Page 16 - 34-1

where ∈ stands for the unit number of the th asset. If ∀ ∈ , then the represents a basket of
�
�
�
where ∈ stands for the unit number of the th asset. If ∀ ∈ , then the represents a basket of
�
underlying assets; if ∃ <0, then the represents a spread. Though the exact distribution of the is
�
�
�
underlying assets; if ∃ <0, then the represents a spread. Though the exact distribution of the is
unknown, its first four moments can be computed and are presented in the Preposition 1.
�
unknown, its first four moments can be computed and are presented in the Preposition 1.

Proposition 1. The first four moments of the are computed as follows:
Proposition 1. The first four moments of the are computed as follows:

�

�
∗
� � �

E � � = � 0
�

� � �
∗
E � � = � 0
��� �

���
�
�

�
�
∗
�� � �� � �� �,� ��
�
∗

E � � = � � 0 0
�
�

�
�� � �� � �� �,� ��
∗
∗

E � � = � � 0 0
��� �
�
���

���
���
�
�
�

�
�
�
∗
∗
�
∗
�� � �� � �� � �� �,� �� �,� �� �,� ��

E � � = � � � 0 0 0
�
�
�
∗
∗
∗

�
E � � = � � � 0 0 0 �� � �� � �� � �� �,� �� �,� �� �,� ��
��� ��� ��� � � �
E � � = ��� ��� ���
�

E � � =
�

Valuation of Spread and Basket Options
� � � � ∗ ∗ ∗ ∗ �� � �� � �� � �� � �� �,� �� �,� �� �,� �� �,� �� �,� �� �,� ��
∑ ��� ∑ ��� ∑ ��� ∑ ��� 0 0 0 0 ,
�
�
�
�
∗
∗
∗
∗
∑ � ∑ � ∑ � ∑ � 0 0 0 0 �� � �� � �� � �� � �� �,� �� �,� �� �,� �� �,� �� �,� �� �,� �� ,
���
���
�
�
�
�
���
���
∗
where 0 = 0 , = , and other notations are defined accordingly.
� ∗ � � �,� �,� � �
where 0 = 0 , �,� = , and other notations are defined accordingly.
Based on Proposition 1 and some statistical computation, the mean (ℳ), variance ( ), skewness ( ), and
� �
�,� � �
), variance ( ),
Based on Proposition 1 and some statistical computation, the mean (
�
Based on Proposition 1 and some statistical computation, the mean (ℳ), variance ( ), skewness ( ), and
), and kurtosis ( ) of the GB(T) can be derived as follows:
kurtosis ( ) of the can be derived as follows:
skewness (
kurtosis ( ) of the can be derived as follows: (4)

ℳ =E � �, ,
ℳ =E � �, (4)

=E � ℳ �, , (5)

�
=E � ℳ �, (5)
�

ℳ �
= E �� ℳ � �, , (6)

�
� �, (6)
= E �� √
√ 4
= E �� ℳ � �, . (7)

4
√
ℳ � �, (7)
= E ��
√
These four characteristics can be exactly computed using present market data.
These four characteristics can be exactly computed using present market data.
These four characteristics can be exactly computed using present market data.

2.3. The Johnson Distribution Family
2.3 The Johnson Distribution Family
2.3. The Johnson Distribution Family
The Johnson (1949) distribution family is a collection of probability distributions,
The Johnson (1949) distribution family is a collection of probability distributions, which are transformed
The Johnson (1949) distribution family is a collection of probability distributions, which are transformed
which are transformed from standard normal distributions via three types of functions with
from standard normal distributions via three types of functions with four parameters. Let stand for a standard
from standard normal distributions via three types of functions with four parameters. Let stand for a standard
four parameters. Let Z stand for a standard normal distribution and X denote a Johnson
normal distribution and denote a Johnson distribution. The relation between and is presented by:
normal distribution and denote a Johnson distribution. The relation between and is presented by:
distribution. The relation between Z and X is presented by:

(8)
= � �, (8)
= � �, ,
where is a location parameter, is a scale parameter, and and are shape parameters. The transformation of

where is a location parameter, is a scale parameter, and and are shape parameters. The transformation of
where a is a location parameter, b is a scale parameter, and c and d are shape parameters.
a standard normal distribution, denoted by , falls into three types: a lognormal system, an unbounded system,
a standard normal distribution, denoted by , falls into three types: a lognormal system, an unbounded system,
The transformation of a standard normal distribution, denoted by ϕ, falls into three types:
and a bounded system, which are specifically presented as follows:
a lognormal system, an unbounded system, and a bounded system, which are specifically
and a bounded system, which are specifically presented as follows:
presented as follows: 5
5 � (log orm l system), ,

�
⁄
( ) = �( �� ) 2 (u bou e system), ,
1 (1 �� ) (bou e system)
⁄
The probability density functions of each system can be derived and presented as follows.
The probability density functions of each system can be derived and presented as
Definition 1. Let denote the Johnson distribution, and , , , and are the four parameters given in equation
follows.
(8). The probability density functions of each system in the Johnson distribution family are given as follows:
Definition 1. Let X denote the Johnson distribution, and a, b, c, and d are the four
(a) Lognormal System (LS)
parameters given in equation (8). The probability density functions of each system in the

�
Johnson distribution family are given as follows: 1 �� �, (9)

( ) =
�� exp � � l �
√2 ( ) 2
where ( ) > 0, ∞ < < ∞, | | = 1, ∞ < < ∞, > 0
⁄
(b) Unbounded System (US)
1 8 1 �
( ) = exp � � si h 1 � �� �, (10)
��
�
√2 �( ) � 2
where ∞ < <∞, ∞< <∞, > 0, ∞ < <∞, >0
(c) Bounded System (BS)
1 1 �
�
( ) = exp � � l � �� �, (11)
��
√2 ( )( ) 2
where < < , ∞ < <∞, >0, ∞< < ∞, >0
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
6
represents all possible pairs of and . For example, the standard normal distribution is known to have
�
=0 and = , which is located at (0, ) and displayed by a circle.
�
The pair ( , ) of the distribution is presented by a curve denoted by . The upper area,
�
��
denoted by , describes the pair ( , ) which can be obtained from the US distribution. The middle
�
��
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
�
distribution family. Therefore, if the pair ( , ) of the of underlying assets belongs to any one of the
�
possible areas, then one of the Johnson distribution family can accurately approximate the target distribution by
matching its first four moments.
Most market data exhibit nonzero skewness and higher kurtosis. This also holds for the , especially in
the cases of higher volatilities, lower correlations among the underlying assets, and longer time to maturity. As
shown in Figure 1, the distribution located in the has relatively higher kurtosis than the distribution in
��
the . Thus, the US distribution is more capable of approximating the distribution. Empirical
��

6 Figure 1 is plotted based on the limiting properties of the skewness and kurtosis of the Johnson distribution family.
6