Power option Power options can be used for the pricing of realized variance and volatility swaps. Let \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\)denote a geometric Brownian motion solution of

\[ d S_{t}=\mu S_{t} d t+\sigma S_{t} d B_{t} \]

where \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion, with \(\mu \in \mathbb{R}\) and \(\sigma>0\).

a) Let \(r \geqslant 0\). Solve the Black-Scholes PDE

\[
\begin{equation*}
r g(x, t)=\frac{\partial g}{\partial t}(x, t)+r x \frac{\partial g}{\partial x}(x, t)+\frac{\sigma^{2}}{2} x^{2} \frac{\partial^{2} g}{\partial x^{2}}(x, t) \tag{6.37}
\end{equation*}
\]

with terminal condition \(g(x, T)=x^{2}, x>0, t \in[0, T]\).

Try a solution of the form \(g(x, t)=x^{2} f(t)\), 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

\[ V_{t}=g\left(S_{t}, tight)=\xi_{t} S_{t}+\eta_{t} A_{t}, \quad 0 \leqslant t \leqslant T \]

hedging the contract with claim payoff \(C=\left(S_{T}ight)^{2}\) at maturity.