# Options Weighted Vega Derivation

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

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.