Let Z be a standard normal random variable. Recall that we can express a geometric Brownian motion as $ S(t) = S_0e^{σ\sqrt{t}Z+(r−σ^2/2)t}, t > 0.$
Show that:
$ e^{−rt}E[S(t) · Z · 1_{{S(t)>k}}] = S_0(Φ'(d1) + σ\sqrt{t}Φ(d1)).$
$ e{−rt}E[S(t) · 1_{S(t)>k}] = S_0Φ(d1).$
We're asked to prove both equations. I'm having a hard time getting started even. Thanks for any potential help.