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?