I'm trying to find an expression for this expectation where g is the "all or nothing" payoff, so the payoff is $P$ if $x>K$ and $0$ if $x<K$.
I have started by expanding replacing S(T) with it's full formula as we do not know the density function for the geometric Brownian Motion but we do know the density function for the Brownian Motion W(T).
So I have: $e^{-rT}E[g(S(0)exp[(mu-(1/2)(sigma)^2)t+(sigma)W(T)]]$
I know that I somehow need to use that the Brownian motion is distributed Normally with mean $0$ and variance T. I also think I'm meant to use the indicator function at some point but I'm unsure how to proceed.