Link between Vega and Gamma

Question

"The vega is the integral of the gamma profits ( ie expected gamma rebalancing P/L) over the duration of the option at one volatility minus the same integral at a different volatility...Mathematically, it is:

$$\text{Vega} = \sigma t S^2 \text{Gamma}$$

where $S$ is the asset price, $t$ the time left to expiration and $\sigma$ the volatility.

This is again from Dynamic Hedging by Taleb. I cannot understand the first sentence because it gives no indication of which volatilities to pick nor what the integrand of the integral would be.

Shortly after Taleb states the formula above, again no justification as to where it came from.

Please could someone explain this better.

Answer 1

Under the Black-Scholes model, \begin{align*} Gamma &= \frac{N'(d_1)}{S \sigma \sqrt{T-t}}\\ Vega &= SN'(d_1) \sqrt{T-t}. \end{align*} Then, it is easy to see that \begin{align*} Vega = S^2 \sigma (T-t) Gamma. \end{align*}

Answer 2

I believe in its most fundamental form it is best to internalize that the expected gamma rebalancing P/L = Option price at one volatility. Thus the difference of the option prices at different volatilities by this interpretation shall be the vega, or the sensitivity of an options price to a change in implied vol.