CC BY
25
THE EXPONENTIATED SKEW LAPLACE DISTRIBUTION: PROPERTIES AND APPLICATIONS
Timothy Kayode Samson1
Statistics Programme, College of Agriculture, Engineering and Science, Bowen University, Iwo,
Nigeria
kayode.samson@bowen.edu.ng1 •
Christian Elendu Onwukwe2, Ekaette Inyang Enang3
Department of Statistics, Faculty of Physical Sciences, University of Calabar, Calabar, Nigeria christianonwukwe@unical.edu.ng2, ekaettenang@unical.edu.ng3
Abstract
In this paper, a 4-parameter Exponentiated Skew Laplace distribution is defined and studied. Various statistical properties including its moment generating function, characteristics function, hazard function, and reliability function of the proposed ESLD were derived. The estimation of its parameters was carried out using the maximum likelihood method of estimation. The performance of the proposed ESLD compared with other similar distributions was demonstrated empirically with daily returns of S & P 500 between 2/02/24 and 28/03/2024 and daily returns of Bitcoin between 2/02/24 and 1/04/24 as obtained from Yahoo Finance. The fitness performance of the proposed distribution was evaluated based on log-likelihood, AIC, and BIC. Results obtained show that the proposed ESLD reported the highest log likelihood as well as the lowest AIC and BIC in the two data sets. This study therefore underscores the superiority of the proposed distribution over the some of the similar existing distributions.
Keywords: Skew Laplace, Exponentiated, Distribution, Reliability function, Maximum Likelihood
I. Introduction
The Laplace distribution also commonly known as the double exponential distribution is named after Pierre Simon Laplace. The growing popularity of the Laplace-based models as described by Lakshmi and Sebastian [1] is due to the properties of the sharp peak at the mode, heavier than normal distribution making it an asymmetric distribution Laplace distribution as described by Nadarajah and Kotz [2] is a tractable lifetime model with applications in various areas including telecommunication, biological sciences, engineering, and life testing among other areas of human endaviours. Despite the significance of this distribution in probabilistic modelling, the Laplace distribution lacks a skewness parameter and hence not be able to account for skewness in data. To overcome this challenge, the Laplace distribution as noted by Kotz et al., [3] and Safavinejad et al.,[4]) was extended by adding a skewness parameter. This extension therefore gave rise to the skew Laplace distribution.
The use Skew-Laplace distribution in Economics, Finance and Engineering has been emphasized by Puig and Stephens [5] while Julia and Vives-Rego [6] used Skew Laplace distribution to analyze bacterial sizes in axenic cultures. The skew Laplace distribution has been applied to different areas of research. This is due to its flexibility to model real data that exhibit skewness. Skew Laplace distribution has also been used in probabilistic modelling of financial data. For instance, Jing et al. [7] applied asymmetric Laplace to currency exchange rates in Australia, Canada, European and United Kingdom. Similarly, Shi et al. [8] used the asymmetric Laplace distribution in portfolio selection. The use of Skew Laplace distribution can also be extended to cryptocurrencies and stock data because of its ability to capture skewed data as cryptocurrencies and stocks have some stylized properties which include departure from normality, price jumps, and high volatility patterns.
In this study, a new form of skew Laplace distribution of Safavinejad et al. [4] is proposed, and some of the statistical properties are proposed with application to real data. The fitness performance of the proposed ESLD compared to some existing distributions will also be investigated in this study. This study adopts the method of exponentiation introduced by Gupta et al. [9] in generating the proposed Exponentiated Skew Laplace Distribution (ESLD). In the exponentiation method, only one shape parameters is introduced into the parent distribution. This method has been used by Agboola et al., [10], Oguntunde et al., [11], Nadarajah and Bakar [12], Datta and Datta [13], Andrade et al., [14], Adubisi et al., [15] among others to generate more flexible distribution.
II. Methods
Exponentiated Skew Laplace Distribution (ESLD)
The probability density function (pdf) and the Cumulative Density Function (CDF) of the Skewed Laplace Distribution (SLD) are defined in (1) and (2) as follows:
f (x;0,e,S) =
1 Ix"¿I *
expl —:— I, x < *
0 + s
0
1 I x -*) >*
expl--I, x > *
0+s
s
Safavinejad et al. [ 4]
(1)
F (x;0,s,S) =
0 I xS) *
expl —:— I, x < *
0+s
1 --
0
"exp!
x-*
, x >*
0 + s v s - œ < x < œ, A> 0, 0,s, > 0, - œ < * < œ
(2)
Safavinejad et al. [4]
where, 0 is a scale parameter, 8 location parameter and skewness parameter.
The probability density function of an Exponentiated generation distribution is given as:
g (X) = AF(x)^1 f (x), A> 0 (3)
The Cumulative Density Function is given as:
G (x ) = [ F ( x)f (4)
Where, f (x) is the density function of the Skewed Laplace Distribution (SLD) and F(x) is its corresponding cumulative density function.
s
Substituting for f (x) in (3) and F(x) in (4) give the pdf and the CDF of the proposed Exponentiated Skew Laplace Distribution.
g (x ) = <
2
0 + s 2
0 + s
exPl
exPl
x-5
~e~\
x-5
0
0+s
exp
x-5 0
2-1
,x <5
s ( x-5 1--expl
2-1
(5)
G(x ) = '
0 ( x-5 exPl
0+s
0
s J\_ 0 + s V s
0, s > 0, 2 > 0, -ro < x < ro,
2-1
,x<5
, x >5
s ( x -5 1--exp l
0+s
s
2-1
(6)
x>5
Theorem 1
A random variable X is said to follow an Exponentiated Skew Laplace Distribution (ESLD), if its probability density function can be expressed as:
g (x H
2 ( x-5 expl
0+s 2
0+s
exp
0 x-5
0 ( x-5 expl
0+s
1-
0+s
0
exp
2-1
,x<5
x-5
2-1
(7)
x>5
Q,s> 0, 2 > 0, x ,
Proof
The purpose of this proof is to show that the proposed ESLD is a probability density function. Since, it a continuous distribution function, then
J g(x)dx
= 1
2 ( x-5 exp
0 + s
0
0 ( x-5 exp
0+s
0
2-i », dx + J(
2 ( x-5 exp
0+s
s
i
1 --
0+s
exp
(8)
x-5
2-1
V 0 (9)
dx
Splitting the expression in equation (9) into two parts as follows:
2 ( x -5 Let gj = J TT— expj
0 + s
0
0 ( x-5 exp
0+s
0
2-1
dx
g2 =
2 0 + s
J exp ) ^TT'1 -0^eXp
0+s
x-5 0
2-1
dx
(10) (11)
Solving for g1.
Let z =
0
x-5 0
dz
1
0+s
e °
x-5
3 0
dx 0+ s
(12)
s
s
s
5
s
when, X = —ro, z = 0 and when, X = 8, z =
e
Then
e+s
(13)
gi =
e
e+s
—o =
e
e+s
(14)
Similarly,
/L w
g2=essi
n J eXP I
e+s i i s
x—8|L s
< 1--exp
I e+s
x—1 fey1
dx = 1 —
Therefore,
J g(x)dx
e
e+s
+1 —
e
e+s
e
=1 .
e+s
(15)
(16)
This implies that the proposed ESLD is a probability density function. When / = 1, the ESLD in (7) reduces to Skew Laplace distribution When X — 0 — 1, the ESLD in (5) reduces to Laplace distribution.
D.9-
-1 0 1
x
Figure 1: Plot ofpdfofthe proposed Exponentiated Skewed Laplace distribution (ESLD).
Figure 2: Plot of the CDF of the proposed distribution ESLD.
Asymptotic Behaviour of the Proposed ESLD Theorem 2
If the random variable X ~ESLD (0,5,s,A), lim g(x) = 0 and lim g(x) = 0.
Proof
g (x )=i
A f x-5 exp
d + s A
0 + s
exPl
0 x-5
0 f x-5 exPl
0 + s
0
A-1
x<5
s f x-5 1 ---expl
0+s
s
A-1
, x >5
For, lim g(x)
A
0 + s
exP
-x-5 0
0 f-x-5" exp
0+s
0
A-1
=0
(17)
Similarly,
lim g (x) =
A ) [ x-51L s f x-5 exP 1--M1 ---exPl
0+s
s
0+s
s
A-1
= 0 (18)
Hence, lim g(x) = 0 and limg(x) = 0
x^-x x^x
Theorem 4
If, X ~ESLD (0,5,s,A), lim G(x) = 0 and lim G(x) = 1.
x^—x x^x
Proof
lim G(x) = lim
0 f x-5' exP
0 + s
0
(19)
8
A
0 + S
Similarly, lim G(x) = lim
0 f-m-S exp
0
A
= 0
(20)
s f x-S 1 ---exp
0 + s
S f m-S|
1 -0+Sexp J
Hence, lim G(x) = 0 and lim G(x) = 1
S
= 1 - 0 = 1
(21)
The proof of the Theorems 3 and 4 indicate that the proposed distribution satisfied the property of the limiting property a probability density function and cumulative density function respectively.
Statistical Properties of the proposed ESLD Moment Generating Function of the Proposed ESLD
Suppose that X is a random variable that follows ESLD with parameters (0,S,S,A), the moment generation function is given as:
Mx (t) = (12)
A
exp
\j f s
T 0
1
:p(tS)Y\A 1|(-1) , , , ,
0 + s j [ j )y \0 + sJl 0 + 1+ j - ts + 1+ j\
Proof
By definition of the moment generating function:
m
Mx (t) = J e* g(x)dx
(22)
A
0 + s
s f x-s
J exp (tx) exp
-m
m ,
J exp (tx) exp I
0j
x-S
0 f x-S exp
0 + s
1-
0
A-1
dx +
s
x-S
0+s
Let p = J ^+~exp(tx)exp [ 0
-m S [
P = J0T7exp(tx) exp I- "
0
1-
0+s s
exp
0+s
f x -SI
exp I 1
[ s J_
A-1
f x_S)] dx
[ 0 J
A-1
f x-S
I S j_
A-1
dx
Let w = x -S , ^ x = S + w, if x = -m, w = -m - m = x=S,w=S-S=0
-m
\j
P = J exp (tw + tS) exp
w I [ 0
exp
0j{0+S
0
P1 = J exp (tw + tS) exp
JF1 *
'w U/
0
1 -
[ 0 + S
0 I A
exp
i A-1
dw
(23)
(24)
(25)
(26)
0
P = J exp (tw + tS) exp
0 j a vj f1 -00S:
exp
JW 0
dw
(27)
A
A
-m
s
S
p=5 A V' (■ -00s
J J exP (tw +15) exP f 00 J exP
jw 0
dw
\J 0
p=555 t"; 1)(-i) j t1 -¿u -1 w++f j *
P exP
P =550fA; 1](-1) ' t1 ' exP (t5)l exP
(t5) J exPftw + 00 + 0\dw
1
P=5 (A; >1)j t1-0s]' «*> 1 , ;
exP
A-1
P =5 ;
;=0 t '
0
t + —+ ^ 00
exP (t5)
00
w | t +1 + ; t 0 0)
w It + 1+ ; I
I 0 0)
dw
(-J1-^) 70MP-0]
0
exP (t5)
P =5A Vf1-00s) 01+;
Similarly,
5 A
0 + s
P2 = j0+-eXP (tx ) exP(
x-5
0
s f x -5
1--exP
0+s
s
A-1
When x = 5, since w = x -5 w = 0, When x = x, u = x
P2 = J exP (tw +15) exP f
w s
1 s f w
1--exP l--
0 + s t s
A-1
dw
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
P2 = J exP (tw +15) exP f—)sf^ • 1](-1)
s
j=
A-1
P2=5 , N
01 j
t s) J=01 J f^x'x
0 + s
0+s
r
I exP
jw t s
J exP (tw +t5) exP - w - — lrfw J t s s )
(37)
(38)
j=
A-1
P2=5 , I(-1)
0tj
A-1
P2=5 , I(-1)
0tj
0 + s, 0 + s
exP
exP
(t5)J
exP
0t
w jw
tw---—
(ts)i
exP
w
t -1 -;
ss
dw
(39)
(40)
P2=5
A- 1
0tj
(-1)j
0+s
exP
( 5)J
exP
-w
1j
-t + - + J-
ss
dw
(41)
0
x
s
s
£
^=I r; v ' fe
exp (tô)
1
1 j
—t + - + — s s
exp
(42)
-2 =I (X ')(—i)
j j exp (tô). [0—11 e+s ) —ts+1+/ J
(43)
P =-lfX —1~1(—1) j f—)j s exp ('ô
2 ¿0[ j / ; [e+s) —ts+1+j X
(') = ^(Pl + P2 )
(44)
" (45)
Mx (') =
X
e+s
¿X ')(—1)j (■—e) ' —
j=0 [ j ) [ e + .) te+1+ j
-I
j=0
fx — j
(—1)j
' s \1 s exp (tô)
e + s ) —ts +1 + j
(46)
Note that, 1 —
X
e
e+.—e
e+s
e+s e+s
fx—1\
M't (' ) = —— exp('ô)I . (—1) e + s
j=0
j
e+s
e
'— +1 + j — ts +1 + j
(47)
(48)
Characteristic Function of the Proposed ESLD
Suppose that X is a random variable that follows ESLD with parameters (0,S,S, a) , the characteristic function, (px (t) is given by:
% (' ) =
X exp(itô) IX 1 )j
T e
1
e+s
— + s ) [ i—+ 1+ j — its + 1+ j _
(49)
Proof
By definition,
m
px (t) = E [exp (itx)1= J exp (itx) g (x)dx
(50)
Since the mgf of the ESLD has been obtained and presented in (40), then, it is easy to obtain its characteristic function and it is given by:
% (' ) =
X exp(itô) II X — V1 )j
T e
1
e+s
— + s ) [ i—+ 1+ j — its + 1+ j _
(51)
to
0
s
s
s
s
s
s
s
Hazard Function of ESLD
The Hazard function by definition is given as:
H ( x) = g (52)
1 — G (x)
Where, H(x) is the hazard function while g(x) and G(x) are the probability density function and cumulative density function respectively.
Hence, the Hazard function of the proposed ESLD can be expressed as:
H (x ) = <
A
0 + s eXPl 0
x-5
0
0 + s eXPl 0
x-5
A-1
0
0 + s eXPl 0
x-5
A-1
, x <5
A f x-5 exPl
0 + s
s
1 -
0 + s
exPl
x-5
A-1
1-
0 + s
exPl
x-5
A-1
, x >5
(53)
Reliability Function
The reliability function of the ESLD is given by:
R( x) = 1 - G( x)
Hence,
(54)
R(x ) = 1 -
0 f x-5 exPl
0+s
1-
0
A-1
x<5
0+s
exPl
x-5
A-1
, x >5
(55)
Parameter Estimation for the ESLD.
The estimation of the parameter of the ESLD defined in (5) will be estimated using the Method of Maximum Likelihood derived below:
Let X1, X2,..., Xn be a random sample of size n from Exponentiated Skew Laplace Distribution (ESLD). Then the likelihood function is given by: For, x <5
A
L(x ) = n\0+-\ H
i=1
0+s
0 f x,-5' exP\
A-1
+s
0
(56)
L{x )=yy H
Taking the natural log of both sides.
log L(x) = n logl —— | + (A - 1)log
A-1
0 + s
n exP
x-5) I 0
0 )\0 + s
exP\
xi~5
A
log L(x ) = n log\^| + (A-1)
+s
, ^ 0 f 2x - 25 log [[—- exPl
=1 +s
0
(57)
(58)
(59)
s
s
s
s
s
s
i=1
A
log L(x ) = n logI —-| + (A- 1)log
0 + s
A
0
0 + s
exp
f n \
2S(x( -S)
i=1_
0
logL(x) = nlogI --| + (A-1)
0 + s
n logI
0
2S(x( -S)
0+s
+
x >S
L(x ) = n(
A
0+s
exp|
x -S
1-
0+s
exp|
0
x -S
A-1
L(x ) = |
A
0+s
jJ exp [ - jj1 - 7s exp|
x -S
A-1
Taking the natural log of both sides.
log L(x) = n log| —— | + (A - 1)log
0+s
A
log L(x ) = n logI --| + (A-1) log
log L(x ) = n log
0+s
' A ' 0 + s
0+s
jM" xiSSSI1 "0+7exp|" xSS.
1 n 2 n
-Sz(x,-s) f s y t(xi-s) " e i=1
—
0+s
| + (A-1)
n
Z(xi-S)-n log
--
0 + s J s i=1
tx-s)
(60)
(61)
(62)
(63)
(64)
(65)
(66)
Hence, the Log Likelihood of the proposed Exponentiated Skew Laplace Distribution (ESLD) is given as:
log L(x) = j
n log n log
A
0+s
'A
0 + S
] + (A-1)
| + (A-1)
n log
0
+-
0 + sj 0£
n
;Z(xi -S)-nlo:
T&-S)
x <S 2
--
0+s J S i=1
Tb-S)
(67)
x >S
Differentiating log L(x) in (57) with respect to each of the parameter in the model and setting them to zero give the estimate of each of the parameter. This was done in R software using the appropriate optimization function.
III. Results
Applications to Real Data
The empirical application of the proposed ESLD was carried out using two sets of real life data. The first data is the daily returns of S& P 500 between 2/02/2024 and 28/03/2024 as obtained from Yahoo finance. The second data set is on the daily returns of Bitcoin between 2/02/2024 and 01/04/2024 which was also obtained from the Yahoo finance website (www.yahoofinance. com). Daily closing returns series were computed from daily closing prices using the formula below:
n
i=1
S
s
s
i=1
s
i=1
1
2
s
i=1
1
s
i=1
DCRt = log
{ DCP ^
DCPt
x100
(68)
t-i J
Where, DCRt is the closing returns at the day t while DCPt and DCPt-1 are the closing prices at the present day and previous day respectively.
The fitness performance of the proposed ESLD was compared with that of similar distributions such as the skew Laplace distribution and Laplace distribution using Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) as defined below:
AIC = 2k - 21n(LL) (69)
BIC = -2LL + k 1og(n) (70)
Where, n is the number of observations, k is the number of parameters and LL is the log-likelihood.
Data set 1
0.46 -0.14 0.10 0.36 0.02 0.25 -0.04 -0.60 0.41 0.25
-0.21 -0.26 0.05 0.91 0.02 -0.16 0.07 -0.07 0.23 0.35
-0.05 -0.44 0.22 0.45 -0.28 -0.05 0.48 -0.08 -0.12 -0.28
0.27 0.24 0.38 0.14 -0.06 -0.13 -0.12 0.37 0.05
Data set 2
0.11 -0.20
1.47 -0.19
-0.45 -0.49
0.79 3.41
0.96 -1.01
0.19 2.10
-0.42 0.08
1.78 0.09
0.71 0.14
-2.98 1.54
-1.23 -2.64
1.73 0.02
0.43 1.23
0.19 -0.42
2.28 2.00
0.54 0.88
2.00 -0.54
-0.33 0.80
0.95 1.73
0.39 -0.29
3.94 -0.92
0.13 0.33
-3.78 4.02
-0.53 -0.15
0.57 0.47
0.42 -0.37
0.87 -0.29
1.91 -0.39
-1.58 -1.15
1.04 -1.01
Table 1: Summary statistics for the data set
Data set
Min.
Max. Mean
SD
Skewness
Data set (S&P 500) Data set (Bitcoin)
1 39 -0.60
II 60
-3.78
0.91 .0763
4.02 .3484
.29451
1.42987
.268
.013
SD- Standard deviation.
n
Table 2: Results of the estimated parameters as well as the goodness fit for the proposed distribution and other
similar distribution
Dataset Distributions 5 9 S A LL AIC BIC
I ESLD -0.060 0.134 0.368 1.052 -5.431 18.862 17.975
SLD -0.061 0.102 0.348 - -7.938 21.876 20.649
LD 0.048 0.233 - - -9.247 22.494 21.677
II ESLD 0.103 0.431 0.462 1.203 -98.346 204.691 203.804
SLD 0.111 0.505 0.442 - -103.041 212.082 211.417
LD 0.189 1.034 - -103.620 211.240 210.796
ESLD- Exponentiated skew Laplace Distribution, SLD- Skew Laplace Distribution, LD- Laplace Distribution, SGED- Skew Generalized Error Distribution.
Result presented in Table 3 show the parameter estimates of the proposed ESLD and compared to other related probability distribution (Skew Laplace distribution and Laplace distribution) as well as their fitness performance. Result of the LogLikelihood shows that among these distributions, the proposed ESLD reported the highest LogLikelihood in the two real data compared with other competing distributions. Similarly, the AIC and BIC reported by the proposed ESLD is lower than that of Skew Laplace distribution and Laplace distribution for both the two data set. These result show better fitness performance of the proposed ESLD than both Skew Laplace distribution and Laplace distribution in modelling financial data.
IV. Discussion
This study improve on the robustness of the Skew Laplace distribution introduced by Safavinejad et al. [4] by introducing additional shape parameter using the method of exponentiation. We derived some of the statistical properties of the proposed Skew Laplace Distribution (ESLD) after ensuring that the proposed ESLD satisfied the properties of a statistical distribution. Some of the statistical properties of the proposed ESLD derived include: moment generating function, characteristic function, hazard function and reliability function. The estimation of its parameters was carried out using the maximum likelihood. The performance of the proposed ESLD compared with other similar distributions were demonstrated empirically with returns from S & P 500 (Dataset 1) and returns from Bitcoin (Dataset 2). The findings suggest that the new distribution outperforms the existing models considered, indicating its better representation and flexibility when compared with some existing models. This findings show that the ELSD outperformed Skew Laplace Distribution (SLD) which indicates that the use of the method of exponentiation improve the performance of distribution. This is corroborated by that of other studies, Agboola et al., [10], Oguntunde et al., [11], Nadarajah and Bakar [12], Datta and Datta [13], Andrade et al., [14] and Adubisi et al., [15] which also found that exponentiated distribution performed better than their parent distributions.
References
[1] Lekshmi, S. and Sebastian, S. (2014). A skewed generalized discrete Laplace distribution. International Journal of Mathematics and Statistics Invention, 2: 95-102.
[2] Nadarajah, S. and Kotz, S. (2003) The Exponentiated Frechet Distribution. Interstat Lectronic Journal,14:1-7.
[3] Kotz, S., Kozubowski, T. J. and Podgorski, K. The Laplace Distribution and Generalizations. Birkh "auser, Berlin, 2001.
[4] Safavinejad, M., Jomhoori, S. and Noughabi, H.A. (2016). Testing skew-Laplace distribution using density-based empirical likelihood approach. Journal of Statistical Research, 13: 124.
[5] Puig, P. and Stephens, A.M. (2007). Goodness of fit tests for the skew-Laplace distribution. SORT, 31: 45-54.
[6] Julia, O. and Vives-Rego, J. (2005). Skew-Laplace Distribution in Gram-negative Bacterial Axenic cultures: New Insights into Intrinsic Cellular Heterogeneity. Microbiology, 151:749- 755.
[7] Jing, H.; Liu, Y.; Zhao, J. (2022). Asymmetric Laplace Distribution Models for Financial Data: VaR and CVaR . Symmetry, 14 :807.
[8] Shi, Y.,Ng, C.T.and Yiu.K.C. (2018). Portfolio selection based on asymmetric Laplace distribution, coherent risk measure, and expectation-maximization estimation. Quantitative Finance and Economics, 2: 776-797.
[9] Gupta, R. C., Gupta, P. L. and Gupta, R. D (1998). Modeling failure time data by Lehman alternatives, Communications in Statistics, Theory and Methods, 27: 887-904.
[10] Agboola, S., Dikko, H.G. and Asiribo, O.E. (2018). On A New Exponentiated Error Innovation Distributions: Evidence of Nigeria Stock Exchange. Journal of Statistics Applications and Probability Letters, 7: 321-331.
[11] Oguntunde, P. E., Adejumo A. O. and Balogun O. S. (2014). Statistical Properties of the Exponentiated Generalized Inverted Exponential Distribution. Applied Mathematics , 4: 47-55.
[12] Nadarajah, S. and Bakar,S. A. A. (2016). An exponentiated geometric distribution. Applied Mathematical Modelling, 40:6775-6787.
[13] Datta, D. and Datta, D. (2013). Comparison of Weibull distribution and exponentiated Weibull distribution based estimation of mean and variance of wind data.International Journal of Energy, Information and Communications, 4: 1-11.
[14] Andrade, T.A. , Rodrigues, H., Bourguignon, M. and Cordeiro, G. (2015). The exponentiated generalized Gumbel distribution. Revista Colombiana de Estadística, 38: 123-143.
[15] Adubisi, O.D, Abdulkadir, A.,Farouk, U.A., and Chiroma, H. (2022). The exponentiated half logistic skew-t distribution with GARCH-type volatility models. Scientific African, 16 : e01253.
