Page 22 - 34-1

We present the mean, variance, skewness, and kurtosis of the US distribution in the following proposition
and their derivations in Appendix B.
Proposition 2. The four characteristics of the US distribution are presented as follows:

�
ℳ ( , , , ) = � sinh(Ω),
��

(13)
( , , , ) = � ( 1)( cosh(2Ω) +1), (12)
��
2
� ( 1)� ( +2) sinh(3Ω) +3 sinh(Ω)�
( , , , ) = , (14)
��
Valuation of Spread and Basket Options �2� cosh(2Ω) +1� �
US ( , , , )
(15)
4
2
2
3
2
� + 2 + 3 3� cosh(4Ω) + 4 ( +2) cosh(2Ω) +3(2 + 1)
= 2 ,
2� cosh(2Ω) +1�
3. Pricing Formula of the GB Options with the US Distribution
where Ω= and = (1 ).
�
⁄
⁄
This section first presents the procedure to find a matching US distribution to
3. Pricing Formula of the GB options with the US Distribution
approximate the GB distribution using the moment-matching method, then derives the
This section first presents the procedure to find a matching US distribution to approximate the GB
pricing formula for the GB options, and finally, show the computation of hedging Greeks.
distribution using the moment-matching method, then derives the pricing formula for the GB options, and
finally, show the computation of hedging Greeks.
3.1. The Moment-Matching Method for the US Distribution
As noted above, the challenge of pricing GB options mainly stems from the lack
3.1. The Moment-Matching Method for the US Distribution
of an exact distribution of the GB; as a result, their pricing formulas can not be derived
As noted above, the challenge of pricing GB options mainly stems from the lack of an exact distribution of
in precisely. To improve the BPW model (Borovkova et al., 2007), we adopt the US
the GB; as a result, their pricing formulas can not be derived in precisely. To improve the BPW model
distribution family with the four correct characteristics presented in equations (4)-(7) to
(Borovkova et al., 2007), we adopt the US distribution family with the four correct characteristics presented in
approximate the GB distribution. To choose a matching US distribution to approximate the
equations (4)-(7) to approximate the GB distribution. To choose a matching US distribution to approximate the
GB distribution, we equalize the first four characteristics of the US distribution to those of
GB distribution, we equalize the first four characteristics of the US distribution to those of the GB distribution,
the GB distribution, and obtain the following equation system:
and obtain the following equation system:
ℳ ( , , , ) = ℳ
��
( , , , ) = (16)
��
( , , , ) =
��
( , , , ) = .
��
By solving this equation system and denoting the solution by � �, , ̅, �, we can determine a matching US
�
̅
By solving this equation system and denoting the solution by , we can
distribution to approximate the GB distribution. 9
9
determine a matching US distribution to approximate the GB distribution.
Because equation (16) is a nonlinear equation system, solving this equation is not straightforward and must
Because equation (16) is a nonlinear equation system, solving this equation is
resort to a numerical method. Tuenter (2001) proposes a root-finding algorithm built on the Newton-Raphson
not straightforward and must resort to a numerical method. Tuenter (2001) proposes a

root-finding algorithm built on the Newton-Raphson method and shows the sufficient
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
conditions for convergence. Therefore, we adopt the method proposed by Tuenter (2001) proposed by
method and shows the sufficient conditions for convergence. Therefore, we adopt the method
Yu (2014), and guaranteed minimum withdrawal benefits, such as in Milevsky and Salisbury (2006) and Yang, Wang, and Liu (2020).
to solve equation (16), and arrange and reduce their results into the following three steps.
Tuenter (2001) to solve equation (16), and arrange and reduce their results into the following three steps.
11
Step 1: Compute the initial value
as follows:
Step 1: Compute the initial value ω as follows:
�

ω = � √ − −
�
Step 2: Set a tolerable error . If | − | < , then � = . Otherwise, continue the following
� ��� �
iteration:
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 Yu (2014), and guaranteed minimum withdrawal benefits, such as in
= ��� − ���
�
�
)
(
Milevsky and Salisbury (2006) and Yang, Wang, and Liu (2020).
���
�
⁄
� (� ��� ����)(� ��� ���� �) � �
= �
��� 14
� (� ��� ���� �)����( ��)(� ��� ��) �� � ��� ��� ��� �
⁄
⁄
���
�� �.
�
where = , and = − �4 � −
��� � � ��� ��� ��
���
Step 3: With � computed in step 2, we can compute Ω, �, and the four parameters � � ̅ � as follows:
̅
�
�
� = ℳ √ � sinh(Ω) (17)
�
�
√
�
= (18)
�
( � − )�
�
̅ = Ω (19)
̅�

̅
= (20)
���( �)
where

� � −
�
Ω = −sign( ) sinh − �� � − �� (21)
� �

3 (22)
�
� = − �4 � � − �
�
� � 3
and ℳ, , , and can be computed by equations (4)-(7) based on current market data.
The Newton-Raphson method can compute � � ̅ � in a fraction of a second, and then determine the
̅
�
matching US distribution to approximate the GB distribution. For the numerical examples presented in Section
0, the Newton-Raphson method converges within five iterations by taking approximately × 0 seconds.
��
Thus, it ensures that the resulting pricing formulas can be instantly computed.

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:
12