I'm reading Paul Wilmott's Introduces Quantitative Finance and stuck a bit with formula $F = S(t)e^{(r-r_f)(T-t)}$ for FX futures pricing. I don't get how to incorporate $r_f$ into the formula, could you please help? There is a description in the book for simple futures:
We know all of $F, S(t), t$ and $T$, but is there any relationship between them? You might think not, since the forward contract entitles us to receive an amount $S(T) − F$ at expiry and this is unknown. However, by entering into a special portfolio of trades now we can eliminate all randomness in the future. This is done as follows.
Enter into the forward contract. This costs us nothing up front but exposes us to the uncertainty in the value of the asset at maturity. Simultaneously sell the asset. It is called going short when you sell something you don’t own. This is possible in many markets, but with some timing restrictions. We now have an amount $S(t)$ in cash due to the sale of the asset, a forward contract, and a short asset position. But our net position is zero. Put the cash in the bank, to receive interest.
When we get to maturity we hand over the amount F and receive the asset, this cancels our short asset position regardless of the value of $S(T)$. At maturity we are left with a guaranteed $−F$ in cash as well as the bank account. The word guaranteed is important because it emphasizes that it is independent of the value of the asset. The bank account contains the initial investment of an amount $S(t)$ with added interest, this has a value at maturity of $$S(t)e^{r(T-t)}$$ Our net position at maturity is therefore $$S(t)e^{r(T-t)} - F$$
Since we began with a portfolio worth zero and we end up with a predictable amount, that predictable amount should also be zero. We can conclude that $$F = S(t)e^{r(T-t)}$$