# Link between spot and forward rates in no-arbitrage world

With reference to the forward exchange rate definition, let be:

• $$S$$: the spot rate

• $$F$$: the forward rate

• $$r_d$$ and $$r_f$$: respectively the domestic and foreign interest rates

• $$DF_d$$ and $$DF_f$$: respectively the domestic and foreign discount factors

Then, by no arbitrage assumption it holds true that:

1) $$S = (1 + r_d)/ (1 + r_f) * F$$ in the discrete case;

2) $$1 + r_d = (1 + r_f) * F/S$$ in the discrete case;

3) $$F = DF_f/ DF_d * S$$ in the continuous case;

4)For the investor who owns in domestic currency a sum X and decides to invest it in the foreign denominated currency, the foreign interest rate $$r_f$$ is perceived as a premium which has to be discounted;

5) All the rest being fixed, $$F$$ is expected to increase as $$r_d$$ decreases

By definition, the FX spot rate is the number of units of domestic currency (also referred to as numéraire) needed to buy one unit of foreign currency at a given time. The FX forward rate is a contract leading to an exchange of notionals in a future time at a pre-specified (forward) rate. The outright forward is related to the FX spot rate via the spot-rates parity: $$f(t,T)=S_te^{(r_d-r_f)(T-t)}$$ By no-arbitrage, any forward value has zero value at inception. At any time $$\tau\in(t,T]$$, for a given exchange rate known as strike rate, the forward value is given by the payoff: $$V_f (t,T)=e^{-r_d (T-t)} (f(t,T)-K)=S_t e^{-r_f (T-t) }-Ke^{-r_d (T-t) }$$