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resulting pricing models can significantly reduce the pricing error, especially in the situations of higher asset
volatilities, lower correlations among underlying asset prices, and a longer time to maturity.

3.3. Hedging Ratio
Hedging the GB options is as important as pricing them for investment banks. Therefore, this subsection
examines how to compute the hedging ratios (or the Greeks) of the GB options. Note that though the pricing
formulas given in equations (27) and (28) are presented in a close form, their Greeks cannot be analytically
derived because �, , ̅, and must be computed via the Newton-Raphson method. To overcome this obstacle,
̅
�
this subsection suggests that the end-users should compute the Greeks directly by their definitions. For
demonstration, the definitions of Greeks are presented as follows.
Definition 2. The Greeks of the GB options can be approximately computed by the following formulas:

( (0) + ) − ( (0))
�
�

�

�
�
�
�
���
=

( + ) − ( )
�
�
���
,
� ��� = ��� ( (0) + ) − ��� ( (0)) , ,
=
�

� + � − � �
��� �,� �,�
= ,
�,�

Valuation of Spread and Basket Options ( + ) − ( )
= ,

( + ) − ( )

= ,

and
( (0) + ) − ( (0))
�
�
��� = , ,
�

��� ( (0) + ) − ��� ( (0))
�
�
��� = � � , ,
�

( + ) − ( )
��� � � , ,
=
�

� + � − � �
��� �,� �,� , ,
�,� =

( + ) − ( )
��� , ,
=

( + ) − ( )
��� , ,
=

where is a sufficiently small number and the other parameters are fixed as a constant in the computation of
where δ is a sufficiently small number and the other parameters are fixed as a constant in
each Greek.
the computation of each Greek.
Based on the pricing formulas presented in Theorem 1, the approximate Greeks of the
GB options can be instantly and accurately computed via the above Greeks computation
14

method. The accuracy depends on the size of δ we choose; that is, the smaller the size of
10
δ, the more accurate the computed Greeks. Note that the size of chosen δ will not affect
the computation time; accordingly, the above Greek formulas can also be viewed as close-
form formulas. As a rule of thumb, we may set δ = 10 (or even smaller) for each case,
-5
which can uniformly yield sufficiently accurate Greeks.
4. Numerical Studies
This section provides some numerical examples to examine the accuracy of the
resulting pricing models and then presents some sensitivity analysis for the Greeks.
10 It is not unreasonable to view the computation of Greeks as a (quasi-) closed-form model since their
solutions generally converges within five iterations with the Newton-Raphson method. For computing
-4
each option value, the presented pricing formula takes approximately 2.33×10 of a second, which
-4
is almost the same as the 1.6×10 taken by the Black and Scholes (1973) formula.
18