(Carr and Lee (2008)) Consider an underlying asset price $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$given by $d S_{t}=r S_{t} d t+\sigma_{t} S_{t} d B_{t}$, where $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$is a standard Brownian motion and $\left(\sigma_{t}ight)_{t \in \mathbb{R}_{+}}$is an (adapted) stochastic volatility process. The riskless asset is priced $A_{t}:=\mathrm{e}^{r t}, t \in[0, T]$. We consider a realized variance swap with payoff $R_{0, T}^{2}=\int_{0}^{T} \sigma_{t}^{2} d t$.

a) Show that the payoff $\int_{0}^{T} \sigma_{t}^{2} d t$ of the realized variance swap satisfies

$$
\begin{equation*}
\int_{0}^{T} \sigma_{t}^{2} d t=2 \int_{0}^{T} \frac{d S_{t}}{S_{t}}-2 \log \frac{S_{T}}{S_{0}} \tag{8.47}
\end{equation*}
$$

b) Show that the price $V_{t}:=\mathrm{e}^{-(T-t) r} \mathbb{E}^{*}\left[\int_{0}^{T} \sigma_{t}^{2} d t \mid \mathcal{F}_{t}ight]$ of the variance swap at time $t \in[0, T]$ satisfies

$$
\begin{equation*}
V_{t}=L_{t}+2(T-t) r \mathrm{e}^{-(T-t) r}+2 \mathrm{e}^{-(T-t) r} \int_{0}^{t} \frac{d S_{u}}{S_{u}} \tag{8.48}
\end{equation*}
$$

where

$$
L_{t}:=-2 \mathrm{e}^{-(T-t) r} \mathbb{E}^{*}\left[\left.\log \frac{S_{T}}{S_{0}} ightvert\, \mathcal{F}_{t}ight]
$$

is the price at time $t$ of the log contract (see Neuberger (1994), Demeterfi et al. (1999)) with payoff $-2 \log \left(S_{T} / S_{0}ight)$, see also Exercises 6.9 and 7.14.

c) Show that the portfolio made at time $t \in[0, T]$ of:
- one $\log$ contract priced $L_{t}$,
- $2 \mathrm{e}^{-(T-t) r} / S_{t}$ in shares priced $S_{t}$,
- $2 \mathrm{e}^{-r T}\left(\int_{0}^{t} \frac{d S_{u}}{S_{u}}+(T-t) r-1ight)$ in the riskless asset $A_{t}=\mathrm{e}^{r t}$,

hedges the realized variance swap.

d) Show that the above portfolio is self-financing.