I have
$$V_{T} = \begin{cases} ( S_{T} - k )_{+}, & \text{$\max_{0 \le t \le T} S_{t} < b$} \\ 0, & \text{otherwise} \end{cases}$$
I need to compute $V_{0}$, the price of the option at the moment $0$. I started:
\begin{align} V_{0} &= \mathbb{E}_{Q} [ e^{-rT} X ] \\ &= e^{-rT} \mathbb{E}_{Q} \left[ ( S_{T} - k )_{+} \cdot \mathbb{1}_{\max_{0 \le t \le T} S_{t} < b} \right] \\ &= e^{-rT} \mathbb{E}_{Q} \left[ ( S_{0} e^{\mu - \frac{\sigma ^2}{2} T + \sigma B_{T}} - k )_{+} \cdot \mathbb{1}_{\max_{0 \le t \le T} S_{0} e^{\sigma W_{t} - \frac{\sigma ^2}{2}t} < b} \right] \end{align}
I want to compute maximum and I started like this:
\begin{align} \max_{0 \le t \le T} S_{0} e^{\sigma W_{t} - \frac{\sigma ^2}{2}t} < b &= \max_{0 \le t \le T} S_{t} \cdot \frac{1}{S_{t}} e^{\sigma W_{t} - \frac{\sigma^{2}}{2}t} < b \\ &= \max_{0 \le t \le T} S_{t} \cdot \frac{e^{- \sigma W_{0} + \frac{\sigma ^2}{2}0}}{S_{0}} \cdot e^{\sigma W_{t} - \frac{\sigma ^2}{2}t} < b \\ &= \max_{0 \le t \le T} \frac{e^{\sigma (W_{t} - W_{0}) - \frac{\sigma ^2}{2}(t-0)}}{S_0} < \frac{b}{S_t} \end{align}
and I stopped here because I am not sure that this is correct.
Thank you for your help!