# Deriving forward rate

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 $$F = ZL_t \frac{(1+i_t^{L})}{(1+i_t^{Z})}$$

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.