Cl,apter 33: Mathematical Considerations for Index Products 651 
The astute reader will notice that the above example can be generalized by 
drawing a three-dimensional graph. The X axis would be the price of ZYX; the Y axis 
would be the dollars of profit in the spread; and instead of "sliding scales," the Z axis 
would be the price of ABX. There is software that can draw 3-dimensional profit 
graphs, although they are somewhat difficult to read. The previous tables would then 
be horizontal planes of the three-dimensional graph. 
This concludes the chapter on riskless arbitrage and mathematical modeling. 
Recall that arbitrage in stock options can affect stock prices. The arbitrage 
techniques outlined here do not affect the indices themselves. That is done by the 
market basket hedges. It was also known that no new models are necessary for 
evaluation. For index options, one merely has to properly evaluate the dividend for 
usage in the standard Black-Scholes model. Future options can be evaluated by set­
ting the risk-free interest rate to 0% in the Black-Scholes model and discounting the 
result, which is the Black model. 
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