Call on the product of two correlated stocks

Two non-dividend stocks follow geometric Brownian motions with volatilities $\sigma_1, \sigma_2$ and correlation $\rho$; the risk-free rate is $r$. Value a European option paying $\max(S_1(T) S_2(T) - X, 0)$ at time $T$.

$V = e^{-rT}[M N(d_1) - X N(d_2)]$ where $M = S_1 S_2\, e^{(2r + \rho\sigma_1\sigma_2)T}$, $\sigma^2 = \sigma_1^2 + \sigma_2^2 + 2\rho\sigma_1\sigma_2$, $d_1 = \frac{\ln(M/X) + \frac12\sigma^2 T}{\sigma\sqrt T}$, $d_2 = d_1 - \sigma\sqrt T$. The product of two correlated lognormals is lognormal with variance $\sigma^2$ per unit time, so the Black formula applies once you compute its risk-neutral mean $M$.