# European Option Pricing under Generalized Tempered Stable Process: Empirical
Analysis

A . H Nzokem

###### Abstract.

The paper investigates the performance of the European option price when the
log asset price follows a rich class of Generalized Tempered Stable (GTS)
distribution. The GTS distribution is an alternative to Normal distribution
and $\alpha$-stable distribution for modeling asset return and many physical
and economic systems. The data used in the option pricing computation comes
from fitting the GTS distribution to the underlying S&P 500 Index return
distribution. The Esscher transform method shows that the GTS distribution
preserves its structure. The extended Black-Scholes formula and the
Generalized Black-Scholes Formula are applied in the study. The 12-point rule
Composite Newton-Cotes Quadrature and the Fractional Fast Fourier (FRFT)
algorithms were implemented and they yield the same European option price at
two decimal places. Compared to the option price under the GTS distribution,
the Black-Scholes (BS) model is underpriced for the Near-The-Money (NTM) and
the in-the-money (ITM) options. However, the BS model and GTS European options
yield the same option price for the deep out-of-the-money (OTM) and the deep-
in-the-money (ITM) options.

Keywords: Generalized Tempered Stable (GTS) Distributions, Black-Scholes (BS)
Model, Fractional Fourier Transform (FRFT), Equivalent Martingale Measure
(EMM), Option Pricing

##  1\. Introduction

Black-Scholes (BS) model [1] is considered the cornerstone of the option
pricing theory. The model relies on the fundamental assumption that the asset
returns have a normal distribution with a known mean and variance. However,
based on empirical studies, the BS model model is inconsistent with a set of
well-established stylized features[2]. The $\alpha$-stable distribution has
been proposed [3, 4] as an alternative to the normal distribution for modeling
asset return and many types of physical and economic systems. In addition to
the theoretical and empirical arguments for modeling with a stable
distribution, the most important argument is that the Central Limit theorem
can be generalized by the stable distribution [5]. Although the stable
distribution allows for varying degrees of tail heaviness and skewness, it has
two major drawbacks [4]: firstly, some lack of closed formulas for density and
cumulative distribution functions; secondly, most of the moments of the stable
distribution are infinite. An infinite variance of the asset return leads to
an infinite price for derivative instruments such as options. The Generalized
Tempered Stable (GTS) distribution was developed to overcome the shortcomings
of the $\alpha$-stable distribution speciﬁcally in modeling high-frequency
asset returns. The tails of the GTS distribution for asset returns are heavier
than the normal distribution but thinner than the stable distribution [6]. See
[7, 5] For more details on Tempered Stable distribution.
There has been considerable research on the stochastic volatility model for
option pricing. In the current study, we contribute to the literature by
empirically investigating the European option pricing under GTS distribution.
The GTS distributions form a seven-parameter family of infinitely divisible
distributions, which cover several well-known subclasses like Variance Gamma
distributions [8, 9], bilateral Gamma distributions [10, 7] and CGMY
distributions [11]. The S&P 500 index historical data was used to fit the GTS
distribution to the underlying data return distribution. The methodology is
based on two computational algorithms, the Fractional Fast Fourier (FRFT) [12,
13] and twelve-point rule Composite Newton–Cotes Quadrature [14, 15].
We organize the paper as follows. We briefly present the GTS distribution’s
theoretical framework in the next section. In section 3, we estimate the seven
parameters of the GTS distribution from the S&P 500 index historical data. In
section 4, the Equivalent Martingale Measure (EMM) is identified and computed;
the Extended Black-Scholes and the Generalized Black-Scholes Formulas are
provided. And the last section presents the empirical analysis of the European
option price computations.

##  2\.  Generalized Tempered Stable (GTS) Process

The Lévy measure of the Generalized Tempered Stable (GTS) distribution
($V(dx)$) is defined (2.3) as a product of a tempering function ($q(x)$) (2.1)
and a Lévy measure of the $\alpha$-stable distribution
($V_{stable}(dx)$)(2.2).

$\displaystyle q(x)$ $\displaystyle=
e^{-\lambda_{+}x}\boldsymbol{1}_{x>0}+e^{-\lambda_{-}|x|}\boldsymbol{1}_{x<0}$
(2.1) $\displaystyle V_{stable}(dx)$
$\displaystyle=\left(\frac{\alpha_{+}}{x^{1+\beta_{+}}}\boldsymbol{1}_{x>0}+\frac{\alpha_{-}}{|x|^{1+\beta_{-}}}\boldsymbol{1}_{x<0}\right)dx$
(2.2) $\displaystyle V(dx)=q(x)V_{stable}(dx)$
$\displaystyle=\left(\frac{\alpha_{+}e^{-\lambda_{+}x}}{x^{1+\beta_{+}}}\boldsymbol{1}_{x>0}+\frac{\alpha_{-}e^{-\lambda_{-}|x|}}{|x|^{1+\beta_{-}}}\boldsymbol{1}_{x<0}\right)dx
$ (2.3)

where $0\leq\beta_{+}\leq 1$, $0\leq\beta_{-}\leq 1$, $\alpha_{+}\geq 0$,
$\alpha_{-}\geq 0$, $\lambda_{+}\geq 0$ and $\lambda_{-}\geq 0$. These
parameters play an essential role in the Levy process. $\beta_{+}$ and
$\beta_{-}$ are the indexes of stability bounded below by 0 and above by 2
[16]. They capture the peakedness of the distribution in a similar way as the
$\beta$-stable distribution, but the distribution tails are tempered. If
$\beta$ increases (decreases), then the peakedness decreases (increases).
$\alpha_{+}$ and $\alpha_{-}$ are the scale parameter, also called the process
intensity [17], they determine the arrival rate of jumps for a given size.
$\lambda_{+}$ and $\lambda_{-}$ control the decay rate on the positive and
negative tails. Additionally, $\lambda_{+}$ and $\lambda_{-}$ are also
skewness parameters. If $\lambda_{+}>\lambda_{-}$ ($\lambda_{+}<\lambda_{-}$),
then the distribution is skewed to the left (right), and if
$\lambda_{+}=\lambda_{-}$, then it is symmetric [5].
The GTS distribution can be denoted by $X\sim
GTS(\textbf{$\beta_{+}$},\textbf{$\beta_{-}$},\textbf{$\alpha_{+}$},\textbf{$\alpha_{-}$},\textbf{$\lambda_{+}$},\textbf{$\lambda_{-}$})$
and $X=X_{+}-X_{-}$ with $X_{+}\geq 0$, $X_{-}\geq 0$. $X_{+}\sim
TS(\textbf{$\beta_{+}$},\textbf{$\alpha_{+}$},\textbf{$\lambda_{+}$})$ and
$X_{-}\sim
TS(\textbf{$\beta_{-}$},\textbf{$\alpha_{-}$},\textbf{$\lambda_{-}$})$.

$\displaystyle\int_{-\infty}^{+\infty}V(dx)=\begin{cases}+\infty&\text{if
}{\beta_{+}\geq 0\vee\beta_{-}\geq 0}\\\
\alpha_{+}{\lambda_{+}}^{\beta_{+}}\Gamma(-\beta_{+})+\alpha_{-}{\lambda_{-}}^{\beta_{-}}\Gamma(-\beta_{-})&\text{if
}{\beta_{+}<0\wedge\beta_{-}<0}\end{cases}$ (2.4)

From (2.4), it results that when $\beta_{+}<0$, TS($\beta_{+}$,
$\alpha_{+}$,$\lambda_{+}$) is of finite activity and can be written as a
Compound Poisson process on the right side ($X_{+}$). we have similar pattern
when $\beta_{-}<0$. However, when $0\leq\beta_{+}\leq 1$, $X_{+}$ is an
infinite activity process with infinite jumps in any given time interval. We
have a similar pattern when $0\leq\beta_{-}\leq 1$. In addition to the
infinite activities process, we have

$\displaystyle\int_{-\infty}^{+\infty}min(1,|x|)V(dx)<+\infty$ (2.5)

By adding the location parameter, the GTS distribution becomes GTS($\mu$,
$\beta_{+}$, $\beta_{-}$, $\alpha_{+}$,$\alpha_{-}$, $\lambda_{+}$,
$\lambda_{-}$) and we have:

$\displaystyle Y=\mu+X=\mu+X_{+}-X_{-}\quad\quad Y\sim
GTS(\mu,\beta_{+},\beta_{-},\alpha_{+},\alpha_{-},\lambda_{+},\lambda_{-}) $
(2.6)

###### Theorem 2.1

Consider a variable $Y\sim
GTS(\textbf{$\mu$},\textbf{$\beta_{+}$},\textbf{$\beta_{-}$},\textbf{$\alpha_{+}$},\textbf{$\alpha_{-}$},\textbf{$\lambda_{+}$},\textbf{$\lambda_{-}$})$,
the characteristic exponent can be written

$\displaystyle\Psi(\xi)=\mu\xi
i+\alpha_{+}\Gamma(-\beta_{+})\left((\lambda_{+}-i\xi)^{\beta_{+}}-{\lambda_{+}}^{\beta_{+}}\right)+\alpha_{-}\Gamma(-\beta_{-})\left((\lambda_{-}+i\xi)^{\beta_{-}}-{\lambda_{-}}^{\beta_{-}}\right)
$ (2.7)

See [10] for theorem 2.1 proof

###### Corollary 2.2

Let $Y=\left(Y_{t}\right)$ be a Lévy process on $\mathbb{R^{+}}$ generated by
$GTS(\textbf{$\mu$},\textbf{$\beta_{+}$},\textbf{$\beta_{-}$},\textbf{$\alpha_{+}$},\textbf{$\alpha_{-}$},\textbf{$\lambda_{+}$},\textbf{$\lambda_{-}$})$.
$\forall t\in\mathbb{R^{+}}$,

$\displaystyle Y_{t}\sim
GTS(t\mu,\beta_{+},\beta_{-},t\alpha_{+},t\alpha_{-},\lambda_{+},\lambda_{-})
$ (2.8)

See [10] for corollary 2.2 proof

##  3\. Parameter Estimation based on daily S&P 500 price

The Standard & Poor’s 500 Composite Stock Price Index, also known as the S&P
500, is a stock index that tracks the share prices of 500 of the largest
public companies in the United States. It is used as a proxy describing the
overall health of the stock market or even the U.S. economy. The daily prices
were adjusted for splits and dividends. The period spans from January 04,
2010, to June 16, 2023 and the data were extracted from Yahoo Finance.

(a) S&P 500 daily price

(b) S&P 500 daily Return ($\%$)

Figure 1. S&P 500 Index Historical Data

The daily price dynamics are provided in Fig 1(a) and the daily return in Fig
1(b). As shown in Fig 1(a), prices have an increasing trend throughout the
period, even after the temporal disruption in the first quarter of 2020 by the
coronavirus pandemic. The volatility (in Fig 1(b)) of the daily return is
higher in the First quarter of 2020 amid the coronavirus pandemic and massive
disruptions in the global economy.

The characteristic function of the GTS process $Y=\\{Y_{t}\\}$ has the
following expression

$\displaystyle\vartheta(\xi,t)=E\left[e^{iY_{t}\xi}\right]=e^{t\Psi(\xi)}$
(3.1)

The GTS density function ($f$) does not have an explicit closed form or an
analytical expression, which makes it challenging to utilize the density
function and its derivatives and to perform the Maximum likelihood method. The
GTS density function ($f$) was derived from Fourier transform ($F(f)$) as
follows:

$F[f](\xi)=\vartheta(-\xi,1)\quad\quad
f(y)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\\!F[f](x)e^{iyx}\,\mathrm{d}x$
(3.2)

The Fractional Fourier Transform (FRFT) technique was used to compute the
probability density function (3.2) and its derivatives
$\left\\{\frac{df(y,V)}{dV_{j}}\right\\}_{1\leq j\leq 7}$ and
$\left\\{\frac{d^{2}f(y,V)}{dV_{k}dV_{j}}\right\\}_{\begin{subarray}{c}1\leq
k\leq 7\\\ 1\leq j\leq 7\end{subarray}}$. See [13, 9, 12] for more details on
FRFT methodologies.

The results of the GTS Parameter Estimation are reported in Table 1. As
expected, the stability index ($\beta$), the process intensity ($\alpha$), and
the decay rate ($\lambda$) are all positive. Both stability indexes are less
than 1, and capture the high peakedness of the underlying distribution. As
shown in Table 1, S&P 500 return is a bit left-skewed distribution
($\lambda_{+}>\lambda_{-}$), and the right-side distribution has a high
arrival rate of jumps ($\alpha_{+}>\alpha_{-}$). The higher arrival rate of
jump on the right-side distribution contributes to the increased nature of the
S&P 500 daily price in Fig 1(a).

Table 1.  FRFT Maximum Likelihood GTS Parameter Estimations Model | $\mu$ | $\beta_{+}$ | $\beta_{-}$ | $\alpha_{+}$ | $\alpha_{-}$ | $\lambda_{+}$ | $\lambda_{-}$
---|---|---|---|---|---|---|---
GTS | -0.693477 | 0.682290 | 0.242579 | 0.458582 | 0.414443 | 0.822222 | 0.727607

The Fractional Fourier Transform (FRFT) was used to estimate
$(\mu,\beta_{+},\beta_{-},\alpha_{+},\alpha_{-},\lambda_{+},\lambda_{-})$
parameter. The Maximum likelihood method was applied to the density function
(3.2) and the second and third derivatives were used to compute the highest
eigenvalues of the hessian Matrix and the value of $||\frac{d\log(ML)}{dV}||$.
The parameter estimations of the last 30 iteration processes are shown in
Table 2; in addition to the seven parameters displayed in the first row, we
have added three statistical indicators: the value of the logarithmic of the
product of the density functions ($\log(ML)$), the norm of the partial
derivative function $||\frac{d\log(ML)}{dV}||$, and the highest value of the
Eigenvalues of the Hessian matrix ($MaxEigenValue$). As shown in the last row
of Table 2, the iteration process stops when the estimated parameter converges
to a stable parameter with a maximum value $\log(ML)=-4659.1914$,
$||\frac{d\log(ML)}{dV}||=0$ and $MaxEigenValue=-1.45426$.

Table 2. GTS Parameters Estimations for S&P 500 data

$Iterations$ | $\mu$ | $\beta_{+}$ | $\beta_{-}$ | $\alpha_{+}$ | $\alpha_{-}$ | $\lambda_{+}$ | $\lambda_{-}$ | $Log(ML)$ | $||\frac{dLog(ML)}{dV}||$ | $MaxEigenValue$
---|---|---|---|---|---|---|---|---|---|---
1 | -0.5266543 | 0.67666185 | 0.43662658 | 0.43109407 | 0.32587754 | 0.85012595 | 0.60145985 | -4664.7647 | 289.206866 | 48.3287566
2 | -0.5459715 | 0.67059496 | 0.42415971 | 0.44989796 | 0.34810329 | 0.80921924 | 0.60289955 | -4660.2156 | 35.9853332 | -6.0431046
3 | -0.7108484 | 0.66676659 | 0.20613817 | 0.46963149 | 0.41246296 | 0.8368645 | 0.73679498 | -4663.5777 | 1082.00304 | 449.252496
4 | -0.6704327 | 0.66891111 | 0.11250993 | 0.46528242 | 0.50028087 | 0.83422813 | 0.86330646 | -4660.5268 | 135.684998 | 15.1103621
5 | -0.739816 | 0.66779246 | 0.09097181 | 0.48302677 | 0.45922815 | 0.85551132 | 0.81320574 | -4660.0208 | 45.7082317 | 10.8419307
6 | -0.6517205 | 0.65580459 | 0.19673172 | 0.47996977 | 0.42743878 | 0.85248358 | 0.75350741 | -4659.833 | 46.3329967 | 11.8534517
7 | -0.8136505 | 0.71997129 | 0.21558026 | 0.44023152 | 0.41952419 | 0.79423596 | 0.74025892 | -4662.4815 | 1187.98211 | 166.006596
8 | -0.7805857 | 0.70635344 | 0.22952102 | 0.44666893 | 0.4166063 | 0.80360771 | 0.7334486 | -4659.7758 | 85.6584355 | -3.4305336
9 | -0.7543747 | 0.69912144 | 0.23468747 | 0.45029318 | 0.41542774 | 0.80935861 | 0.73077041 | -4659.1944 | 1.07446211 | -0.7993855
10 | -0.7533784 | 0.69885561 | 0.23480071 | 0.45042677 | 0.41541378 | 0.80956671 | 0.73072468 | -4659.1943 | 1.03706137 | -0.8136858
11 | -0.752414 | 0.69859801 | 0.2349108 | 0.45055614 | 0.41540021 | 0.8097682 | 0.73068024 | -4659.1942 | 1.00184387 | -0.8273569
12 | -0.7514794 | 0.69834808 | 0.23501796 | 0.45068156 | 0.41538701 | 0.80996353 | 0.73063702 | -4659.1942 | 0.96860649 | -0.8404525
13 | -0.7496917 | 0.69786927 | 0.23522417 | 0.4509216 | 0.41536164 | 0.81033731 | 0.73055392 | -4659.194 | 0.90739267 | -0.8650984
14 | -0.7471903 | 0.69719768 | 0.23551545 | 0.45125776 | 0.41532584 | 0.81086063 | 0.73043671 | -4659.1938 | 0.82669598 | -0.8987475
15 | -0.7463993 | 0.69698491 | 0.23560822 | 0.45136413 | 0.41531445 | 0.8110262 | 0.73039942 | -4659.1937 | 0.80232744 | -0.9091961
16 | -0.7456279 | 0.69677723 | 0.23569899 | 0.45146791 | 0.41530331 | 0.8111877 | 0.73036294 | -4659.1937 | 0.77907591 | -0.9193009
17 | -0.7434222 | 0.69618233 | 0.23596025 | 0.45176484 | 0.41527126 | 0.81164974 | 0.73025805 | -4659.1935 | 0.71530396 | -0.9477492
18 | -0.7427203 | 0.69599272 | 0.2360439 | 0.45185938 | 0.41526101 | 0.81179683 | 0.73022449 | -4659.1934 | 0.69583386 | -0.9566667
19 | -0.7376152 | 0.69460887 | 0.23665992 | 0.45254797 | 0.41518555 | 0.81286777 | 0.72997772 | -4659.193 | 0.56549166 | -1.0197019
20 | -0.7353516 | 0.69399264 | 0.23693733 | 0.45285383 | 0.41515159 | 0.81334323 | 0.72986678 | -4659.1929 | 0.51375646 | -1.0466686
21 | -0.7342715 | 0.69369811 | 0.23707047 | 0.45299987 | 0.41513529 | 0.81357023 | 0.72981356 | -4659.1928 | 0.47702329 | -1.0633442
22 | -0.727748 | 0.69191195 | 0.23788938 | 0.45388257 | 0.41503509 | 0.81494168 | 0.72948683 | -4659.1924 | 0.30254549 | -1.1564316
23 | -0.7269375 | 0.69168915 | 0.23799284 | 0.45399234 | 0.41502243 | 0.81511217 | 0.72944561 | -4659.1924 | 0.28659447 | -1.1671214
24 | -0.7261524 | 0.69147312 | 0.23809343 | 0.45409871 | 0.41501011 | 0.81527736 | 0.72940554 | -4659.1923 | 0.27235218 | -1.177299
25 | -0.7074205 | 0.68625723 | 0.24059125 | 0.45665159 | 0.41470173 | 0.81923664 | 0.72841073 | -4659.1916 | 0.15127988 | -1.3705213
26 | -0.7028887 | 0.68497607 | 0.24122265 | 0.45727561 | 0.4146222 | 0.82020261 | 0.72815815 | -4659.1915 | 0.12096032 | -1.4033087
27 | -0.6987925 | 0.68381109 | 0.24180316 | 0.45784257 | 0.4145479 | 0.82107955 | 0.72792474 | -4659.1914 | 0.07862954 | -1.4283679
28 | -0.6934505 | 0.68228736 | 0.24256739 | 0.45858391 | 0.41444901 | 0.82222574 | 0.72761631 | -4659.1914 | 0.75324649 | -1.6442249
29 | -0.6934765 | 0.68229002 | 0.24257963 | 0.45858236 | 0.41444404 | 0.82222285 | 0.72760762 | -4659.1914 | 9.93E-06 | -1.4542674
30 | -0.6934774 | 0.68229032 | 0.24257975 | 0.45858219 | 0.41444396 | 0.8222226 | 0.7276075 | -4659.1914 | 7.4193E-08 | -1.4542632

See [10] for more details and methodologies on Fitting GTS distribution.

##  4\. Pricing European Options under GTS Process

###  4.1. GTS Process: Risk Neutral Esscher Measure

The method of Esscher transform was introduced by [18] as an efficient
technique for pricing derivative securities if a Lévy process models the
logarithms of the underlying asset prices. An Esscher transform of a stock-
price process provides an equivalent martingale measure. The existence of the
equivalent Esscher transform measure is not always guaranteed, and the issue
of the unicity of the equivalent martingale measure remains recurrent when
pricing an option.
From the characteristic function $\vartheta(\xi,t)$ in (3.1), the Moment
generating function $M(h,t)$ of the Generalized Tempered Stable (GTS)
distribution can be written as follows.

$\displaystyle M(h,t)= \vartheta(-ih,t)=e^{t\Psi(-ih)}\quad\hbox{with
$-\lambda_{-}<h<\lambda_{+}$}$ (4.1)

Under the Esscher transform with parameter h, the probability density of
$Y=\\{Y_{t}\\}$ becomes:

$\displaystyle\hat{f}(x,t,h)= \frac{e^{hx}{f(x,t)}}{M(h,t)}$ (4.2)

The Moment generating function $M(h,t)$ becomes M(z,t,h)

$\displaystyle M(z,t,h)=
E^{h}\left[e^{zX_{t}}\right]=\int_{0}^{+\infty}e^{zx}\hat{f}(x,t,h)dx=\left(\frac{M(h+z,1)}{M(h,1)}\right)^{t}=M(z,1,h)^{t}$
(4.3) $\displaystyle
M(z,1,h)=e^{\Psi_{h}(z)}\quad\quad\Psi_{h}(z)=\Psi(-i(h+z))-\Psi(-i(h)))$
(4.4)

Given the process $\left\\{e^{-r\tau}S(\tau)\right\\}_{\tau\geq 0}$ with $r$
the constant risk-free rate of interest, we look into the conditions to have
$h^{*}$ such as

$\displaystyle E^{h^{*}}\left[e^{-r\tau}S(\tau)\right]=S(0)$ (4.5)

$S(\tau)=S(0)e^{Y_{\tau}}$, with $Y_{\tau}$ is the GTS process described by
the characteristic exponent (2.7). The equation (LABEL:eq:l71) becomes

$\displaystyle
e^{r\tau}=E^{h^{*}}\left[e^{Y_{\tau}}\right]=M(1,1,h^{*})^{\tau}=e^{\tau\Psi_{h^{*}}(1)}\quad\quad\Psi_{h^{*}}(1)=r\quad\hbox{
with $-\lambda_{-}<h^{*}<\lambda_{+}$ }$ (4.6)

The existence and unicity of $h^{*}$ in (4.10) were studied empirically using
the parameter data from Table 1. Over the interval
$]-\lambda_{-};\lambda_{+}[$, $\Psi_{h}(1)$ is a strictly increasing function,
as shown in Fig 2(a). Fig 2(b) provides the solution $h^{*}$ of equation
(4.10) for free interest rate less than 10%. The solution $h^{*}$ increases
with free interest rate $r$.

(a) $\Psi_{h}(1)$

(b) $\Psi_{h^{*}}(1)=r$

Figure 2. $\mu=-0.693477$, $\beta_{+}=0.682290$, $\beta_{-}=0.242579$,
$\alpha_{+}=0.458582$, $\alpha_{-}=0.414443$, $\lambda_{+}=0.822222$,
$\lambda_{-}=0.727607$

###### Theorem 4.1

(GTS Esscher transform distribution)
Esscher transform of GTS process $Y=\\{Y_{t}\\}_{t\geq 0}$ with parameter
$(t\mu,\beta_{+},\beta_{-},t\alpha_{+},\alpha_{-},\lambda_{+},\lambda_{-})$ is
also a is also a GTS process $Y^{*}=\\{Y^{*}_{t}\\}_{t\geq 0}$ with parameter
$(t\mu,\beta_{+},\beta_{-},t\alpha_{+},t\alpha_{-},\tilde{\lambda}_{+},\tilde{\lambda}_{-})$

$\displaystyle\tilde{\lambda}_{+}=\lambda_{+}-h^{*}\quad\quad\tilde{\lambda}_{-}=\lambda_{-}+h^{*}$
(4.7)

###### Proof.

we recall the function $\Psi_{h^{*}}(z)$ in (4.4).

$\displaystyle\Psi_{h^{*}}(z)$
$\displaystyle=\Psi(-i(h^{*}+z))-\Psi(-i(h^{*})))$ (4.8) $\displaystyle=\mu
z+\alpha_{+}\Gamma(-\beta_{+})\left(((\lambda_{+}-h^{*})+z)^{\beta_{+}}-{(\lambda_{+}-h^{*})}^{\beta_{+}}\right)+$
$\displaystyle\alpha_{-}\Gamma(-\beta_{-})\left(((\lambda_{-}+h^{*})+z)^{\beta_{-}}-{(\lambda_{-}+h^{*})}^{\beta_{-}}\right)$

Using the Esscher transform method, the moment generating function for GST
process $Y=\\{Y_{t}\\}_{t\geq 0}$ becomes:

$\displaystyle M(z,t,{h^{*}})=
E^{{h^{*}}}\left[e^{zY_{t}}\right]=M(z,1,{h^{*}})^{t}=e^{t\Psi_{h^{*}}(z)}\quad
\hbox{with $-\lambda_{-}-h^{*}\leq z\leq\lambda_{+}-h^{*}$ }$ (4.9)

The Esscher transform method preserves the structure of the GTS process. We
have a new GST process generated by the GST distribution
$GST(\mu,\beta_{+},\beta_{-},\alpha_{+},\alpha_{-},\lambda_{+}-h^{*},\lambda_{-}+h^{*})$.
∎

###  4.2. Extended Black-Scholes Formula

Under the EMM, $f(\xi,\tau,h^{*})$ is the probability density of GST
distribution with parameter
$(\tau\mu,\beta_{+},\beta_{-},\tau\alpha_{+},\tau\alpha_{-},\tilde{\lambda}_{+},\tilde{\lambda}_{-})$.
($\tilde{\lambda}_{+},\tilde{\lambda}_{-}$) is defined in (4.7).

###### Corollary 4.2

(Extended Black-Scholes)
Let $r$ a continuously compounded risk-free rate of interest;
$Y=\\{Y_{t}\\}_{t\geq 0}$, a GST Process with parameter
$(t\mu,\beta_{+},\beta_{-},t\alpha_{+},t\alpha_{-},\tilde{\lambda}_{+},\tilde{\lambda}_{-})$;
and $(S(0)e^{X_{T}}-K)^{+}$ the terminal payoff for a contingent claim with
the expiry date $T$. Then at time $t<T$, the arbitrage price of the European
call option with the strike price $K$ can be written as follows.

$\displaystyle F^{GTS}_{call}(S_{t},t)$
$\displaystyle=S(t)\left[1-F(log(\frac{K}{S(t)}),\tau,h^{*}+1)\right]-Ke^{-r\tau}\left[1-F(log(\frac{K}{S(t)}),\tau,h^{*})\right]$
(4.10) $\displaystyle
F(log(\frac{K}{S(t)}),\tau,h^{*})=\int_{-\infty}^{log(\frac{K}{S(t)})}f(\xi,\tau,h^{*})d\xi$
(4.11)

Where $\tau=T-t$, $F(k,\tau,h^{*})$ and $F(k,\tau,h^{*}+1)$ are the cumulative
distribution of GST distribution.

See [9] for corollary 4.2 proof

The GTS distribution function’s probability density and cumulative functions
were obtained through the inverse Fourier transform. See [13, 12, 9] for
details.

$\displaystyle f(\xi,\tau,h^{*})=\frac{1}{2\pi}\int_{-\infty}^{+\infty}e^{i\xi
z+\tau\Psi_{h^{*}}(-z)}dz\quad\quad
F(\xi,\tau,h^{*})=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{e^{i\xi
z+\tau\Psi_{h^{*}}(-z)}}{iz}dz+\frac{1}{2}$ (4.12)

(a) $\tau=\frac{1}{12}$ years

(b) $\tau=0.25$ years

(c) $\tau=0.5$ years

Figure 3. Estimations: $f(\xi,\tau,h^{*})$ versus $f(\xi,\tau,h^{*}+1)$

Based on parameter data from Table 1, we added a 6% risk-free interest rate
and the corresponding Esscher transform parameter ($h^{*}=-2.4448$) in Fig
2(b). FRFT technique was used to compute the density and cumulative functions
in (4.12).

(a) $\tau=\frac{1}{12}$ years

(b) $\tau=0.25$ years

(c) $\tau=0.5$ years

Figure 4. Estimations: $F(\xi,\tau,h^{*})$ versus $F(\xi,\tau,h^{*}+1)$

As shown in Fig 3 and 4, the discrepancy between both functions increases with
the period ($\tau$). The estimation results will be used to compute the option
pricing (4.11).

###  4.3.  Generalized Black-Scholes Formula

Under the EMM, $f(\xi,\tau,h^{*})$ can be written as follows:

$\displaystyle
f(\xi,\tau,h^{*})=\frac{1}{2\pi}\int_{-\infty}^{+\infty}e^{-i\xi
z-\tau\varphi(z)}dz\quad\quad\varphi(z)=-\Psi_{h^{*}}(z)$ (4.13)

with $\Psi_{h^{*}}(z)$ defined in (4.3).

###### Theorem 4.3

Let $r$ a continuously compounded risk-free rate of interest;
$Y=\\{Y_{t}\\}_{t\geq 0}$, a Variance-Gamma Process with parameter $(\mu
t,\delta,\sigma,\alpha t,\theta)$; and $(S(0)e^{X_{T}}-K)^{+}$, the terminal
payoff for a contingent claim with the expiry date $T$. Then at time $t<T$,
the arbitrage price of the European call option with the strike price $K$ can
be written as follows.

$\displaystyle
F^{GTS}_{call}(S_{t},t)=\frac{K}{2\pi}\int_{-\infty+iq}^{+\infty+iq}{\frac{e^{\left(i\xi
log(\frac{S(t)}{K})-\tau(r+\varphi(\xi))\right)}}{i\xi(i\xi-1)}d\xi}$ (4.14)

Where $\varphi(z)$ is the characteristic exponent of the VG model, $\tau=T-t$,
and $q<-1$.

See [9] for theorem 4.3 proof

To compute $F^{GTS}_{call}$ in (4.14), the numerical integration technique,
called the Newton-Cotes formula of 12 degrees, will be implemented as follows.

$\displaystyle
g^{x}(\xi)=\frac{exp\left[{i(\xi+iq)x-\tau(r-\Psi_{h^{*}}(\xi+iq))}\right]}{2\pi
i(\xi+iq)(i(\xi+iq)-1)}\quad\widehat{F^{GTS}_{call}}=\frac{b}{n}\sum_{p=0}^{\frac{n}{Q}-1}\sum_{j=0}^{Q}W_{j}g^{log(\frac{S(t)}{K})}(\xi_{Qp+j})$
(4.15)

In order to perform the integral (4.15), the following parameter values were
used: $a=0$, $b=20$, $Q=12$, $n=5000Q$, $n_{0}=5000$. See [15, 14, 19] for
more details on the choice of parameter values and the twelve-point rule
Composite Newton–Cote’s methodology.

###  4.4. Parameter $q$ Evaluation

Let us consider the stock or index price $S=S_{0}e^{Y}$ and the strike price
$K$; the call payoff can be written as follows.

$\displaystyle(S(0)e^{Y_{T}}-K)^{+}$
$\displaystyle=S(t)(e^{Y_{\tau}}-k)^{+}=S(t)g(Y_{\tau},k)\ \ \ \
k=\frac{K}{S_{t}}$ (4.16)

The Fourier transform of the call payoff in (4.16) can be derived as follows

$\displaystyle\mathscr{F}[g](y,k)$
$\displaystyle=\int_{0}^{+\infty}e^{-iyx}g(x,k)dx=\frac{ke^{-iylog(k)}}{iy(iy-1)}\quad
\hbox{for $\Im(y)<-1$}$ (4.17)

Call payoff in (4.16) can be recovered from the inverse of Fourier and labeled
$\check{g}(x,k)$ with

$\displaystyle\check{g}(x,k)=\frac{1}{2\pi}\int_{-\infty+iq}^{+\infty+iq}e^{iyx}\mathscr{F}[g](y,k)dy\hskip
14.22636pt \hbox{for $q<-1$}$ (4.18)

The payoff ($\check{g}(x,k)$) in (4.18) depends on the $q$ parameter as shown
in Fig 5(a). The inverse Fourier ($\check{g}(x,k)$) produces poor results if
$q=-2$ (color red).

To find the $q$ value with a high level of accuracy, we define the error
function ($ER(k,q)$) between the call payoff in (4.16) and the inverse Fourier
payoff in (4.18).

$\displaystyle
ER(k,q)=\sqrt{\frac{1}{m}\sum_{j=1}^{m}\left[(e^{x_{j}}-k)^{+}-\check{g}(x_{j},k)\right]^{2}}\hskip
14.22636pt \hbox{with $-M\leq x_{j}\leq M$}$ (4.19)

(a) $(X-k)^{+}$ versus $\check{g}(x,k)$

(b) $ER(k,q)$ and Optimal parameter (q)

Figure 5. Optimal parameter ($q$) and minimum Error value ($ER(k,q)$)

Fig 5(b) displays (in color blue) the $ER(k,q)$ minimum value as a function of
the strike price $k$; and the optimal correspondent parameter $q$ as a
function of the strike price $k$ (in color red). Both graphs display almost a
constant function with respect to the strike price.

##  5\. Empirical Analysis: A case of S&P 500 return

Based on parameter data from Table 1, we added a 6% risk-free interest rate
and the corresponding Esscher transform parameter ($h^{*}=-2.4448$) in Fig
2(b). The GTS option pricing will be computed across maturity and option
moneyness using Extended and Generalized Black-Scholes formulas. The closed-
form Black-Scholes model [20] was added to the analysis as a benchmark.

$\displaystyle F^{BS}_{call}(S_{t},\tau)$
$\displaystyle=S_{t}N(d_{1})-Ke^{-r\tau}N(d_{2})$ (5.1) $\displaystyle
d_{1}=\frac{Ln(\frac{S_{t}}{K})+(r+\frac{1}{2}\sigma^{*2})\tau}{\sigma^{*}\sqrt{\tau}}\quad\quad
d_{2}$ $\displaystyle=d_{1}-\sigma^{*}\sqrt{\tau}\quad\quad
N(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}\exp\left(-\frac{t^{2}}{2}\
\right)\,dt$ (5.2)

The variance $\sigma^{*}=0.1848$ is the annualized variance computed from the
variance in Table 2.
Moneyness describes the intrinsic value of an option in its current state,
which indicates whether the option would make money if exercised immediately.
Option moneyness can be classified into three categories: At-the-money (ATM)
option ($k=\frac{S_{t}}{K}=1$), Out-of-the-money (OTM) option
($k=\frac{S_{t}}{K}<1$), and In-the-money (ITM) option
($k=\frac{S_{t}}{K}>1$).
On August 15, 2023, the S&P 500 market price closed at $\$4,437.86$. We
compute the GTS call option price on S&P 500 with the spot price ($S_{0}$)
$\$4,437.86$. Table 3 summarizes the price computations as a function of time
to maturity ($\tau$) and option moneyness ($k=\frac{S_{t}}{K}$).

Table 3. European Call Price on S&P 500

Strike Price | Moneyness($\frac{S_{t}}{K}$) | BSM | GTS(4.14) | GTS(4.7) | BSM | GTS(4.14) | GTS(4.7) | BSM | GTS(4.14) | GTS(4.7) | BSM | GTS(4.14) | GTS(4.7) | BSM | GTS(4.14) | GTS(4.7) | BSM | GTS(4.14) | GTS(4.7)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Period ( in year) | 0.0625 | 0.125 | 0.25 | 0.5 | 0.75 | 1
2,689.61 | 1.65 | 1,758.315 | 1,758.320 | 1,758.315 | 1,768.345 | 1,768.349 | 1,768.345 | 1,788.291 | 1,788.296 | 1,788.291 | 1,827.756 | 1,827.782 | 1,827.776 | 1,866.803 | 1,866.905 | 1,866.898 | 1,905.614 | 1,905.833 | 1,905.826
2,773.66 | 1.60 | 1,674.579 | 1,674.584 | 1,674.579 | 1,684.922 | 1,684.927 | 1,684.922 | 1,705.492 | 1,705.498 | 1,705.493 | 1,746.215 | 1,746.258 | 1,746.252 | 1,786.624 | 1,786.778 | 1,786.772 | 1,826.924 | 1,827.229 | 1,827.221
2,863.14 | 1.55 | 1,585.441 | 1,585.445 | 1,585.441 | 1,596.118 | 1,596.122 | 1,596.118 | 1,617.352 | 1,617.359 | 1,617.354 | 1,659.443 | 1,659.517 | 1,659.511 | 1,701.402 | 1,701.634 | 1,701.628 | 1,743.414 | 1,743.833 | 1,743.826
2,958.57 | 1.50 | 1,490.361 | 1,490.364 | 1,490.361 | 1,501.393 | 1,501.397 | 1,501.393 | 1,523.336 | 1,523.347 | 1,523.343 | 1,566.952 | 1,567.080 | 1,567.075 | 1,610.728 | 1,611.072 | 1,611.066 | 1,654.742 | 1,655.312 | 1,655.305
3,060.59 | 1.45 | 1,388.723 | 1,388.726 | 1,388.723 | 1,400.135 | 1,400.140 | 1,400.136 | 1,422.842 | 1,422.865 | 1,422.861 | 1,468.225 | 1,468.444 | 1,468.439 | 1,514.200 | 1,514.700 | 1,514.695 | 1,560.590 | 1,561.354 | 1,561.347
3,169.90 | 1.40 | 1,279.825 | 1,279.828 | 1,279.825 | 1,291.645 | 1,291.650 | 1,291.647 | 1,315.189 | 1,315.242 | 1,315.238 | 1,362.746 | 1,363.110 | 1,363.105 | 1,411.460 | 1,412.173 | 1,412.167 | 1,460.696 | 1,461.702 | 1,461.695
3,287.30 | 1.35 | 1,162.861 | 1,162.864 | 1,162.861 | 1,175.120 | 1,175.130 | 1,175.127 | 1,199.631 | 1,199.756 | 1,199.752 | 1,250.063 | 1,250.643 | 1,250.638 | 1,302.261 | 1,303.246 | 1,303.241 | 1,354.907 | 1,356.202 | 1,356.196
3,413.74 | 1.30 | 1,036.899 | 1,036.903 | 1,036.900 | 1,049.637 | 1,049.670 | 1,049.667 | 1,075.413 | 1,075.692 | 1,075.688 | 1,129.906 | 1,130.791 | 1,130.786 | 1,186.555 | 1,187.872 | 1,187.867 | 1,243.245 | 1,244.872 | 1,244.866
3,550.29 | 1.25 | 900.861 | 900.871 | 900.869 | 914.166 | 914.285 | 914.282 | 941.956 | 942.524 | 942.521 | 1,002.394 | 1,003.666 | 1,003.663 | 1,064.634 | 1,066.328 | 1,066.323 | 1,126.008 | 1,127.992 | 1,127.986
3,698.22 | 1.20 | 753.494 | 753.552 | 753.550 | 767.731 | 768.107 | 768.105 | 799.315 | 800.337 | 800.335 | 868.330 | 870.037 | 870.033 | 937.303 | 939.382 | 939.378 | 1,003.876 | 1,006.213 | 1,006.208
3,859.01 | 1.15 | 593.480 | 593.800 | 593.799 | 610.219 | 611.153 | 611.152 | 649.077 | 650.650 | 650.648 | 729.588 | 731.693 | 731.690 | 806.084 | 808.496 | 808.492 | 878.047 | 880.690 | 880.686
4,034.42 | 1.10 | 420.919 | 422.039 | 422.037 | 444.934 | 446.566 | 446.564 | 495.718 | 497.690 | 497.687 | 589.513 | 591.851 | 591.848 | 673.405 | 676.021 | 676.016 | 750.358 | 753.204 | 753.199
4,226.53 | 1.05 | 245.429 | 247.061 | 247.060 | 283.281 | 284.940 | 284.939 | 347.774 | 349.635 | 349.633 | 453.122 | 455.394 | 455.391 | 542.703 | 545.313 | 545.310 | 623.363 | 626.252 | 626.249
4,437.86 | 1.00 | 100.299 | 100.154 | 100.153 | 146.757 | 147.206 | 147.205 | 217.356 | 218.448 | 218.447 | 326.853 | 328.694 | 328.692 | 418.344 | 420.686 | 420.683 | 500.326 | 503.053 | 503.049
4,671.43 | 0.95 | 23.028 | 21.981 | 21.981 | 56.504 | 55.743 | 55.743 | 116.478 | 116.527 | 116.526 | 217.593 | 218.712 | 218.711 | 305.239 | 307.061 | 307.059 | 385.051 | 387.398 | 387.395
4,930.96 | 0.90 | 2.309 | 2.146 | 2.146 | 14.527 | 13.845 | 13.845 | 51.103 | 50.510 | 50.510 | 130.982 | 131.319 | 131.318 | 208.118 | 209.260 | 209.259 | 281.521 | 283.307 | 283.305
5,221.01 | 0.85 | 0.078 | 0.090 | 0.090 | 2.209 | 2.034 | 2.034 | 17.375 | 16.800 | 16.800 | 69.543 | 69.311 | 69.311 | 130.530 | 130.995 | 130.994 | 193.324 | 194.458 | 194.456
5,547.33 | 0.80 | 0.001 | 0.002 | 0.002 | 0.173 | 0.163 | 0.163 | 4.290 | 4.021 | 4.021 | 31.596 | 31.153 | 31.153 | 73.869 | 73.823 | 73.822 | 122.955 | 123.470 | 123.469
5,917.15 | 0.75 | 0.000 | 0.000 | 0.000 | 0.006 | 0.007 | 0.007 | 0.712 | 0.645 | 0.646 | 11.837 | 11.474 | 11.474 | 36.829 | 36.533 | 36.533 | 71.172 | 71.223 | 71.222
6,339.80 | 0.70 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.072 | 0.064 | 0.064 | 3.493 | 3.304 | 3.304 | 15.702 | 15.392 | 15.392 | 36.685 | 36.488 | 36.488
6,827.48 | 0.65 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.004 | 0.004 | 0.004 | 0.767 | 0.702 | 0.703 | 5.514 | 5.310 | 5.310 | 16.377 | 16.132 | 16.132
7,396.43 | 0.60 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.116 | 0.103 | 0.103 | 1.520 | 1.428 | 1.428 | 6.111 | 5.932 | 5.932
8,068.84 | 0.55 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.011 | 0.009 | 0.009 | 0.309 | 0.281 | 0.281 | 1.819 | 1.729 | 1.729

(4.14): 12-point rule Composite Newton-cotes Quadrature (4.7): FRFT-CDF

The Fractional Fourier Transform (FRFT) algorithm is used to perform the GTS
cumulative function in the extended Black-Scholes formula (4.7), whereas the
twelve-point rule Composite Newton–Cotes Quadrature algorithm is used to
perform the Generalized Black-Scholes Formula (4.14). As expected, both
algorithms produce the same estimation for European option prices at two
decimal places.
To generalize the analysis and account for a large range of option moneyness
and maturity, The error (5.3) was computed as the difference between VG and BS
option prices:

$\displaystyle
Error(k,\tau)=\frac{F^{GTS}_{call}(S_{t},\tau)}{K}-\frac{F^{BS}_{call}(S_{t},\tau)}{K}\quad\quad\hbox{$(k=\frac{S_{t}}{K})$}$
(5.3)

Fig 6 displays the error ($Error(k,\tau)$) as a function of the time to
maturity ($\tau$) and the option moneyness ($k$). The spot price ($S_{t}$) is
a constant, and the option moneyness depends on the strike price. The Black-
Scholes (BS) model and GTS European option produce different pricing results.
Compared to the option price under the GTS distribution, the BS model is
underpriced for the at-the-money (ATM) and the in-the-money (ITM) options as
shown in Fig 6.

(a) Error(k,$\tau$)

(b) Error(k,$\tau$) (top view)

Figure 6. Combined Effects of time to Maturity and Option Moneyness

For the out-the-money (OTM) option, the Black-Scholes (BS) model is slightly
overpriced (dark blue color in Fig 6). However, the BS model and GTS European
options yield the same option price for the deep out-of-the-money (OTM) and
the deep-in-the-money (ITM) options.
The findings can be compared to the European option pricing under the
Variance-Gamma process [9, 21]. In both studies, we have the same pattern. the
BS model is underpriced for the in-the-money (ITM) option, whereas the BS
model is overpriced for the out-the-money (ITM) option. However, The error
($Error(k,\tau)$) between the BS model and GTS European options has a slightly
different magnitude compared to the magnitude error($Error(k,\tau)$) between
the BS model and option pricing under the Variance-Gamma process [9].

##  6\. Conclusion

The paper examines the pricing of the European option when the log asset price
follows a rich class of Generalized Tempered Stable (GTS) distribution. The
S&P 500 data is used to illustrate the pricing results. The Equivalent
Martingale Measure (EMM) of the GTS distribution exists, and the Esscher
transform method preserves the structure of the GTS distribution.
The extended Black-Scholes formula was computed based on the cumulative
distribution function (CDF) generated by the Fractional Fast Fourier (FRFT)
algorithms. In addition, The Generalized Black-Scholes Formula was computed
using the 12-point rule Composite Newton-Cotes Quadrature algorithms. Both
algorithms yield the same European option price at two decimal places.
The Black-Scholes (BS) model and GTS European options produce different
pricing results. Compared to the option price under the GTS distribution, the
BS model is underpriced for the near-the-Money (NTM) and the in-the-money
(ITM) options. However, BS model and GTS European options yield the same
option price for the deep out-of-the-money (OTM) and the deep-in-the-money
(ITM) options.

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