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Taking volatility swap payoff as $$( \sigma_F - \sigma_S ) * volatility~notional $$

and Taking variance swap payoff as $$( \sigma_F^2 - \sigma_S^2 ) * variance~notional $$

I am trying to understand the origin of the relationship;

$$variance~notional = \frac{vega}{(2\sigma_s)}$$

I understand that vega is volatility notional as $\frac{\delta f} {\delta \sigma_F}$ is the change of payoff with respect to volatility point

I understand that variance notional is $\frac{\delta f} {\delta \sigma_F^2}$ as this is the change of payoff with respect to variance point

and $2\sigma_s$ is obviously the derivative of $\sigma_s^2$

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Look at the infinitesimal version of the change in variance: $$ d\sigma^2 = 2\sigma d\sigma + (d \sigma)^2 $$ The Ito term $(d\sigma)^2$ is non-zero for stochastic processes, and is of order $dt$, but if we ignore that then we get the approximate relation $$ d\sigma^2 \approx 2 \sigma d\sigma $$ which is where the factor $2 \sigma$ comes from in the translation between variance and vega notional.

As AlexC wrote it is based on a linearization of the P/L (the Ito term is a "convex" term if you will)

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