This formula for new VIX is not intuitive, and is quite complicated, but I will explain both old VIX formula and new VIX formula in simple geometric terms. Let's create a chart of options prices, calls and puts, vs their strikes. You will end up with a chart that looks something like this -
If options are relatively expensive the lines would be higher; if options are relatively cheaper the the lines would be lower. At one point, these lines will intersect, and the strike where they intersect will be very close to where the underlying index is trading. And the height of the point where these lines intersect is (approximately) proportional to the old VIX value. If volatility is high, then options are expensive, and the height of intersection point will be higher. If volatility is low, the height will be proportionally lower.
The math behind this is based on approximation for the price of ATM straddle ≈ 0.8 * index price * volatility * √ time to expiration In our case, volatility ≈ height of the intersection point (0.4 * index price * √ time to expiration )
New VIX is calculated in a very different way, but also has the place on the chart. Take the area under the call and put curves, that looks like a curved pyramid.
The area under the curve is (approximately) proportional to the square of the new VIX! This is something that I covered in a previous post, so for the sake of not repeating myself I will just summarize:
old VIX is proportional to the height of the pyramid, new VIX is proportional to the square of the area of the pyramid.
This connects two ideas, and also shows how new VIX uses information from all options, as opposed to the original VIX that uses only ATM options.
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