Consider an underlying asset price process \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\)under a risk-neutral measure \(\mathbb{P}^{*}\) with risk-free interest rate \(r\).

a) Show that the price at time \(t\) of the European call option with strike price \(K\) and maturity \(T\) is lower bounded by the positive part \(\left(S_{t}-K \mathrm{e}^{-(T-t) r}ight)^{+}\)of the corresponding forward contract price, i.e. we have the model-free bound

\[
\mathrm{e}^{-(T-t) r} \mathbb{E}^{*}\left[\left(S_{T}-Kight)^{+} \mid \mathcal{F}_{t}ight] \geqslant\left(S_{t}-K \mathrm{e}^{-(T-t) r}ight)^{+}, \quad 0 \leqslant t \leqslant T
\]

b) Show that the price at time \(t\) of the European put option with strike price \(K\) and maturity \(T\) is lower bounded by \(K \mathrm{e}^{-(T-t) r}-S_{t}\), i.e. we have the model-free bound

\[
\mathrm{e}^{-(T-t) r} \mathbb{E}^{*}\left[\left(K-S_{T}ight)^{+} \mid \mathcal{F}_{t}ight] \geqslant\left(K \mathrm{e}^{-(T-t) r}-S_{t}ight)^{+}, \quad 0 \leqslant t \leqslant T .
\]