In this section, we want to discuss how the market for foreign exchange is (and isn’t) organized
and provide a brief description of the various contracts that are bought and sold. We want then
discuss how various arbitrage relationships constrain prices. Finally, we want to
introduce two importance concepts: covered interest rate parity and uncovered
interest rate parity.
The simplest forex transaction is a contract for the immediate
exchange of one currency for another between two parties.
This is known as a spot contract, and the exchange rate for
this transaction is often called the spot exchange rate.
In this class, the use of the term “exchange rate” always refers to the spot rate.
Technology today reduces the risk of one party failing to deliver on
its side of the transaction (default risk or settlement risk) is
essentially zero.
The spot contract is the most common type of trade and appears in
almost 90% of all forex transactions.
There are many derivative contracts in the foreign exchange market, of which the following are the most common.
Forwards With a forward contract the two parties make the contract today, but the settlement date for the
delivery of the currencies is in the future, or forward. The time to delivery, or maturity, varies.
Swaps A swap contract combines a spot sale of foreign currency with a forward repurchase of the same
currency. This is a common contract for counterparties dealing in the same currency pair over and over again.
Futures A futures contract is a promise that the two parties holding the contract will deliver currencies
to each other at some future date at a prespecified exchange rate, just like a forward contract. Unlike the
forward contract, however, futures contracts are standardized, mature at certain regular dates, and can be traded
on an organized futures exchange.
Options An option provides one party, the buyer, with the right to buy (call) or sell (put) a currency in
exchange for another at a prespecified exchange rate at a future date. The buyer is under no obligation to trade
and, in particular, will not exercise the option if the spot price on the expiration date turns out to be more
favorable.
Derivatives allow investors to engage in hedging (selling of risk) and speculation (buying of risk).
Example 1: Hedging. As chief financial officer of a U.S. firm, you expect to receive payment of € 1 million
in 90 days for exports to France. The current spot rate is $1.20 per €. Your firm will incur losses on the
deal if the dollar weakens to less than $1.10 per euro. You advise that the firm buy € 1 million in call
options on dollars at a rate of $1.15 per euro, ensuring that the firm’s euro receipts will sell for at least
this rate. This locks in a minimal profit even if the spot rate falls below $1.15. This is hedging.
Derivatives allow investors to engage in hedging (risk avoidance) and
speculation (risk taking).
Example 2: Speculation. The market currently prices one-year euro
futures at $1.30, but you think the dollar will weaken to $1.43 in
the next 12 months. If you wish to make a bet, you would buy these futures,
and if you are proved right, you will realize a 10% profit. Any level
above $1.30 will generate a profit. If the dollar is at or below $1.30 a
year from now, however, your investment in futures will be a total loss.
This is speculation.
Fig. 6 Arbitrage With Two Currencies and Two Locations (FT figure 2-6)¶
Arbitrage ensures that the trade of currencies in New York along the path AB occurs at the same exchange
rate as via London along path ACDB. At B the pounds received must be the same, regardless of the route taken to get to B:
Triangular arbitrage ensures that the direct trade of currencies along the path AB occurs at the same
exchange rate as via a third currency along path ACB. The pounds received at B must be the same on both paths:
If the direct trade from dollars to pounds has a better rate:
\(E_{£/\$} > E_{£/\text{€}} E_{\text{€}/\$}\).
If the indirect trade has a better rate:
\(E_{£/\$} < E_{£/\text{€}} E_{\text{€}/\$}\).
The two trades have the same rate and yield the same result:
\(E_{£/\$} = E_{£/\text{€}} E_{\text{€}/\$}\).
Only in the last case are there no profit opportunities. This is the no-arbitrage condition:
The majority of the world’s currencies trade directly with only one or two of the major currencies, such as the
dollar, euro, yen, or pound.
Many countries do a lot of business in major currencies such as the U.S. dollar, so individuals always have the
option to engage in a triangular trade at the cross rate.
When a third currency, such as the U.S. dollar, is used in these transactions, it is called a vehicle currency because it is not the home currency
of either of the parties involved in the trade and is just used for
intermediation.
Basic Idea: An risk-neutral investor should be indifferent (i.e. should expect to earn the same return)
investing domestically versus investing abroad.
no arbitrage argument
Invest $1 domestically
earn domestic interest rate
Invest $1 abroad
convert into foreign currency,
earn foreign interest rate
convert back into dollars
Exchange Rate Risk A trader in New York cares about returns in U.S. dollars. A dollar deposit pays a
known return, in dollars. But a euro deposit pays a return in euros, and one year from now we cannot know for
sure what the dollar-euro exchange rate will be.
Contracts to exchange euros for dollars in one year’s time carry an exchange rate of \(F_{\$/\text{€}}\) dollars
per euro. This is known as the forward exchange rate.
If you invest in a dollar deposit, your $1 placed in a U.S. bank account will be worth \((1 + i_{\$})\) dollars
in one year’s time. The dollar value of principal and interest for the U.S. dollar bank deposit is called the
dollar return.
If you invest in a euro deposit, you first need to convert the dollar to euros. Using the spot exchange rate,
$1 buys \(1/E_{\$/\text{€}}\) euros today.
These \(1/E_{\$/\text{€}}\) euros would be placed in a euro account earning \(i_{\text{€}}\) , so in a year’s time they
would be worth \((1 + i_{\text{€}})/E_{\$/\text{€}}\) euros.
To fully hedge, you engage in a forward contract today to make the future transaction at a forward rate \(F_{\$/\text{€}}\).
The \((1 + i_{\text{€}})/E_{\$/\text{€}}\) euros you will have in one year’s time can then be exchanged for
\((1 + i_{\text{€}})\frac{F_{\$/\text{€}}}{E_{\$/\text{€}}}\) dollars, or the dollar return on the euro bank deposit.
\[\underbrace{1+i_{\$}}_{\text{dollar return on dollar deposits}} =
\underbrace{(1+i_{\text{€}})\frac{F_{\$/\text{€}}}{E_{\$/\text{€}}}}_{\text{dollar return on euro deposits}}\]
This is called covered interest parity (CIP) because all exchange rate risk on the euro side has been “covered”
by use of the forward contract.
Uncovered interest parity is a no-arbitrage condition that describes an equilibrium in which investors
are indifferent between the returns on unhedged interest-bearing bank deposits in two currencies.
In this case, traders face exchange rate risk and must make a forecast of the future spot rate, which we call
the expected exchange rate : \(E^e_{\$/\text{€}}\)
Based on the forecast, you expect that the \((1+i_{\text{€}})/E_{\$/\text{€}}\) euros you will have in one year’s time
will be worth \((1+i_{\text{€}}) \times \frac{E^e_{\$/\text{€}}}{E_{\$/\text{€}}}\) when converted into dollars; this is the
expected dollar return on euro deposits. The expression for uncovered interest parity (UIP) is:
\[\begin{split}\underbrace{1+i_{\$}}_{\text{dollar return on dollar deposits}} =
\underbrace{(1+i_{\text{€}})\frac{E^e_{\$/{\text{€}}}}{E_{\$/{\text{€}}}}}_{\substack{\text{expected dollar return}\\\text{on euro deposits}}}\end{split}\]
We can rearrange the terms in the uncovered interest parity expression to solve for the spot rate:
If the forward rate equals the expected spot rate, the expected rate of depreciation equals the forward premium
(the proportional difference between the forward and spot rates):
If UIP and CIP hold, the 12-month forward premium should equal the 12-month expected rate of depreciation.
A scatterplot showing these two variables should be close to the diagonal 45-degree line.
Using evidence from surveys of individual forex traders’ expectations over the period 1988 to 1993, UIP finds
some support, as the line of best fit is close to the diagonal
Uncovered Interest Parity: A Useful Approximation¶
(1)¶\[\begin{split}\underbrace{i_{\$}}_{\substack{\text{interest rate}\\\text{on dollar deposits}}} =
\underbrace{\underbrace{i_{\text{€}}}_{\substack{\text{interest rate on}\\\text{euro deposits}}} +
\underbrace{\frac{\Delta E^e_{\$/\text{€}}}{E_{\$/{\text{€}}}}}_{\substack{\text{expected depreciation}
\\\text{rate of the dollar}}}}_{\substack{\text{expected dollar rate of return}\\\text{on euro deposits}}}\end{split}\]
This approximate equation for UIP says that the home interest rate equals the foreign interest rate plus the
expected rate of depreciation of the home currency.
Suppose the dollar interest rate is 4% per year and the euro 3%. If UIP is to hold, the expected rate of
dollar depreciation over a year must be 1%. The total dollar return on the euro deposit is approximately equal
to the 4% that is offered by dollar deposits.
