Margrabe price for swapping one asset into another

Two assets follow geometric Brownian motions with volatilities $\sigma_1,\sigma_2$ and correlation $\rho$, and pay no dividends. Price a European option that at time $T$ lets the holder give up asset 2 and receive asset 1, i.e. with payoff $\max(S_1(T)-S_2(T),0)$.

$S_1N(d_1)-S_2N(d_2)$, with $d_{1,2}=\frac{\ln(S_1/S_2)\pm\tfrac12\hat\sigma^2T}{\hat\sigma\sqrt T}$ and $\hat\sigma^2=\sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2$. Using asset 2 as numeraire, the ratio $S_1/S_2$ is a driftless GBM with volatility $\hat\sigma$, and the payoff becomes $S_2(T)\max(S_1/S_2-1,0)$: a call struck at $1$ with zero rate. The interest rate drops out.