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## Keywords

Currencies, Derivatives, Foreign Exchange Rates

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# Definition

Probably the most simple form of derivatives, forward contracts are agreements to buy or sell at a certain point in the future an asset for a pre-specified price. Forwards are a common instrument to hedge the currency risk, when the investor is expecting to receive or pay a certain amount of money expressed in foreign currency in the near future. There are no costs in entering into a forward agreement apart from the differences in the bid-ask spread proposed by the financial institutions.

A very important feature of the forward contracts is that these are binding contracts (contrary to options), and therefore both parts are obliged to honour the contract and deliver the asset at that price. This condition impacts the calculation of the payoff of the forward.

Forward and spot prices are closely related. Assuming continuous compounding, we can determine the forward price of an asset as:

(1)where $F$ and $S$ are the forward and the spot prices of the underlying asset respectively, $r$ is the risk free rate and $T$ is the period of time for compounding. This formulation implies there are no arbitrage opportunities. Note that if $F > Se^{rT}$, speculators may buy the asset now and enter into a short forward agreement, fixing now the sale price at a later date. The inverse applies if $F < Se^{rT}$. If the underlying asset is an investment asset which provides a known dividend yield, the above formula must be corrected to account for the rate of return of the investment, $q$:

(2)The initial cost of a forward contract is zero. Luenberger [3]: “The forward price is the price that applies at delivery. This price is negotiated so that the initial payment is zero; that is, the value of the contract is zero when it is initiated”. At each point in time, until maturity, the value of a forward contract $f$ with a delivery price of $F$ is calculated as:

(3)When the contract is first established, the price of the forward contract $K$ is determined so as to ensure that its value $f$ is zero.

# Forward exchange rate contracts

We can also define forward contracts on foreign currencies, where the underlying asset is the exchange rate, or a certain number of units of a foreign currency. We start be defining $S$ as the spot exchange rate and $F$ as the forward price, both expressed as units of the base currency per unit of foreign currency. Holding currency provides the investor with an interest gain at the risk-free rate prevailing in the respective country. If we take $r$ as the domestic risk-free rate and $r_f$ as the risk-free rate in the foreign country, the forward price is then as before:

(4)This is known as the ‘interest-rate parity’. If this was not the case arbitrage opportunities would arise, forcing the prices back to equilibrium. If $F > Se^{(r-r_f)T}$, a profit could be obtained by:

- Borrowing $Se^{(-r_fT)}$ in domestic currency at rate $r$ for time $T$
- Buying $e^{-r_fT}$ of the foreign currency and invest this at the rate $r_f$
- Short sell a forward contract on one unit of the foreign currency

At time $T$, the arbitrageur will receive one unit of foreign currency from the deposit, which he sells at the forward price $F$. From this results, he is able to repay the loan $Se^{(r-r_f)T}$ and still obtain a net profit of $F - Se^{(r-r_f)T}$.

# Hedging

Forwards on exchange rates are a commonly used hedging instrument for its simplicity and practicality. Suppose an US investor is expecting to receive in three months time the amount of Eur 150,000. Not wanting to take the risk of a depreciation in the exchange rate, he can enter into a forward rate agreement to sell Eur 150,000 in three months at the forward price of $F$, therefore fixing the foreign exchange rate at the level $F$. Forward contracts are quite effective when the agent knows exactly the amount he/she expects to receive at a future date, and allow him/her to hedge completely the risk of a change in the exchange rate. Nevertheless, while eliminating completely the risk of a currency depreciation, in the opposite case the potential gain is lost as the exchange rate is already determined and is binding.

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1. Garman, M.; and Kohlhagen, S.
Foreign Currency Option Values. Journal of International Money and Finance, 2, 231-237, 1983.2. Hull, John C.
Options, Futures and Other Derivatives. Pearson International Edition, 2006.3. Luenberger, D.
Investment Science. Oxford University Press, 1998.4. Wystup, U.
FX options and structured products. Wiley Finance, 2006. |

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