Page 17 - 34-1

� (log orm l (log orm l system),
�
s
y
em)
,
st
) 2 (u bou e system),
� �� ) 2 (u bou
�
��
⁄
( ) = �(
⁄ e system),
( ) = �(
� (log orm l (log orm l system),
�
st
,
y
s
em)
(bou e s
1 (1 �� 1 (1 �� ) (bou e system)
em)
⁄ )

st
y
⁄
) 2 (u bou e system),
�
��
�
��
( ) = �( ( ) = �( � ) 2 (u bou (log orm l system),
⁄
⁄ e system),
�
s
y
NTU Management Review Vol. 34 No. 1 Apr. 2024

(log orm l
,
em)
st
n be
The probabili ty d ensit y funct ions o f ea ch s yst �� 1 (1 e r iv ) ed a nd pr em) d as fol l ows.
esente
c
e
The probability density functions of each system can be derived and presented as follows.
a
m
d ��
) 2 (u bou e system),
1 (1
(bou e system)
y
st
(bou e s
⁄ ) �
⁄ �
��
( ) = �( ( ) = �( �� ) 2 (u bou
⁄
⁄ e system),
th
Le
para
,
meters g
d
,
iven
n
t
eq
e
istribu
are
.
Definition 1 Definition 1. Let denote the Johnson distribution, and , , , and are the four parameters given in equation
in
an
uati
t
r
ou
the f
Johnso
on
denot
e

i
fol
ed

y
ions
funct
pr

d
as
The probability density functions of each system can be derived and presented as follows.
nd
ensit
r
m
ows.
ea
yst
The probabili
l
d
e
ty d
ch s
iv
c
a
esente
f

, ��

��
d ⁄
e 1⁄ )
)
on, and (1
a (bou e system) system)
n be (bou e
o 1 (1

ys
ty den
in
he
(8). The pro (8). The probability density functions of each system in the Johnson distribution family are given as follows:
u
fun
tem
ven as foll
J
g
li
o
babi
s
ch s
on fam
ti
son distrib
i
are
of ea
ows:
ily
ion
sit
h
n
ct
y
t

LeThe probability density functions of each system can be derived and presented as follows. ability density functions of each system can be derived and presented as follows.
e
r

ou
iven
meters g
denot
The prob
,
d
an
istribu
i
on, and
,
on
uati
,
t
e
Definition 1 Definition 1. Let denote the Johnson distribution, and , , , and are the four parameters given in equation
in
.
the f
t
th
eq
n
d
are
Johnso
para

(a) Lognormal System (LS)
(a) Lognormal System (LS ) ct ion s of ea t i on, and , , , an d are the f ou r para meters g iven in eq uati on
(8). The probability density functions of each system in the Johnson distribution family are given as follows:
(a) Lognormal System (LS) ch system in the Johnson distribution family are given as follows:
li
babi
y
ty den
(8). The pro
fun
sit
Definition 1
.
d
istribu
denot
th
e

e
Le
Johnso
t
n
Definition 1. Let denote the Johnson distribution, and , , , and are the four parameters given in equation

are
sit
he
ily
fun
J
g
y
babi
ty den
i
(8). The pro
li
(8). The probability density functions of each system in the Johnson distribution family are given as follows:

(a) Lognormal System (LS ) ct ion s of ea ch s ys tem in t 1 o h n son distrib u ti on fam �� �, , ven as foll ows: (9)
�
�
(a) Lognormal System (LS)
1

(9)
exp � � l �
( ) = ( ) = exp � � l � �� �,
��
��

2
(a)
(a) Lognormal System (LS √2 ( ) √2 ( ) 2 � � (9) (9)
Lognormal System (LS) )
1
1

exp � � l ��
( ) =
where ( ) > 0, ∞ < < ∞, | | = 1, ∞ < < ∞, > 0
( ) = exp � � l
⁄
��
��

where ( ) > 0, ∞ ⁄< < ∞, | | = 1, ∞ < < ∞, > 0 �� �, �� �,

2
2
�
�
√2 ( )
√2 ( )

1
1
exp � � l ��
(b) Unbounded System (US) �� ( ) = exp � � l �� �, �� �, (9) (9)
(b) Unbounded System (US)
( ) =

��
)⁄
|
∞ √2 ( )
∞, | √2 ( )
⁄
,

∞
,
<

0

=
>
<
0
1,
<
∞
where ( where ( ) > 0, ∞ < < ∞, | | = 1, ∞ < < ∞, > 0 2 > 2

<

(b) Unbounded System (US) 1 1 1 1 > 0 � �
ded
(
System ⁄< < ∞, | | = 1, ∞ < < ∞, > 0
1

(b) Unboun where ( ) > 0, ∞ <where ( (b) Unbounded System (US) < ∞, | | = 1, ∞ < < ∞, ) > 0, ∞ U S) exp � � si h �� �, �� �, (10) (10)
⁄
1
( ) = ( ) = exp � � si h � �
��
��

(b)
(b) Unbounded Unbounded System (US) System (US) √2 �( ) � � � 2 � 2 1 1 � ,
√2 �( )
�
1
1
1
1
exp � � si h �� �,
( ) =
exp � �
( ) =
where ∞ < <∞, ∞< <∞, > 0, ∞ < <∞, >0 si h � � � �� �, (10) (10)
��
��

where ∞ < <∞, ∞< <∞, > 0, ∞ < <∞, >0
2
� 2

�
�
�
�

1
1

√2 �( ) 11
√2 �( )
exp � � si h �� �,
( ) =
( ) =
(c) Bounded System (BS)
(c) Bounded System (BS) √2 �( ) � � exp � � si h 1 � 1 � �� �, (10) (10)
��
��
where ∞ < <∞, ∞< <∞, > 0, ∞ < <∞, >0
where ∞ < <∞, ∞< <∞, > 0, ∞ < <∞, >0
2
� 2
�
√2 �( )
1 1 1 1 � �
�
where ∞ < ∞< <∞, > 0, ∞ < <∞, >0
�
(c) Bounded System (BS) <∞, ∞< where ∞ < �� �, �� �, (11) (11)
(c) Bounded System (BS) <∞, > 0, ∞ < <∞, >0 <∞,
( ) = ( ) = exp � � l
exp � � l ��
��
��
(c)
(c) Bound (c) Bounded System (BS)) ed System (BS) √2 ( √2 ( )( ))( ) 2 2 �
Bounded System (BS
�
�
�
1

1
1
1
where < < , ∞ < <∞, >0, ∞< < ∞, >0 2 �� �, �� �, (11) (11)
exp � � l
( ) =
( ) =
where < < , ∞ < <∞, >0, ∞< < ∞, >0
exp � � l ��

��

��

√2 ( )( ))( ) ��
√2 (
2
�
�

1

1
1
1
exp � � l � �� �,
The advantage of the Johnson distribution family lies in its rich pair of skewness and kurtosis. To express
The advantage of the Johnson distribution family lies in its rich pair of skewness and kurtosis. To express
,
exp � � l �
( ) =
( ) =
where < < , ∞ < <∞, >0, ∞< < ∞, >0 2 �� �, (11) (11)
��
��
where < < , ∞ < <∞, >0, ∞< < ∞, >0
√2 (
√2 ( )( ))( )
2
this feature more explicitly, we present Figure 1 with the vertical axis representing the kurtosis ( ) and
this feature more explicitly, we present Figure 1 with the vertical axis representing the kurtosis ( ) and
The advantage of the Johnson distribution family lies in its rich pair of skewness and kurtosis. To express
where < < , ∞ < <∞, >0, ∞< < ∞, >0
The advantage of the Johnson distribution family lies in its rich pair of skewness and kurtosis. To express
where < < , ∞ < <∞, >0, ∞< < ∞, >0
this feature more explicitly, we present Figure 1 with the vertical axis representing the kurtosis ( ) and
The advantage of the Johnson distribution family lies in its rich pair of skewness and kurtosis. To express
�
this feature more explicitly, we present Figure 1 with the vertical axis representing the kurtosis ( ) and
horizontal axis representing the square of skewness ( ), and its coordinate is denoted by ( , ). Figure 1 , ). Figure 1
�
�
�
The advantage of the Johnson distribution family lies in its rich pair of skewness and kurtosis. To express
horizontal axis representing the square of skewness ( ), and its coordinate is denoted by (
6
represents all possible pairs of and . For example, the standard normal distribution is known to have and . For example, the standard normal distribution is known to have
6
this feature more explicitly, we present Figure 1 with the vertical axis representing the
this feature more explicitly, we present Figure 1 with the vertical axis represen kurtosis ( ) and
�
horizontal axis representing the square of skewness ( ), and its coordinate is denoted by ( , ). Figure 1 , ). Figure 1 is ( ) and
�
horizontal axis representing the square of skewness ( ), and its coordinate is denoted by ( ting the kurtos
represents all possible pairs of
�
�
�
The advantage of the Johnson distribution family lies in its rich pair of skewness
�
=0 and � � 6 � 6 � � � �
=0 and = , which is located at (0, ) and displayed by a circle. = , which is located at (0, ) and displayed by a circle.
�
represents all possible pairs of and . For example, the standard normal distribution is known to have and . For example, the standard normal distribution is known to have
horizontal axis representing the square of skewness ( ), and its coordinate is denoted by ( , ). Figure 1 , ). Figure 1
and kurtosis. To express this feature more explicitly, we present Figure 1 with the vertical
horizontal axis representing the square of skewness ( ), and its coordinate is denoted by (
represents all possible pairs of
The pair ( , ) of the distribution is presented by a curve denoted by . The upper area, , ) of the distribution is presented by a curve denoted by . The upper area,
=0 and = , which is located at (0, ) and displayed by a circle. = , which is located at (0, ) and displayed by a circle. the standard normal and .
represents all
�� is known to have to have
represents all possible pairs of and . For example, For example, the standard normal distribution
�
�
The pair (
axis representing the kurtosis ( ) and horizontal axis representing the square of skewness
�
�
=0 and possible pairs of � 6 � 6 ��distribution is known
denoted by , describes the pair ( , ) which can be obtained from the US distribution. The middle , describes the pair ( , ) which can be obtained from the US distribution. The middle
�
�� pair ( t= , which is located at (0, ) and displayed by a circle. = , which is located at (0, ) and displayed by a circle.
�
denoted by �� The pair ( =0 and =0 and The , ) of �� , ) of the distribution is presented by a curve denoted by . The upper area, he distribution is presented by a curve denoted by . The upper area,
�
. Figure 1 represents all possible pairs of
�
, and its coordinate is denoted by

��
��
area, denoted by , shows the pair ( , ) which can be obtained by the BS distribution. The bottom , shows the pair ( , ) which can be obtained by the BS distribution. The bottom
, describ
�� of the distribution is presented by a curve denoted by . The is presented by a curve denoted by . The upper area,
The
area, denoted by �� 6 , describes the pair ( , ) which can be obtained from the US distribution. The middle es the pair ( , ) which can be obtained from the US distribution. The middle upper area, of the distribution � � � �� = 0 and
�
�
�
and . For example, the standard normal distribution is known to have
denoted by pair ( , )
denoted by
The pair ( , )

��
��
��
area, denoted by Impossible Area, depicts the pair ( , ) which cannot be captured by the Johnson , ) which cannot be captured by the Johnson
area, denoted by , shows the pair ( , ) which can be obtained by the BS distribution. The bottom , shows the pair ( , ) which can be obtained by the BS distribution. The bottom
denoted by , describes the pair ( , ) which can be obtained from the US distribution. The middle , describes the pair ( , ) which can be obtained from the US distribution. The middle
area, denoted by Impossible Area, depicts the pair ( � �
�
�
�
�
= 3, which is located at (0,3) and displayed by a circle.
area, denoted by
denoted by
��
��
��
��
distribution family. Therefore, if the pair ( , ) of the of underlying assets belongs to any one of the , ) of the of underlying assets belongs to any one of the
area, denoted by , shows the pair ( , ) which can be obtained by the BS distribution. The bottom
area, denoted by Impossible Area, depicts the pair ( , ) which cannot be captured by the Johnson , ) which cannot be captured by the Johnson
distribution family. Therefore, if the pair ( � � � � � �
The pair
of the f distribution is presented by a curve denoted by Curve .
area, denoted by
area, denoted by Impossible Area, depicts the pair ( , ) which can be obtained by the BS distribution. The bottom , shows the pair (
��
��
LS
e
i
n

oximate
b
ge

r

ar
r
t
n
t
i
he
,

f
i
c
n
t
e
y
possible are possible areas, then one of the Johnson distribution family can accurately approximate the target distribution by
n

by
o

on
o
th
m
a
t
l

o
a
f
h
J
o
app
y
i
e
s

stribu
ccu
e
s

l
h
d
o
t
r
n
d
ti
i
s
a
a
n

t
area, denoted by Impossible Area, depicts the pair ( , ) which cannot be captured by which cannot be captured by the Johnson
area, denoted by Impossible Area, depicts
at �
� �
u�
distribution family. Therefore, if the pair ( the pair ( which can be obtained
distribution family. Therefore, if the pair ( , ) of the of underlying assets belongs to any one of the , ) of the of underlying assets belongs to any one of the the Johnson , )
The upper area, denoted by Area , describes the pair
matching its matching its first four moments. r i b u US i, ) of ithe of r at e l y app runderlying assets assets belongs to t belongs to any one of the any one tiof the

t four moments.
firs
by
e

s
c, ) of the of underlying
i
he
possible aredistribution distribution family. Therefore,

l
a
n
possible areas, then one of the Johnson distribution family can accurately approximate the target distribution by
f
n

d
n
o

m
y
ccu
o
ge
i

d
h
oximate
s
on
n
n
h

a
t
s

,
t
n
ar
tif the
f
family. if
stribu
th
o
a
e
t
t
o
e
a
�
�
JTherefore, the pair (

opair (
from the US distribution. The middle area, denoted by Area , shows the pair
Most market
o
xhibit
ta
e
a
d
ewness
sk
r
onze
n
Most market data exhibit nonzero skewness and higher kurtosis. This also holds for the , especially in
matching its first four moments. ibution family can accurately approximate the target distribution by
matching its possible areas, then one of the Johnson distribution family can accurately approximate the target distribution by possible areas, then one of the Johnson distr a nd h i gher ku r to s i s . This BS a lso holds for the , e s p eciall y i n
t four moments.

firs
which can be obtained by the BS distribution. The bottom area, denoted by Impossible
er
g
r
i
ng
e

.
m
an
m

e
ity
t
ssets,
un
o
a

l
t
t
ng
o
atu
As
derly
d
h

i

a
i
l
r
l
o
i
a
li
m
s

a
v
r
r
e

c
h
o
t

o
,
lo

e
h
i
the cases of the cases of higher volatilities, lower correlations among the underlying assets, and longer time to maturity. As
t
s
e
e
g
r
o
t
w
i
n
n
matchi
ku
holds
p
d
eciall
ewness
.
lso
s
i
xhibit
n
Most market data exhibit nonzero skewness and higher kurtosis. This also holds for the , especially in
onze
a
This
r
e
r
sk
s
the
,
s
o
e
to
a
for
gher

y
a
match
i
i
n
h
nd

ta
Most market ng its first four moments. ing its first four moments.

which cannot be captured by the Johnson distribution
Area, depicts the pair
hMost market data exhibit nonzero skewness and higher kurtosis. This also holds for the , especially in data exhibit nonzero skewness and higher kurtosis. This also holds for the , especially in
Most market
e
an
m
the cases of shown in Figure 1, the distribution located in the has relatively higher kurtosis than the distribution in has relatively higher kurtosis than the distribution in

t
v
w
r
i
.
t
r
lo
g
e
derly
t
o
o
m
i
l
i
o
ng
t
r
i
li
o
ity
h
As
c
r
o
e

a
i
a
a
i
l
n
s
m
er
t
o
e
un
d
t
a
s

e
,

h

r

g

atu
ng
n
the cases of higher volatilities, lower correlations among the underlying assets, and longer time to maturity. As
l
ssets,
e
shown in Figure 1, the distribution located in the
��
��
of the GB of underlying assets belongs to any
family. Therefore, if the pair
r
l
US
ma
i
t
istr
b
t
i
ibuti
t
ng
a
u
i
d
e
. Thus,
e

t
or

s
.

E
f
r
s
d
p
h
e
i
i
on

the the
prox
c
i
capabl
o

m
. Thus, the US distribution is more capable of approximating the distribution. Empirical
i
m
a
on
p
i

i
t
d
h
l
t
er
l
e
o
i
v
li
r
t

a
an
ng
o

a
un
e
l
i
r
t

atu
o
derly

m
r
the cases of higher volatilities, lower correlations among the underlying assets, and longer time to maturity. As

r
a
m
t

As
n
o
g
.
ity
n
o
e
s

the cases of
h
e
e
s
ng
i
a
w

lo
g
,
e
m
ssets,
i
o

c
r
t
h
i

the ��shown in Figure 1, the distribution located in the has relatively higher kurtosis than the distribution in has relatively higher kurtosis than the distribution in

shown in Figure 1, the distribution located in the
��
one of the possible areas, then one of the Johnson distribution family can accurately
��
��
shown in Figure 1,
the shown in Figure 1, the distribution located in the has relatively higher kurtosis than the distribution in has relatively higher kurtosis than the distribution in
. Thus, the distribution located i
d
US
. Thus, istr
more capable of approximating the distribution. Empirical
the the
the US distribution is more capable of approximating the distribution. Empirical ibution is n the
approximate the target distribution by matching its first four moments.
��
��
��
��
l
a
ot
m
se
6 Figure 1 is p 6 Figure 1 is plotted based on the limiting properties of the skewness and kurtosis of the Johnson distribution family. i c a l

ily
o
.
b
ti
n fa
ted
o
e limiting p
kewn
h
e s
e J
on t

r
hnson dist
h
and ku

ti
o
f
rt
es o
r

t
sis of
th
oper

ess
d
i
bu

o
u
ibuti
or
p
ma
i
d
e
i
i
istr

b
t
.
the the
a
on

m

i
i
prox
s
f
E
. Thus,
m
on
p

. Thus, the US distribution is more capable of approximating the distribution. Empirical
t
ng
t
US
i
h
d

i
s
e
capabl
r
t
r
e

the

��
��
6
6
ti
on t
i
f
l
sis of

d
n fa
es o
r
m
o
rt
o
a
ti

e s
.
kewn
t
b

hnson dist
h
se
h

ted
ily
oper
o
ot
e limiting p
and ku
bu
6 Figure 1 is p 6 Figure 1 is plotted based on the limiting properties of the skewness and kurtosis of the Johnson distribution family.
ess

th
e J
r

e s
and ku
kewn

f
th
6 Figure 1 is p 6 Figure 1 is plotted based on the limiting properties of the skewness and kurtosis of the Johnson distribution family.
ess
b

a
on t
hnson dist
e limiting p
m
h
o
ti
i
bu
d
se

n fa
o
r
oper
l
r

ot
rt
es o
ti
sis of
o
h
t
ted
e J
ily
.
6
6
6
Figure 1 is plotted based on the limiting properties of the skewness and kurtosis of the Johnson
distribution family. 6 6

9