Coupons, Dividends and FX forwards

Suppose now that the asset provides a known income in cash. This could well be a stream of coupons when valuing a T-bond forward. It is easy to show that the arbitrage free value of the forward will be

$$F\left( t,\tau \right) =\left[ S\left( t\right) -I\left( t,\tau \right) %% \right] e^{r\left( \tau -t\right) }$$,

where $I\left( t,\tau \right)$ is the present value of all incomes between $%% t$ and $\tau$. The same arbitrage opportunities as above arise, if this relationship does not hold.

We now turn to an asset that provides a known dividend yield $\delta$. It is shown trivially that in this case

$$F\left( t,\tau \right) =S\left( t\right) e^{\left( r-\delta \right) \left( \tau -t\right) }\text{.}$$

If we consider forwards on currencies, we have to keep in mind that by holding a foreign currency we receive the foreign rate of return $r_{f}$. This results into the relationship

$$F\left( t,\tau \right) =S\left( t\right) e^{\left( r-r_{f}\right) \left( \tau -t\right) }\text{.}$$

The above relationships will result into the value of a forward contract at a point in time $t^{\star }$ after it has been issued. Remember that the initial value of the contract is equal to zero. If we denote this value with $f\left( t^{\star },\tau \right)$, the following will hold

$$\begin{tabular}{ll} $f\left( t^{\star },\tau \right) =S\left( t^{... ...e^{-r\left( \tau -t^{\star }\right) }$ & , known dividend yield \end{tabular}$$