Deriving forward rate

Question

I want to price a 1 year future under the condition of no arbitrage and based on LOOP. At time T, I sell currency Z and buy currency L. At time $t$, we define the exchange rate as $ZL_t$. The 1 year risk free rates are yearly compounded to $(1+i_t^{Z})$ and $(1+_t^{L})$ respectively. We don't want to exchange money at time $t$ so we need to agree on the value $K_t$; another condition is that we need to calculate $K_t$ such that the future equals 0 at $t$.

Now, I've had some courses where we mainly used stocks as example and then we need to satisfy the condition $K_t = S_te^{r(T-t)}$. However I am confused how to derive $ZL_t$ under the above mentioned conditions, as this is an exchange rate and I am having a bit difficulty to wrap my head around it. So we basically enter $F = ZL_t \frac{(1+i_t^{L})}{(1+i_t^{Z})}$

Answer 1

For a bit more clarity, I'll replace $ZL_t$ with $X_t^{ZL}=X_t$ with the meaning: at time $t$, $1$ unit of currency $Z$ (asset, foreign, overZee) can be bought with $X_t$ units of currency $L$ (numeraire, domestic, Local).

If $$ K < X_{t_0}(1+ i^L)(1+i^Z)^{-1}, $$

then, at time $t_0$, one can

• go long the forward contract that allows one to buy $1+i^{Z}$ units of $Z$ currency at $K$ exchange rate, at time $T$ (one year from $t_0$ to keep formulas cleaner),
• borrow $1$ unit of $Z$ currency at $i^Z$ interest rate and convert it to $L$ currency, and
• lend the $X_{t_0}$ units of $L$ currency obtained from conversion at $i^L$ interest rate.

At $t_0$, the value of this portfolio is $0$, but at time $T$ its value (in currency $L$)

$$ (1+i^Z)\cdot (X_T -K) -(1+i^Z)\cdot X_T + (1+i^L)\cdot X_{t_0} $$ $$ = -(1+i^Z)K + X_{t_0} (1+i^L) $$ is strictly positive.

The reversed inequality can't hold either based on mirroring arbitrage arguments.