Convexity adjustment (§ 2.3 of Broadie and Jain (2008)).

a) Using Taylor's formula

$$
\sqrt{x}=\sqrt{x_{0}}+\frac{x-x_{0}}{2 \sqrt{x_{0}}}-\frac{\left(x-x_{0}ight)^{2}}{8 x_{0}^{3 / 2}}+o\left(\left(x-x_{0}ight)^{2}ight)
$$

find an approximation of $R_{0, T}=\sqrt{R_{0, T}^{2}}$ using $\sqrt{\mathbb{E}\left[R_{0, T}^{2}ight]}$ and correction terms.

b) Find an (approximate) relation between the variance swap price $\mathbb{E}^{*}\left[R_{0, T}^{2}ight]$ and the volatility swap price $\mathbb{E}^{*}\left[R_{0, T}ight]$ up to a correction term.