Let $0<$ a $<$ b$.$ A European derivative pays $1$ if S$($T$)$ lies in the interval $($a$,$b$)$ and $0$ otherwise. $($i$)$ Compute the Black$-$Scholes price pieY$($t$)$ of the derivative and $($ii$)$ find the value of S$(0)$ for which pieY$($O$)$ is maximal.