Black's Futures Contract Model

To estimate the premium of a call option on a futures contract, one can use Black's model, shown here:

\begin{align} C = \frac{F}{e^{rt}} \cdot N(d) - \frac{K}{e^rt}} \cdot N(d - \sigma\sqrt{t}) \end{align}


\begin{align} d = \frac {ln(\frac{F}{K})+ \frac{\sigma^2}{2}t} {\sigma\sqrt{t}} \end{align}
C call premium
F futures underlying option
K Exercise price
t Time until call expires
$\sigma$ underlying asset's standard deviation

The first step in estimating the call premium is to calculate the value for $d.$ After you have this value you may use a spreadsheet such as Excel to find $N(d)$ or you may look up the value for $N(d)$ in a statistical table. $N(d)$ is the cumulative normal distribution from negative infinity to $d.$ You can find $N(d)$ as the NORMSDIST function in Excel, or read it from Cumulative Normal Distribution tables.

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