Page 48 - 34-1

 The skewness of ( ( ))
��

�
� ( )�
( ) = ))

 The skewness of ( (
��
��
�

�
�V( )��
� ( )�

( ) =
��
�
�
�
�
( ) 3 ( ) ( ) + 2 ( )

�V( )��
=
�
�
�
�
�V( )��
( ) 3 ( ) ( ) + 2 ( )

=
�
� ( 1)� ( +2) sinh(3Ω) +3 sinh(Ω)�
�V( )��

=
�
�2� cosh(2Ω) +1� sinh(Ω)�
� ( 1)� ( +2) sinh(3Ω) +3

=
where the numerator part of the skewness is
�
�2� cosh(2Ω) +1�

E( ) 3E( )E( ) + 2 ( )
�
�
�
where the numerator part of the skewness is

1 3E( )E( ) + 2 ( )
�
�
�
E( )
3
�
3
�
�
�sinh(3Ω) + �sinh(Ω) + ( cosh(2Ω) 1) � �sinh(Ω)�
=
4 �
2
3
3
1
4 �
�
=
�sinh(3Ω) + �sinh(Ω) + ( cosh(2Ω) 1) � �sinh(Ω)�
4
� 4
2
2 �sinh (Ω)
�
�
2 �sinh (Ω)
�
1
3
3
�
�
�
�sinh(Ω) �sinh(3Ω) + �sinh(3Ω) �sinh(Ω) �sinh(Ω)
=
�
�
1
�
3
3
�
3
4
4
4
4
2
�sinh(Ω) �sinh(3Ω) + �sinh(3Ω) �sinh(Ω) �sinh(Ω)
=
4
4
2
4
4
�
3
1
�
�sinh(3Ω) + �sinh(Ω)
3
1
�
�
2 �sinh(3Ω) + �sinh(Ω)
2

2
2
�
1
�
3
= sinh(3Ω) �( 3 + 2) sinh(Ω) �( 2 1)
�
�
�
3
1
4 sinh(3Ω) �( 3 + 2) sinh(Ω) �( 2 1)
�
�
4
=
4
4
1
�
= �( 1) � ( +2)sinh(3Ω) + 3sinh(Ω)� 3 � � � � 3 � 3 �
�
1
�
�
= 4 �( 1) � ( +2)sinh(3Ω) + 3sinh(Ω)�
4
where
where
� 1 1
�sinh(3Ω) 3sinh(Ω)�
Valuation of Spread and Basket Options sinh (Ω) = �sinh(3Ω) 3sinh(Ω)�
sinh (Ω) =
�
4 4
1 1
sinh(Ω)cosh(2Ω) = �sinh(3Ω) sinh(Ω)�
sinh(Ω)cosh(2Ω) = �sinh(3Ω) sinh(Ω)�
2 2
and
and
1 1 1 � �
1

� � �� �
�
�
E( ) =E �� � � � �� �
E( ) =E �� �
2 2 2 2
1 1 3 3 3 3 3 3 3
3
= = � � E � � �� � � E � � �� + � � E � � �� ��
� E � � �� + � � E � �
�� �
�
� E � �
8 8 8
8 8 8
1 3 3
��
1
3
� � E � � 3 ��
� � E � �
8
8
1 � 3 � 3 � 1 �
1 �
�
�
3 �
= 1 ��� � 3 �� � + � � �� �
�
8
= 8 ��� � �� � + � �� �
8
8
8 8 8 8
1 � 3 �
3
= 1 �sinh(3Ω) + �sinh(Ω) ,
�
�
= 4 �sinh(3Ω) + �sinh(Ω)
4
4
Ω= , and = (1 ). 4
�
⁄
⁄
�
Ω= , and = (1 ).
⁄
⁄

 The kurtosis of ( ( ))

��
� 35
� ( )� 35
( ) =
�� �
�V( )�
�
�
�
�
�
( ) 4 ( ) ( ) + 6 ( ) ( ) 3 ( )
=
�
�V( )��
�
�
�
�
�
( + 2 + 3 3) cosh(4Ω) + 4 ( +2) cosh(2Ω) +3(2 + )
= ,
2� cosh(2Ω) + � �
where the numerator part of the skewness is
( ) 4 ( ) ( ) + 6 ( ) ( ) 3 ( )
�
�
�
�
�
3
�
�
�
�
= cosh(4Ω) cosh(2Ω) + sinh(3Ω)sinh(Ω) + 3 sinh (Ω)
8 2 8
+ 3 sinh (Ω)cosh(2Ω) 3 sinh (Ω) 3 sinh (Ω)
�
�
�
�
�
3
�
�
�
�
= cosh(4Ω) cosh(2Ω) + cosh(4Ω) + cosh(2Ω)
8 2 8 2 2
3 3 3 3 3
�
�
+ cosh(2Ω) cosh(4Ω) cosh(2Ω) + �
2 2 4 2 4

9
3
3
3
3
cosh(2Ω) + cosh(4Ω) + cosh(2Ω)
�
�
�
2 2 8 2 8

�
�
�
�
�
= cosh(4Ω)( 4 + 6 3) + cosh(2Ω)( 3 +2)
8 2
3
+ (2 3 + )
�
�
8

�
�
�
�
�
�
= ( ) � ( + 2 + 3 3) cosh(4Ω) + 4 ( +2) cosh(2Ω)
8
+3(2 + )� 40
where

sinh (Ω) = �cosh(2Ω) �
�
2

�
sinh (Ω) = �cosh(4Ω) 4cosh(2Ω) +3�
8

sinh(3Ω)sinh(Ω) = �cosh(4Ω) cosh(2Ω)�
2

�
sinh (Ω)cosh(2Ω) = �cosh(4Ω) 2cosh(2Ω) + �
4
and
�
� �� �
E( ) =E �� exp � � exp �
2 2
36