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FX Forward Rates from Spot Rates and Interest Rate Parity

Article Quant Q&A · Author: Pietro Scaglione
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Summary

The document presents the no-arbitrage relationship between foreign exchange spot and forward rates. With the spot quote defined as domestic currency per unit of foreign currency, the forward rate reflects the difference between domestic and foreign interest rates over the contract period. It gives both a discrete-rate relationship and a continuous-compounding expression using discount factors.

It also describes the value of an existing forward contract at an intermediate time: the discounted value of receiving foreign currency at spot is offset by the discounted domestic-currency payment at the agreed strike. The forward has zero value when initiated under the stated no-arbitrage setup. These formulas rely on consistent currency quotation, interest-rate conventions, and discounting assumptions. The text does not discuss transaction costs, funding frictions, collateral, or market basis, which can cause observed forward prices to depart from the simplified parity relationship.

Key ideas

  • The forward exchange rate is linked to spot by the interest-rate differential under no arbitrage.
  • The document gives discrete and continuous compounding formulations of spot-forward parity.
  • A newly entered forward has zero value under the stated assumptions.
  • An existing contract's value depends on spot and discounted domestic and foreign cash flows.

Tags

ForexArbitrageDerivatives pricing
Full text
# Link between spot and forward rates in no-arbitrage world


# 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

## Answer by FunnyBuzer (score 0, accepted)

https://quant.stackexchange.com/a/51326

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: $$V_f (t,T)=e^{-r_d (T-t)} (f(t,T)-K)=S_t e^{-r_f (T-t) }-Ke^{-r_d (T-t) }$$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.