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Unformatted text preview: INTERNATIONAL
FINANCIAL
MANAGEMENT
Fifth Edition
Instructor’s Notes: Students must
augment these materials for their
own study purposes! EUN / RESNICK
Adapted by M.D. Griffiths Lecture 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 72 Lecture 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 73 Forwards and Futures: definition
Contracts that specify today the price for a future
delivery of an asset or a commodity at a specific
time.
In general they follow equilibrium, noarbitrage
pricing conditions.
Forward and futures differ for technical reasons but
essentially satisfy the same need and follow the
same rules.
But: Are Forward and futures price are different? Futures Contracts: Preliminaries A futures contract is like a forward contract: A futures contract is different from a forward
contract: 75 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 Tbills held in a street
name at your brokerage). 76 Currency Futures Markets
The Chicago Mercantile Exchange (CME) is by far the largest
followed by Singapore Exchange (SIMEX) Others include: The Philadelphia Board of Trade (PBOT)
The Tokyo International Financial Futures Exchange
The London International Financial Futures Exchange Contracts on the CME market generally trade according to the
following rules Expiry cycle: March, June, September, December.
Delivery date third Wednesday of delivery month.
Last trading day is the 2nd business day preceding the delivery day. CME After Hours
Extendedhours 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. 78 Futures Contracts: Preliminaries
Some definitions apply: Futures price is the market price of the contract Forward price is the market price for a range of delivery
dates Delivery price is the contracted upon price for the
contract Settlement price is the price at the end of each trading
day before delivery
BE CAREFUL: you contract on something but then the
market fluctuates and the delivery price remains constant Convergence of Futures to Spot
The closer the delivery date, the more the futures contract
will converge to the market spot price.
Mkt
Forward
Price Spot Price
Mkt
Forward
Price Spot Price
Time (a) (b) Time Forward Contracts vs Futures
Contracts
FORWARDS FUTURES Private contract between 2 parties Exchange traded Nonstandard contract Standard contract Usually 1 specified delivery date
Settled at maturity
Delivery or final cash
settlement usually occurs Range of delivery dates
Settled daily
Contract usually closed out
prior to maturity Futures Contracts Available on a wide range of underlying assets
Exchange traded
Specifications need to be defined: What can be delivered,
Where it can be delivered,
When it can be delivered Settled daily Futures Contracts: Preliminaries Standardizing Features: Contract Size
Delivery Month
Daily resettlement
Margins requirements Most contracts are closed out before maturity Delivery If a contract is not closed out before maturity, it usually
settled by delivering the assets underlying the contract.
When there are alternatives about what is delivered,
where it is delivered, and when it is delivered, the party
with the short position chooses. A few contracts (for example, those on stock indices and
Eurodollars) are settled in cash Margins A margin is cash or marketable securities deposited by an
investor with his or her broker
Initial performance bond (about 2 percent of contract
value, cash or Tbills held in a street name at your
brokerage). Used for margins requirement. Depends on
leverage. 2%=50:1
The balance in the margin account is adjusted to reflect
daily settlement
Margins minimize the possibility of a loss through a
default on a contract Some additional definitions Open interest: the total number of contracts outstanding equal to number of long positions or number of short
positions 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.
Volume of trading: the number of trades in 1 day Reading Currency Futures Quotes
OPEN HIGH LOW SETTLE CHG LIFETIME
HIGH LOW OPEN
INT Euro/US Dollar (CME)—€125,000; $ per €
Mar 1.3136 1.3167 1.3098 1.3112 .0025 1.3687 1.1363 159,822
Jun 1.3170 1.3193 1.3126 1.3140 .0025 1.3699 1.1750 10,096 Closing price Highest and lowest
prices over the life
Daily Change
of the contract.
Opening price Lowest price that day
Number of open contracts
Highest price that day
Expiry month Reading Currency Futures Quotes
OPEN HIGH LOW SETTLE CHG LIFETIME
HIGH LOW OPEN
INT Euro/US Dollar (CME)—€125,000; $ per €
Mar 1.3136 1.3167 1.3098 1.3112 .0025 1.3687 1.1363 159,822
Jun 1.3170 1.3193 1.3126 1.3140 .0025 1.3699 1.1750 10,096
Sept 1.3202 1.3225 1.3175 1.3182 .0025 1.3711 1.1750
600 Notice that open interest is larger the closer the delivery
date, in this case March, 2005.
In general, open interest typically decreases with term to
maturity of most futures contracts. Basic Currency
Futures Relationships
OPEN HIGH LOW SETTLE CHG LIFETIME
HIGH LOW OPEN
INT Euro/US Dollar (CME)—€125,000; $ per €
Mar 1.3136 1.3167 1.3098 1.3112 .0025 1.3687 1.1363 159,822
The holder of a long position is committing himself to pay $1.3112 per
euro for €125,000—a $163,900 position.
As there are 159,822 such contracts outstanding, this represents a
notational principal of over $26 billion (NxFXxSize)! Basic Currency
Futures Relationships
OPEN HIGH LOW SETTLE CHG LIFETIME
HIGH LOW OPEN
INT Euro/US Dollar (CME)—€125,000; $ per €
Mar 1.3136 1.3167 1.3098 1.3112 .0025 1.3687 1.1363 159,822 Notice that if you had been smart or lucky enough to open
a long position at the lifetime low of $1.1363 by now your
gains would have been
$21,862.50 = ($1.3112/€ – $1.1363/€) × €125,000
REMEMBER: it’s a zero sum game! Someone took the
short position at $1.1363 and lost the same amount! Basic Currency
Futures Relationships
OPEN HIGH LOW SETTLE CHG LIFETIME
HIGH LOW OPEN
INT Euro/US Dollar (CME)—€125,000; $ per €
Mar 1.3136 1.3167 1.3098 1.3112 .0025 1.3687 1.1363 159,822 If you had been smart or lucky enough to open a short
position at the lifetime high of $1.3687 by now your gains
would have been:
$7,187.50 = ($1.3687/€ – $1.3112/€) × €125,000 Reading Currency Futures Quotes
OPEN HIGH LOW SETTLE CHG LIFETIME
HIGH LOW OPEN
INT Euro/US Dollar (CME)—€125,000; $ per €
Mar 1.3136 1.3167 1.3098 1.3112 .0025 1.3687 1.1363 159,822
Jun 1.3170 1.3193 1.3126 1.3140 .0025 1.3699 1.1750 10,096
Sept 1.3202 1.3225 1.3175 1.3182 .0025 1.3711 1.1750
600 Recall, our interest rate parity condition:
1 + i$
F($/€)
=
1 + i€
S($/€) Reading Currency Futures Quotes
OPEN HIGH LOW SETTLE CHG LIFETIME
HIGH LOW OPEN
INT Euro/US Dollar (CME)—€125,000; $ per €
Mar 1.3136 1.3167 1.3098 1.3112 .0025 1.3687 1.1363 159,822
Jun 1.3170 1.3193 1.3126 1.3140 .0025 1.3699 1.1750 10,096
Sept 1.3202 1.3225 1.3175 1.3182 .0025 1.3711 1.1750
600 From June to September the euro is at a premium (weaker
dollar) therefore we should expect higher interest rates in
dollar denominated accounts: if we find a higher rate in a
euro denominated account, we may have found an arbitrage. 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 onedayshorter
maturity.
724 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. 725 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 Gain/Loss Account Balance
$1.31
$1,250 = ($1.31 – $1.30)$6,500 + $1,250
$7,750 = ×125,000
$1.30
–$1,250
$6,500
$1.27
–$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.
726 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
$1,250
$1.30
–$1,250
$1.27
–$3,750
$1.26
–$1,250
$1.24
–$2,500
727 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)
728 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)
729 Sample Problem The March 2008 Mexican peso futures contract has a price
of $0.90975 per 10MXN. You believe the spot price in
March will be $0.97500 per 10MXN.
What speculative position would you enter into to attempt
to profit from your beliefs?
Calculate your anticipated profits, assuming you take a
position in three contracts. What is the size of your profit
(loss) if the futures price is indeed an unbiased predictor
of the future spot price and this price materializes? Sample Problem If you expect the Mexican peso to rise from $0.90975 to
$0.97500 per 10 MXN, you would take a long position in
futures since the futures price of $0.90975 is less than your
expected spot price. You expect the peso to appreciate
and want to receive the benefit in the margin account. Your anticipated profit from a long position in three
contracts is: 3 x ($0.097500  $0.090975) x MP500,000 =
$9,787.50, where MXN500,000 is the contractual size of
one MXN contract. Sample Problem If the futures price is an unbiased predictor of the expected
spot price, the expected spot price is the futures price of
$0.90975 per 10 MXN. If this spot price materializes, you will not have any
profits or losses from your long position in three futures
contracts: 3 x ($0.090975  $0.090975) x MP500,000 = 0 Other futures: 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
90day Eurodollar deposit—the contract is cash
settled. Traded on the CME and the Singapore
International Monetary Exchange. The contract trades in the March, June, September
and December cycle. Reading Eurodollar Futures Quotes
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 threemonth
LIBOR calculated as F = 100 – LIBOR. Lenders buy the rate!
The closing price for the June contract is 96.56 thus the implied yield
is 3.44 percent = 100 – 96.56
Since it is a 3month contract one basis point corresponds to a $25 price
change: .01 percent of $1 million represents $100 on an annual basis. Other futures: Eurodollar Interest Rate Futures Contracts Eurodollar futures are a way for companies and banks to
lock in an interest rate today, for money it intends to
borrow or lend in the future.
CME Eurodollar futures prices are determined by the
market’s forecast of the 3month USD Libor interest rate
expected to prevail on the settlement date. The settlement
price of a contract is defined to be 100.00 minus the
official British Bankers Association fixing of 3month
Libor on the contract settlement date. For example, if 3month Libor sets at 5.00% on the contract settlement date,
the contract settles at a price of 95.00 What do we want? Amy wants to lock in an interest rate so that the
correct amount of money will be ready when we need it.
Therefore, we need the money at the instrument’s maturity. But, the
amount available at maturity is a function of the interest rate which is
determined at the beginning of the investment period . So, for certainty in July, we want the future to mature in April!
And we want to trade it back there!
M A M J J A S O
Cash Flows
For a 3mth instrument,
the interest rate is known
with certainty here! We need the money here so,
this is when the instrument has
to mature! Making Futures Work Buying a future is an obligation to buy, selling a future is an
obligation to sell.
To hedge with futures, borrowers sell to lock in their interest
rate. When hedging with futures, investors buy to lock in their
rates.
If interest rates rise, then what will happen to the futures price?
Amy is worried about interest rates falling, so does she want to
buy or sell a futures contract? Why is she worried about interest rates falling? What month futures contract does she want? The Variation Margin Let’s say Amy buys 100 June futures contract at 93.51. If we look at the first
week of trading, it might look like this: Day
Mon trade
Mon close
Tue close
Wed close
Thur close
Fri close Closing Price
93.51
93.54
93.53
93.48
93.50
93.53 Tick Change
0
+3
1
5
___
+2
___
+3
___ Margin Change
0
7,500
2,500
12,500
______
______
5,000
7,500
______ Number of ticks * tick value * Number of contracts = Margin change Did the Hedge Work?
Fixed rate of interest _____%
6.49
Futures profit/loss
_____MM
Return on Investment _____MM
Total return
_____MM
Rate of return
_____% Rate that futures locks in to. If rate this = lock rate then
hedge worked. Amy was concerned in Mar that when she received the money in June
interest rates would have declined, so she bought a futures contract to
fix the interest rate from June to September.
Let’s say, Amy bought the future at $93.51, so she locked in what rate? Did the Hedge Work?
6.49
Fixed rate of interest _____%
0.235
Futures profit/loss
_____MM
Return on Investment _____MM
Total return
_____MM
Rate of return
_____% Rate that futures locks in to. If this rate = lock rate then
hedge worked. Assume that in June, the futures contract settles at 94.45. What is the profit or loss
on the contract?
Number of ticks * Tick value * No. of contract = Profit/loss
94
$25
100
_______
_____
_______
_$235,000
________ Did the Hedge Work?
Fixed rate of interest _____%
6.49
0.235
Futures profit/loss
_____MM
Return on Investment _____MM
Total return
_____MM
Rate of return
_____% Rate that futures locks in to. If this rate = lock rate then
hedge worked. Assume the June futures contract settles at 94.45
Now let’s look at the cash flow on the investment: When Amy gets the money in June,
she immediately invests it at the then current 3month money market rate which is
_5.55%
______ Did the Hedge Work?
Fixed rate of interest _____%
6.49
0.235
Futures profit/loss
_____MM
1.3875
Return on Investment _____MM
Total return
_____MM
Rate of return
_____% Rate that futures locks in to. If this rate = lock rate then
hedge worked. Assume the June futures contract settles at 94.45
Based on this, Amy’s cash flow from the money market would be: $100,000,000 * 5.55% * 3
= $1,387,500
12 Did the Hedge Work?
6.49
Fixed rate of interest _____%
0.235
Futures profit/loss
_____MM
1.3875
Return on Investment _____MM
1.6225
Total return
_____MM
Rate of return
_____% Rate that futures locks in to. If this rate = lock rate then
hedge worked. Now, we can figure out what Amy’s total return would be:
Futures profit/loss + ROI
= $235,000 + $1,387,500 = $1,622,500 Did the Hedge Work?
Fixed rate of interest _____%
6.49
0.235
Futures profit/loss
_____MM
1.3875
Return on Investment _____MM
1.6225
Total return
_____MM
6.49
Rate of return
_____% Rate that futures locks in to. If this rate = lock rate then
hedge worked. Now, we can figure out what Amy’s rate of return would be: $1,622,500 12
*
= 6.49%
$100,000,000 3 Hedging with Futures Interest rate futures are quoted as 100 minus the rate of
interest. This is known as backtofront pricing.
Backtofront pricing affects the decision of whether to
buy or sell.
A borrower is concerned about interest rates rising. When
hedging with FRAs, borrowers buy to lock in their interest
rate. To hedge with futures, borrowers sell to lock in their
interest rate. Since the pricing of futures is inverse, so is
the trading.
When hedging with futures, investors buy to lock in their
rates. Trading irregularities Futures Markets are also a great place to launder
money 747 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.
748 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 or sell a given quantity of an
asset in the future, at prices agreed upon today. Calls vs. Puts 749 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 750 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 Inthemoney Atthemoney 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. Outofthemoney 751 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
752 = Intrinsic
Value + Speculative
Value Currency Options Markets
PHLX HKFE 20hour trading day. OTC volume is much bigger than exchange
volume. Trading is in six major currencies against the U.S.
dollar. 753 PHLX Currency Option Specifications
Currency
Australian dollar
British pound
Canadian dollar
Euro
Japanese yen
Swiss franc
754 Contract Size
AD10,000
£10,000
CAD10,000
€10,000
¥1,000,000
SF10,000 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 inthemoney, it is worth S – E.
T If the call is outofthemoney, it is worthless.
CaT = CeT = Max[ST  E, 0] 755 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 inthemoney, it is worth E  S .
T If the put is outofthemoney, it is worthless.
PaT = PeT = Max[E – ST, 0] 756 Basic Option Profit Profiles
Profit Owner of the call If the call is inthemoney, it is worth
ST – E.
If the call is outofthemoney, it is
worthless and the –c
0
E + c0
buyer of the call
E
loses his entire
investment of c0.
Outofthemoney Inthemoney
loss
757 Long 1 call ST Basic Option Profit Profiles
Profit Seller of the call If the call is inthemoney, the writer
loses ST – E.
If the call is outofthemoney, the
writer keeps the
option premium. c0 ST
E
loss 758 Outofthemoney E + c0 Inthemoney short 1
call Basic Option Profit Profiles
Profit If the put is inthemoney, it is E – p
0
worth E – ST.
The maximum
gain is E – p0 If the put is outofthemoney, it
– p0
is worthless and
the buyer of the
put loses his
entire investment
loss
of p0.
759 Owner of the put ST
E – p0 long 1 put E Inthemoney Outofthemoney Basic Option Profit Profiles
If the put is inthemoney, it is
worth E –ST. The
maximum loss is
– E + p0 Profit Seller of the put p0 If the put is outofthemoney, it
is worthless and
the seller of the
put keeps the
option premium– E + p
0
of p0.
loss
760 ST
E – p0 E short 1 put Example
Profit Consider a call
option on €31,250.
The option premium
is $0.25 per €
The exercise price is
$1.50 per €.
–$0.25 Long 1 call
on 1 euro
ST
$1.75 $1.50
loss
761 Example
Profit Consider a call
option on €31,250.
The option premium
is $0.25 per €
The exercise price is
$1.50 per €.
–$7,812.50 Long 1 call
on €31,250
ST
$1.75 $1.50
loss
762 Example
Profit What is the maximum gain on this put option?
$42,187.50 $42,187.50 = € 31,250 × ($1.50 – $0.15)/ € 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
763 $1.50 Long 1 put
on €31,250 $4,687.50 = € 31,250 × ($0.15)/ € PutCall Parity Relationship (equities) An investor has two ways to achieve limited downside risk with
unlimited upside protection: Purchase a stock today (S) and buy a put option (p) at a given
exercise price (X).
S+p Buy a risk free zero coupon bond with a maturity value of X and
buy a call option (c) at a given exercise price (X).
X∗ er∗ t + c Relation Between Prices of a Put
and a Call with Same Features?
Strategy 1
Buy Stock; Buy a PUT
Investment:
Expiration:
S>X
S<X S+p Strategy 2
Buy Call; Invest PV(X)
c + PV(X) Value = S
Value = X Value = S
Value = X Since the Outcomes are the Same , the Investments must be Equal, i.e., [S
+ p = c + PV(X)] or p = c  S + PV(X)
This is Known as PutCall Parity PutCall Parity Relationship The putcall parity equation implies the
following: The greater the exercise price the more expensive is the put premium. The higher the riskfree rate the less
expensive is the put premium. The higher the purchase price of the stock
the less expensive is the put premium PutCall Parity: Example
Suppose We Knew a 1 Year Call Option with an
Exercise Price of $25 was Worth $5 Today given
10% Interest Rates. If the Stock Price is Currently
$21 What is the Put Option Worth? p=
=
=
= c  S + Xert
Xe
$5  $21 + $25*(0.904)
$5
$5  $21 + $22.62
$5
$6.62
$6.62 Putcall parity: example On June 20th, ABC Oct 65 puts are quoted at 12,
and the Oct 65 calls are quoted at 8.5. If ABC
pays no dividend and rates are 7% per annum
(continuously compounded), at what stock price
were these options quoted. (June 20th is exactly
4 months from Oct expiry). Assume European
style options.
c  p = S  XerT
S = 8.5  12 + 65e0.07x0.3333
= 60.00 Arbitrages with PutCall Parity Stock ABC trades at $35, and 3 month 30 strike calls trade at 7.
If rates are 6% (continuously compounded), where should the put
trade?
p = c + X erT  S
= 7 + 30x0.98  35
= 1.40
If the put trades at 3 (and everything else is the same) how do you
take advantage of the mispricing? Theoretically there should be arbitrage profit of $1.60 Sell the LHS of the above equation, and buy the RHS Arbitrages with Putcall parity Sell the LHS, means sell a put at 3, and invest the proceeds at the
riskfree rate. Buy the RHS means buy a call at 7, a riskfree zero coupon bond
at Xert = 29.40, and sell the stock at 35. Notice the $1.40 we are
short is more than offset by the put sale. On day 0, we have $1.60 in our pocket. Arbitrages with Putcall parity At expiry, if the put finishes in the money, we will have the
stock “put” to us at 30. If the call finishes in the money, we
will exercise to buy the stock at 30. In either case we are
buying the stock at 30. Given this accounting, we can ignore
any residual option value. (This is how brokers account for
options expiry) The zero coupon bond we purchased for 29.40 is now worth
$30 and just covers the cost of the stock purchase We keep the $1.60 + interest. PutCall Parity Relationship
S0(1+ ρ)T + Pe = Ce + X(1+r)T Where ρ is the foreign interest rate and the spot rate is
being quoted in units of the foreign currency
Note similarity to putcall parity for options on stocks.
Also, a call to buy one currency denominated in the other
with exercise price of X is equivalent to a put to sell X
units of the latter currency denominated in the former
currency with exercise price of 1/X. American Option Pricing
Relationships With an American option, you can do everything
that you can do with a European option AND you
can exercise prior to expiry—this option to
exercise early has value, thus:
CaT > CeT = Max[ST  E, 0]
PaT > PeT = Max[E  ST, 0]
773 Market Value, Time Value and Intrinsic
Value for an American Call
Profit
The red line shows the
payoff at maturity, not
profit, of a call option. Ma et V
rk Note that even an outofthemoney option
has value—time
value. 774 Long 1 call Intrinsic value
Time value
Outofthemoney loss e
alu E Inthemoney ST European Option Pricing Relationships
Consider two investments
1
Buy a European call option on the British pound
futures contract. The cash flow today is –Ce
Replicate the upside payoff of the call by 2 Borrowing the present value of the dollar exercise
price of the call in the U.S. at i$ 1 The cash flow today is
2 Lending the present value of ST at i£
The cash flow today is 775 E
(1 + i$)
– ST
(1 + i£) European Option Pricing Relationships
When the option is inthemoney both strategies
have the same payoff.
When the option is outofthemoney it has a
higher payoff than the borrowing and lending
strategy.
Thus:
ST
E
Ce > Max (1 + i ) – (1 + i ) , 0
£
$
776 European Option Pricing Relationships
Using a similar portfolio to replicate the upside
potential of a put, we can show that: E
ST
Pe > Max (1 + i ) – (1 + i ) , 0
$
£ 777 What is the Relation Between Price Changes
on Stock and Call? The Call Option is Exchangeable into the Underlying Stock
The Value of the Call is Driven (Largely) by the Value of
the Underlying Stock
When the Stock Price Goes Up so Does the Call Price; When
the Stock Price Goes Down so Does the Call Price
Perfect Positive Correlation
Potential to be Able to Create a Perfect Hedge Valuation European Call  Under Uncertainty
Current Stock Price = $ 20; Exercise Price = $ 22.50; Time to
Expiration = 1 YR; Risk Free Rate = 10%; Stock Price in One Year is
Either $ 25 or $ 15 $25 Option = $ 2.50 $15 Option = $ 0.00 $20 Eliminate Uncertainty? How? The Delta Hedge! Form a Hedge Portfolio
Long the Stock and Short the Call;
(Reverse Correlation with Respect to Impact on Wealth)
25 H  2.50 = 15 H =>
H = 0.25
Long 0.25 Shares and Short 1 Call Option
If Stock Price Moves Up to $ 25;
Portfolio = 25 * 0.25  2.50 = $ 3.75
If Stock Price Moves Down to $ 15;
Portfolio = 15 * 0.25  0 = $ 3.75
Hedge Ratio = 0.25; Regardless of Whether Stock Goes Up or
Down  the Value of Hedge Portfolio is $ 3.75 Value of Call Option Today ?
Value Riskless Portfolio, Absent Arbitrage Opportunities,
Must Earn the RiskFree Rate of Interest Value of the Portfolio Today =
PV(3.75) = 3.75 * e ( 0.1 )( 1)= $ 3.39 The Value of the Stock Today is $20; The Value of the
Portfolio Today is 20 * 0.25  C = 5  C = $ 3.39
or C = $ 1.61 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 1year 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%
782 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 dollareuro
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
783 $1.20 Binomial Option Pricing Model
A call option on the euro with exercise price S0($/€)
= $1.50 will have the following payoffs.
By exercising the call option, you can buy €1 for $1.50.
If S1($/€) = $1.875/€ the option is inthemoney: S0($/€) S1($/€)
$1.875 $1.50 …and if S1($/€) = $1.20/€ the option is
outofthemoney: 784 $1.20 C1($/€)
$.375 $0 Binomial Option Pricing Model
We can replicate the payoffs of the call option.
By taking a position in the euro along with some
judicious borrowing and lending.
S0($/€) S1($/€)
$1.875 C1($/€)
$.375 $1.20 $0 $1.50
785 Binomial Option Pricing Model
Borrow the present value (discounted at i$) 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
786 $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
787 $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
788 $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($/€) $1.50
789 S1($/€) debt portfolio C1($/€)
$1.875 – $1.20 = $.675 $.375
If i$ = 3% and i€ = 2% the call is worth
5
€1.00 $1.50 – $1.20
$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.
the Many derivative securities can be valued by
valuing portfolios of primitive securities when
those portfolios have the same payoffs as the
derivative securities.
790 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=
S1up – S1 down
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. 791 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
$0.375 – $0
Cup – C down
=
=
H=
=
S1up – S1down
$1.875 – $1.20 $0.675 The delta of a put option is negative.
Deltas change through time.
792 5
9 Creating a Riskless Hedge
The standard size of euro options on the PHLX is €10,000.
In our simple world where the dollareuro exchange rate is
S0($/€) = $1.50/€ today and in the next year, S1($/€) is either
$1.875/€ or $1.20/€ an atthemoney call on €10,000 has
these payoffs:
$1.875
If the exchange rate at maturity goes up
× 10,000 = $18,750
€
€1.00
– $15,000
to S1($/€) = $1.875/€ then the option
up
C1 = $3,750
finishes inthemoney.
$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
793 $1.20
× 10,000
€
€1.00 = $12,000
down
C1 = $0 Creating a Riskless Hedge
Consider a dealer who has just written 1 atthemoney
call on €10,000. He calculates the hedge ratio as 5/9:
$3,750
C up– C down = $3,750 – 0
=
=
H=
$18,750 – $12,000
$6,750
S1up – S1 down
He can hedge his position with three trades:
1. If i$ = 3% then he could borrow $6,472.49
today and owe $6,666.66 in one period.
5
$12,000 ×
9
2. = $6,666.66 $6,666.66
$6,472.49 =
1.03 Then buy the present value of €5,555.56 = €10,000 ×
(buy euro at spot exchange rate,
€5,555.56
compute PV at i€ = 2%), €5,446.62 =
1.02 3. 5
9 Invest €5,446.62 at i€ = 2%.
794 5
9 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
$1,697.44 =
Step 3
Invest €5,446.62 at i€ = 2%
Net cost = $1,697.44
795 5 × €10,000
$1.20
$1.50
–
×
9
(1.02)
€1.00
(1.03) $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. 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
796 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
$1.20
× €10,000
$12,000 =
€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
797 Risk Neutral Valuation of Options
Solving for p gives the riskneutral 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
798 Risk Neutral Valuation of Options
Now we can value the call option as the present value
(discounted at the USD riskfree rate) of the expected
value of the option payoffs, calculated using the riskneutral probabilities. $1.875
× €10,000 ←value of €10,000
$18,750 =
€1.00
$3,750 = payoff of right to buy €10,000 for $15,000 €10,000 = $15,000
$1,697.44 C0 = $1,697.44 =
799 $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
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
$1,697.44 this should be easy in the sense that we
already know the right answer.
As before, i$ = 3%, i€ = 2%, S0($/€) = $1.50
€1.00
$1.50×1.03
$1.5147
F1($/€) = €1.00×1.02 = €1.00
7 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
€9,902.91– €8,000
p=
€12,500 – €8,000
7 p = .4229 Test Your Intuition (continued)
€1.00
× $15,000 ←value of $15,000
€12,500 =
$1.20
0 = payoff of right to sell $15,000 for €10,000
€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 1.50
$P0 = $1,697.44 = €1,131.63 × $1.00
€
7 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 = $1,697.44
7 TakeAway Lessons
Convert future values from one currency to another
using forward exchange rates.
Convert present values using spot exchange rates.
Discount future values to present values using the
correct interest rate, e.g. i$ discounts dollar amounts
and i€ discounts amounts in euro.
To find the riskneutral probability, set the forward
price derived from IRP 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 Finding Risk Neutral Probabilities
down
F1($/€) = p × S1 ($/€) + (1 – p) × S1 ($/€)
up
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
p= $15,147.06 – $12,000
$1.5147 – $1.20
=
= .4662
$18,750 – $12,000
$1.875 – $1.20 For a put on $15,000 with a strike price of €10,000 we solved
€9,902.91 = p × €12,500 + (1 – p) × €8,000
€9,902.91– €8,000
€0.6602– €.5333
p=
=
= .4229
€12,500 – €8,000
€.8333 – €.5333 Currency Futures Options
Are an option on a currency futures contract. Exercise of a currency futures option results in a
long futures position for the holder of a call or the
writer of a put. Exercise of a currency futures option results in a
short futures position for the seller of a call or the
buyer of a put. If the futures position is not offset prior to its
expiration, foreign currency will change hands. 7 Currency Futures Options
Why a derivative on a derivative? Transactions costs and liquidity. For some assets, the futures contract can have
lower transactions costs and greater liquidity than
the underlying asset. Tax consequences matter as well, and for some
users an option contract on a future is more tax
efficient. The proof is in the fact that they exist. Binomial Futures Option Pricing
A 1period atthemoney call option on euro futures has a
strike price of F1($€) = $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 Binomial Futures Option Pricing
Consider the Portfolio: long Η 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
Η = 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 = $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
p= 1.03
– .80
1.02
1.25 – 0.80 Find the value of an atthemoney call
and a put on €1 with
Strike Price = $1.50
i$ = 3%
i€ = 2%
u = 1.25
d = .8 C0 = C0 = $.169744
P0 = $0.11316 .4662× $0.375
1.03 $1.50 = $.169744 = .4662 $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 = .4229 × $0.30
1.03 = $0.11316 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? Η= $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. $1.20 = 0.8 × $1.50
$0 = Call payoff Hedging a Call Using the Spot Market
T=0
Η= $0.375 – $0 $1.875 – $1.20 S0($€) = $1.50/€ Cash Flows T = 1
= 5/9 S1($€) = $1.875
C1= $.375
Call finishes inthemoney,
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 = $6.00 Go long PV of €5.
S1($€) = $1.20
€5 $1.50
C1= $0
×
= $7.3529
Cost today =
1.02 €1.00
Call finishes outofthemoney, so we
Write 9 calls:
can sell our nowsurplus €5 at $1.20.
Cash inflow = 9 × $0.169744 = $1.5277
Cash inflow = 5 × $1.20 = $6.00
Portfolio cash flow today = –$5.8252
Handy thing to notice: $5.8252 × 1.03 = $6.00 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? Η= $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. Hedging a Put Using the Spot Market
T=0
Η= $0 – $0.30 $1.875 – $1.20 Cash Flows T = 1
= – 4/9 S0($€) = $1.50/€
Borrow the PV of €4 at i€ = 2%.
€4 $1.50
×
= $5.8824
Inflow =
1.02 €1.00
Write 9 puts:
Cash inflow = 9 × $0.15555 = $1.3992
Portfolio Inflow today = $7.2816 S1($€) = $1.875
Put finishes outofthemoney.
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 inthemoney, 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 = $7.50 Hedging a Call Using Futures
S1($€) =
$1.875 S0($€) =
$1.50/€ Futures contracts matures: buy 5 units at
forward price. Cost = 5× $1.5147 = $7.5735
Call finishes inthemoney, we must buy 4 additional units of
underlying at S1($/€) = $1.875. Cost = 4 × $1.875 = $7.50
Option cash inflow = 9 × $1.50 = $13.50
Portfolio cash flow = –$1.5735
S ($€) =
1 Go long 5 futures contracts. $1.20 Futures contracts matures: buy 5 units at
forward price. Cost = 5× $1.5147 = $7.5735
Cost today = 0
$1.50 1.03
×
= $1.5147Call finishes outofthemoney, 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
Handy thing to notice: $1.5277 × 1.03 = $1.5735 Hedging a Put Using Futures
S0($€) =
$1.50/€ S1($€) =
$1.875
Futures contracts matures: sell €5 at forward price.
Loss = 4× [$1.875 – $1.5147] = $1.4412
Put finishes outofthemoney. Option cash flow = 0
Portfolio cash flow = –$1.4412 S1($€) =
Go short 4 futures contracts. $1.20
Put finishes inthemoney, 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
Handy thing to notice: $1.3992 × 1.03 = $1.4412 2Period Options
Value a 2period 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 Supup = $2.3438
2
Cupup = $0.8468
2 S = $1.875
C = $1.0609
up
1
up
1 S0 = $1.50/€
C0 = $0.4802 Supdown = $1.50
2
Cupdown = $0
2
Sdown $1.20
1= down
.4662× $0.8468
C1 = $0
C=
= $1.06
1.03
.4662× $1.0609
C0 =
= $0.4802
7
1.03
up
1 Sdowndown $0.96
=
2
Cdowndown = $0
2 ...
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This note was uploaded on 12/10/2011 for the course FIN 417 taught by Professor Griffith during the Spring '11 term at Miami University.
 Spring '11
 Griffith
 International Finance

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