Garman Kohlhagen Currency Model

To estimate the premium of a call option on a currency contract, one can use the Garman Kohlhagen model, shown here:

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


\begin{align} d = \frac {{ln(\frac{S}{K})+(r_d-r_f+ \frac{\sigma^2}{2})t}} {\sigma\sqrt{t}} \end{align}
C call premium
S spot exchange rate underlying option
K Exercise exchange rate
t Time until call expires
$\sigma$ underlying asset's standard deviation
rd Domestic interest rate
rf Foreign interest rate

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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