Consider the Black-Scholes model, i.e. a market model with one risky asset with price $S_{t}$ at time $t$ given by

$$S_{t}=S_{0} \exp \left(\sigma B_{t}+\mu t\right),$$

where $\left(B_{t}\right)_{t \geqslant 0}$ denotes a Brownian motion on $(\Omega, \mathcal{F}, \mathbb{P}), \mu>0$ the constant growth rate, $\sigma>0$ the constant volatility and $S_{0}>0$ the initial price of the asset. Assume that the riskless rate of interest is $r \geqslant 0$.

(a) Consider a European option $C=f\left(S_{T}\right)$ with expiry $T>0$ for any bounded, continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$. Use the Cameron-Martin theorem to characterize the equivalent martingale measure and deduce the following formula for the price $\pi_{C}$ of $C$ at time 0 :

$$\pi_{C}=e^{-r T} \int_{-\infty}^{\infty} f\left(S_{0} \exp \left(\sigma \sqrt{T} y+\left(r-\frac{1}{2} \sigma^{2}\right) T\right)\right) \frac{1}{\sqrt{2 \pi}} e^{-y^{2} / 2} d y$$

(b) Find the price at time 0 of a European option with maturity $T>0$ and payoff $C=\left(S_{T}\right)^{\gamma}$ for some $\gamma>1$. What is the value of the option at any time $t \in[0, T] ?$ Determine a hedging strategy (you only need to specify how many units of the risky asset are held at any time $t$ ).