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Lecture 6

Futures and Foreign Exchange

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 The Forward Contract
Consider a farmer growing a single crop, such as wheat,
whose entire planting season’s revenue depends critically on
the highly volatile crop price.
Miller who must purchase wheat for processing faces a risk
management problem that is the mirror image of the
farmer’s.
Both parties can hedge their risk now by entering into a
forward contract now: at harvest time,
• the farmer will deliver the wheat to the miller
• the miller will pay the farmer a pre-specified price.

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 Futures Markets
Forward contracts traded on an exchange are called futures
contracts.
Futures exchanges have several notable features:
• They list the terms for many specific, though standard
contracts: commodity, size of contract, maturity date.
• Standard contracts implies less flexibility, but more
liquidity.
• Marking to market: Daily settling up of any gains or losses
on the contract.
• Exchange guarantees the contract, bearing all default risk.

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 Figure 22.1 The Futures Contract, WSJ October
2, 2021 1

Note: Prices are for

October 1, 2021.
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 The Basics of Futures Contracts 1

Futures contract obliges traders to purchase or sell an

asset at an agreed-upon futures price at contract maturity.
No money actually changes hands when contract is entered
into. But terminology is similar, as follows:
Trader taking the long position commits to purchasing the
commodity on the delivery date.
• Said to “buy” a contract.

Trader taking the short position commits to delivering the

commodity at contract maturity.
• Said to “sell” a contract.

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 Profits of Futures Contracts
Suppose you “bought” a future on corn with a futures price of
541.50 cents per bushel, and at maturity the spot price of
corn turned out to be 546.50.
Then at maturity, you can buy corn for the futures price of
541.50, but you own a commodity worth 546.50. So your
profit is 546.50 – 541.50 = 5 cents per bushel.
Profit to long = spot price of commodity at maturity − futures
price at the time of the contract.
Profit to short = original futures price − spot price at maturity.
• The futures contract is a zero-sum game, which means
gains and losses net out to zero.

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 Table 22.1 Existing Contracts 1

Foreign Currencies Agricultural Metals and Energy Interest Rate Futures Equity Indexes
British pound Corn Copper Eurodollar S&P 500 index
Canadian dollar Oats Aluminum Euroyen Dow Jones Industrials
Japanese yen Soybeans Gold Euro-denominated bond S&P Midcap 400
Euro Soybean meal Platinum Euroswiss NASDAQ 100
Swiss franc Soybean oil Palladium Sterling NYSE index
Australian dollar Wheat Silver British government bond Russell 2000 index
Mexican peso Barley Crude oil German government bond Nikkei 225 (Japanese)
Brazilian real Flaxseed Heating oil Italian government bond FTSE index (British)
Palm oil Gas oil Canadian government bond CAC-40 (French)
Rye Natural gas Treasury bonds DAX-30 (German)
Cattle Gasoline Treasury notes All ordinary (Australian)
Hogs Propane Treasury bills Toronto 35 (Canadian)
Pork bellies Kerosene LIBOR Dow Jones Euro STOXX 50
Cocoa Fuel oil EURIBOR Industry indexes:
Coffee Iron ore Interest rate swaps • Banking
Cotton Electricity Federal funds rate • Telecom
Milk Weather Bankers' acceptance • Utilities

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 Trading Mechanics 1

Trading is overwhelmingly conducted through electronic

networks, particularly for financial futures.
Once a trade is agreed to, the clearinghouse enters the
picture.
• Rather than having the long and short traders hold
contracts with each other, the clearinghouse guarantees to
be seller for the long position and the buyer for the short
position.
• As such, the net position of the clearinghouse is zero.
• The clearinghouse bears the risk that any party either
doesn’t deliver or doesn’t pay. But they reduce this risk by
requiring margin from traders (more on that later).

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 Figure 22.3 Trading Without a Clearinghouse;
Trading With a Clearinghouse

Figure 22.3 Panel A, Trading without a clearinghouse. Panel B, Trading with a

clearinghouse.

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 Trading Mechanics
When the contract is first entered into, the exchange requires
each trader to establish a margin account.
• If the initial margin required for corn is 10%, and the price
is $5.415 per bushel, with 5000 bushels per contract, then
buyer and seller must put up 10%*5.415*5000 = $2707.5
per contract.
Traders will often want to exit their position before the
maturity of the future. If so, they can still realize gains and
losses.
How? By a process called marking to market:
• each day, the clearinghouse adds or subtracts money from
the margin accounts by the amount the futures price has
risen or fallen
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 Marking to Market
Suppose the price of corn rises from $5.415 to $5.435 per
bushel. Then the clearinghouse
• credits the margin account for the long position by (5000
bushels) x (2 cents per bushel) = $100 per contract.
• reduces the margin account for the short position by $100
per contract.

If one day this process causes a trader’s margin to fall below

a critical value called the maintenance margin, then a
margin call is triggered:
• Trader must add money to the margin, or reduce the size
of her position.
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 Marking to Market
Suppose that some
futures price changes
daily according to the
table to the right.

Then the
clearinghouse will
adjust the margin of
the long trader as in
the next table. The
short margin will
have opposite
adjustments.

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 Early Profits and Convergence Property

If a long trader exits the contract early (say at time t), she
closes her margin account and pockets the money. Her total
profit, then, is the total amount by which the margin has
increased (due to marking to market): Ft – F0.
If short trader exits at time t, her total profit is F0 – Ft.
Convergence property: as time approaches the maturity of
the future, the futures price converges to the spot price of the
underlying commodity. Why?
Otherwise there would be an arbitrage:
Suppose a corn future matures on December 1st. If corn is
$5 on Dec 1st, but the corn futures price is $4.50, buy the
corn future just before it matures and sell the corn itself,
earning 50 cents.
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 Forwards vs Futures: recap
Futures differ from forwards in that futures are:
• standardized contracts,
• traded on an exchange,
but most importantly
• marked to market daily, so you can exit contract early and
still realize profits and/or losses without exchanging the
commodity
Forwards, on the other hand, are
• not traded on an exchange
• always held to maturity, which is therefore when profits
and losses are realizes.

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 Profit equivalence
Profit is the same under these three circumstances.
Suppose at time T, spot price on corn is ST .
1. You buy a forward contract on corn with forward price F 0
and maturity T, and you hold it to maturity.
• At time T, contract lets you buy corn for F0, then you
can sell it at the spot price ST . Profit is ST – F0.
2. You buy a futures contract with futures price F 0 and
maturity T, but you exit just before maturity time T.
• When you exit, margin has changed by F T – F0. By
convergence property, this equals S T – F0.
3. (Rare) Buy the contract in (2) and let it mature at time T.
• Clearinghouse gives you corn and your initial margin,
you pay them F0 and can sell corn for ST. Profit is ST –
F0.
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 Futures Markets Strategies:
Speculation
How rare is #3, where futures contracts mature and
commodity is delivered through the clearinghouse?
1–3%
Most contracts are liquidated early. Why? Much trading is
speculation.

Speculators: Have no fundamental interest in the

commodity (to buy or sell). Instead just trying to profit off
price movements.
Example: You think that oil prices will rise. Buy a future with
futures price F0. When oil rises, oil futures price will rise too
(convergence property), say to Ft . When you liquidate,
margin has risen by Ft – F0 , which is your profit.
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 Speculation (continued)
But then why not just buy oil directly and sell it later?
• Futures markets have low transaction costs and allow high
leverage (through margin).

Example: Suppose initial margin for oil future is 10%.

• If futures price is $72 per barrel, contract is 1000 barrels,
then initial margin is 72 x 1000 x 10% = $7200.
• If oil rises by $1 per barrel (1.39%), and the futures price
also rises by $1 per barrel, then contract now worth $1 x
1000 = $1000 more.
• This is an increase of $1000/$7200 = 13.9%, which is 10
times the percentage increase in the oil price.

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 Hedging
Hedgers: have a fundamental interest in the commodity (to
buy or sell), and want to eliminate price risk.
Key difference from speculators is that hedgers are already
planning to buy or sell commodity as part of their business.
Example: Oil distributor plans to sell 100,000 barrels of oil in
February, wants to lock in price of $72. Could write a forward
with buyer, but what if buyer doesn’t want to? Then just sell
a future on the exchange, liquidate just before maturity.

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 Basis Risk and Hedging 1

Basis is the difference between the futures price Ft and the

current spot price Pt (not the spot price PT when the future
matures at time T).
• On the maturity date of a contract, the basis must be zero,
as convergence property implies that FT − PT = 0

Before maturity, however, the futures price Ft for later

delivery may differ substantially from the current spot price
Pt .
• That is, if the hedger wants to sell before maturity (say at
time t < T), she bears basis risk: Ft is unequal to Pt , so
liquidating a short future and selling commodity would net
F0 – Ft + Pt, which is not equal to F0.
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 Speculating on the Basis
Suppose that gold is $1691 per ounce, and the futures price
for June delivery is $1696 per ounce (basis is - $5).

If you buy gold and short the future, then you’ll profit if the
spread narrows (likely, due to convergence):
• If gold rises to $1695, and the June futures prices rises to
$1699, then basis narrows to -$4.
• Gain on gold is $1695 - $1691 = $4, loss on futures
position is $1699 - $1696 = $3
• Net gain is the narrowing of the basis: $1 per ounce.

Q: If you hold position to maturity, is your profit riskless?

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 Speculating on the Basis
In practice, hedge funds
engage in massive basis
trading in Treasury
bonds: buy bonds, short
the futures.

Basis is tiny, so they lever up

by funding bond purchases
with repos (overnight loans
from banks), which are rolled
over daily.

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 Regulators Are Worried
• In a market downturn (i.e., COVID), banks may not rollover repo loans or
may raise their interest rate, or
• clearinghouse may suddenly raise margin requirements.

May force hedge funds to exit position early, suddenly, and simultaneously,
selling Treasuries back and causing their price to plummet.
So the SEC is proposing to impose more oversight on hedge funds, reducing
their basis trading.

Ken Griffin (Citadel) objects to

proposal. Says basis trading:
• Offers liquidity to Treasury
futures buyers (favored by
asset managers, due to margin
leverage).
• Keeps interest low, which helps
corporations and govt.

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 Spot-Futures Parity Theorem 1

Spot-futures parity theorem describes the theoretically

correct relationship between spot and future prices.
Note that if the hedge has no risk, the return should equal the
risk-free rate.
• Violation of the parity relationship gives rise to arbitrage
opportunities.

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 Hedge Example
S&P 500 trades at 1,000; investor holds $1,000 in an
indexed fund.
Dividends of $20 will be paid on the indexed portfolio at year-
end.
A futures price for year-end delivery of the contract is $1,010.
The investor hedges by selling or shorting one contract.
Final value of stock portfolio, ST $ 970 $ 990 $ 1, 010 $ 1, 030 $ 1, 050 $ 1, 070

Payoff from short futures position

40 20 0 −20 −40 −60
( equals F0 − FT = $1, 010 − ST )
Dividend income 20 20 20 20 20 20

Total $ 1, 030 $ 1, 030 $ 1, 030 $ 1, 030 $ 1, 030 $ 1, 030

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 Spot-Futures Parity Theorem 2

In our example, the overall position is perfectly hedged.

• Perfect hedge should earn the riskless rate of return.
This relationship can be used to develop the futures pricing
relationship.

( F0 + D ) − S0 (1,010 + 20) − 1,000

= = 3%
S0 1,000

Caveat: dividend payouts are not perfectly riskless, but:

• they’re highly predictable over short periods, especially
for diversified portfolios
• uncertainty is extremely small compared to stock price
movements
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 Spot-Futures Parity Theorem 3

( F0 + D) − S0
= rf
S0

Rearranging terms yields.

F0 = S0 (1 + rf ) − D = S0 (1 + rf − d )
where
d = dividend yield = D S0
Example: If S&P 500 is at S0=4000, expected dividend yield
is d = 2%, and risk-free rate is 1%, then one year future
should have futures price

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 Arbitrage Possibilities

If spot-futures parity theorem is not observed, then arbitrage

is possible.
• If the futures price is too high, short the futures and acquire
the stock by borrowing the money at the risk-free rate.
• If the futures price is too low, go long futures, short the
stock and invest the proceeds at the risk-free rate.

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 Alternatively, “cost of carry”
Parity relationship also is called the cost-of-carry
relationship:
Compared to buying a future which delivers the stock later
(costs F0), buying now and carrying to the future imposes

• time value of money cost S0(1+ rf ) (you’ve tied up your

funds in the stock), but offset by
• dividend yield of d (same as dividend of S0d).

Cost of both approaches must be equal:

For longer maturities, just compound the cost over periods:

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 Spreads Between Different Maturities
F (T ) = S  (1 + r − d )
T1

1 0 f

F (T ) = S  (1 + r − d )
T2

2 0 f

implies:
F (T ) = F (T )  (1 + r − d )
( T 2 −T 1 )

2 1 f

If rf  d , then the futures price will be higher on longer-

maturity contracts.
If rf  d , longer-maturity futures prices will be lower.

For futures contracts on commodities that pay no dividend, d

= 0, F must increase as time to maturity increases.

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 Figure 22.6 Gold Futures Prices, September
2021

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 Futures Prices Versus Expected Spot
Prices
Expectations hypothesis: F0 = E ( PT )
Normal backwardation (Keynes and Hicks): short position
is typically a hedger (think farmer or oil company selling their
commodity), and must offer a discount to the long speculator
bearing risk: F0 < E(PT)
Contango: long position is typically a hedger (wheat
processors buying wheat), so offer a premium to short
speculators bearing risk: F0 > E(PT)
Modern portfolio theory (consensus view): depends on
whether beta of underlying is positive or negative
𝑇
1 + 𝑟𝑓
𝐹0 = 𝑃0 (1 + 𝑟𝑓 )𝑇 = 𝐸(𝑃𝑇 )
1 + 𝐸(𝑟𝑖 )

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 Figure 22.7 Futures Price Over Time, Special
Case

Figure 22.7 Futures price over time, in the special case that the expected spot
price remains unchanged.

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 Foreign Exchange Futures: The Markets

Foreign exchange risk can be hedged through currency

futures or forward markets.
Forward market in foreign exchange is informal.
• Consists of a network of banks and brokers.
• Allow customers to purchase or sell currency in the future
at a currently agreed-upon rate of exchange.
• No clearinghouse, so you’ll only find trading partners if
you’re creditworthy.

Currency futures are traded on the CME or the London

International Financial Futures Exchange.

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 Figure 23.1 Foreign Exchange Futures

Source: The Wall Street Journal, October 2, 2021.

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 Figure 23.1 Foreign Exchange Futures

Source: The Wall Street Journal, October 2, 2021.

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 Foreign Exchange Futures: Interest Rate
Parity
T
 1 + rUS 
Interest rate parity theorem: F0 = E0  
 1 + rUK 

Where rUS and rUK are the risk-free rates in US and UK,
E0 is the current direct exchange rate: number of dollars
required for purchasing one pound.
F0 is today’s forward rate. number of dollars agreed today
for purchase of one pound at time T.
Intuition: Start with one pound. Convert to E0 dollars today
and invest at rUS until T, or invest now at rUK until T and
convert to locked-in F0 dollars. Should give same number:
𝐸0 (1 + 𝑟𝑈𝑆 )𝑇 = 𝐹0 (1 + 𝑟𝑈𝐾 )𝑇

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 Using Futures to Manage Exchange Rate
Risk
Suppose a US firm exports most of its product to the UK.
If the direct rate falls (pound depreciates), then the pound
revenues from selling in the UK convert to fewer US dollars,
reducing profits.

The firm can buy futures to offset this. If firm shorts a three
month futures contract with futures price of $1.40 per pound,
and direct exchange rate in three months is $1.30 per pound,
then the firm profits F0 – FT = $1.40 - $1.30 = $.10 per pound.

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 How many futures for hedging?
Suppose dollar value of profits falls by $200,000 for every
$.10 depreciation of the pound. You want to hedge with an
opposite exposure.
Suppose you short a future that promises deliver Y pounds in
exchange for F0 dollars. How large so Y be so that you profit
by exactly $200,000?
When pound depreciates by $.10, then because futures
prices move similarly, you’ll have F0 – FT = $.10. Futures
profits then are
$(F0 – FT ) x Y = $.10 x Y = $200000,
implying Y is 2M pounds.

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 Using Futures to Manage Exchange Rate
Risk
Hedge ratio: The number of hedging vehicles (e.g., futures contracts)
needed to offset the risk of a particular unprotected position.
• Interpreted as a ratio of sensitivities to the underlying source of
uncertainty.

Change in value of unprotected position for a given change in exchange rate

H=
Profit derived from one futures position for the same change in exchange rate

In our example, numerator is $200,000, denominator is $.10, so

H = 2M pounds.
If each pound-futures contract calls for delivery of 62,500 pounds, you
would sell 2M/62,500 = 32 contracts.

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 Figure 23.3 Profits as a Function of the
Exchange Rate

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