(Black-Scholes-Merton formula). Let the interest rate $r$ and the volatility $\sigma>0$ be constant. Let
$$
S(t)=S(0) e^{\left(r-\frac{1}{2} \sigma^2\right) t+\sigma W(t)}
$$
be a geometric Brownian motion with mean rate of return $r$, where the initial stock price $S(0)$ is positive. Let $K$ be a positive constant. Show that, for $T>0$,
$$
\mathbb{E}\left[e^{-r T}(S(T)-K)^{+}\right]=S(0) N\left(d_{+}(T, S(0))\right)-K e^{-r T} N\left(d_{-}(T, S(0))\right),
$$
where
$$
d_{ \pm}(T, S(0))=\frac{1}{\sigma \sqrt{T}}\left[\log \frac{S(0)}{K}+\left(r \pm \frac{\sigma^2}{2}\right) T\right],
$$
and $N$ is the cumulative standard normal distribution function
$$
N(y)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^y e^{-\frac{1}{2} z^2} d z=\frac{1}{\sqrt{2 \pi}} \int_{-y}^{\infty} e^{-\frac{1}{2} z^2} d z .
$$