Consider an asset price $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$with log-return dynamics

$$
d \log S_{t}=\mu d t+Z_{N_{t^{-}}} d N_{t}, \quad t \geqslant 0
$$

i.e. $S_{t}:=S_{0} \mathrm{e}^{\mu t+Y_{t}}$ in a pure jump Merton model, where $\left(N_{t}ight)_{t \in \mathbb{R}_{+}}$is a Poisson process with intensity $\lambda>0$ and $\left(Z_{k}ight)_{k \geqslant 0}$ is a family of independent identically distributed Gaussian $\mathcal{N}\left(\delta, \eta^{2}ight)$ random variables. Compute the price of the log-return variance swap

$$
\begin{aligned}
\mathbb{E}\left[\int_{0}^{T}\left(d \log S_{t}ight)^{2} d N_{t}ight] & =\mathbb{E}\left[\int_{0}^{T}\left(\mu d t+Z_{N_{t^{-}}} d N_{t}ight)^{2} d N_{t}ight] \\
& =\mathbb{E}\left[\int_{0}^{T}\left(Z_{N_{t^{-}}} d N_{t}ight)^{2} d N_{t}ight] \\
& =\mathbb{E}\left[\int_{0}^{T}\left(\log \frac{S_{t}}{S_{t^{-}}}ight)^{2} d N_{t}ight] \\
& =\mathbb{E}\left[\sum_{n=1}^{N_{T}}\left(\log \frac{S_{T_{k}}}{S_{T_{k-1}}}ight)^{2}ight]
\end{aligned}
$$

using the smoothing lemma Proposition 20.10.