a) Solve the Black-Scholes PDE

\[
\begin{equation*}
r g(t, x)=\frac{\partial g}{\partial t}(t, x)+r x \frac{\partial g}{\partial x}(t, x)+\frac{\sigma^{2}}{2} x^{2} \frac{\partial^{2} g}{\partial x^{2}}(t, x) \tag{6.38}
\end{equation*}
\]

with terminal condition \(g(T, x)=1, x>0\).

Try a solution of the form \(g(t, x)=f(t)\) and find \(f(t)\).

b) 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

\[
V_{t}=g\left(t, S_{t}ight)=\xi_{t} S_{t}+\eta_{t} A_{t}
\]

hedging the contract with claim payoff \(C=\$ 1\) at maturity.