The last part of the solution to 2.6

[sin.tutyou]

I am confused on why suddenly Black's formula comes into picture, and why S_T^2 can be write in as that.
Would you please explain or point to some reference related?

[mj]

the Black formula always applies when you have a call option on a lognormal quantity.

A square of a log-normal is also log-normal. So all we have to do is rewrite it as its mean
times a log-normal with mean 1 in order to fit it into the Black formula.