Black-Scholes formula

For a non-dividend-paying stock, a European call with strike $K$ and remaining maturity $\tau$ has value

For $v,m\ge 0$, define $F(v,m)=\mathbb{E}[(e^{-v/2+\sqrt{v}Y}-m{)}^{+}]$, where $Y$ has the standard normal distribution. For $v,m>0$, completing the square in the standard normal density giveswhere $\Phi$ is the standard normal distribution function. The boundary values are $F(0,m)=(1-m{)}^{+}$ and $F(v,0)=1$. Multiplying by the current stock price produces a zero-interest Black-Scholes formula with total variance $v$ and relative strike $m$.