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
Primary vs. Derivative Products


Primary Financial Products: their values are
determined by their own cash flows. E.g.,
stocks, bonds, currencies, (real & artificial)
commodities, etc
Derivative Products (Derivatives, Contingent
Claims): their values are derived from the value
of the underlying primary security. E.g.,
forward, futures, options, swaps, insurance
products, etc.
Futures Contracts: Preliminaries

A futures contract is like a forward contract:


A futures contract is different from a forward
contract:

7-2
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:







CFTC
Contract size ~ originally determined around $100k
Delivery months ~ 3, 6, 9, 12
Daily settlement (mark to market)
Trading costs ~ Commission
A daily price limit
Initial performance bond (=initial margin, about
2 percent of contract value, cash or T-bills held
in a street name at your brokerage).
7-3
Daily Settlement: 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.
7-4
Daily Settlement: 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.
7-5
Daily Settlement: An Example


With futures, we have daily settlement 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
=> A zero-sum game!
After the daily settlement, each party has a new
contract at the new price with one-day-shorter
maturity.
7-6
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.

7-7
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 Settlement: 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.
7-8
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
7-9
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
7-10
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)
7-11
Currency Futures Markets


The Chicago Mercantile Exchange (CME) is by
far the largest.
Others include:




7-12
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.
7-13
CME After Hours



Extended-hours trading on GLOBEX runs from
17:00 p.m. to 16:00 p.m CST.
The Singapore Exchange (SIMEX) offers
interchangeable contracts.
There are other markets, but none are close to
CME and SIMEX trading volume.
7-14
Reading Currency Futures Quotes, 2/10
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
7-15
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.
7-16
Reading Currency Futures Quotes, 2/10
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.
7-17
Basic Currency Futures Relationships, 2/10
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!
7-18
Reading Currency Futures Quotes, 2/10
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
(arbitrage possibility if there is a sizable disparity):
1 + i$
1 + i€
7-19
=
F($/€)
So($/€)
Reading Currency Futures Quotes, 2/10
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 20xx, 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.
7-20
Eurodollar Interest Rate
Futures Contracts


Widely used futures contract for hedging shortterm U.S. dollar interest rate risk.
The underlying asset is a hypothetical $1,000,000
90-day Eurodollar deposit—the contract is cash
settled.
7-21
Reading Eurodollar Futures Quotes, 2/10
OPEN HIGH LOW SETTLE CHG
YLD
CHG
OPEN
INT
Eurodollar (CME)—1,000,000; pts of 100%
Jun 96.56
96.58
96.55
96.56
-
3.44
-
1,398,959
Eurodollar futures prices are stated as an index number of three-month LIBOR
calculated as F = 100 - LIBOR. The implied yield 3.44% (=100 – 96.56)
Since it is a 3-month contract one basis point corresponds to a $25 price change:
.01 percent of $1 million represents $100 on an annual basis.
If you to secure 3.44% (APR) at minimum for a 3 month investment starting
June of this year, you want to take a long position to avoid F going high (or
avoid LIBOR getting low).
7-22
Trading irregularities

Futures Markets are also a great place to launder
money

7-23
The zero sum nature of futures is the key to
laundering the money.
Money Laundering: Hillary
Clinton’s Cattle Futures
James B. Blair,
Outside Counsel to
Tyson Foods Inc.,
Arkansas' largest
employer, gets
Hillary’s
discretionary
order.
7-24
winners
losers
Submits
identical long
and short trades
Robert L. "Red" Bone,
(Refco broker), allocates
trades ex post facto.
Options Contracts: Preliminaries

An option gives the holder the right, but not the
obligation, to buy (Call) or sell (Put) a given
quantity of an asset in the future, at prices
agreed upon today.
7-25
Options Contracts: Preliminaries

European vs. American options



7-26
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 is more valuable
(greater flexibility), American options are worth more
than European options.
Options Contracts: Preliminaries

In-the-money


At-the-money


It is profitable to exercise the option.
Indifferent
Out-of-the-money

7-27
It is not profitable to exercise.
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
7-28
=
Intrinsic
Value
+
Speculative
Value
Currency Options Markets



PHLX (Philadelphia Stock Exchange)
OTC volume is much bigger than exchange
volume.
Trading is in six major currencies against the
U.S. dollar.
7-29
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
7-30
Call Option Pricing Relationships at Expiry




Call option holder has two prices to choose
from, ST or E.
Since a call is used to buy, you look for a lower
price to buy. If ST > E (in the money), then
Exercise. If ST < E (out of the money), then Do
Not Exercise.
If the call is in-the-money, it is worth ST – E.
If the call is out-of-the-money, it is worthless.
CT = Max[ST - E, 0]
7-31
Put Option Pricing Relationships at Expiry




Put option holder has two prices to choose from,
ST or E.
Since a put is used to sell, you look for a higher
price to sell. If ST < E (in the money), then
Exercise. If ST > E (out of the money), then Do
Not Exercise.
If the put is in-the-money, it is worth E - ST.
If the put is out-of-the-money, it is worthless.
PT = Max[E – ST, 0]
7-32
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
7-33
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
7-34
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
7-35
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
7-36
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
7-37
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
7-38
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
7-39
$1.50
Long 1 put
on €31,250
$4,687.50 = €31,250×($0.15)/€
American Option Pricing
Relationships

With an American option, you can do
everything that you can do with a European
option AND you can exercise prior to expiry—
this option to exercise early has value, thus:
CaT > CeT = Max[ST - E, 0]
PaT > PeT = Max[E - ST, 0]
7-40
Market Value, Time Value and Intrinsic
Value for an American Call
Profit
The red line shows the
payoff at maturity, not
profit, of a call option.
Long 1 call
Note that even an outof-the-money option
has value—time value.
Intrinsic value
Time value
Out-of-the-money
loss
7-41
In-the-money
E
ST
European Option Pricing
Relationships
Consider two investments
1
Buy a European call option on the British pound
futures contract. The cash flow today is –Ce
2
Replicate the upside payoff of the call by
Borrowing the present value of the dollar exercise
price of the call in the U.S. at i$
1
E
The cash flow today is
(1 + i$)
2
Lending the present value of ST at i£
The cash flow today is
7-42
–
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
£
$
7-43
European Option Pricing
Relationships
Using a similar portfolio to replicate the upside
potential of a put, we can show that:
Pe > Max
7-44
E
ST
–
,0
(1 + i$) (1 + i£)
A Brief Review of CIRP
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%
7-45
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
7-46
$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
7-47
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
7-48
$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
7-49
$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
7-50
$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
7-51
$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($/€)
$1.50
7-52
S1($/€) 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.
7-53
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.
7-54
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.
7-55
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:
$1.875
If the exchange rate at maturity goes up
× €10,000 = $18,750
– $15,000
to S1 = $1.875/€ then the option finishes €1.00
in-the-money.
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
7-56
$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%.
7-57
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
7-58
$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.




7-59
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 CIRP
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×
= $15,147.06 = p × $18,750 + (1 – p) × $12,000
€1.00
7-60
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
7-61
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 =
7-62
1.03
Test Your Intuition
Use risk neutral valuation to find the value of a put
option on $15,000 with a strike price of €10,000.
Hint: given that we just found that the value of a call
option on €10,000 with a strike price of $15,000 was
$1,697.44 this should be easy in the sense that we
already know the right answer.
$1.50
As before, i$ = 3%, i€ = 2%,
S0 =
€1.00
$1.50×1.03
$1.5147
F1($/€) = €1.00×1.02 = €1.00
7-63
Test Your Intuition (continued)
$1.50×1.03
$1.5147
F1($/€) = €1.00×1.02 = €1.00
€12,500 =
€1.00
× $15,000 ←value of $15,000
$1.20
€10,000 = $15,000
€1.00
× $15,000 ←value of $15,000
€8,000 =
$1.875
€1.00
$15,000 ×
= €9,902.91
$1.5147
€9,902.91 = p × €12,500 + (1 – p) × €8,000
p = .4229
7-64
Test Your Intuition (continued)
€1.00
× $15,000 ←value of $15,000
$1.20
0 = payoff of right to sell $15,000 for €10,000
€12,500 =
€10,000 = $15,000
€1,131.63
€P0 = €1,131.63 =
€1.00
× $15,000 ←value of $15,000
€8,000 =
$1.875
€2,000 = payoff of right to sell $15,000 for €10,000
.4229×€0 + (1–.4229)×€2,000
1.02
$P0 = $1,697.44 = €1,131.63 × $1.50
€1.00
7-65
Test Your Intuition (continued)
The value of a call option on €10,000 with a strike
price of $15,000 is $1,697.44
The value of a put option on $15,000 with a strike
price of €10,000 is €1,131.63
At the spot exchange rate these values are the
same:
$1.50
€1,131.63 × €1.00 = $1,697.44
7-66
Take-Away Lessons
Convert future values from one currency to another
using forward exchange rates.
Convert present values using spot exchange rates.
Discount future values to present values using the
correct interest rate, e.g. i$ discounts dollar amounts
and i€ discounts amounts in euro.
To find the risk-neutral probability, set the forward
price derived from CIRP equal to the expected value
of the payoffs.
To find the option value discount the expected value
of the option payoffs calculated using the risk neutral
probabilities at the correct risk free rate.
7-67
Finding Risk Neutral Probabilities
up
down
F1 = p × S1 + (1 – p) × S1
For a call on €10,000 with a strike price of $15,000 we solved
$15,147.06 = p × $18,750 + (1 – p) × $12,000
$15,147.06 – $12,000
$1.5147 – $1.20
p=
=
= .4662
$18,750 – $12,000
$1.875 – $1.20
For a put on $15,000 with a strike price of €10,000 we solved
€9,902.91 = p × €12,500 + (1 – p) × €8,000
p=
7-68
€9,902.91– €8,000
€0.6602– €.5333
=
= .4229
€12,500 – €8,000
€.8333 – €.5333
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.
7-69
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.
7-70
Binomial Futures Option Pricing
A 1-period at-the-money call option on euro futures has a
strike price of F1 = $1.5147/€
$1.875×1.03 $1.8934
=
F1($|€) =
€1.00×1.02
€1.00
$1.5147
F1 = $1.50×1.03 =
€1.00×1.02
€1.00
Option Price =
?
Call Option Payoff = $0.3787
$1.20×1.03 $1.2118
=
F1($|€) =
€1.00×1.02
€1.00
Option Payoff = $0
When a call futures option is exercised the holder acquires
1. A long position in the futures contract
2. A cash amount equal to the excess of the futures price over the strike
price
7-71
Binomial Futures Option Pricing
Consider the Portfolio:
long H futures contracts
short 1 futures call option
$1.5147
F1($|€) = $1.50×1.03 =
€1.00×1.02
€1.00
$1.875×1.03 $1.8934
=
F1($|€) =
€1.00×1.02
€1.00
Futures Call Payoff = –$0.3787
Futures Payoff = H × $0.3603
Portfolio Cash Flow =
H × $0.3603 – $0.3787
Option Price = $0.1714
Portfolio is riskless when the
portfolio payoffs in the “up” state
equal the payoffs in the “down”
state:
H×$0.3603 – $0.3787 = –H×$0.3147
The “right” amount of futures
contracts is
7-72
H = 0.5610
$1.20×1.03 $1.2118
=
F1($|€) =
€1.00×1.02
€1.00
Futures Payoff = –H×$0.3147
Option Payoff = $0
Portfolio Cash Flow =
–H×$0.3147
Binomial Futures Option Pricing
The payoffs of the portfolio are
–$0.1766 in both the up and
down states.
$1.5147
F1($|€) = $1.50×1.03 =
€1.00×1.02
€1.00
There is no cash flow at initiation
with futures.
Without an arbitrage, it must be the
case that the call option income is
equal to the present value of
$0.1766 discounted at i$ = 3%
C0 = $0.1714 =
7-73
$0.1766
1.03
$1.875×1.03 $1.8934
=
F1($|€) =
€1.00×1.02
€1.00
Call Option Payoff = –$0.3787
Futures Payoff = H × $0.3603
Portfolio Cash Flow =
0.5610 × $0.3603 – $0.3787
= –$0.1766
$1.20×1.03 $1.2118
=
F1($|€) =
€1.00×1.02
€1.00
Futures Payoff = –0.5610×$0.3147
Option Payoff = $0
Portfolio Cash Flow =
–0.5610×$0.3147 = –$0.1766
Option Pricing
1.03
– .80
1.02
p=
1.25 – 0.80
Find the value of an at-the-money call
and a put on €1 with
Strike Price = $1.50
i$ = 3%
$1.50
i€ = 2%
u = 1.25
C0 = $.169744
d = .8
.5338 × $0.30
.4662× $0.375
7-74
1.03
$1.875 = 1.25 × $1.50
$0.375 = Call payoff
$0 = Put payoff
$1.20 = 0.8 × $1.50
$0 =Call payoff
$0.30 = Put payoff
P0 = $0.15555
C0 =
= .4662
= $.169744
P0 =
1.03
= $0.15555
Hedging a Call Using the Spot Market
We want to sell call options. How many units of the
underlying asset should we hold to form a riskless portfolio?
H=
$0.375 – $0
$1.875 – $1.20
= 5/9
$1.875 = 1.25 × $1.50
$0.375 = Call payoff
$1.50
Sell 1 call option; buy 5/9 of the
underlying asset to form a riskless
portfolio.
If the underlying is indivisible, buy 5
units of the underlying and sell 9 calls.
7-75
$1.20 = 0.8 × $1.50
$0 = Call payoff
Hedging a Call Using the Spot Market
T=0
Cash Flows
T=1
S1($|€) = $1.875
C1= $.375
= 5/9
H=
Call finishes in-the-money,
$1.875 – $1.20
so we must buy an additional €4 at $1.875.
Cost = 4 × $1.875 = $7.50
S0($|€) = $1.50/€
Cash inflow call exercise = 9 × $1.50 = $13.50
Portfolio cash flow = $6.00
Go long PV of €5.
S1($|€) = $1.20
€5 $1.50
C1= $0
×
= $7.3529
Cost today =
1.02 €1.00
Call finishes out-of-the-money, so we
Write 9 calls:
can sell our now-surplus €5 at $1.20.
Cash inflow = 9 × $0.169744 = $1.5277
Cash inflow = 5 × $1.20 = $6.00
Portfolio cash flow today = –$5.8252
$0.375 – $0
Handy thing to notice: $5.8252 × 1.03 = $6.00
7-76
Hedging a Put Using the Spot Market
We want to sell put options. How many units of the
underlying asset should we hold to form a riskless portfolio?
H=
$0 – $0.30
$1.875 – $1.20
= – 4/9
S1($|€) = $1.875
Put payoff = $0.0
S0 = $1.50/€
S1($|€) = $1.20
Put payoff = $0.30
Sell 1 put option; short sell 4/9 of the underlying asset to form a
riskless portfolio. If the underlying is indivisible, short 4 units of
the underlying and sell 9 puts.
7-77
Hedging a Put Using the Spot Market
T=0
H=
$0 – $0.30
$1.875 – $1.20
Cash Flows
= – 4/9
S0 = $1.50/€
Borrow the PV of €4 at i€ = 2%.
€4 $1.50
Inflow =
×
= $5.8824
1.02 €1.00
Write 9 puts:
Cash inflow = 9 × $0.15555 = $1.3992
Portfolio Inflow today = $7.2816
T=1
S1 = $1.875
Put finishes out-of-the-money.
To repay loan buy €4 at $1.875.
Cost = 4 × $1.875 = $7.50
Option cash inflow = 0
Portfolio cash flow = $7.50
S1 = $1.20
put finishes in-the-money, so we
must buy 9 units of underlying at
$1.50 each = 9×1.50 = $13.50
use 4 units to cover short sale, sell
remaining 5 units at $1.20 = $6.00
Handy thing to notice: $7.2816 × 1.03 = $7.50
Portfolio cash flow = $7.50
7-78
Hedging a Call Using Futures
S1 = $1.875
Futures contracts matures: buy 5 units at
forward price. Cost = 5× $1.5147 = $7.5735
S0 =
Call finishes in-the-money, we must buy 4 additional units of
$1.50/€
underlying at S1($/€) = $1.875. Cost = 4 × $1.875 = $7.50
Option cash inflow = 9 × $1.50 = $13.50
Portfolio cash flow = –$1.5735
S1 = $1.20
Futures contracts matures: buy 5 units at
Go long 5 futures contracts.
forward price. Cost = 5× $1.5147 = $7.5735
Cost today = 0
$1.50 1.03
×
= $1.5147Call finishes out-of-the-money, so we
Forward Price =
€1.00 1.02
Write 9 calls:
sell our 5 units of underlying at $1.20.
Cash inflow = 9 × $0.169744 = $1.5277
Cash inflow = 5 × $1.20 = $6.00
Portfolio cash flow today = $1.5277
Portfolio cash flow = –$1.5735
7-79
Handy thing to notice: $1.5277 × 1.03 = $1.5735
Hedging a Put Using Futures
S1 = $1.875
S0 = $1.50/€
Futures contracts matures: sell €5 at forward price.
Loss = 4× [$1.875 – $1.5147] = $1.4412
Put finishes out-of-the-money. Option cash flow = 0
Portfolio cash flow = –$1.4412
S1 = $1.20
Go short 4 futures contracts.
Put finishes in-the-money, we must
Cost today = 0
buy €9 at $1.50/€ = 9×1.50 = $13.50
$1.50 1.03
×
= $1.5147 Futures contracts matures: sell €4 at
Forward Price =
€1.00 1.02
forward price $1.5147/€
Write 9 puts:
4× $1.5147 = $6.0588
Cash inflow = 9 × $0.15555 = $1.3992
sell remaining €5 at $1.20 = $6.00
Portfolio Inflow today = $1.3992
Portfolio cash flow = –$1.4412
7-80
Handy thing to notice: $1.3992 × 1.03 = $1.4412
2-Period Options
Value a 2-period call option on
€1 with a strike price = $1.50/€
i$ = 3%; i€ = 2%
u = 1.25; d = .8
1.03
– .80
1.02
= .4662
p=
1.25 – 0.80
$2.3438
$0.8468
up
S1 = $1.875
up
C1 = $1.0609
up-down
S2
= $1.50
up-down
C2
= $0
S0 = $1.50/€
C0 = $0.4802
down
S1 = $1.20
down
C1 = $0
.4662× $0.8468
C1 =
= $1.06
1.03
.4662× $1.0609
C0 =
= $0.4802
7-81
1.03
up
up-up
S2 =
up-up
C2 =
down-down
S2
= $0.96
down-down
C2
= $0