Page 19 - 34-1

NTU Management Review Vol. 34 No. 1 Apr. 2024

Figure 1 Pairs of

Note: Figure 1 depicts the pair that can be yielded by the US, LS, and BS distributions,
respectively.

We present the mean, variance, skewness, and kurtosis of the US distribution in the
following proposition and their derivations in Appendix B.
n
nd
on
ibuti
i
ku
f
o
llo
a
t
variance,
he
e
m
o
the
t
f
US

h
istr
rt
ean,

osis
d

kewness,
s

We present We present the mean, variance, skewness, and kurtosis of the US distribution in the following proposition
posi
pro
tion
wing
and their de riva ti ons in A ppendi x B.
and their derivations in Appendix B.
o
n

as follows:
r
Propositio Proposition 2. The four characteristics of the US distribution are presented as follows:
c
.
the
f
ted
er
i
c
T
bution
a
r
four
he
cs
Proposition 2. The four characteristics of the US distribution are presented as follows:
US di

t
hara
e presen

t
s
isti
2
ℳ ( , , , ) = � sinh(Ω), � (12) (12)

�
ℳ ( , , , ) = � sinh(Ω), ,
�� ��
� � (13) (13)
( 1)( cosh(2Ω) +1), 2Ω) +1),
( , , ( , , , ) =, ) = ( 1)( cosh( ,
��
��
2 2
� ( 1)� ( +2) sinh(3Ω) +3 sinh(Ω)
� ( 1)� ( +2) sinh(3Ω) +3 sinh(Ω)��
( , , ( , , , ) = , ) = , , , (14) (14)
��
��
�2� cosh(2Ω) +1��
�2� cosh(2Ω) +1 � �
US ( , , , )
( , , , )
US
(15) (15)
2
2
4
4
3
2
23
2
2
� + 2 + 3 3� cosh(4Ω) + 4 ( +2) cosh(2Ω) +3(2 + 1)) + 4 ( +2) cosh(2Ω) +3(2 + 1)
� + 2 + 3 3� cosh(4Ω
= = , , ,
2� cosh(2Ω) +1�) +1�
2� cosh(2Ω 2 2
where Ω= and = (1 ).
�
⁄
⁄
where Ω= and ⁄ = (1 ). ⁄ �

n
t

pt
ions with the US Distribution
F
g

GB
3. Prici 3. Pricing Formula of the GB options with the US Distribution
or
h
e
o
a
m
of
ul
11
h
t
e
G
ion

r
o

p
e
s
to
B

e
te
s
roxim
t
d
e
This sectio This section first presents the procedure to find a matching US distribution to approximate the GB

a
a
r
n
to
he
pp
c
u
t
f
p
t
S
n

d
r
g
c

m
a
U

a
st
f
t
bu
h
i
e
in
i
distri
r
n
c
p
t
ptions,

g
v
t
h
i
s
h
d
e

n
an
e
o
for
for
e
t
h
ng
h
,

t
ula
d

m
i
o
d
m
e
i
r

the
c
n
e
r
i

e
e

g

t
t
o
n
m
h
m
m
n
-
u
distribution distribution using the moment-matching method, then derives the pricing formula for the GB options, and
a

s
i
B
G
finally, sho finally, show the computation of hedging Greeks.
w th
s
mputati
.
ng Greek
on of hedgi
e co

3.1. The 3.1. The Moment-Matching Method for the US Distribution
US Distr
i
Mome

Me
thod
bution
t-Mat
for

ching
the
n
cha
As noted a As noted above, the challenge of pricing GB options mainly stems from the lack of an exact distribution of
an
G

v
distribu
t
B

e
g
st
xac
om
option
llenge

b

o
i
on
f
ems

,
pricin
s
of
a
ack
h
nly
t
t
of
o
m
fr
e
t
e
l
i
he

the GB; as a result, their pricing formulas can not be derived in precisely. To improve the BPW model
the GB; as a r esult, t h e i r p r i cing formul a s c an n ot b e derived in p re ci sel y . T o i mpro ve t he B PW m odel
et
ib
mily
orrect
a
har
w
),
c
on
acteristics

fa
w

r
uti
2007

c
h
l.,
p
US
it
fou

n
pt
the
h
dist
ado
r
i
e
(Borovkov (Borovkova et al., 2007), we adopt the US distribution family with the four correct characteristics presented in
t
e
a
resented
the
m
t
i
)
t
m

app
e
equations ( equations (4)-(7) to approximate the GB distribution. To choose a matching US distribution to approximate the
r
B
ibuti
ose
o

T
r
)
e

di
at
oxi

n
dist
p
rox
he
a
tio
ma
g
a
p
-
o
bu
t
s
n.
G
US

atchi
on
tri
7
to
(
4
cho
GB distributi on, w e equ a li z e t he f irst f our c haract eri s ti cs o f t h e U S di s t r i b ut io n to t h os e of th e G B d ist r ibut i o n,
GB distribution, we equalize the first four characteristics of the US distribution to those of the GB distribution,
and obtain the follo wing equati on sy stem:
and obtain the following equation system:
ℳ ( , , , = ℳ = ℳ
ℳ ( , , , ))
��
��
( , , , ))
( , , , = =
�� �� (16) (16)
( , , , ))
( , , , = =
��
��
( , , , ))
( , , , = =
�� ��
i
b
t
we
n

,
h
e

i
o
s
n
By solving By solving this equation system and denoting the solution by � �, , ̅, �, we can determine a matching US
ot
m
e
a

s
o
n
g
i
n

i
a
t
t
a
US
n
m

e
d
ing
ch
y
s

d
ne
a
e
o
et
c
d
u
s
n
a

ermi
t
h
t
t

u
l
q
̅
̅
�
�
, ̅

,
y �
,
�
�
distribution to approximate the GB distribution. 9 9
distribution to approximate the GB distribution.
w
raight
for
n
st
i
no
mu
Because eq Because equation (16) is a nonlinear equation system, solving this equation is not straightforward and must
t

st
s
equati
ard and
o
s

syst
(
e
sol
n

em,
non
i
near
v
ng
16)
io
io
i
q
li
hi
uat
u
s
n
a

at
t

b
me
uil
(
son
uenter
a
t
u
T
Newt
p

resort to a resort to a numerical method. Tuenter (2001) proposes a root-finding algorithm built on the Newton-Raphson
)
on-Raph
pro

he
t
n
2
o
oses
001
t
hod.
n
r
oo
i
t-f
nding
l
hm
a
ori

merical
g
t

almer
g
1
)
i
ed

al
(201
at
ach
ing
,
n

s
ch
pricin
d
as

is

Lo,

9 The momen 9 The moment-matching approach is also used for the pricing of Asian options, such as in Chang and Tsao (2011) and Lo, Palmer, and

P
the

a
such
u
n

A
sao
io
opt
m
a

nd
an
so

sia
T
,
d
for
g
o
C

ppro
an
a
t-
h

n
f
ns
d
i
t
a
n
Yu (2014), an Yu (2014), and guaranteed minimum withdrawal benefits, such as in Milevsky and Salisbury (2006) and Yang, Wang, and Liu (2020).

h
a
0
.
w
2
b
l
)

r
Liu (
e
20
w
d
e
Mil
g

u
ev
d
n
i

s
m
nd

e
d
n
an
Salisb

a
ar
ry (2006) a
e
sky

u
t

,
g
s
u
an
m
i
f
m
t
s
u
, an
c
n
d
a
i
a
Y
W
h
,

ng
i
11 11