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Appendix B. Proof of Proposition 2
���
Based on the definition in equation (8), and � � where is a standard normal random
Valuation of Spread and Basket Options �
variable, , , c , and . The four characteristics of the US distribution given in Proposition 2
can be obtained by the following derivations.

 The mean of (ℳ ( ))
��
ℳ ( ) E( ) E( ) � (Ω) ,
�
��
where
1 1
E( ) E � � � � ��
2 2
1 1
� � E � � �� � � E � � ��
2 2
1 � 1 �
�
�� � �
2 2
1 �
�
�( �� )
2
�
� (Ω) ,
Ω , and (1 ).
�
⁄
⁄

 The variance of ( ( ))
��
( ) V( )
��
V( )
�
� ( ) ( ) �
�
�
�
1
� ( co (2Ω) 1) (Ω)�
�
�
�
2
1
�
�
� ( co (2Ω) 1) �co (2Ω) 1��
2 2
�

( 1)(ωco (2Ω) 1) ,
2
where
1 1 �
�
E( ) E �� � � � �� �
2 2
1 2 2 1 2 2 1
� � E � � �� � � E � � ��
4 4 2
1 1 1
�
�
��� ��
4 4 2
1
�
( co (2Ω) 1).
2

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