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Valuation of Spread and Basket Options

Most market data exhibit nonzero skewness and higher kurtosis. This also holds
for the GB, especially in the cases of higher volatilities, lower correlations among the
underlying assets, and longer time to maturity. As shown in Figure 1, the distribution
located in the Area has relatively higher kurtosis than the distribution in the Area .
US
BS
Thus, the US distribution is more capable of approximating the GB distribution. Empirical
examinations with market data show that located in the Area may only occur
BS
in a very extreme and unreal situation. Therefore, this study does not adopt the BS
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distribution to fit the GB distribution.
Since the US distribution has one more flexible parameter than the LS distribution,
Curve lies on the edge of Area . Therefore, the US distribution is much more versatile
US
LS
and can fit the GB distribution better than the LS distribution. Nonetheless, Borovkova
et al. (2007) adopt the LS distribution to approximate the GB distribution; consequently,

their resulting model has limited capacity to capture a variety of real skewness and
kurtosis. The aforementioned mismatch with the real skewness and kurtosis may cause
some pricing error, especially in the cases of higher volatilities, lower correlations among

underlying assets, and longer time to maturity. This phenomenon is illustrated by the
examples presented in Figures 2 and 3, which show that the US distribution can fit the
GB distribution better than the LS distribution. In addition, this study matches the first
four moments of the four-parameter US distribution with the GB distribution. Thus, the
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US distribution can approximate the GB distribution. In summary, to enhance the pricing
accuracy and retain computational efficiency, this study adopts the US distribution to
approximate the GB distribution.

7 Thanks to the anonymous reviewers for the suggestions about the empirical examination of the BS
distribution. Appendix A provides the pair of of the GB distribution based on the numerical
examinations from Tables 3 to 8.
8 Thanks to the anonymous reviewers for the suggestions about the theoretical foundation for the
US distribution as an approximate distribution for the GB distribution. Based on the theoretical
foundation of the Edgeworth series expansion method, matching the second or higher-order moments
of both the underlying and approximating distributions shows that the underlying distribution can be
approximated by the approximating distribution in terms of an Edgeworth series expansion. For more
information, refer to Jarrow and Rudd (1982).

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