I am struggling with this question:
Let $B$ be a standard Brownian motion. In a Black-Scholes model, at time $t$, the stock price is given by \begin{equation} S_t = \exp \{ \sigma B_t + ( r- \frac{1}{2} \sigma^2 ) t \}. \end{equation} where $\sigma >0$ and $r$ are constants. Let $a>0$. We want to calculate the time-0-price of an exotic option which will pay $1$ at the time $\tau = \inf \{ t \in [0,T]: S_t > e^{\sigma a} \}$ if the time happens before the expiry $T>0$, otherwise it pays nothing.
The following fact is given: For any $c \in \mathbb{R}$, $a>0$, \begin{equation} \mathbb{P} \, ( \inf \{ t \in [0,T]: B_t + ct =a \} \leq T ) = 1- \Phi \bigg( \frac{a-cT}{\sqrt{T}} \bigg) + e^{2ac} \Phi \bigg( \frac{-a-cT}{\sqrt{T}} \bigg), \end{equation} where $\Phi$ denotes the cumulative distribution function of the $N(0,1)$ distribution.
I was only taught about the pricing of European options. What do we need to do in this case?