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Valuation of Spread and Basket Options

3.2. Pricing Formula of the Basket/Spread Options with the US Distribution
Basket/Spread options are financial contracts on the basket/spread of multiple

underlying assets whose final payoffs can be jointly defined as follows: (23)
Max − , 0 ,
Max − , 0 ,
Max − , 0 , , (23) (23)
Max − ,0 , (24) (24)
Max − ,0 ,
Max − ,0 , , (24)
where represents the strike price, and GBC and GBP denote the call and put options on the GB, respectively.
where represents the strike price, and GBC and GBP denote the call and put options on the GB, respectively.

(23)
where K represents the strike price, and GBC and GBP denote the call and put options on
Max − , 0 ,
where represents the strike price, and GBC and GBP denote the call and put options on the GB, respectively.
The generalized basket (GB) is defined as follows:
The generalized basket (GB) is defined as follows:
the GB, respectively. The generalized basket (GB) is defined as follows:

Max − ,0 ,
The generalized basket (GB) is defined as follows: (24)
� �

where represents the strike price, and GBC and GBP denote the call and put options on the GB, respectively.

� , ∈ �0, �,
�

� , ∈ �0, �, ,
� �
� , ∈ �0, �,
The generalized basket (GB) is defined as follows: � �
� �
���
���
where ∈ represents the unit number of the th asset. If ∀ ∈ , then , then the GB
� the GB represents a basket of

���
�
�
represents the unit number of the ith asset. If
where
where ∈ represents the unit number of the th asset. If ∀ ∈ , then the GB represents a basket of

�
�
� , ∈ �0, �,
where ∈ represents the unit number of the th asset. If ∀ ∈ , then the GB represents a basket of
� �
�
underlying assets; if ∃ < 0, then the GB represents a spread. � , then the GB represents a spread.
�
� �
represents a basket of underlying assets; if
underlying assets; if ∃ < 0, then the GB represents a spread.
�
underlying assets; if ∃ < 0, then the GB represents a spread. �
���
�
Based on the martingale pricing method, the pricing formulas of the GB options can be derived by
Based on the martingale pricing method, the pricing formulas of the GB options can
�
where ∈ represents the unit number of the th asset. If ∀ ∈ , then the GB represents a basket of
Based on the martingale pricing method, the pricing formulas of the GB options can be derived by
�
�
Based on the martingale pricing method, the pricing formulas of the GB options can be derived by
underlying assets; if ∃ < 0, then the GB represents a spread.
computing the following expectations:
be derived by computing the following expectations:
computing the following expectations:
�
computing the following expectations:
the pricing formulas of the GB options can be derived by
Based on the martingale pricing method, (25)
�Max − , 0 �, ,
���

computing the following expectations: ��� ��� �Max − , 0 �, (25) (25)

�Max − , 0 �,

��� �Max . (26) (26)
− ,0 �,

�Max − ,0 �,
���
(25)
�Max − ,0 �, (26)
�Max − , 0 �,
��� ���

However, as mentioned above, the distribution of the GB(T) is unknown, resulting above
However, as mentioned above, the distribution of the is unknown, resulting in the
However, as mentioned above, the distribution of the is unknown, resulting in the above
(26)

�Max − ,0 �,
However, as mentioned above, the distribution of the is unknown, resulting in the above
���

in the above expectations cannot be analytically derived. Instead, the US distribution is
expectations cannot be analytically derived. Instead, the US distribution is employed to approximate the GB
expectations cannot be analytically derived. Instead, the US distribution is employed to approximate the GB
However, as mentioned above, the distribution of the is unknown, resulting in the above
expectations cannot be analytically derived. Instead, the US distribution is employed to approximate the GB
employed to approximate the GB distribution and then to derive the approximate pricing
distribution and then to derive the approximate pricing formula of the GB option. Once the matching US
distribution and then to derive the approximate pricing formula of the GB option. Once the matching US
expectations cannot be analytically derived. Instead, the US distribution is employed to approximate the GB
distribution and then to derive the approximate pricing formula of the GB option. Once the matching US
formula of the GB option. Once the matching US distribution is obtained following the
distribution is obtained following the procedure outlined in section 0, the approximate pricing formulas of the
distribution and then to derive the approximate pricing formula of the GB option. Once the matching US
distribution is obtained following the procedure outlined in section 0, the approximate pricing formulas of the
distribution is obtained following the procedure outlined in section 0, the approximate pricing formulas of the
procedure outlined in section 3.1, the approximate pricing formulas of the GB options can
GB options can be derived and presented as follows. The derivation is presented in Appendix C.
distribution is obtained following the procedure outlined in section 0, the approximate pricing formulas of the
GB options can be derived and presented as follows. The derivation is presented in Appendix C.
GB options can be derived and presented as follows. The derivation is presented in Appendix C.
be derived and presented as follows. The derivation is presented in Appendix C.
GB options can be derived and presented as follows. The derivation is presented in Appendix C.

Theorem 1. The pricing formulae of the GB call and put options are as follows:

Theorem 1. The pricing formulae of the GB call and put options are as follows:
Theorem 1. The pricing formulae of the GB call and put options are as follows:
Theorem 1. The pricing formulae of the GB call and put options are as follows:
Theorem 1. The pricing formulae of the GB call and put options are as follows: 1
̅
�

1+2 ̅
̅
�
��� �ℳ ��� − + − � + � 1+2 ̅ ̅� � + � 1
1+2 ̅
� �
1
̅

1+2 ̅
̅1
� � + �
2
2

��� ��� �ℳ − + − � + � ̅� � � + � � ̅ (27)
̅�
2
2 � � +

�ℳ − + − � + � �
�ℳ − + − � +
̅ ̅
2 2 2 2 (27)
̅� ̅�
(27)
�
� 1 − 2 ̅ ̅ ̅ 1 (27)
1
− � 1 − 2 ̅ ̅ � � − ��,
1 − 2 ̅ 1 − 2 ̅
� �
̅� � − ��, 1
1
� − ��, ,
̅
− � ̅�
2
�
−
− � 2 2 � � � − ��, ̅ ̅
̅�
2 ̅�
2 2 2 ̅� 2 ̅
1+2 ̅ ̅ 1
�
̅
�
�
̅
��� � ��� 1+2 ̅ ̅� � + � 1 1
1+2 ̅
1+2 ̅
� �
��� − � +
1

� � + �
̅ + �
� − � + � ̅�
2
2
� − � + �
��� � − � + � 2 ̅� 2 � � ̅ ̅ (28)
̅�
� � + �
2
2
2 2 ̅� ̅ (28) (28)
�
̅
� 1 − 2 ̅ ̅ ̅ 1 1 (28)
1 − 2 ̅
�
̅� � − ��, − ��, ,

� � − ��,
− − � ̅� 1 − 2 ̅ � 1 � ̅ 1
1 − 2 ̅
� � �
−
2
̅�

2
− � 2 2 2 � � − ��, ̅ ̅
̅�
2
2 2 ̅� ̅ �� �
��� � ��� �
,
�� � and �, , ̅, and are
�
�� � �
−1 ̅
̅ ̅
�
� �
where ℳ is defined in equation (4),
�,
where ℳ is defined in equation (4), ̅ + sinh sinh −1 −1 �, � � � � � � � � � , and �, , ̅, and are
̅ ̅ +
�
�
, and �, , ̅, and are
��� �
−1 sinh � � 16
�
�
��
where ℳ is defined in equation (4), ̅ + ̅ ��� � � � � �, √�� √�� �� � , and �, , ̅, and are ̅
� �
� ��
�
�
̅
where ℳ is defined in equation (4), ̅ + sinh � �, � �� √�� � ̅
� �
�
given in equations (17)-(20).
given in equations (17)-(20). � � �� √��
given in equations (17)-(20).
given in equations (17)-(20).
With inheriting from the merits of the BPW model (Borovkova et al., 2007), the derived pricing models
With inheriting from the merits of the BPW model (Borovkova et al., 2007), the derived pricing models
With inheriting from the merits of the BPW model (Borovkova et al., 2007), the derived pricing models
With inheriting from the merits of the BPW model (Borovkova et al., 2007), the derived pricing models
given in equations (27) and (28) can together price both basket and spread options, and thus, the pricing and
given in equations (27) and (28) can together price both basket and spread options, and thus, the pricing and
given in equations (27) and (28) can together price both basket and spread options, and thus, the pricing and
hedging of the two options can be managed consistently and efficiently. Furthermore, the pricing models
given in equations (27) and (28) can together price both basket and spread options, and thus, the pricing and
hedging of the two options can be managed consistently and efficiently. Furthermore, the pricing models
hedging of the two options can be managed consistently and efficiently. Furthermore, the pricing models
improve the pricing capacity of the BPW model (Borovkova et al., 2007) by incorporating one more flexible
hedging of the two options can be managed consistently and efficiently. Furthermore, the pricing models
improve the pricing capacity of the BPW model (Borovkova et al., 2007) by incorporating one more flexible
parameter, which can capture the features of the first four moments of the GB distribution. Therefore, the
improve the pricing capacity of the BPW model (Borovkova et al., 2007) by incorporating one more flexible
improve the pricing capacity of the BPW model (Borovkova et al., 2007) by incorporating one more flexible
parameter, which can capture the features of the first four moments of the GB distribution. Therefore, the
parameter, which can capture the features of the first four moments of the GB distribution. Therefore, the
13
parameter, which can capture the features of the first four moments of the GB distribution. Therefore, the
13
13 13