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Valuation of Spread and Basket Options

is assumed to be 1.5%. The weight of each company makes the underlying general basket
belong to the basket options if all weights are positive, including GB1, GB2, and GB4, and
belong to the spread options if some weights are negative, including GB3, GB5, and GB6.
All parameters are summarized in Table 9.

Table 10 presents the pricing results of various GB computed by three different
pricing methods: USD represents the pricing model based on the unbounded system of the
Johnson distribution family, SLN represents the pricing model developed by Borovkova

et al. (2007), and MC denotes the Monte Carlo simulation method based on 100,000
simulation paths with the variance reduction technique named the antithetic variates
method. The standard error of the Monte Carlo simulation method is denoted by “se”. All
pricing results show that our pricing model can accurately price both the basket and spread
options based on the market data.

4.3. Sensitivity Analysis
Since basket and spread options do not have close-form pricing models, their

pricing models in the early literature are developed independently under various model
assumptions. This may lead to inconsistency, and cause pricing and hedging errors between
basket and spread options. However, our pricing model can price both basket and spread
options, and thus, it can eliminate the pricing errors. In addition, the Greeks of basket and
spread options are derived from the same pricing formulas; in consequence their Greek

risks can be integrated to help traders manage and hedge their option portfolios.
As indicated by Figures 4, 5, and 6, the correlation coefficient ρ substantially affects
the Greeks of both basket and spread options. The humped-shape figure of the correlation

vega shows that ρ positively affects the basket-option value, which decreases with increasing
ρ. On the contrary, ρ negatively affects the spread-option value, which increases with
decreasing ρ. The behavior of vega (υ) and delta (Δ) of an asset is affected by ρ, moneyness,
and the (long or short) position of the asset. These Greeks can help financial institutions
construct hedging strategies to manage the risks of issuing basket/spread options.

Next, we present some numerical examples to demonstrate the sensitivity analysis
of basket and spread options based on the Greek formulas provided in Definition 2. To
save space, we only show the delta (Δ), vega (υ), and correlation vega of both options in

Figures 4, 5, and 6. Other Greeks can also be easily examined by using Definition 2. For

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