Under the Black-Scholes-Merton Model the price of a European vanilla put option at time with underlying stock , strike \(K\) and maturity \(T\) is given by \[ \begin{align}P(S_{t},t)=N(-d_{-})Ke^{-r(T-t)}-N(-d_{+})S_{t}\end{align} \] 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 \[ P(S_{t},t)=Ke^{-r(T-t)}-S_{t}+C(S_{t},t). \]