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Does anyone have a good reference on how to derive time weighted vega for options? The only literature I found was in this presentation:

http://www.topquants.nl/wordpress/wp-content/uploads/2015/01/Van-Gulik-Risk-management-at-Optiver.pdf

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Unfortunately, I don't quite understand how the author got from step 2 to 3. Why is perfect correlation assumed? I did the derivation for $\rho = 1$, and I did not seem to get the 3rd equation. Also, it is also unclear on why there needs to be a floor for the time conversion in the 4th equation.

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