A2. Let $\bar{S}_t := \max_{0 < u < t} S_u$. Consider the path-dependent contingent claim Y given by
$\qquad Y = \begin{cases} 1, & \text{if } \bar{S}_T \in [a, b), \\ 0, & \text{otherwise}, \end{cases}$
where $a, b$ are constants with $b > a > 0$. Compute the price of this contingent claim at
time $t \in [0, T]$. Note you need to consider different cases based on $\bar{S}_t$.