Editing Black–Scholes model (section)

===Alternative formulation===
Introducing auxiliary variables allows for the formula to be simplified and reformulated in a form that can be more convenient (this is a special case of the [[Black model|Black '76 formula]]):
:<math>\begin{align}
  C(F, \tau) &= D \left[ N(d_+) F - N(d_-) K \right] \\
       d_+ &=
         \frac{1}{\sigma\sqrt{\tau}}\left[\ln\left(\frac{F}{K}\right) + \frac{1}{2}\sigma^2\tau\right] \\
       d_- &= d_+ - \sigma\sqrt{\tau}
\end{align}</math>

where:

<math>D = e^{-r\tau}</math> is the discount factor

<math>F = e^{r\tau} S = \frac{S}{D}</math> is the [[forward price]] of the underlying asset, and <math>S = DF</math>

Given put–call parity, which is expressed in these terms as:
:<math>C - P = D(F - K) = S - D K</math>

the price of a put option is:
:<math>P(F, \tau) = D \left[ N(-d_-) K - N(-d_+) F \right]</math>