1. Futures and Options on Foreign Exchange
This chapter discusses exchange-traded currency futures contracts, options contracts, and options on currency futures.
Futures and Options on Foreign Exchange
Chapter 7
Copyright © 2018 by the McGraw-Hill Companies, Inc. All rights reserved.

2. Chapter Outline Futures Contracts: Preliminaries
Currency Futures Markets
Basic Currency Futures Relationships
Options Contracts: Preliminaries
Currency Options Markets
Currency Futures Options
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
Summary
BIG chapter
Copyright © 2018 by the McGraw-Hill Companies, Inc. All rights reserved.
7-2

3. Futures Contracts: Preliminaries
A futures contract is like a forward contract in that 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 in that futures are standardized contracts trading on organized exchanges with daily resettlement through a clearinghouse.
Copyright © 2018 by the McGraw-Hill Companies, Inc. All rights reserved.
7-3

4. Futures Contracts: Preliminaries (continued)
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)
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7-4

5. Currency Futures Markets
The CME Group (formerly Chicago Mercantile Exchange) is by far the largest currency futures market.
The Singapore Exchange offers interchangeable contracts.
There are other markets, but none are close to CME and SIMEX trading volume.
Expiry cycle: March, June, September, December.
The delivery date is the third Wednesday of delivery month.
The last trading day is the second business day preceding the delivery day.
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7-5

6. EXHIBIT 7.3 CME Group Currency Futures Contract Quotations
Open interest refers to the number of contracts outstanding for a particular delivery month—it’s a good proxy for demand for a contract. Notice that open interest is greatest in the nearby contract.
In general, open interest typically decreases with term to maturity of most futures contracts.
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7-6

7. Daily Resettlement
With futures contracts, we have daily resettlement of gains and 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-day-shorter maturity.
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7-7

8. Daily Resettlement: An Example
Consider a long position in a CME British pound futures contract.
It is written on £62,500 and price is quoted in dollars and cents per £, out to 4 decimal points.
The minimum price increment is $ per British pound (corresponds to $6.25/contract)
We went long the futures contract at a price of $ per £.
At initiation of the contract, the long posts an initial performance bond of $1,625.
The maintenance performance bond is $1,300.
This means that you get a margin call when your position erodes by $325.
Longs get margin calls when the settle price falls.
Shorts get margin calls when the settle price rises.
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7-8

9. 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 $1,625.
The maintenance level is $1,300.
If this investor loses more than $325, he has a decision to make; he can maintain his long position only by adding more funds, and if he fails to do so his position will be closed out with an offsetting short position.
By the way a $325 loss would correspond to a price movement of only 52 x or a bit more than half a penny
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7-9

10. Daily Resettlement: first 3 days
Over the first 3 days, the pound strengthens then depreciates in dollar terms:
Settle
Gain/Loss
Account Balance
$1.3010
$62.50
= ($1.3010/£ – $1.3000/£) × £62,500
$1,687.50
= $1,625 + $62.50
$62.50 = ($ $1.3000) x 62,500
–$187.50
$1,500
$1.2980
$1.2948
–$200.00
$1,300
On day three we receive a margin call, having lost $325.
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7-10

11. Toting Up
At the end of this adventure, we have three ways of computing gains and losses:
Sum of daily gains and losses.
– $325 = $62.50 – $ – $200
Contract size times the difference between initial contract price and last settlement price.
– $325 = ($1.2948/£ – $1.3000/£) × £62,500
Ending balance on the account minus beginning balance on the account, adjusted for deposits or withdrawals.
– $325 = $1,300 – $1,625
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7-11

12. Options Contracts: Calls vs. Puts
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:
Call options give 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 give 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.
The holder of the option bought that right by paying the seller (aka the “writer”) of the option at initiation of the contract an option premium.
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7-12

13. Options Contracts: Preliminaries
European versus American options:
European options can only be exercised on the expiration date while American options can be exercised at any time up to and including the expiration date.
American options are usually worth more than European options, other things equal.
Moneyness
If immediate exercise is profitable, an option is “in the money.”
Out of the money options can still have value.
There’s also Asian options, by the way
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7-13

14. PHLX Currency Option Specifications
Contract Size
Australian dollar
AUD 10,000
British pound
GBP 10,000
Canadian dollar
CAD 10,000
Euro
EUR 10,000
Japanese yen
JPY 1,000,000
New Zealand dollar
NZD 10,000
Swiss franc
CHF 10,000
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7-14

15. Basic Option Pricing Relationships at Expiry
At expiry, an American 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]
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]
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7-15

16. Basic Call Option Profit Profiles
Long 1 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. ST
Consider a call option on €10,000.
The option premium is $0.25 per €
The exercise price is $1.50 per €
The option costs $2,500 today and the break-even exchange rate is $1.75 per €
–c0
E + c0
Short 1 call E
Out-of-the-money
In-the-money
Loss
Out-of-the-money
In-the-money
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7-16

17. Basic Put Option Profit Profiles
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
Short 1 put
Consider a put option on €10,000.
The option premium is $0.15 per €.
The exercise price is $1.50 per euro.
What is the maximum gain on this put option? $13,500 = €10,000×($1.50 – $0.15)/€
At what exchange rate do you break even?
$1.35 per euro
E – p0 E ST
– p0
Long 1 put
In-the-money
Out-of-the-money
In-the-money
Out-of-the-money
Loss
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7-17

18. 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.
Note that even an out-of-the-money option has value—time value.
Long 1 call
Market Value
Intrinsic value ST
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]
Time value
Out-of-the-money
In-the-money
Out-of-the-money
In-the-money
Loss E
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7-18

19. European Option Pricing Relationships
Consider two investments:
Buy a European call option on the British pound futures contract. The cash flow today is –Ce.
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$ , the cash flow today is
Lending the present value of ST at i£, the cash flow today is E
(1 + i$) ST
(1 + i£) –
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7-19

20. European Option Pricing Relationships (concluded)
Ce > Max ST E
(1 + i£)
(1 + i$) –
, 0
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,
Using a similar portfolio to replicate the upside potential of a put, we can show that:
Pe > Max
Contrast this with the exact pricing equations found later in this presentation.
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7-20

21. Binomial Option Pricing
Imagine a world where the spot exchange rate is S0($/€) = $1.50/€ today and in the next year S1($/€) is either $1.80/€ or $1.20/€. €10,000 will change from $15,000 to either $18,000 or $12,000. A call option on €10,000 with strike price S0($/€) = $1.50 will payoff either $3,000 or zero. If S1($/€) = $1.800/€ the option is in-the-money since you can buy €10,000 (worth $18,000 at S1($/€) = $1.80/€ ) for only $15,000.
$15,000
$18,000 = €10,000 ×
$1.80
€1.00
$12,000 = €10,000 ×
$1.20
At $1.20/€ the option is out-of-the-money because no one would pay $15,000 for an asset only worth $12,000
By the way, the standard size option contract on the Philadelphia exchange is €10,000, but at NYBoT it’s €31,250 and other amounts on other exchanges.
C1up = $3,000
C1down = $0
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7-21

22. Binomial Option Pricing Model
We can replicate the payoffs of the call option by taking a long position in a bond with FV = €5,000 along with the right amount of dollar-denominated borrowing (in this case borrow the PV of $6,000). The portfolio payoff in one period matches the option payoffs:
$9,000
– $6,000
= $3,000
= C1up
Where do the €5,000 and $6,000 come from? The Hedge Ratio. (see later slide) The point is that the portfolio has precisely the call option’s payoff, so the portfolio is worth the same as the option today.
$15,000
$6,000
– $6,000
= $0
= C1down
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7-22

23. Binomial Option Pricing Replicating Portfolio
The replicating portfolio’s dollar cost today is the sum of today’s dollar cost of the present value of €5,000 less the cash inflow from borrowing the present value of $6,000:
$1.50
$6,000
(1 + i$)
€1.00
€5,000 × –
(1 + i€ )
When S0($/€) = $1.50/€, i$ = 7.1%, and i€ = 5%, the most a willing buyer should pay for the call option is $1, That’s what it would cost him today to build a portfolio that perfectly replicates the call option payoffs—why pay more to buy a ready-made option?
Here we see that the most a willing buyer should pay for the call option is $1, – you can also prove to yourself that the least that a willing seller will accept is $1,540.62
$1, = $7, − $5,602.24
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7-23

24. The Hedge Ratio
We replicated the payoffs of the call option with a levered position in the underlying asset (in this case, borrowing $5, to buy €4, 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
$5, = $6,000 / 1.071
€4, = €5,000 / 1.05
To replicate the call, first calculate the hedge ratio as ½--this tells us that the call option only moves half as fast as the underlying asset.
With a hedge ratio of ½ we need a long position in ½ of €10,000 or €5,000.
Now we have something that’s worth either $9,000 or $6,000 in one period when the spot $1.80 or $1.20/€ respectively.
At the down state rate of $1.20 we need to owe $6,000 to replicate the zero payoff of the call.
At the up state exchange rate of $1.80 our portfolio also perfectly replicates the call payoff of $3,000.
Pretty cool—eh?
This ratio gives the number of units of the underlying asset we should hold and the amount of borrowing in order to create a replicating portfolio.
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7-24

25. Hedge Ratio (continued)
This practice of constructing a riskless hedge is sometimes called delta hedging.
The hedge ratio of a call option is positive.
Recall from the example:
H =
C1 – C1
S1 – S1
down up
$3,000 – $0
$18,000 – $12,000 1 2 =
The hedge ratio of a put option is negative.
These hedge ratios change through time.
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7-25

26. Currency Futures Options
Currency futures options are options 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.
Copyright © 2018 by the McGraw-Hill Companies, Inc. All rights reserved.
7-26

27. Binomial Futures Option Pricing
A 1-period at-the-money call option on euro futures has a strike price of F1($|€) = $1.5300/€
$1.50×1.071
€1.00×1.05
F1($|€) =
$1.5300
€1.00 =
$1.80×1.071
$1.8360
$1.20×1.071
$1.2240
Call Option Payoff = $0.3060
Option Price = ?
A cash amount equal to the excess of the futures price over the strike price is like a daily resettlement.
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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7-27

28. Binomial Futures Option Pricing (continued)
Consider the portfolio: Long H futures contracts Short 1 futures call option
$1.80×1.071
€1.00×1.05
F1($|€) =
$1.8360
€1.00 =
$1.50×1.071
€1.00×1.05
F1($|€) =
$1.5300
€1.00 =
Futures Call Payoff = –$0.3060
Futures Payoff = H × $0.2700
Option Price = $0.1714
Portfolio Cash Flow =
H × $ – $0.3060
Portfolio is riskless when the portfolio payoffs in the “up” state equal the payoffs in the “down” state:
H × $ – $ = –H × $0.3300
The “right” amount of futures contracts is H =
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.
In this case the holder of the futures call receives $ = $ $
When you go long in a futures contract at $1.53 you settle up if the underlying goes up or down:
if S1 = $1.80, you receive $0.27 = $ $1.53.
If S1 = $1.20, you lose $.33 = $ $1.20.
$1.20×1.071
€1.00×1.05
F1($|€) =
$1.2240
€1.00 =
Portfolio Cash Flow = –H × $0.3300
Option Payoff = $0
Futures Payoff = –H × $0.3300
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7-28

29. Binomial Futures Option Pricing (conclusion)
$1.80×1.071
€1.00×1.05
F1($|€) =
$1.8360
€1.00 =
The payoffs of the portfolio are –$ in both the up and down states.
Call Option Payoff = –$0.3060
$1.50×1.071
€1.00×1.05
F1($|€) =
$1.5300
€1.00 =
Futures Payoff = × $0.2700
Portfolio Cash Flow =
0.510 × $ – $0.3060
= –$0.1683
With futures there is no cash flow at initiation. Without an arbitrage, it must be the case that the call option income today is equal to the present value of $ discounted at i$ = 7.10%:
$1.20×1.071
€1.00×1.05
F1($|€) =
$1.2240
€1.00 =
Futures Payoff = –0.510 × $0.3300
$0.1683
1.071
C0 = $ =
Option Payoff = $0
Portfolio Cash Flow =
–0.510×$ = –$0.1683
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7-29

30. Risk Neutral Valuation
Calculating the hedge ratio is vitally important if you are going to use options.
The seller needs to know the hedge ratio if he wants to protect his profits or eliminate his downside risk.
The buyer needs to know the hedge ratio to decide how many options to buy.
Knowing what the hedge ratio is isn’t especially important if you are only trying to value options.
Risk Neutral Valuation is a very handy shortcut to valuation.
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7-30

31. Risk Neutral Valuation of Options: Example
We can safely assume that IRP holds:
F1($/€) =
$1.5300
€1.00
$1.50×(1.071)
€1.00×(1.05) =
€10,000 = $15,000
$1.20
€1.00
€10,000 ×
$12,000 =
$1.80
$18,000 =
Set the value of €10,000 bought forward at $1.5300/€ equal to the expected value of the two possibilities shown above:
$15,300 = q × $18,000 + (1 – q) × $12,000
€10,000 ×
$1.5300
€1.00 =
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7-31

32. Risk Neutral Probability
Solving for q gives the risk-neutral probability of an “up” move in the exchange rate: $15,300 = q × $18,000 + (1 – q) × $12,000
q =
$15,300 – $12,000
$18,000 – $12,000
q = 11/20 =
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7-32

33. Risk Neutral Valuation of Options (continued)
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 =
$1.80
$18,000 =
←value of €10,000
11/20
$3,000= payoff of right to buy €10,000 for $15,000
$1,540.62
9/20
$0 = payoff of right to buy €10,000 for $15,000
C0 = $1, =
(11/20) × $3,000 + (9/20)×$0
1.071
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7-33

34. Duality of Calls and Puts on Currencies
The value of a call option on €10,000 with a strike price of $15,000 is $1,540.62
The value of a put option on $15,000 with a strike price of €10,000 is also $1,540.62
Call Option Seller
Call Option Owner
€10,000
$15,000
Put Option Seller
Put Option Owner
€10,000
$15,000
If the options finish in-the-money they have the same cash flows. So they should have the same value today.
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7-34

35. 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 the value of a call option on €10,000 with a strike price of $15,000 to be $1,540.62, this should be easy in the sense that we already know the right answer.
$1.50
€1.00
S0($/€) =
As before, i$ = 7.1%, i€ = 5%,
$1.50×1.071
€1.00×1.05
F1($/€) =
$1.5300
€1.00 =
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7-35

36. Test Your Intuition (continued)
$1.50×1.071
€1.00×1.05
F1($/€) =
$1.5300
€1.00 =
€10,000 = $15,000
€1.00
$1.80
$15,000 ×
€8, =
$1.20
€12,500 =
←€ value of $15,000
when S1 = $1.20/€
when S1 = $1.80/€
6/17
11/17
Notice that this risk-neutral probability is different from the risk neutral probability that we calculated earlier.
$15,000 ×
$1.5300
€1.00
= €9,803.92
€ 9, = q × €12,500 + (1 – q) × €8,333.33
q =
€9,803.92– €8,333.33
€12,500 – €8,333.33
q = 6/17
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7-36

37. Test Your Intuition (concluded)
€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
6/17
€1,027.08
€1.00
$1.80
$15,000 ×
←value of $15,000
€8, =
11/17
€1,666.67= payoff of right to sell $15,000 for €10,000
The value of a call option on €10,000 with a strike price of $15,000 is $1,540.62
The value of a put option on $15,000 with a strike price of €10,000 is €1,027.08
At the spot exchange rate these values are the same: $1, = €1, × $1.50/€1
€P0 = €1, =
6/17× €0 + (11/17)×€1,666.67
1.05
$P0 = $1, = €1, ×
$1.50
€1.00
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7-37

38. Risk Neutral Valuation Practice
Use risk neutral valuation to value a PUT option on £8,000 with a strike price of €10,000. S0(£/€) = £0.80/€, i£ = 15½% and i€ = 5% In the next year, there are two possibilities: S1(£/€) = £1.00/€ or S1(£/€) = £0.75/€
€10,666.67
€8,000
€10,000
9/22
p1 = €0 up
Step 1: Calculate risk neutral probabilities.
13/22
Step 2: Calculate option value as the present value of the expected value of the option payoffs.
By the way, a call option on €10,000 with a strike price of £8,000 is worth £ (the risk-neutral probability for the call is .52)
AND £ = €1, × £.80/€
Sweet.
Also recall that the IRP forward price is £.88/€
p = €2,000
down
€9, = q × €10, (1 – q) × €8,000
p0 = €1,125.54=
13/22 × €2,000
1.05
q =
€9,090.91– €8,000
€10,666.67– €8,000
= 9/22
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7-38

39. Risk Neutral Valuation Practice (continued)
Check your work by finding the value of an at-the-money 1-period call option on €10,000 with a strike of £8,000.
S0(£/€) = £0.80/€, i£ = 15½%, i€ = 5%, so F1(£/€) = £0.8800/€
In the next year, there are two possibilities:
S1(£/€) = £1.00/€ or S1(£/€) = £0.75/€
£10,000 = × €10,000
£1.00
€1.00
0.52
c1 = £2,000 u
c0 = £900.43
€10,000 × = £8,000
£0.80
€1.00
= max[0, £10,000 − £8,000]
0.48
q × c1 + (1 – q) × c1
1 + r₤
c0 = d u
£7,500 = × €10,000
£0.75
€1.00
c1 = £0 d
q = =
£.88/€ − £.75/€
£1.0/€ − £.75/€
= max[0, £7,500 − £8,000]
0.52 × £2, × £0
1.155
c0 =
= £900.43
= €1, × £0.80/€
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7-39

40. Things to be Careful About
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 IRP equal to the expected value of the payoffs calculated using q and solve for q.
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.
Copyright © 2018 by the McGraw-Hill Companies, Inc. All rights reserved.
7-40

41. Black–Scholes Pricing Formulae
The Black-Scholes formulae for the price of a European call and a put written on currency are:
In the early 1970s, Fischer Black, Myron Scholes, and Robert Merton made a major breakthrough in the pricing of options. In 1997, the importance of this work was recognized with the Nobel Prize. Basically, the EOPM uses the same intuitions as the BOPM, but uses the normal approximation to the binomial process.
N(d) = Probability that a standardized, normally distributed, random variable will be less than or equal to d.
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42. Black–Scholes Pricing Formula
Use the European option pricing formula to find the value of a six-month call option on Japanese yen. The strike price is $1 = ¥100. The volatility is 25 percent per annum; r$ = 5.5% and r¥ = 6%.
N(d) was calculated using NORMSDIST in excel
And none of the intermediate steps were rounded—use the memory function of your calculator
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43. Summary
Forward, futures, and options contracts are derivative securities.
Their value is derived from the value of the asset that underlies these securities.
Forward and futures contracts are similar instruments, but there are differences.
Both are contracts to buy or sell a certain quantity of a specific underlying asset at some specific price in the future.
Futures contracts, however, are exchange-traded, and there are standardized features that distinguish them from the tailor-made terms of forward contracts.
The two main standardized features are contract size and maturity date.
Futures contracts are marked-to-market on a daily basis at the new settlement price.
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44. Summary (continued)
A futures market requires speculators and hedgers to effectively operate. Hedgers attempt to avoid the risk of price change of the underlying asset, and speculators attempt to profit from anticipating the direction of future price changes.
The CME Group and the NASDAQ OMX Futures Exchange are the two largest currency futures exchanges.
The pricing equation typically used to price currency futures is the IRP relationship, which is also used to price currency forward contracts.
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45. Summary (concluded)
An option is the right, but not the obligation, to buy or sell the underlying asset for a stated price over a stated time period.
Call options give the owner the right to buy, put options the right to sell.
American options can be exercised at any time during their life; European options can only be exercised at maturity.
Exchange-traded options with standardized features are traded on two exchanges.
Options on spot foreign exchange are traded at the NASDAQ OMX PHLX, and options on currency futures are traded at the CME.
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