Black-Scholes formulae

From WilmottWiki

The Black-Scholes formulae give the theoretical values of financial options in the Black-Scholes world in which the underlying asset follows a lognormal random walk.

The value of a European call option with asset price $S$ , at time $t$ , with strike $K$ , expiration $T$ , asset volatility $\sigma$ and risk-free interest rate $r$ is

$SN\left(d_1\right)-Ke^{-r(T-t)}N\left(d_2\right)$

where $d_1=\frac{\ln(S/K)+\left(r+\frac{1}{2}\sigma^2\right)(T-t)}{\sigma \sqrt{T-t}}$ , $d_2=d_1-\sigma \sqrt{T-t}$ and $N(x)=\frac{1}{\sqrt{2 \pi}}\int_{-\infty}^x e^{-\frac{1}{2}s^2}ds.$