Dividends and FX options

When a stock pays a continuous dividend at a rate of $\delta$, or if one considers FX options where the foreign interest rate is $r^{f}$ [which is equivalent to a dividend payment], the Greeks can be computed easily following the same steps. The same is also true for European put options. We now summarize the results in the following tables:

Non dividend paying stock

$$\begin{tabular}{l\vert ll} & \textbf{Calls} & \textbf{Puts} \h... ...) $ & $-KTe^{-rT}%% \mathcal{N}\left( -d_{2}\right) $ \hline \end{tabular}$$

Dividend paying stock [index]

$$\begin{tabular}{l\vert ll} & \textbf{Calls} & \textbf{Puts} \h... ...) $ & $-KTe^{-rT}%% \mathcal{N}\left( -d_{2}\right) $ \hline \end{tabular}$$

FX

All formulas are the same as the ones for the dividend paying stocks, where $\delta$ is replaced by $r^{f}$. There is an extra rho calculated with respect to the foreign interest rate, given by

$$\begin{tabular}{l\vert ll} & \textbf{Calls} & \textbf{Puts} \h... ... & $%% STe^{-r^{f}T}\mathcal{N}\left( -d_{1}\right) $ \hline \end{tabular}$$