Let $S_1$ and $S_2$ be two risky assets. The market also has a riskfree asset, and only one driving Wiener process. The parameters are as in the Black SCholes, with $\mu_1, \mu_2, \sigma_1, \sigma_2$.
$dS_i = \mu_1S_idt + \sigma_i S_idW^P$
I wish to price the option at time $t$ which gives us $1$ if $S_1 > S_2$ at time $T$, otherwise $0$.
My method: calculate $S_1(T) - S_2(T)$ explicitly. Then we need the $Q$-probability that this is strictly positive. However, assuming this is the correct method, I get stuck on calculating this $Q$-probability.
EDIT: I got an answer, but it depends on the cumulative distribution of a normal distribution. Does this look right guys?