# relationship between notional amounts of volatility swaps and variance swaps

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 $$\sigma_s^2$$

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.