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The formula for calculating the VIX from SPX options is given on page 4 of this document:

https://www.cboe.com/micro/vix/vixwhite.pdf

My question is, why is the option price at each strike weighted by the inverse of the strike price squared? This would seem to overweight OTM put options (with low strikes) and underweight OTM call options (with high strikes).

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As the other comments already suggest this topic has been discussed many times and the references / links that are provided are far more detailed than my quick solution below.

In the construction / replication of a variance swap one tries to achieve a CONSTANT dollar gamma. This is done by buying a strip of calls and puts, weighted by 1/K^2 as you suggested. Now why did I all-cap the word CONSTANT? because the dollar gamma = $\gamma S^2$. As you can see the black scholes gamma is scaled by the squared level of the stock price. So to get a constant dollar gamma you need to have more of the lower strike options else they won't generate enough dollar gamma because the level of the stock is too low.

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