Under the Black-Scholes-Merton Model the price of a European vanilla call option at time \(t\) with underlying stock \(S_t\), strike \(K\) and maturity \(T\) is given by \[ C(S_{t},t)=N(d_{+})S_{t}-N(d_{-})Ke^{-r(T-t)}, \] where: - \(d_{+}=\frac{1}{\sigma \sqrt{ T-t }}\left( \log\left( \frac{S_{t}}{K} \right)+\left( r + \frac{\sigma^2}{2} \right)(T-t) \right)\) - \(d_{-}=d_{+}-\sigma \sqrt{ T-t }\).
The price of European put and call options are linked by [[put-call-parity]] which states that \[ P(S_{t},t)=Ke^{-r(T-t)}-S_{t}+C(S_{t},t). \]