(x_{j-1})}|$,$(2)=|\hat{F(x_{j})}-\hat{F_{n}(x_{j})}|$,
$n=2755$

$x_{j}$ | $n\\_\\{j\\}$ | $\hat{F_{n}(x_{j})}$ | $\hat{F(x_{j})}$ | (1) | (2) | $x_{j}$ | $n\\_\\{j\\}$ | $\hat{F_{n}(x_{j})}$ | $\hat{F(x_{j})}$ | (1) | (2) | $x_{j}$ | $n\\_\\{j\\}$ | $\hat{F_{n}(x_{j})}$ | $\hat{F(x_{j})}$ | (1) | (2)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
-5.01 | 1 | 0.000362976 | 0.00053753 | 0.00053753 | 0.00017456 | -1.685 | 7 | 0.04827586 | 0.0350006 | 0.01073443 | 0.01327526 | 1.64 | 4 | 0.96442831 | 0.95534423 | 0.00763218 | 0.00908409
-4.975 | 0 | 0.000362976 | 0.00057061 | 0.00020763 | 0.00020763 | -1.65 | 11 | 0.0522686 | 0.03665193 | 0.01162393 | 0.01561667 | 1.675 | 9 | 0.9676951 | 0.95760224 | 0.00682607 | 0.01009286
-4.94 | 0 | 0.000362976 | 0.00057987 | 0.00021689 | 0.00021689 | -1.615 | 3 | 0.05335753 | 0.03840489 | 0.01386372 | 0.01495264 | 1.71 | 4 | 0.96914701 | 0.959677 | 0.0080181 | 0.00947001
-4.905 | 0 | 0.000362976 | 0.00056565 | 0.00020267 | 0.00020267 | -1.58 | 4 | 0.05480944 | 0.04030396 | 0.01305358 | 0.01450548 | 1.745 | 5 | 0.97096189 | 0.9615774 | 0.0075696 | 0.00938449
-4.87 | 0 | 0.000362976 | 0.00053125 | 0.00016827 | 0.00016827 | -1.545 | 5 | 0.05662432 | 0.042371 | 0.01243843 | 0.01425332 | 1.78 | 4 | 0.97241379 | 0.96333629 | 0.0076256 | 0.0090775
-4.835 | 0 | 0.000362976 | 0.00048258 | 0.0001196 | 0.0001196 | -1.51 | 6 | 0.05880218 | 0.04462677 | 0.01199755 | 0.01417541 | 1.815 | 3 | 0.97350272 | 0.9649836 | 0.00743019 | 0.00851912
-4.8 | 1 | 0.000725953 | 0.00042749 | 6.45E-05 | 0.00029847 | -1.475 | 11 | 0.06279492 | 0.04708394 | 0.01171824 | 0.01571098 | 1.85 | 3 | 0.97459165 | 0.9665451 | 0.00695762 | 0.00804655
-4.765 | 0 | 0.000725953 | 0.00037483 | 0.00035112 | 0.00035112 | -1.44 | 8 | 0.06569873 | 0.04974601 | 0.01304891 | 0.01595272 | 1.885 | 2 | 0.9753176 | 0.96805709 | 0.00653456 | 0.00726051
-4.73 | 0 | 0.000725953 | 0.00033343 | 0.00039252 | 0.00039252 | -1.405 | 4 | 0.06715064 | 0.05260725 | 0.01309148 | 0.01454339 | 1.92 | 6 | 0.97749546 | 0.96953709 | 0.00578052 | 0.00795838
-4.695 | 0 | 0.000725953 | 0.00031096 | 0.00041499 | 0.00041499 | -1.37 | 10 | 0.0707804 | 0.05565387 | 0.01149677 | 0.01512653 | 1.955 | 3 | 0.97858439 | 0.97100001 | 0.00649546 | 0.00758439
-4.66 | 0 | 0.000725953 | 0.00031305 | 0.0004129 | 0.0004129 | -1.335 | 6 | 0.07295826 | 0.05886627 | 0.01191413 | 0.01409199 | 1.99 | 3 | 0.97967332 | 0.97245259 | 0.00613181 | 0.00722074
-4.625 | 2 | 0.001451906 | 0.00034253 | 0.00038342 | 0.00110938 | -1.3 | 11 | 0.076951 | 0.06222216 | 0.0107361 | 0.01472884 | 2.025 | 3 | 0.98076225 | 0.97389748 | 0.00577584 | 0.00686477
-4.59 | 0 | 0.001451906 | 0.00039908 | 0.00105283 | 0.00105283 | -1.265 | 5 | 0.07876588 | 0.06570024 | 0.01125076 | 0.01306564 | 2.06 | 1 | 0.98112523 | 0.97531775 | 0.00544451 | 0.00580748
-4.555 | 0 | 0.001451906 | 0.00047921 | 0.0009727 | 0.0009727 | -1.23 | 9 | 0.08203267 | 0.06929436 | 0.00947152 | 0.0127383 | 2.095 | 2 | 0.98185118 | 0.97669941 | 0.00442581 | 0.00515177
-4.52 | 1 | 0.001814882 | 0.00057667 | 0.00087523 | 0.00123821 | -1.195 | 8 | 0.08493648 | 0.07297576 | 0.00905691 | 0.01196072 | 2.13 | 3 | 0.98294011 | 0.97803303 | 0.00381815 | 0.00490708
-4.485 | 0 | 0.001814882 | 0.00068316 | 0.00113172 | 0.00113172 | -1.16 | 16 | 0.0907441 | 0.07674627 | 0.00819021 | 0.01399783 | 2.165 | 3 | 0.98402904 | 0.97929655 | 0.00364356 | 0.00473249
-4.45 | 0 | 0.001814882 | 0.00078932 | 0.00102556 | 0.00102556 | -1.125 | 8 | 0.09364791 | 0.08064272 | 0.01010138 | 0.01300519 | 2.2 | 4 | 0.98548094 | 0.98047385 | 0.00355519 | 0.00500709
-4.415 | 2 | 0.002540835 | 0.00088588 | 0.000929 | 0.00165495 | -1.09 | 7 | 0.09618875 | 0.08468243 | 0.00896548 | 0.01150631 | 2.235 | 1 | 0.98584392 | 0.98155588 | 0.00392506 | 0.00428804
-4.38 | 0 | 0.002540835 | 0.0009648 | 0.00157603 | 0.00157603 | -1.055 | 11 | 0.10018149 | 0.08890488 | 0.00728386 | 0.0112766 | 2.27 | 2 | 0.98656987 | 0.98253026 | 0.00331366 | 0.00403961
-4.345 | 0 | 0.002540835 | 0.00102027 | 0.00152057 | 0.00152057 | -1.02 | 13 | 0.10490018 | 0.09335839 | 0.0068231 | 0.01154179 | 2.305 | 1 | 0.98693285 | 0.98339832 | 0.00317155 | 0.00353453
-4.31 | 1 | 0.002903811 | 0.00104947 | 0.00149136 | 0.00185434 | -0.985 | 11 | 0.10889292 | 0.09809628 | 0.00680391 | 0.01079665 | 2.34 | 1 | 0.98729583 | 0.98417215 | 0.0027607 | 0.00312368
-4.275 | 1 | 0.003266788 | 0.00105301 | 0.0018508 | 0.00221378 | -0.95 | 13 | 0.11361162 | 0.10317228 | 0.00572064 | 0.01043934 | 2.375 | 1 | 0.9876588 | 0.98486216 | 0.00243367 | 0.00279664
-4.24 | 0 | 0.003266788 | 0.00103485 | 0.00223193 | 0.00223193 | -0.915 | 13 | 0.11833031 | 0.10863572 | 0.00497589 | 0.00969459 | 2.41 | 2 | 0.98838475 | 0.98548587 | 0.00217293 | 0.00289889
-4.205 | 0 | 0.003266788 | 0.00100196 | 0.00226483 | 0.00226483 | -0.88 | 15 | 0.12377495 | 0.11452697 | 0.00380334 | 0.00924798 | 2.445 | 1 | 0.98874773 | 0.98606329 | 0.00232146 | 0.00268444
-4.17 | 0 | 0.003266788 | 0.00096346 | 0.00230332 | 0.00230332 | -0.845 | 16 | 0.12958258 | 0.1208738 | 0.00290115 | 0.00870877 | 2.48 | 1 | 0.98911071 | 0.98661629 | 0.00213144 | 0.00249441
-4.135 | 0 | 0.003266788 | 0.00092963 | 0.00233716 | 0.00233716 | -0.81 | 12 | 0.13393829 | 0.12768918 | 0.0018934 | 0.00624912 | 2.515 | 2 | 0.98983666 | 0.9871586 | 0.00195211 | 0.00267806
-4.1 | 0 | 0.003266788 | 0.00091061 | 0.00235617 | 0.00235617 | -0.775 | 18 | 0.14047187 | 0.13497076 | 0.00103247 | 0.00550111 | 2.55 | 2 | 0.99056261 | 0.98770945 | 0.00212721 | 0.00285317
-4.065 | 0 | 0.003266788 | 0.00091524 | 0.00235155 | 0.00235155 | -0.74 | 17 | 0.14664247 | 0.14270246 | 0.00223059 | 0.00394001 | 2.585 | 0 | 0.99056261 | 0.98827661 | 0.002286 | 0.002286
-4.03 | 0 | 0.003266788 | 0.00094992 | 0.00231686 | 0.00231686 | -0.705 | 26 | 0.15607985 | 0.15085789 | 0.00421542 | 0.00522196 | 2.62 | 0 | 0.99056261 | 0.9888634 | 0.00169921 | 0.00169921
-3.995 | 0 | 0.003266788 | 0.00101785 | 0.00224893 | 0.00224893 | -0.67 | 21 | 0.16370236 | 0.15940559 | 0.00332573 | 0.00429677 | 2.655 | 1 | 0.99092559 | 0.98946709 | 0.00109552 | 0.0014585
-3.96 | 0 | 0.003266788 | 0.00111857 | 0.00214821 | 0.00214821 | -0.635 | 29 | 0.17422868 | 0.16831548 | 0.00461312 | 0.0059132 | 2.69 | 1 | 0.99128857 | 0.99008131 | 0.00084428 | 0.00120725
-3.925 | 0 | 0.003266788 | 0.00124799 | 0.0020188 | 0.0020188 | -0.6 | 15 | 0.17967332 | 0.17756601 | 0.00333734 | 0.00210731 | 2.725 | 0 | 0.99128857 | 0.99069034 | 0.00059822 | 0.00059822
-3.89 | 0 | 0.003266788 | 0.00139882 | 0.00186797 | 0.00186797 | -0.565 | 32 | 0.19128857 | 0.18715116 | 0.00747784 | 0.0041374 | 2.76 | 1 | 0.99165154 | 0.99127971 | 8.86E-06 | 0.00037184
-3.855 | 2 | 0.00399274 | 0.00156146 | 0.00170533 | 0.00243128 | -0.53 | 21 | 0.19891107 | 0.19708652 | 0.00579795 | 0.00182455 | 2.795 | 0 | 0.99165154 | 0.99183786 | 0.00018632 | 0.00018632
-3.82 | 0 | 0.00399274 | 0.00172515 | 0.00226759 | 0.00226759 | -0.495 | 24 | 0.2076225 | 0.2074137 | 0.00850263 | 0.0002088 | 2.83 | 2 | 0.9923775 | 0.99234968 | 0.00069813 | 2.78E-05
-3.785 | 1 | 0.004355717 | 0.00187926 | 0.00211349 | 0.00247646 | -0.46 | 22 | 0.21560799 | 0.21820243 | 0.01057993 | 0.00259445 | 2.865 | 1 | 0.99274047 | 0.99280499 | 0.00042749 | 6.45E-05
-3.75 | 0 | 0.004355717 | 0.00201464 | 0.00234108 | 0.00234108 | -0.425 | 36 | 0.22867514 | 0.22954979 | 0.01394181 | 0.00087466 | 2.9 | 1 | 0.99310345 | 0.99319919 | 0.00045871 | 9.57E-05
-3.715 | 0 | 0.004355717 | 0.00212481 | 0.00223091 | 0.00223091 | -0.39 | 41 | 0.24355717 | 0.24157648 | 0.01290134 | 0.00198069 | 2.935 | 1 | 0.99346642 | 0.9935296 | 0.00042615 | 6.32E-05
-3.68 | 1 | 0.004718693 | 0.00220677 | 0.00214894 | 0.00251192 | -0.355 | 39 | 0.25771325 | 0.25442008 | 0.01086292 | 0.00329316 | 2.97 | 2 | 0.99419238 | 0.99380157 | 0.00033515 | 0.00039081
-3.645 | 0 | 0.004718693 | 0.00226154 | 0.00245715 | 0.00245715 | -0.32 | 53 | 0.276951 | 0.26822579 | 0.01051254 | 0.00872521 | 3.005 | 0 | 0.99419238 | 0.99402612 | 0.00016626 | 0.00016626
-3.61 | 1 | 0.00508167 | 0.00229408 | 0.00242461 | 0.00278759 | -0.285 | 34 | 0.2892922 | 0.28313512 | 0.00618412 | 0.00615708 | 3.04 | 0 | 0.99419238 | 0.99421539 | 2.30E-05 | 2.30E-05
-3.575 | 0 | 0.00508167 | 0.0023128 | 0.00276887 | 0.00276887 | -0.25 | 41 | 0.30417423 | 0.29927354 | 0.00998135 | 0.00490069 | 3.075 | 1 | 0.99455535 | 0.99438437 | 0.00019199 | 0.00017099
-3.54 | 0 | 0.00508167 | 0.00232865 | 0.00275302 | 0.00275302 | -0.215 | 54 | 0.32377495 | 0.31673807 | 0.01256384 | 0.00703689 | 3.11 | 0 | 0.99455535 | 0.99454829 | 7.07E-06 | 7.07E-06
-3.505 | 2 | 0.005807623 | 0.00235387 | 0.0027278 | 0.00345375 | -0.18 | 52 | 0.34264973 | 0.33558577 | 0.01181081 | 0.00706396 | 3.145 | 0 | 0.99455535 | 0.99472139 | 0.00016604 | 0.00016604
-3.47 | 0 | 0.005807623 | 0.00240051 | 0.00340711 | 0.00340711 | -0.145 | 71 | 0.36842105 | 0.35582438 | 0.01317465 | 0.01259667 | 3.18 | 1 | 0.99491833 | 0.99491285 | 0.0003575 | 5.48E-06
-3.435 | 0 | 0.005807623 | 0.00247905 | 0.00332858 | 0.00332858 | -0.11 | 59 | 0.38983666 | 0.37740585 | 0.0089848 | 0.01243081 | 3.215 | 1 | 0.99528131 | 0.99513078 | 0.00021245 | 0.00015053
-3.4 | 3 | 0.006896552 | 0.00259707 | 0.00321056 | 0.00429949 | -0.075 | 62 | 0.4123412 | 0.40022352 | 0.01038686 | 0.01211768 | 3.25 | 0 | 0.99528131 | 0.99537645 | 9.51E-05 | 9.51E-05
-3.365 | 0 | 0.006896552 | 0.00275835 | 0.00413821 | 0.00413821 | -0.04 | 71 | 0.43811252 | 0.42411332 | 0.01177213 | 0.0139992 | 3.285 | 4 | 0.99673321 | 0.99564687 | 0.00036556 | 0.00108634
-3.33 | 0 | 0.006896552 | 0.00296235 | 0.0039342 | 0.0039342 | -0.005 | 79 | 0.46678766 | 0.44885915 | 0.01074663 | 0.01792851 | 3.32 | 0 | 0.99673321 | 0.99593476 | 0.00079845 | 0.00079845
-3.295 | 2 | 0.007622505 | 0.00320427 | 0.00369228 | 0.00441823 | 0.03 | 81 | 0.49618875 | 0.47420195 | 0.00741429 | 0.0219868 | 3.355 | 0 | 0.99673321 | 0.99623048 | 0.00050274 | 0.00050274
-3.26 | 0 | 0.007622505 | 0.00347555 | 0.00414695 | 0.00414695 | 0.065 | 66 | 0.52014519 | 0.49985209 | 0.00366334 | 0.0202931 | 3.39 | 1 | 0.99709619 | 0.99651915 | 0.00021407 | 0.00057704
-3.225 | 1 | 0.007985481 | 0.00376493 | 0.00385758 | 0.00422055 | 0.1 | 56 | 0.54047187 | 0.52550399 | 0.0053588 | 0.01496788 | 3.425 | 0 | 0.99709619 | 0.9967907 | 0.00030549 | 0.00030549
-3.19 | 0 | 0.007985481 | 0.00405981 | 0.00392567 | 0.00392567 | 0.135 | 58 | 0.5615245 | 0.55085199 | 0.01038012 | 0.01067251 | 3.46 | 0 | 0.99709619 | 0.99703301 | 6.32E-05 | 6.32E-05
-3.155 | 2 | 0.008711434 | 0.00434785 | 0.00363763 | 0.00436359 | 0.17 | 51 | 0.5800363 | 0.57560617 | 0.01408167 | 0.00443013 | 3.495 | 0 | 0.99709619 | 0.99723748 | 0.00014129 | 0.00014129
-3.12 | 0 | 0.008711434 | 0.00461851 | 0.00409292 | 0.00409292 | 0.205 | 64 | 0.60326679 | 0.59950707 | 0.01947077 | 0.00375972 | 3.53 | 0 | 0.99709619 | 0.99739926 | 0.00030307 | 0.00030307
-3.085 | 5 | 0.010526316 | 0.00486449 | 0.00384694 | 0.00566182 | 0.24 | 54 | 0.62286751 | 0.62233802 | 0.01907123 | 0.00052949 | 3.565 | 0 | 0.99709619 | 0.99751804 | 0.00042185 | 0.00042185
-3.05 | 2 | 0.011252269 | 0.00508274 | 0.00544358 | 0.00616953 | 0.275 | 53 | 0.64210526 | 0.64393434 | 0.02106682 | 0.00182907 | 3.6 | 0 | 0.99709619 | 0.99759689 | 0.0005007 | 0.0005007
-3.015 | 2 | 0.011978221 | 0.00527498 | 0.00597729 | 0.00670324 | 0.31 | 49 | 0.65989111 | 0.66418868 | 0.02208342 | 0.00429757 | 3.635 | 0 | 0.99709619 | 0.99764367 | 0.00054748 | 0.00054748
-2.98 | 0 | 0.011978221 | 0.00544774 | 0.00653048 | 0.00653048 | 0.345 | 47 | 0.676951 | 0.68305233 | 0.02316123 | 0.00610134 | 3.67 | 0 | 0.99709619 | 0.99766906 | 0.00057287 | 0.00057287
-2.945 | 0 | 0.011978221 | 0.00561168 | 0.00636654 | 0.00636654 | 0.38 | 46 | 0.69364791 | 0.70058043 | 0.02362944 | 0.00693252 | 3.705 | 0 | 0.99709619 | 0.99768495 | 0.00058876 | 0.00058876
-2.91 | 2 | 0.012704174 | 0.00578052 | 0.0061977 | 0.00692365 | 0.415 | 42 | 0.70889292 | 0.71672986 | 0.02308195 | 0.00783694 | 3.74 | 1 | 0.99745917 | 0.99770357 | 0.00060738 | 0.00024441
-2.875 | 0 | 0.012704174 | 0.00596947 | 0.0067347 | 0.0067347 | 0.45 | 42 | 0.72413793 | 0.73160823 | 0.02271531 | 0.0074703 | 3.775 | 0 | 0.99745917 | 0.9977359 | 0.00027673 | 0.00027673
-2.84 | 1 | 0.013067151 | 0.00619354 | 0.00651064 | 0.00687361 | 0.485 | 49 | 0.74192377 | 0.74542577 | 0.02128784 | 0.003502 | 3.81 | 0 | 0.99745917 | 0.99779057 | 0.0003314 | 0.0003314
-2.805 | 3 | 0.01415608 | 0.00646575 | 0.0066014 | 0.00769033 | 0.52 | 36 | 0.75499093 | 0.75827955 | 0.01635577 | 0.00328862 | 3.845 | 0 | 0.99745917 | 0.99787202 | 0.00041286 | 0.00041286
-2.77 | 0 | 0.01415608 | 0.00679564 | 0.00736044 | 0.00736044 | 0.555 | 33 | 0.76696915 | 0.77031477 | 0.01532384 | 0.00334562 | 3.88 | 0 | 0.99745917 | 0.99798187 | 0.00052271 | 0.00052271
-2.735 | 0 | 0.01415608 | 0.00718809 | 0.00696799 | 0.00696799 | 0.59 | 33 | 0.77894737 | 0.78166917 | 0.01470003 | 0.0027218 | 3.915 | 0 | 0.99745917 | 0.998117 | 0.00065784 | 0.00065784
-2.7 | 2 | 0.014882033 | 0.00764271 | 0.00651337 | 0.00723932 | 0.625 | 37 | 0.7923775 | 0.79246369 | 0.01351632 | 8.62E-05 | 3.95 | 0 | 0.99745917 | 0.99827094 | 0.00081178 | 0.00081178
-2.665 | 1 | 0.015245009 | 0.00815393 | 0.00672811 | 0.00709108 | 0.66 | 34 | 0.80471869 | 0.8027957 | 0.0104182 | 0.00192299 | 3.985 | 0 | 0.99745917 | 0.99843458 | 0.00097541 | 0.00097541
-2.63 | 1 | 0.015607985 | 0.00871162 | 0.00653338 | 0.00689636 | 0.695 | 23 | 0.81306715 | 0.81273525 | 0.00801655 | 0.0003319 | 4.02 | 1 | 0.99782214 | 0.9985973 | 0.00113813 | 0.00077515
-2.595 | 4 | 0.017059891 | 0.00930242 | 0.00630557 | 0.00775747 | 0.73 | 26 | 0.82250454 | 0.82232427 | 0.00925711 | 0.00018027 | 4.055 | 0 | 0.99782214 | 0.99874832 | 0.00092618 | 0.00092618
-2.56 | 6 | 0.01923775 | 0.00991136 | 0.00714853 | 0.00932639 | 0.765 | 28 | 0.83266788 | 0.83157862 | 0.00907409 | 0.00108925 | 4.09 | 1 | 0.99818512 | 0.99887802 | 0.00105588 | 0.0006929
-2.525 | 1 | 0.019600726 | 0.01052387 | 0.00871388 | 0.00907686 | 0.8 | 30 | 0.84355717 | 0.84051747 | 0.00784959 | 0.0030397 | 4.125 | 0 | 0.99818512 | 0.9989791 | 0.00079398 | 0.00079398
-2.49 | 0 | 0.019600726 | 0.01112771 | 0.00847302 | 0.00847302 | 0.835 | 19 | 0.85045372 | 0.84906839 | 0.00551122 | 0.00138533 | 4.16 | 0 | 0.99818512 | 0.99904741 | 0.0008623 | 0.0008623
-2.455 | 3 | 0.020689655 | 0.01171472 | 0.00788601 | 0.00897494 | 0.87 | 29 | 0.86098004 | 0.85722732 | 0.0067736 | 0.00375272 | 4.195 | 0 | 0.99818512 | 0.99908247 | 0.00089735 | 0.00089735
-2.42 | 3 | 0.021778584 | 0.0122821 | 0.00840755 | 0.00949648 | 0.905 | 21 | 0.86860254 | 0.8649413 | 0.00396127 | 0.00366124 | 4.23 | 1 | 0.99854809 | 0.99908738 | 0.00090226 | 0.00053929
-2.385 | 2 | 0.022504537 | 0.01283309 | 0.00894549 | 0.00967144 | 0.94 | 18 | 0.87513612 | 0.87222815 | 0.0036256 | 0.00290797 | 4.265 | 0 | 0.99854809 | 0.99906853 | 0.00052044 | 0.00052044
-2.35 | 2 | 0.02323049 | 0.01337696 | 0.00912758 | 0.00985353 | 0.975 | 17 | 0.88130672 | 0.87904889 | 0.00391277 | 0.00225783 | 4.3 | 1 | 0.99891107 | 0.99903489 | 0.0004868 | 0.00012382
-2.315 | 1 | 0.023593466 | 0.01392822 | 0.00930227 | 0.00966525 | 1.01 | 27 | 0.89110708 | 0.88540103 | 0.00409432 | 0.00570605 | 4.335 | 0 | 0.99891107 | 0.99899639 | 8.53E-05 | 8.53E-05
-2.28 | 5 | 0.025408348 | 0.01450523 | 0.00908823 | 0.01090311 | 1.045 | 16 | 0.8969147 | 0.89129736 | 0.00019028 | 0.00561735 | 4.37 | 1 | 0.99927405 | 0.99896338 | 5.23E-05 | 0.00031067
-2.245 | 3 | 0.026497278 | 0.0151283 | 0.01028005 | 0.01136898 | 1.08 | 19 | 0.90381125 | 0.89676548 | 0.00014922 | 0.00704577 | 4.405 | 0 | 0.99927405 | 0.99894501 | 0.00032904 | 0.00032904
-2.21 | 2 | 0.02722323 | 0.01581745 | 0.01067983 | 0.01140578 | 1.115 | 12 | 0.90816697 | 0.90184565 | 0.0019656 | 0.00632132 | 4.44 | 0 | 0.99927405 | 0.99894827 | 0.00032578 | 0.00032578
-2.175 | 4 | 0.028675136 | 0.01659023 | 0.010633 | 0.01208491 | 1.15 | 9 | 0.91143376 | 0.9065871 | 0.00157987 | 0.00484666 | 4.475 | 0 | 0.99927405 | 0.99897717 | 0.00029687 | 0.00029687
-2.14 | 3 | 0.029764065 | 0.01745976 | 0.01121537 | 0.0123043 | 1.185 | 13 | 0.91615245 | 0.91104355 | 0.00039021 | 0.0051089 | 4.51 | 0 | 0.99927405 | 0.99903235 | 0.0002417 | 0.0002417
-2.105 | 1 | 0.030127042 | 0.01843324 | 0.01133082 | 0.0116938 | 1.22 | 18 | 0.92268603 | 0.91526835 | 0.0008841 | 0.00741767 | 4.545 | 1 | 0.99963702 | 0.99911103 | 0.00016302 | 0.000526
-2.07 | 2 | 0.030852995 | 0.01951117 | 0.01061587 | 0.01134182 | 1.255 | 17 | 0.92885662 | 0.91932123 | 0.00336479 | 0.00953539 | 4.58 | 0 | 0.99963702 | 0.99920743 | 0.0004296 | 0.0004296
-2.035 | 3 | 0.031941924 | 0.02068741 | 0.01016559 | 0.01125452 | 1.29 | 12 | 0.93321234 | 0.92321878 | 0.00563785 | 0.00999357 | 4.615 | 0 | 0.99963702 | 0.99931353 | 0.00032349 | 0.00032349
-2 | 2 | 0.032667877 | 0.02195002 | 0.0099919 | 0.01071785 | 1.325 | 12 | 0.93756806 | 0.92700085 | 0.00621149 | 0.01056721 | 4.65 | 0 | 0.99963702 | 0.99942007 | 0.00021695 | 0.00021695
-1.965 | 4 | 0.034119782 | 0.02328294 | 0.00938493 | 0.01083684 | 1.36 | 13 | 0.94228675 | 0.9306828 | 0.00688526 | 0.01160395 | 4.685 | 0 | 0.99963702 | 0.99951794 | 0.00011908 | 0.00011908
-1.93 | 4 | 0.035571688 | 0.02466813 | 0.00945165 | 0.01090356 | 1.395 | 4 | 0.94373866 | 0.9342573 | 0.00802945 | 0.00948136 | 4.72 | 0 | 0.99963702 | 0.99959806 | 3.90E-05 | 3.90E-05
-1.895 | 6 | 0.037749546 | 0.02608817 | 0.00948352 | 0.01166138 | 1.43 | 12 | 0.94809437 | 0.93773667 | 0.00600199 | 0.01035771 | 4.755 | 0 | 0.99963702 | 0.9996549 | 1.79E-05 | 1.79E-05
-1.86 | 2 | 0.038475499 | 0.0275288 | 0.01022075 | 0.0109467 | 1.465 | 12 | 0.95245009 | 0.94109407 | 0.0070003 | 0.01135602 | 4.79 | 0 | 0.99963702 | 0.99968469 | 4.77E-05 | 4.77E-05
-1.825 | 7 | 0.041016334 | 0.02898126 | 0.00949423 | 0.01203507 | 1.5 | 8 | 0.9553539 | 0.94430824 | 0.00814185 | 0.01104566 | 4.825 | 0 | 0.99963702 | 0.99968708 | 5.01E-05 | 5.01E-05
-1.79 | 6 | 0.043194192 | 0.03044404 | 0.01057229 | 0.01275015 | 1.535 | 5 | 0.95716878 | 0.94735687 | 0.00799703 | 0.00981191 | 4.86 | 0 | 0.99963702 | 0.99966487 | 2.78E-05 | 2.78E-05
-1.755 | 1 | 0.043557169 | 0.03192378 | 0.01127042 | 0.01163339 | 1.57 | 11 | 0.96116152 | 0.95022032 | 0.00694846 | 0.0109412 | 4.895 | 1 | 1 | 0.9996235 | 1.35E-05 | 0.0003765