0. 7 Futures and Options on Foreign Exchange Chapter Objective:
This chapter discusses exchange-traded currency futures contracts, options contracts, and options on currency futures. 7
Chapter Seven
Futures and Options on Foreign Exchange
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1. Primary vs. Derivative Products
Primary Financial Products: their values are determined by their own cash flows. E.g., stocks, bonds, currencies, (real, financial, and 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.

2. (Currency) Futures Contracts: Preliminaries
A futures contract is like a forward contract:
It specifies that a certain currency will be exchanged for another at a specified time in the future at prices specified today.
A futures contract is different from a forward contract:
Futures are standardized contracts trading on organized exchanges with daily resettlement through a clearinghouse.
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3. 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 ~ commissions for a round trip
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).
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4. Daily Settlement: An Example
Consider a long position in the CME EUR/USD 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 $1,350.
The maintenance performance bond is $1,000.
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5. 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.
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6. 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.
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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 $1,350.
The maintenance level is $1,000.
If this investor loses more than $350 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.
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8. Daily Settlement: An Example
Over the first 3 days, the euro strengthens then depreciates in dollar terms (with $1,500 initial balance):
Settle
Gain/Loss
Account Balance
$1.31
$1,250
= ($1.31 – $1.30)×125,000
$2,750
= $1,500 + $1,250
$1.30
–$1,250
$1,500 = $1,750 - $1,250
$1.29
–$1,250
$250
= $1,500 - $1,250
On third day suppose our investor keeps his long position open by posting an additional $1,100 at minimum to achieve the initial margin requirement of $1,350. Otherwise, his account will be closed out with $250 left for him. A total cost of $1,250 from $1,500 initial balance.
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9. Toting Up
At the end of the 3rd day, our investor has three ways of computing his gains and losses:
Sum of daily gains and losses
– $1,250 = $1,250 – $1,250 – $1,250
Contract size times the difference between initial contract price and last settlement price.
– $1,250 = ($1.29/€ – $1.30/€) × €125,000
Ending balance on account minus beginning balance on account, adjusted for deposits or withdrawals.
– $1, 250 = $ $1,500
7-9

10. Currency Futures Markets
The Chicago Mercantile Exchange (CME) is by far the largest.
Others include:
The Philadelphia Board of Trade (PBOT)
The MidAmerica Commodities Exchange
The Tokyo International Financial Futures Exchange
The London International Financial Futures Exchange
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11. 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.
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12. 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.
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13. Reading Currency Futures Quotes, 2/10/11
OPEN INT
OPEN
HIGH
LOW
SETTLE
CHG
Euro/US Dollar (CME)—€125,000; $ per €
Mar
1.4748
1.4830
1.4700
1.4777
.0028
172,396
Jun
1.4737
1.4818
1.4693
1.4763
.0025
2,266
Closing price?
Expiry month
Thursday March 3, 2005
Daily Change
Opening price
Lowest price that day
Number of open contracts
Highest price that day
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14. 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.
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15. Reading Currency Futures Quotes, 2/10/11
OPEN
HIGH
LOW
SETTLE
CHG
OPEN INT
Euro/US Dollar (CME)—€125,000; $ per €
1.4748
1.4830
1.4700
1.4777
.0028
Mar
172,396
1.4737
1.4818
1.4693
1.4763
.0025
Jun
2,266
Notice that open interest is greatest in the nearby contract, in this case March, 2011.
Thursday March 3, 2005
In general, open interest typically decreases with term to maturity of most futures contracts.
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16. Reading Currency Futures Quotes, 2/10/11
OPEN
HIGH
LOW
SETTLE
CHG
OPEN INT
Euro/US Dollar (CME)—€125,000; $ per €
1.4748
1.4830
1.4700
1.4777
.0028
Mar
172,396
1.4737
1.4818
1.4693
1.4763
.0025
Jun
2,266
Recall from chapter 6, our interest rate parity condition (arbitrage possibility if there is a sizable disparity):
Thursday March 3, 2005
1 + i€
1 + i$ F So =
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17. Reading Currency Futures Quotes, 2/10
OPEN
HIGH
LOW
SETTLE
CHG
OPEN INT
Euro/US Dollar (CME)—€125,000; $ per €
1.4748
1.4830
1.4700
1.4777
.0028
Mar
172,396
1.4737
1.4818
1.4693
1.4763
.0025
Jun
2,266
From March to June 2011, 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.
Thursday March 3, 2005
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18. Eurodollar Interest Rate Futures Contracts
Widely used futures contract for hedging short-term U.S. dollar interest rate risk.
The underlying asset is a hypothetical $1,000, day Eurodollar deposit—the contract is cash settled.
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19. Reading Eurodollar Futures Quotes, 2/10/11
OPEN
HIGH
LOW
SETTLE
CHG
OPEN INT
YLD
Eurodollar (CME)—1,000,000; pts of 100%
96.56
96.58
96.55 -
3.44
Jun
1,398,959
Eurodollar futures prices are stated as an index number of three-month LIBOR calculated as F = 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 the F going high (or avoid LIBOR getting low).
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20. Trading irregularities
Futures Markets are also a great place to launder money
The zero sum nature of futures is the key to laundering the money.
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21. Money Laundering: Hillary Clinton’s Cattle Futures
winners
              
James B. Blair, Outside Counsel to Tyson Foods Inc., Arkansas' largest employer, gets Hillary’s discretionary order.
losers
Submits identical long and short trades
Hillary Clinton Futures Trades Detailed
By Charles R. Babcock Washington Post Staff Writer Friday, May 27, 1994; Page A01
Robert L. "Red" Bone, (Refco broker), allocates trades ex post facto.
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22. 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.
A real-life example of a call option is a rain check. A real-life example of a put option is Ray Lewis contract (Ray is selling his service to the Ravens).
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23. Options Contracts: Preliminaries
European vs. American options
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 an option early is more valuable (greater flexibility), American options are worth more than European options.
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24. Options Contracts: Preliminaries
In-the-money
It is profitable to exercise the option.
At-the-money
Indifferent
Out-of-the-money
It is not profitable to exercise.
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25. Options Contracts: Preliminaries
Intrinsic Value (price on the rain check vs future spot price => savings!)
The difference between the exercise price of the option and the future spot price of the underlying asset.
Time Value (pay more than the intrinsic value?)
The difference between the option premium and the intrinsic value of the option (this time value is different from TVM).
Option Premium
Intrinsic Value
Time Value + =
7-25

26. 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.
7-26

27. PHLX Currency Option Specifications
Contract Size
Australian dollar
AD10,000
British pound
£10,000
Canadian dollar
CAD10,000
Euro
€10,000
Japanese yen
¥1,000,000
Swiss franc
SF10,000
7-27

28. 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]
7-28

29. 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]
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30. Basic Option Profit Profiles
Owner of the call
If the call is in-the-money, it is worth ST – E.
If the call is out-of-the-money, it is worthless and the buyer of the call loses his entire investment of c0.
Long 1 call ST
–c0
E + c0 E
Out-of-the-money
In-the-money
loss
7-30

31. Basic Option Profit Profiles
Seller of the call
If the call is in-the-money, the writer loses ST – E.
If the call is out-of-the-money, the writer keeps the option premium. c0 E
E + c0 ST
short 1 call
loss
7-31

32. Basic Option Profit Profiles
Owner of the put
If the put is in-the-money, it is worth E – ST. The maximum gain is E – p0
If the put is out-of-the-money, it is worthless and the buyer of the put loses his entire investment of p0.
E – p0
E – p0 E ST
– p0
long 1 put
In-the-money
Out-of-the-money
loss
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33. Basic Option Profit Profiles
Seller of the put
If the put is in-the-money, it is worth E –ST. The maximum loss is – E + p0
If the put is out-of-the-money, it is worthless and the seller of the put keeps the option premium of p0. p0 E
E – p0 ST
short 1 put
– E + p0
loss
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34. Example Long 1 call on 1 pound Consider a call option on €31,250.
Profit
Consider a call option on €31,250.
The option premium is $0.25 per €
The exercise price is $1.50 per €.
Long 1 call on 1 pound
$1.75 ST
–$0.25
$1.50
loss
7-34

35. Example Long 1 call on €31,250 Consider a call option on €31,250.
Profit
Consider a call option on €31,250.
The option premium is $0.25 per €
The exercise price is $1.50 per €.
Long 1 call on €31,250
$1.75 ST
–$7,812.50
$1.50
loss
7-35

36. Example Long 1 put on €31,250 $4,687.50 = €31,250×($0.15)/€
Profit
What is the maximum gain on this put option?
At what exchange rate do you break even?
$42, = €31,250×($1.50 – $0.15)/€
$42,187.50
Consider a put option on €31,250.
The option premium is $0.15 per €
The exercise price is $1.50 per euro.
$1.35 ST
–$4,687.50
Long 1 put on €31,250
$1.50
$4, = €31,250×($0.15)/€
loss
7-36

37. 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:
(note T > t)
Cat > Cet
Pat > Pet
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38. Market Value, Time Value and Intrinsic Value for an European Call at t
Profit
The red line shows the payoff at maturity, not profit, of a call option.
Note that even an out-of-the-money option has value—time value.
Long 1 call
Market Value
Intrinsic value ST
Time value
Out-of-the-money
In-the-money
loss E
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39. European Call Option Pricing relationship (determine the call price using the “rep” portfolio & no arbitrage. e.g., $1.1 for $2-B, $1.4 for $3-R, what about $2-B & $6-R? )
Consider two investments in a Call or a “Rep” Portfolio
Buy a European call option on the British pound futures contract. The cash flow today is – Ce with CT = Max[ST - E, 0]
Replicate the upside payoff of the call by
Borrowing the present value of the dollar exercise price of the call in the U.S. at i$ E
(1 + i$)
The cash flow today is
Lending the present value of One BP (FV=One BP=$ST) at i£ ST
(1 + i£)
The cash flow today is –
7-39

40. European Call Option Pricing Relationships
When the option is in-the-money both strategies have the same payoff at T (i.e., ST – E).
When the option is out-of-the-money the call has a higher payoff (0) than the borrowing and lending strategy (ST – E, which is negative).
Thus, the present price is:
Ce > Max ST E
(1 + i£)
(1 + i$) –
, 0
7-40

41. European Put Option Pricing Relationships
Using a similar portfolio to replicate the upside potential of a put, we can show that:
Pe > Max ST E
(1 + i£)
(1 + i$) –
, 0
7-41

42. A Brief Review of CIRP $1.5147 F1 = €1.00
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/€ (note that this diagram is sideway) 1
Borrow $1.5m at i$ = 3%
Owe $1.545m
F1 =
$1.5147
€1.00
Exchange $1.5m for €1m at spot
Invest €1m at i€ = 2%
Receive €1.02 m
7-42

43. Binomial Call 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
$1.20
7-43

44. Binomial Option Pricing Model
A call option on the euro with exercise price E = $1.50 (=S0) 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:
$1.20
$1.875 S1 S0
$1.50 C1
$.375 $0
…and if S1 = $1.20/€ the option is out-of-the-money:
7-44

45. 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.
$1.20
$1.875 S1 S0
$1.50 C1
$.375 $0
7-45

46. 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
$1.20
– $1.20
= $0 $0
7-46

47. 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
$.375
1.80 = S0
debt
portfolio C1 S1
$1.20
$1.875
$1.50
– $1.20
= $.675
= $0
$.375 $0
7-47

48. Binomial Option Pricing Model
The replicating portfolio’s dollar value today (how much it costs to make the portfolio) is the sum of today’s dollar value of the present value of one euro less the present value of a $1.20 debt:
$1.50
$1.20
(1 + i$)
€1.00
(1 + i€) × – S0
debt
portfolio C1 S1
$1.20
$1.875
$1.50
– $1.20
= $.675
= $0
$.375 $0
7-48

49. Binomial Option Pricing Model
We can value the call option as 5/9 of the value of the replicating portfolio: C0 = × 5 9
$1.50
$1.20
(1 + i$)
€1.00
(1 + i€) – S0 S1
debt
portfolio C1
$1.875
– $1.20
= $.675
$.375
If i$ = 3% and i€ = 2% the call is worth
$ = × 5 9
$1.50
$1.20
(1.03)
€1.00
(1.02) –
$1.50
$1.20
– $1.20
= $0 $0
7-49

50. 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

51. 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:
H =
C – C
S1 – S1
down up
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.
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52. 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:
H =
C – C
S1 – S1
down up
$0.375 – $0
$1.875 – $1.20
$0.375
$0.675 5 9 =
The delta of a put option is negative.
Deltas change through time.
7-52

53. 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
€1.00
$1.20 ×
If the exchange rate at maturity goes up to S1 = $1.875/€ then the option finishes in-the-money.
€10,000 =
$18,750
– $15,000
C1 = $3,750 up
× €10,000 = $15,000
$1.50
€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
€10,000 =
$12,000
C = $0
down
7-53

54. 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:
H =
C – C
S1 – S1
down up =
$3,750 – 0
$18,750 – $12,000
$3,750
$6,750 5 9
He can hedge his position with three trades:
If i$ = 3% then he could borrow $6, today and owe $6, in one period.
Then buy the present value of €5, (buy euro at spot exchange rate,
compute PV at i€ = 2%),
Invest €5, at i€ = 2%.
$6, =
$6,666.66
1.03
€5, =
€5,555.56
1.02
$12,000 × = $6,666.66 5 9
Net cost of hedge = $1,697.44
= €10,000 ×
7-54

55. Replicating Portfolio Call on €10,000 K = $1.50/€
$6,666.67
$10,416
FV € investment in $
€5,555.56
FV € investment ×
– $ 6,666.67
– $6,666
Service Loan
$1.875
S1($|€)
€1.00
$1.20
T = 0
T = 1
Borrow $6, at i$ = 3%
Step 1
= $3,750
= 0
Buy €5, at S0 = $1.50/€
Step 2
the replicating portfolio payoffs and the call option payoffs are the same so the call is worth
$1, = × 5 9
$1.50
$1.20
(1.03)
€1.00
€10,000
(1.02) –
€10,000 = $15,000
Invest €5, at i€ = 2%
Step 3
Net cost = $1,697.44
7-55

56. Risk Neutral Valuation of Options
Calculating the hedge ratio is vitally important if you are going to use options.
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.
7-56

57. Risk Neutral Valuation of Options
We can safely assume that CIRP holds:
F1 =
$1.5147
€1.00
$1.50×(1.03)
€1.00×(1.02) =
€10,000 = $15,000
$1.20
€1.00
€10,000 ×
$12,000 =
$1.875
$18,750 =
Set the value of €10,000 bought forward at $1.5147/€ equal to the expected value of the two possibilities shown above:
$15, = p × $18,750 + (1 – p) × $12,000
€10,000×
$1.5147
€1.00 =
7-57

58. Risk Neutral Valuation of Options
Solving for p gives the risk-neutral probability of an “up” move in the exchange rate:
$15, = p × $18,750 + (1 – p) × $12,000
p =
$15, – $12,000
$18,750 – $12,000
p = .4662
7-58

59. 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 risk-neutral probabilities.
€10,000 = $15,000
$1.20
€1.00
€10,000 ×
$12,000 =
$1.875
$18,750 =
←value of €10,000
$3,750 = payoff of right to buy €10,000 for $15,000
$1,697.44
$0 = payoff of right to buy €10,000 for $15,000
C0 = $1, =
.4662×$3,750 + (1–.4662)×0
1.03
7-59

60. 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, this should be easy in the sense that we already know the right answer.
$1.50
€1.00
S0 =
As before, i$ = 3%, i€ = 2%,
$1.50×1.03
€1.00×1.02
F1 =
$1.5147
€1.00 =
7-60

61. Test Your Intuition (continued)
$1.50×1.03
€1.00×1.02
F1 =
$1.5147
€1.00 =
€10,000 = $15,000
€1.00
$1.875
$15,000 ×
€8,000 =
$1.20
€12,500 =
←value of $15,000
$15,000 ×
$1.5147
€1.00
= €9,902.91
€9, = p × €12,500 + (1 – p) × €8,000
p = .4229
7-61

62. Test Your Intuition (continued)
€1.00
$1.20
$15,000 ×
€12,500 =
←value of $15,000
0 = payoff of right to sell $15,000 for €10,000
€10,000 = $15,000
€1,131.63
€1.00
$1.875
$15,000 ×
€8,000 =
←value of $15,000
€2,000 = payoff of right to sell $15,000 for €10,000
€P0 = €1, =
.4229×€0 + (1–.4229)×€2,000
1.02
$P0 = $1, = €1, ×
$1.50
€1.00
7-62

63. 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, ×
€1.00
$1.50
= $1,697.44
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64. Take-Away Lessons
Convert future values from one currency to another using forward exchange rates.
Convert present values using spot exchange rates.
Discount future values to present values using the correct interest rate, e.g. i$ 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.
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65. Finding Risk Neutral Probabilitiesup
F1 = p × S1 + (1 – p) × S1
down
For a call on €10,000 with a strike price of $15,000 we solved
$15, = p × $18,750 + (1 – p) × $12,000
p =
$15, – $12,000
$18,750 – $12,000
= .4662 =
$ – $1.20
$1.875 – $1.20
For a put on $15,000 with a strike price of €10,000 we solved
€9, = p × €12,500 + (1 – p) × €8,000
p =
€9,902.91– €8,000
€12,500 – €8,000
= .4229 =
€0.6602– €.5333
€.8333 – €.5333
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66. 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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67. 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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68. Binomial Futures Option Pricing
A 1-period at-the-money call option on euro futures has a strike price of F1 = $1.5147/€
$1.50×1.03
€1.00×1.02
F1 =
$1.5147
€1.00 =
$1.875×1.03
$1.8934
$1.20×1.03
F1($|€) =
$1.2118
Call Option Payoff = $0.3787
Option Price = ?
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
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69. Binomial Futures Option Pricing
$1.875×1.03
€1.00×1.02
F1 =
$1.8934
€1.00 =
Consider the Portfolio:
long H futures contracts
short 1 futures call option
Futures Call Payoff = –$0.3787
Futures Payoff = H × $0.3603
$1.50×1.03
€1.00×1.02
F1($|€) =
$1.5147
€1.00 =
Portfolio Cash Flow =
H × $ – $0.3787
Option Price = $0.1714
Portfolio is riskless when the portfolio payoffs in the “up” state equal the payoffs in the “down” state:
H×$ – $ = –H×$0.3147
The “right” amount of futures contracts is
H =
$1.20×1.03
€1.00×1.02
F1($|€) =
$1.2118
€1.00 =
Portfolio Cash Flow =
–H×$0.3147
Option Payoff = $0
Futures Payoff = –H×$0.3147
7-69

70. Binomial Futures Option Pricing
$1.875×1.03
€1.00×1.02
F1($|€) =
$1.8934
€1.00 =
The payoffs of the portfolio are –$ in both the up and down states.
Call Option Payoff = –$0.3787
Futures Payoff = H × $0.3603
Portfolio Cash Flow =
× $ – $0.3787
= –$0.1766
$1.50×1.03
€1.00×1.02
F1($|€) =
$1.5147
€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 $ discounted at i$ = 3%
$1.20×1.03
€1.00×1.02
F1($|€) =
$1.2118
€1.00 =
Futures Payoff = –0.5610×$0.3147
Option Payoff = $0
$0.1766
1.03
C0 = $ =
Portfolio Cash Flow =
–0.5610×$ = –$0.1766
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71. Option Pricing $0.375 = Call payoff $0 = Put payoff C0 = $.169744
– .80
p =
1.25 – 0.80
= .4662
1.03
1.02
Find the value of an at-the-money call and a put on €1 with
Strike Price = $1.50
i$ = 3%
i€ = 2%
u = 1.25
d = .8
$1.50
$1.875 = 1.25 × $1.50
$1.20 = 0.8 × $1.50
$0.375 = Call payoff
$0 = Put payoff
C0 = $
P0 = $
$0 =Call payoff
$0.30 = Put payoff
.4662× $0.375
C0 =
1.03
= $
.5338 × $0.30
P0 =
1.03
= $
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72. 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?
$0.375 – $0
H =
$1.875 – $1.20
= 5/9
$1.50
$1.875 = 1.25 × $1.50
$1.20 = 0.8 × $1.50
$0.375 = Call payoff
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.
$0 = Call payoff
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73. Hedging a Call Using the Spot Market
Cash Flows
T = 1
S1 = $1.875
$0.375 – $0
H =
$1.875 – $1.20
= 5/9
C1= $.375
Call finishes in-the-money,
so we must buy an additional €4 at $1.875.
Cost = 4 × $1.875 = $7.50
Cash inflow call exercise = 9 × $1.50 = $13.50
Portfolio cash flow = $6.00
S0 = $1.50/€
Go long PV of €5.
Cost today = €5
1.02
× = $7.3529
$1.50
€1.00
S1 = $1.20
C1= $0
Call finishes out-of-the-money, so we can sell our now-surplus €5 at $1.20.
Cash inflow = 5 × $1.20 = $6.00
Write 9 calls:
Cash inflow = 9 × $ = $1.5277
Portfolio cash flow today = –$5.8252
Handy thing to notice: $ × 1.03 = $6.00
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74. 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?
$0 – $0.30
H =
$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.
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75. Hedging a Put Using the Spot Market
Cash Flows
T = 1
S1 = $1.875
$0 – $0.30
H =
$1.875 – $1.20
= – 4/9
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
S0 = $1.50/€
Borrow the PV of €4 at i€ = 2%.
Inflow = €4
1.02
× = $5.8824
$1.50
€1.00
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
Portfolio cash flow = $7.50
Write 9 puts:
Cash inflow = 9 × $ = $1.3992
Portfolio Inflow today = $7.2816
Handy thing to notice: $ × 1.03 = $7.50
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76. Hedging a Call Using Futures
Futures contracts matures: buy 5 units at forward price. Cost = 5× $ = $7.5735
Call finishes in-the-money, we must buy 4 additional units of underlying at S1($/€) = $ Cost = 4 × $1.875 = $7.50
Option cash inflow = 9 × $1.50 = $13.50
Portfolio cash flow = –$1.5735
Go long 5 futures contracts.
Cost today = 0
Forward Price =
Futures contracts matures: buy 5 units at forward price. Cost = 5× $ = $7.5735
× = $1.5147
1.03
1.02
$1.50
€1.00
Call finishes out-of-the-money, so we sell our 5 units of underlying at $1.20.
Cash inflow = 5 × $1.20 = $6.00
Portfolio cash flow = –$1.5735
Write 9 calls:
Cash inflow = 9 × $ = $1.5277
Portfolio cash flow today = $1.5277
Handy thing to notice: $ × 1.03 = $1.5735
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77. Hedging a Put Using Futures
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
Go short 4 futures contracts.
Cost today = 0
Forward Price =
Put finishes in-the-money, we must buy €9 at $1.50/€ = 9×1.50 = $13.50
Futures contracts matures: sell €4 at forward price $1.5147/€
4× $ = $6.0588
sell remaining €5 at $1.20 = $6.00
Portfolio cash flow = –$1.4412
× = $1.5147
1.03
1.02
$1.50
€1.00
Write 9 puts:
Cash inflow = 9 × $ = $1.3992
Portfolio Inflow today = $1.3992
Handy thing to notice: $ × 1.03 = $1.4412
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78. – .80
p =
1.25 – 0.80
= .4662
1.03
1.02
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
S2 = $2.3438
S = $1.50
up-down
up-up
C2 = $0.8468
up-up
C1 = $1.0609 up
S0 = $1.50/€
S1 = $1.875
S1 = $1.20
down up
C0 = $0.4802
S = $0.96
down-down
C = $0
up-down
C = $0
down
.4662× $0.8468
C1 =
1.03
= $1.06 up
1.03
.4662× $1.0609
C0 =
= $0.4802
C = $0
down-down
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