Consider a price process (left(S_{t} ight)_{t in mathbb{R}_{+}})given by (d S_{t}=r S_{t} d t+sigma S_{t} d B_{t})

Consider a price process \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\)given by \(d S_{t}=r S_{t} d t+\sigma S_{t} d B_{t}\) under the risk-neutral probability measure \(\mathbb{P}^{*}\), where \(r \in \mathbb{R}\) and \(\sigma>0\), and the option with payoff

\[S_{T}\left(S_{T}-K\right)^{+}=\operatorname{Max}\left(S_{T}\left(S_{T}-K\right), 0\right)\]

at maturity \(T\).

a) Show that the option payoff can be rewritten as

\[\left(S_{T}\left(S_{T}-K\right)\right)^{+}=N_{T}\left(S_{T}-K\right)^{+}\]

for a suitable choice of numéraire process \(\left(N_{t}\right)_{t \in[0, T]}\).

b) Rewrite the option price \(\mathrm{e}^{-(T-t) r} \mathbb{E}^{*}\left[\left(S_{T}\left(S_{T}-K\right)\right)^{+} \mid \mathcal{F}_{t}\right]\) using a forward measure \(\widehat{\mathbb{P}}\) and a change of numéraire argument.

c) Find the dynamics of \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\)under the forward measure \(\widehat{\mathbb{P}}\).

d) Price the option with payoff

\[S_{T}\left(S_{T}-K\right)^{+}=\operatorname{Max}\left(S_{T}\left(S_{T}-K\right), 0\right)\]

at time \(t \in[0, T]\) using the Black-Scholes formula.