Call struck on the squared stock price

In the Black-Scholes model with rate $r$ and volatility $\sigma$, find the price of a European contract paying $\max(S_T^2-K,0)$ at time $T$.

A Black-Scholes call on $S^2$ with volatility $2\sigma$ and forward $F=S_0^2e^{(2r+\sigma^2)T}$. Since $S_T^2$ is lognormal, the price is $e^{-rT}[FN(d_1)-KN(d_2)]$ with $d_1=\tfrac{\ln(F/K)+2\sigma^2T}{2\sigma\sqrt T}$ and $d_2=d_1-2\sigma\sqrt T$.