Compute the variance swap rate

$$
\mathrm{VS}_{T}:=\frac{1}{T} \mathbb{E}\left[\lim _{n ightarrow \infty} \sum_{k=1}^{n}\left(\frac{S_{k T / n}-S_{(k-1) T / n}}{S_{(k-1) T / n}}ight)^{2}ight]=\frac{1}{T} \mathbb{E}\left[\int_{0}^{T} \frac{1}{S_{t}^{2}}\left(d S_{t}ight)^{2}ight]
$$

on the index whose level $S_{t}$ is given in the following two models.

a) Heston (1993) model. Here, $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$is given by the system of stochastic differential equations

$$
\left\{\begin{array}{l}
d S_{t}=\left(r-\alpha v_{t}ight) S_{t} d t+S_{t} \sqrt{\beta+v_{t}} d B_{t}^{(1)} \\
d v_{t}=-\lambda\left(v_{t}-might) d t+\gamma \sqrt{v_{t}} d B_{t}^{(2)}
\end{array}ight.
$$

where $\left(B_{t}^{(1)}ight)_{t \in \mathbb{R}_{+}}$and $\left(B_{t}^{(2)}ight)_{t \in \mathbb{R}_{+}}$are standard Brownian motions with correlation $ho \in$ $[-1,1]$ and $\alpha \geqslant 0, \beta \geqslant 0, \lambda>0, m>0, r>0, \gamma>0$.

b) SABR model with $\beta=1$. The index level $S_{t}$ is given by the system of stochastic differential equations

$$
\left\{\begin{aligned}
d S_{t} & =\sigma_{t} S_{t} d B_{t}^{(1)} \\
d \sigma_{t} & =\alpha \sigma_{t} d B_{t}^{(2)}
\end{aligned}ight.
$$

where $\alpha>0$ and $\left(B_{t}^{(1)}ight)_{t \in \mathbb{R}_{+}}$and $\left(B_{t}^{(2)}ight)_{t \in \mathbb{R}_{+}}$are standard Brownian motions with correlation $ho \in[-1,1]$.