Futures and Options
on Foreign Exchange
7
Chapter Seven
Chapter Objective:
This chapter discusses exchange-traded currency
futures contracts, options contracts, and options on
currency futures.
7-0
Chapter Outline







Futures Contracts: Preliminaries
Currency Futures Markets
Basic Currency Futures Relationships
Eurodollar Interest Rate Futures Contracts
Options Contracts: Preliminaries
Currency Options Markets
Currency Futures Options
7-1
Chapter Outline (continued)






Basic Option Pricing Relationships at Expiry
American Option Pricing Relationships
European Option Pricing Relationships
Binomial Option Pricing Model
European Option Pricing Model
Empirical Tests of Currency Option Models
7-2
Futures Contracts: Preliminaries

A futures contract is like a forward contract:


A futures contract is different from a forward
contract:

7-3
It specifies that a certain currency will be
exchanged for another at a specified time in the
future at prices specified today.
Futures are standardized contracts trading on
organized exchanges with daily resettlement
through a clearinghouse.
Futures Contracts: Preliminaries

Standardizing Features:




Contract Size
Delivery Month
Daily resettlement
Initial performance bond (about 2 percent of
contract value, cash or T-bills held in a street
name at your brokerage).
7-4
Daily Resettlement: An Example





Consider a long position in the CME Euro/U.S.
Dollar contract.
It is written on €125,000 and quoted in $ per €.
The strike price is $1.30; the maturity is 3
months.
At initiation of the contract, the long posts an
initial performance bond of $6,500.
The maintenance performance bond is $4,000.
7-5
Daily Resettlement: An Example




Recall that an investor with a long position gains
from increases in the price of the underlying asset.
Our investor has agreed to BUY €125,000 at
$1.30 per euro in three months time.
With a forward contract, at the end of three
months, if the euro was worth $1.24, he would
lose $7,500 = ($1.24 – $1.30) × 125,000.
If instead at maturity the euro was worth $1.35,
the counterparty to his forward contract would
pay him $6,250 = ($1.35 – $1.30) × 125,000.
7-6
Daily Resettlement: An Example


With futures, we have daily resettlement of
gains an losses rather than one big settlement at
maturity.
Every trading day:



if the price goes down, the long pays the short
if the price goes up, the short pays the long
After the daily resettlement, each party has a
new contract at the new price with one-dayshorter maturity.
7-7
Performance Bond Money




Each day’s losses are subtracted from the
investor’s account.
Each day’s gains are added to the account.
In this example, at initiation the long posts an
initial performance bond of $6,500.
The maintenance level is $4,000.

7-8
If this investor loses more than $2,500 he has a
decision to make: he can maintain his long position
only by adding more funds—if he fails to do so, his
position will be closed out with an offsetting short
position.
Daily Resettlement: An Example

Over the first 3 days, the euro strengthens then
depreciates in dollar terms:
Settle
$1.31
$1.30
$1.27
Gain/Loss
Account Balance
$1,250 = ($1.31$7,750
– $1.30)×125,000
= $6,500 + $1,250
–$1,250
$6,500
–$3,750
$2,750 + $3,750 = $6,500
On third day suppose our investor keeps his long
position open by posting an additional $3,750.
7-9
Daily Resettlement: An Example

Over the next 2 days, the long keeps losing money
and closes out his position at the end of day five.
Settle
Gain/Loss
$1.31
$1.30
$1.27
$1.26
$1.24
$1,250
–$1,250
–$3,750
–$1,250
–$2,500
7-10
Account Balance
$7,750
$6,500
$2,750 + $3,750 = $6,500
$5,250 = $6,500 – $1,250
$2,750
Toting Up
At
the end of his adventures, our investor has
three ways of computing his gains and losses:
Sum
of daily gains and losses
– $7,500 = $1,250 – $1,250 – $3,750 – $1,250 – $2,500
Contract size times the difference between initial
contract price and last settlement price.
– $7,500 = ($1.24/€ – $1.30/€) × €125,000
Ending balance on account minus beginning balance on
account, adjusted for deposits or withdrawals.
– $7,500 = $2,750 – ($6,500 + $3,750)
7-11
Daily Resettlement: An Example
Settle
Gain/Loss
Account Balance
$1.30
–$–
$6,500
$1.31
$1,250
$7,750
$1.30
–$1,250
$6,500
$1.27
–$3,750
$2,750 + $3,750
$1.26
–$1,250
$5,250
$1.24
–$2,500
$2,750
Total loss = – $7,500 = ($1.24 – $1.30) × 125,000
= $2,750 – ($6,500 + $3,750)
7-12
Currency Futures Markets


The Chicago Mercantile Exchange (CME) is by
far the largest.
Others include:




7-13
The Philadelphia Board of Trade (PBOT)
The MidAmerica Commodities Exchange
The Tokyo International Financial Futures Exchange
The London International Financial Futures Exchange
The Chicago Mercantile Exchange




Expiry cycle: March, June, September,
December.
Delivery date third Wednesday of delivery
month.
Last trading day is the second business day
preceding the delivery day.
CME hours 7:20 a.m. to 2:00 p.m. CST.
7-14
CME After Hours



Extended-hours trading on GLOBEX runs from
2:30 p.m. to 4:00 p.m dinner break and then
back at it from 6:00 p.m. to 6:00 a.m. CST.
The Singapore Exchange offers interchangeable
contracts.
There are other markets, but none are close to
CME and SIMEX trading volume.
7-15
Reading Currency Futures Quotes
OPEN
HIGH
LOW
SETTLE
CHG
OPEN
INT
.0028
.0025
172,396
2,266
Euro/US Dollar (CME)—€125,000; $ per €
Mar
Jun
1.4748
1.4737
1.4830
1.4818
1.4700
1.4693
1.4777
1.4763
Closing price
Expiry
month
Daily Change
Opening price
Lowest price that day
Number of open contracts
Highest price that day
7-16
Basic Currency Futures Relationships



Open Interest refers to the number of contracts
outstanding for a particular delivery month.
Open interest is a good proxy for demand for a
contract.
Some refer to open interest as the depth of the
market. The breadth of the market would be
how many different contracts (expiry month,
currency) are outstanding.
7-17
Reading Currency Futures Quotes
OPEN
HIGH
LOW
SETTLE
CHG
OPEN
INT
.0028
.0025
172,396
2,266
Euro/US Dollar (CME)—€125,000; $ per €
Mar
Jun
1.4748
1.4737
1.4830
1.4818
1.4700
1.4693
1.4777
1.4763
Notice that open interest is greatest in the nearby contract,
in this case March, 2008.
In general, open interest typically decreases with term to
maturity of most futures contracts.
7-18
Basic Currency Futures Relationships
OPEN
HIGH
LOW
SETTLE
CHG
OPEN
INT
.0028
.0025
172,396
2,266
Euro/US Dollar (CME)—€125,000; $ per €
Mar
Jun
1.4748
1.4737
1.4830
1.4818
1.4700
1.4693
1.4777
1.4763
The holder of a long position is committing himself to pay
$1.4777 per euro for €125,000—a $184,712.50 position.
As there are 172,396 such contracts outstanding, this
represents a notational principal of over $31.8 billion!
7-19
Reading Currency Futures Quotes
OPEN
HIGH
LOW
SETTLE
CHG
OPEN
INT
.0028
.0025
172,396
2,266
Euro/US Dollar (CME)—€125,000; $ per €
Mar
Jun
1.4748
1.4737
1.4830
1.4818
1.4700
1.4693
1.4777
1.4763
Recall from chapter 6, our interest rate parity condition:
1 + i$
F($/€)
=
1 + i€
S($/€)
7-20
Reading Currency Futures Quotes
OPEN
HIGH
LOW
SETTLE
CHG
OPEN
INT
.0028
.0025
172,396
2,266
Euro/US Dollar (CME)—€125,000; $ per €
Mar
Jun
1.4748
1.4737
1.4830
1.4818
1.4700
1.4693
1.4777
1.4763
From March to June 2008 we should expect lower interest
rates in dollar denominated accounts: if we find a higher rate in
a euro denominated account, we may have found an arbitrage.
7-21
Options Contracts: Preliminaries


An option gives the holder the right, but not the
obligation, to buy or sell a given quantity of an
asset in the future, at prices agreed upon today.
Calls vs. Puts


7-22
Call options gives the holder the right, but not the
obligation, to buy a given quantity of some asset at
some time in the future, at prices agreed upon today.
Put options gives the holder the right, but not the
obligation, to sell a given quantity of some asset at
some time in the future, at prices agreed upon today.
Options Contracts: Preliminaries

European vs. American options



7-23
European options can only be exercised on the
expiration date.
American options can be exercised at any time up to
and including the expiration date.
Since this option to exercise early generally has value,
American options are usually worth more than
European options, other things equal.
Options Contracts: Preliminaries

In-the-money


At-the-money


The exercise price is less than the spot price of the
underlying asset.
The exercise price is equal to the spot price of the
underlying asset.
Out-of-the-money

7-24
The exercise price is more than the spot price of the
underlying asset.
Options Contracts: Preliminaries

Intrinsic Value


The difference between the exercise price of the
option and the spot price of the underlying asset.
Speculative Value

The difference between the option premium and the
intrinsic value of the option.
Option
Premium
7-25
=
Intrinsic
Value
+
Speculative
Value
Currency Options Markets





PHLX
HKFE
20-hour trading day.
OTC volume is much bigger than exchange
volume.
Trading is in six major currencies against the
U.S. dollar.
7-26
PHLX Currency Option
Specifications
Currency
Australian dollar
British pound
Canadian dollar
Euro
Japanese yen
Swiss franc
Contract Size
AD10,000
£10,000
CAD10,000
€10,000
¥1,000,000
SF10,000
http://www.phlx.com/products/xdc_specs.htm
7-27
Basic Option Pricing
Relationships at Expiry



At expiry, an American call option is worth the
same as a European option with the same
characteristics.
If the call is in-the-money, it is worth ST – E.
If the call is out-of-the-money, it is worthless.
CaT = CeT = Max[ST - E, 0]
7-28
Basic Option Pricing
Relationships at Expiry



At expiry, an American put option is worth the
same as a European option with the same
characteristics.
If the put is in-the-money, it is worth E - ST.
If the put is out-of-the-money, it is worthless.
PaT = PeT = Max[E – ST, 0]
7-29
Basic Option Profit Profiles
Profit
Owner of the call
If the call is in-themoney, it is worth
ST – E.
Long 1 call
If the call is out-ofthe-money, it is
worthless and the
–c0
buyer of the call
loses his entire
investment of c0.
loss
7-30
ST
E + c0
E
Out-of-the-money
In-the-money
Basic Option Profit Profiles
Profit
Seller of the call
If the call is in-themoney, the writer
loses ST – E.
If the call is out-ofc0
the-money, the writer
keeps the option
premium.
ST
E + c0
E
loss
7-31
Out-of-the-money
In-the-money
short 1
call
Basic Option Profit Profiles
Profit
If the put is inthe-money, it is E – p
0
worth E – ST.
The maximum
gain is E – p0
If the put is outof-the-money, it
is worthless and – p0
the buyer of the
put loses his
entire investment
of p0.
loss
7-32
Owner of the put
ST
E – p0
long 1 put
E
In-the-money
Out-of-the-money
Basic Option Profit Profiles
If the put is inthe-money, it is
worth E –ST. The
maximum loss is
– E + p0
Profit
p0
If the put is outof-the-money, it
is worthless and
the seller of the
put keeps the
option premium
– E + p0
of p0.
loss
7-33
Seller of the put
ST
E – p0
E
short 1 put
Example



Profit
Consider a call
option on €31,250.
Long 1 call
on 1 pound
The option premium
is $0.25 per €
The exercise price is
$1.50 per €.
–$0.25
ST
$1.75
$1.50
loss
7-34
Example



Profit
Consider a call
option on €31,250.
Long 1 call
on €31,250
The option premium
is $0.25 per €
The exercise price is
$1.50 per €.
–$7,812.50
ST
$1.75
$1.50
loss
7-35
Example
Profit What is the maximum gain on this put option?
$42,187.50 = €31,250×($1.50 – $0.15)/€
$42,187.50


Consider a put
option on €31,250.
The option premium
is $0.15 per €
At what exchange rate do you break even?
ST
–$4,687.50
$1.35

The exercise price is
$1.50 per euro.
loss
7-36
$1.50
Long 1 put
on €31,250
$4,687.50 = €31,250×($0.15)/€
American Option Pricing
Relationships

With an American option, you can do
everything that you can do with a European
option AND you can exercise prior to expiry—
this option to exercise early has value, thus:
CaT > CeT = Max[ST - E, 0]
PaT > PeT = Max[E - ST, 0]
7-37
Market Value, Time Value and Intrinsic
Value for an American Call
Profit
The red line shows the
payoff at maturity, not
profit, of a call option.
Long 1 call
Note that even an outof-the-money option
has value—time value.
Intrinsic value
Time value
Out-of-the-money
loss
7-38
In-the-money
E
ST
European Option Pricing
Relationships
Consider two investments
1
Buy a European call option on the British pound
futures contract. The cash flow today is –Ce
2
Replicate the upside payoff of the call by
Borrowing the present value of the dollar exercise
price of the call in the U.S. at i$
1
E
The cash flow today is
(1 + i$)
2
Lending the present value of ST at i£
The cash flow today is
7-39
–
ST
(1 + i£)
European Option Pricing
Relationships
When the option is in-the-money both strategies
have the same payoff.
When the option is out-of-the-money it has a
higher payoff than the borrowing and lending
strategy.
Thus:
ST
E
Ce > Max (1 + i ) – (1 + i ) , 0
£
$
7-40
European Option Pricing
Relationships
Using a similar portfolio to replicate the upside
potential of a put, we can show that:
Pe > Max
7-41
E
ST
–
,0
(1 + i$) (1 + i£)
A Brief Review of IRP
Recall that if the spot exchange rate is S0($/€) = $1.50/€,
and that if i$ = 3% and i€ = 2% then there is only one
possible 1-year forward exchange rate that can exist
without attracting arbitrage: F1($/€) = $1.5147/€
0
1. Borrow $1.5m at i$ = 3%
2. Exchange $1.5m for €1m at spot
3. Invest €1m at i€ = 2%
7-42
1
4. Owe $1.545m
$1.5147
F1($/€) =
€1.00
5. Receive €1.02 m
Binomial Option Pricing Model
Imagine a simple world where the dollar-euro
exchange rate is S0($/€) = $1.50/€ today and in the
next year, S1($/€) is either $1.875/€ or $1.20/€.
S0($/€)
S1($/€)
$1.875
$1.50
7-43
$1.20
Binomial Option Pricing Model
A call option on the euro with exercise price
S0($/€) = $1.50 will have the following payoffs.
By exercising the call option, you can buy €1 for $1.50.
If S1($/€) = $1.875/€ the option is in-the-money:
S0($/€)
$1.50
7-44
S1($/€)
$1.875
C1($/€)
$.375
…and if S1($/€) = $1.20/€ the option is
out-of-the-money:
$1.20
$0
Binomial Option Pricing Model
We can replicate the payoffs of the call option.
By taking a position in the euro along with some
judicious borrowing and lending.
S0($/€)
S1($/€)
$1.875
C1($/€)
$.375
$1.50
7-45
$1.20
$0
Binomial Option Pricing Model
Borrow the present value (discounted at i$) of $1.20
today and use that to buy the present value
(discounted at i€) of €1. Invest the euro today and
receive €1 in one period. Your net payoff in one
period is either $0.675 or $0.
S0($/€)
S1($/€) debt portfolio C1($/€)
$1.875 – $1.20 = $.675 $.375
$1.50
7-46
$1.20 – $1.20 = $0
$0
Binomial Option Pricing Model
The portfolio has 1.8 times the call option’s payoff
so the portfolio is worth 1.8 times the option value.
$.675
1.80 =
$.375
S0($/€)
S1($/€) debt portfolio C1($/€)
$1.875 – $1.20 = $.675 $.375
$1.50
7-47
$1.20 – $1.20 = $0
$0
Binomial Option Pricing Model
The replicating portfolio’s dollar value today is the
sum of today’s dollar value of the present value of
one euro less the present value of a $1.20 debt:
€1.00 × $1.50 – $1.20
(1 + i€) €1.00
(1 + i$)
S0($/€)
S1($/€) debt portfolio C1($/€)
$1.875 – $1.20 = $.675 $.375
$1.50
7-48
$1.20 – $1.20 = $0
$0
Binomial Option Pricing Model
We can value the call option as 5/9 of the
value of the replicating portfolio:
5
€1.00 × $1.50 – $1.20
×
C0 = 9
(1 + i€) €1.00
(1 + i$)
S0($/€)
$1.50
7-49
S1($/€) debt portfolio C1($/€)
$1.875– $1.20 = $.675 $.375
If i$ = 3% and i€ = 2% the call is worth
5
$1.20
€1.00
$1.50
$0.1697 = ×
–
×
9
(1.02) €1.00 (1.03)
$1.20 – $1.20 = $0
$0
Binomial Option Pricing Model

The most important lesson from the binomial
option pricing model is:
the replicating portfolio intuition.

Many derivative securities can be valued by
valuing portfolios of primitive securities when
those portfolios have the same payoffs as the
derivative securities.
7-50
The Hedge Ratio

In the example just previous, we replicated the
payoffs of the call option with a levered
position in the underlying asset. (In this case,
borrowing dollars to buy euro at the spot.)
The hedge ratio of a option is the ratio of change in
the price of the option to the change in the price of
the underlying asset:
C up– C down
H = up
S1 – S1down
This ratio gives the number of units of the underlying
asset we should hold for each call option we sell in
order to create a riskless hedge.
7-51
Hedge Ratio


This practice of the construction of a riskless
hedge is called delta hedging.
The delta of a call option is positive.
Recall from the example:
$0.375
5
$0.375 – $0
C up– C down
=
=
H = up
down =
9
S1 – S1
$1.875 – $1.20 $0.675

The delta of a put option is negative.
Deltas change through time.
7-52
Creating a Riskless Hedge
The standard size of euro options on the PHLX is €10,000.
In our simple world where the dollar-euro exchange rate is
S0($/€) = $1.50/€ today and in the next year, S1($/€) is either
$1.875/€ or $1.20/€ an at-the-money call on €10,000 has
these payoffs:
If the exchange rate at maturity goes up
to S1($/€) = $1.875/€ then the option
finishes in-the-money.
$1.875
× €10,000 = $18,750
€1.00
– $15,000
C1up = $3,750
$1.50
× €10,000 = $15,000
€1.00
If the rate goes down, the option
finishes out of the money. No one will
pay $15,000 for €10,000 worth $12,000
7-53
$1.20
× €10,000 = $12,000
€1.00
C1down = $0
Creating a Riskless Hedge
Consider a dealer who has just written 1 at-the-money
call on €10,000. He calculates the hedge ratio as 5/9:
$3,750
5
C up– C down = $3,750 – 0
=
=
H = up
down
$18,750
–
$12,000
$6,750
9
S1 – S1
He can hedge his position with three trades:
1. If i$ = 3% then he could borrow $6,472.49
today and owe $6,666.66 in one period.
5
$12,000 ×
= $6,666.66
9
2.
$6,666.66
$6,472.49 =
1.03
Then buy the present value of €5,555.56 = €10,000 × 5
9
(buy euro at spot exchange rate,
€5,555.56
compute PV at i€ = 2%), €5,446.62 =
1.02
3.
Invest
€5,446.62 at i€ = 2%.
7-54
Net cost of hedge = $1,697.44
Service Loan
FV € investment in $
T=0
FV € investment
S1($|€)
Replicating Portfolio
Call on €10,000
K($/€) = $1.50/€
T=1
Step 1
Borrow $6,472.49 at i$ = 3% $1.875
×€5,555.56 = $10,416 – $6,666 = $3,750
€1.00
Step 2
the replicating portfolio payoffs and the call
Buy €5,446.62 at
option payoffs are the same so the call is worth
S0($|€) = $1.50/€
€10,000 = $15,000
5 × €10,000
$1.20
$1.50
$1,697.44 =
–
×
9
(1.02)
€1.00
(1.03)
Step 3
Invest €5,446.62 at i€ = 2%
Net cost = $1,697.44
7-55
$1.20
× €5,555.56 = $6,666.67 – $ 6,666.67 = 0
€1.00
Risk Neutral Valuation of Options

Calculating the hedge ratio is vitally important if
you are going to use options.




7-56
The seller needs to know it if he wants to protect his
profits or eliminate his downside risk.
The buyer needs to use the hedge ratio to inform his
decision on how many options to buy.
Knowing what the hedge ratio is isn’t especially
important if you are trying to value options.
Risk Neutral Valuation is a very hand shortcut
to valuation.
Risk Neutral Valuation of Options
We can safely
assume that IRP
holds:
$1.5147 $1.50×(1.03)
=
F1($/€) =
€1.00×(1.02)
€1.00
$1.875
× €10,000
$18,750 =
€1.00
€10,000 = $15,000
$12,000 =
$1.20
× €10,000
€1.00
Set the value of €10,000 bought forward at $1.5147/€ equal to
the expected value of the two possibilities shown above:
$1.5147
€10,000× €1.00 = $15,147.06 = p × $18,750 + (1 – p) × $12,000
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Risk Neutral Valuation of Options
Solving for p gives the risk-neutral probability of
an “up” move in the exchange rate:
$15,147.06 = p × $18,750 + (1 – p) × $12,000
$15,147.06 – $12,000
p=
$18,750 – $12,000
p = .4662
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Risk Neutral Valuation of Options
Now we can value the call option as the present value
(discounted at the USD risk-free rate) of the expected
value of the option payoffs, calculated using the riskneutral probabilities.
$1.875
× €10,000 ←value of €10,000
€1.00
$3,750 = payoff of right to buy €10,000 for $15,000
$18,750 =
€10,000 = $15,000
$1,697.44
$1.20
× €10,000 ←value of €10,000
$12,000 =
€1.00
$0 = payoff of right to buy €10,000 for $15,000
.4662×$3,750 + (1–.4662)×0
C0 = $1,697.44 =
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1.03
Currency Futures Options




Are an option on a currency futures contract.
Exercise of a currency futures option results in a
long futures position for the holder of a call or
the writer of a put.
Exercise of a currency futures option results in a
short futures position for the seller of a call or
the buyer of a put.
If the futures position is not offset prior to its
expiration, foreign currency will change hands.
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Currency Futures Options





Why a derivative on a derivative?
Transactions costs and liquidity.
For some assets, the futures contract can have
lower transactions costs and greater liquidity
than the underlying asset.
Tax consequences matter as well, and for some
users an option contract on a future is more tax
efficient.
The proof is in the fact that they exist.
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Exercises
EOC Problems 8-12
End Chapter Seven
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