When looking at the explanation of CBOE S&P 500 Implied Correlation Indices available here, it is written that such indices:

[...] "may be used to provide trading signals for a strategy known as volatility dispersion (or correlation) trading. For example, a long volatility dispersion trade is characterized by selling at-the-money index option straddles and purchasing at-the-money straddles in options on index components."

However, it is not obvious to see the behavior of this strategy (i.e. short ATM Index straddles - long ATM Index components straddles) with reference to correlation. First, I guess we are talking about the correlation between the S&P 500 and its individual constituents, am I right?

Then comes my question:

Could someone provide me with a mathematical derivation of the payoff of this strategy exhibiting the aforementioned (implied) correlation?

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    $\begingroup$ Note that options on index are sensitive to the correlation between components $-$ because the volatility of the index depends on the covariance of its components $-$ whereas a basket of options on the constituents will not be $-$ because it is a plain linear combination of stock options. From there you see that by holding option positions on the index and its components you can trade correlation. $\endgroup$ – Daneel Olivaw Jul 25 '17 at 13:33
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    $\begingroup$ If time permits I'll try to write down a more detailed answer. Also check out @Quantuple's answer here, you will see how correlation impacts the price of a basket option. $\endgroup$ – Daneel Olivaw Jul 25 '17 at 13:39

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