(3p) Recall that $\tilde{S}_t = S_t / e^{rt}$ is a martingale under the pricing measure $\mathbb{Q}$ . Using the pricing formula, rewriting, and applying Doob's inequality (for $p = 1$ ; see the formula sheet) we get

\begin{array}{l} \Pi_ {0} \left(\max _ {0 \leq t \leq T} S _ {t}\right) = \mathbb {E} ^ {\mathbb {Q}} \left[ \frac {\max _ {0 \leq t \leq T} S _ {t}}{e ^ {r T}} \right] \\ \leq \mathbb {E} ^ {\mathbb {Q}} \left[ \max _ {0 \leq t \leq T} \frac {S _ {t}}{e ^ {r t}} \right] \leq \frac {e}{e - 1} \left(\mathbb {E} ^ {\mathbb {Q}} \left[ \tilde {S} _ {T} \ln \tilde {S} _ {T} \right] + 1\right). \end{array}

Now $\tilde{S}_T = e^{-rT}e^{(r - \sigma^2 /2)T + \sigma \tilde{W}_T} = e^{-T\sigma^2 /2 + \sigma \tilde{W}_T}$ , with $\tilde{W}_T\sim^{\mathbb{Q}}\mathcal{N}(0,T)$ . Hence,

\begin{array}{l} \mathbb {E} ^ {\mathbb {Q}} \left[ \tilde {S} _ {T} \ln \tilde {S} _ {T} \right] = \mathbb {E} ^ {\mathbb {Q}} \left[ e ^ {- T \sigma^ {2} / 2 + \sigma \tilde {W} _ {T}} \left(- T \sigma^ {2} / 2 + \sigma \tilde {W} _ {T}\right) \right] \\ = e ^ {- T \sigma^ {2} / 2} \left(- T \sigma^ {2} / 2 \mathbb {E} ^ {\mathbb {Q}} \left[ e ^ {\sigma \tilde {W} _ {T}} \right] + \sigma \mathbb {E} ^ {\mathbb {Q}} \left[ \tilde {W} _ {T} e ^ {\sigma \tilde {W} _ {T}} \right]\right). \end{array}

The first expectation is easy to compute using the MGF (see the formula sheet): $\mathbb{E}^{\mathbb{Q}}[e^{\sigma \tilde{W}_T}] = e^{T\sigma^2 /2}$ . For the second expectation, note that writing $p(x)$ for the density of $\mathcal{N}(0,T)$ , we have

\mathbb {E} ^ {\mathbb {Q}} \left[ \tilde {W} _ {T} e ^ {\sigma \tilde {W} _ {T}} \right] = \int_ {- \infty} ^ {\infty} x e ^ {\sigma x} p (x) \mathrm {d} x = \frac {\partial}{\partial \sigma} \int_ {- \infty} ^ {\infty} e ^ {\sigma x} p (x) \mathrm {d} x = \frac {\partial}{\partial \sigma} \mathbb {E} ^ {\mathbb {Q}} \left[ e ^ {\sigma \tilde {W} _ {T}} \right] = T \sigma e ^ {T \sigma^ {2} / 2}.