Log contracts can be used for the pricing and hedging of realized variance swaps.

a) Solve the PDE \[
\begin{equation*}
0=\frac{\partial g}{\partial t}(x, t)+r x \frac{\partial g}{\partial x}(x, t)+\frac{\sigma^{2}}{2} x^{2} \frac{\partial^{2} g}{\partial x^{2}}(x, t) \tag{6.39}
\end{equation*}
\]
with the terminal condition \(g(x, T):=\log x, x>0\).
Try a solution of the form \(g(x, t)=f(t)+\log x\), and find \(f(t)\).

b) Solve the Black-Scholes PDE \[
\begin{equation*}
r h(x, t)=\frac{\partial h}{\partial t}(x, t)+r x \frac{\partial h}{\partial x}(x, t)+\frac{\sigma^{2}}{2} x^{2} \frac{\partial^{2} h}{\partial x^{2}}(x, t) \tag{6.40}
\end{equation*}
\]
with the terminal condition \(h(x, T):=\log x, x>0\).
Try a solution of the form \(h(x, t)=u(t) g(x, t)\), and find \(u(t)\).

c) Find the respective quantities \(\xi_{t}\) and \(\eta_{t}\) of the risky asset \(S_{t}\) and riskless asset \(A_{t}=A_{0} \mathrm{e}^{r t}\) in the portfolio with value \[
V_{t}=g\left(S_{t}, tight)=\xi_{t} S_{t}+\eta_{t} A_{t}
\]
hedging a \(\log\) contract with claim payoff \(C=\log S_{T}\) at maturity.