international financial management

Irwin/McGraw-Hill Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.

9-1

INTERNATIONALFINANCIAL

MANAGEMENT

EUN / RESNICKSecond Edition

9Chapter Nine

Futures and Options on Foreign Exchange

Chapter Objective:

This chapter discusses exchange-traded currency futures contracts, options contracts, and options on currency futures.


9-2

Chapter Outline

Futures Contracts: Preliminaries Currency Futures Markets Basic Currency Futures Relationships Eurodollar Interest Rate Futures Contracts Options Contracts: Preliminaries Currency Options Markets Currency Futures Options


9-3

Chapter Outline (continued)

Basic Option Pricing Relationships at Expiry American Option Pricing Relationships European Option Pricing Relationships Binomial Option Pricing Model European Option Pricing Model Empirical Tests of Currency Option Models


9-4

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.


9-5

Futures Contracts: Preliminaries

Standardizing Features: Contract Size Delivery Month Daily resettlement

Initial Margin (about 4% of contract value, cash or T-bills held in a street name at your brokers).


9-6

Daily Resettlement: An Example

Suppose you want to speculate on a rise in the $/¥ exchange rate (specifically you think that the dollar will appreciate).

Currently $1 = ¥140. The 3-month forward price is $1=¥150.

Currency per U.S. $ equivalent U.S. $

Wed Tue Wed TueJapan (yen) 0.007142857 0.007194245 140 1391-month forward 0.006993007 0.007042254 143 1423-months forward 0.006666667 0.006711409 150 1496-months forward 0.00625 0.006289308 160 159


9-7


Currently $1 = ¥140 and it appears that the dollar is strengthening.

If you enter into a 3-month futures contract to sell ¥ at the rate of $1 = ¥150 you will make money if the yen depreciates. The contract size is ¥12,500,000

Your initial margin is 4% of the contract value:

¥1150

$10¥112,500,0.04 $3,333.33


9-8


If tomorrow, the futures rate closes at $1 = ¥149, then your position’s value drops.

Your original agreement was to sell ¥12,500,000 and receive $83,333.33

But now ¥12,500,000 is worth $83,892.62

¥1149

$100¥112,500,062.892,83$

You have lost $559.28 overnight.


9-9


The $559.28 comes out of your $3,333.33 margin account, leaving $2,774.05

This is short of the $3,355.70 required for a new position.

¥149

$10¥12,500,00.04 $3,355.70

Your broker will let you slide until you run through your maintenance margin. Then you must post additional funds or your position will be closed out. This is usually done with a reversing trade.


9-10

Daily Resettlement: Short Position *

Suppose you want to speculate on a rise in the $/¥ exchange rate (specifically you think that the dollar will appreciate).

Currently $1 = ¥140. The 3-month forward price is $1=¥150.

Currency per U.S. $ equivalent U.S. $

Wed Tue Wed TueJapan (yen) 0.007142857 0.007194245 140 1391-month forward 0.006993007 0.007042254 143 1423-months forward 0.006666667 0.006711409 150 1496-months forward 0.00625 0.006289308 160 159


9-11

What do we have ? Spot rate for yen:

$1= ¥ 140 (or, ¥1=$1/140) 3-month futures contract (¥12,500,000 each

contract) quotes at:$1= ¥150 (or, ¥1=$1/150)

If you expect that dollar will appreciate (that is, yen will depreciate), and wish to make profit from futures speculation, what will you do?

Daily Resettlement: An Example *Daily Resettlement: Short Position *


9-12

Daily Resettlement: An Example *

You will sell ¥ at the rate of $1= ¥150 today. This means that you agree to sell ¥12,500,000 and receive $83,333.33=¥12,500,000×($1/ ¥150) three months later.

You deposit your initial margin (4% of the contract value=$83,333.33×0.04=$3,333.33) to meet you margin requirement.

On the next day, the futures rate closes at $1=¥149 (¥ appreciates), what happens to your investment?

This means that if you want to close your position, you shall buy ¥ at $1=¥ 149. That is, you should pay $83,892.62=¥ 12,500,000×($1/¥ 149).

Daily Resettlement: Short Position *


9-13

Your profit will be $83,333.33-$83,892.62= -$559.28 if you close your position today.

This means that your “opportunity cost” is $559.28 if you do not really close your position. (Or, think it in another way, those who sell ¥ futures today can receive $83,892.62 while you can only receive $83,333.33)

This loss in value is than reflected in your margin account, with its new balance $2,774.05=$3,333.33-$559.28.

On the next day, if the settlement price is $1=¥ 146, and assume that the maintenance margin is $2,500, what happens then?



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Your profit will be ¥12,500,000×[($1/¥149)-($1/¥146)]=-1,723.82(note that by daily resettlement, your contract rate is renewed to $1=¥149 yesterday, not $1=¥150!)

Again, this loss in value will be reflected in your margin account, with its new balance $1,050.23=$2,774.05-$1,723.82.(This also equals to 3,333.33+12,500,000[(1/150)-(1/146)].)

Now, your margin account has its balance lower than the maintenance margin ($2,500). You should deposit money into your account until it meets the margin requirement which is now ¥12,500,000×($1/¥146) ×0.04=$3424.66.(You have to deposit $3,424.66-$1,723.82=$1,700.84 into your account.)



9-15

What happens if you expect that ¥ will appreciate? *

Other things being equal, how will you speculate using futures contract if you expect that ¥ will be appreciating, instead of depreciating?

You will buy ¥ at the rate of $1= ¥150 today. This means that you agree to pay $83,333.33=¥12,500,000×($1/ ¥150) to buy ¥12,500,000 three months later.

You deposit your initial margin (4% of the contract value=$83,333.33×0.04=$3,333.33) to meet you margin requirement.

On the next day, the futures rate closes at $1=¥149 (¥ appreciates, as you expected), what happens to your investment?

This means that if you want to close your position, you shall sell ¥ at $1=¥ 149. That is, you will receive $83,892.62=¥ 12,500,000×($1/¥ 149).

Daily Resettlement: Long Position *


9-16

Your profit will be $83,892.62-$83,333.33= $559.28 if you close your position today.

This means that your “opportunity profit” is $559.28 if you do not really close your position. (Or, think it in another way, those who buy ¥ futures today should pay $83,892.62 while you only have to pay $83,333.33)

This gain in value is than reflected in your margin account, with its new balance $3,892.61=$3,333.33+$559.28.

On the next day, if the settlement price is $1=¥146, and assume that the maintenance margin is $2,500, what happens then?

What happens if you expect that ¥ will appreciate? *Daily Resettlement: Long Position *


9-17

Your profit will be ¥12,500,000×[($1/¥146)-($1/¥149)]= $1,723.82(Again, by daily resettlement, your contract rate is renewed to $1=¥149 yesterday, not $1=¥150!)

This gain in value will be reflected into your margin account, with its new balance $5,616.43= $3,892.61 +$1,723.82.(This also equals to 3,333.33+12,500,000[(1/146)-(1/150)].)

Now, your margin account balance is still greater than the maintenance margin. You do not have to do anything to your margin account. Or, you can withdraw your profit as long as your margin account balance is still greater than its maintenance margin.

What happens if you expect that ¥ will appreciate? *Daily Resettlement: Long Position *


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What do we learn from this example? *

Comparing the gain or loss made by both cases, you will learn that as the textbook stated, futures trading between the long and the short is a zero-sum game. That is, the gain of the long is exactly the loss of the short, and vice versa.

Short position Long position

profit Margin account profit Margin account

Day 0($1=¥150)

$3,333.33 $3,333.33

Day 1($1=¥149)

-$559.28 $2774.05 $559.28 $3,892.61

Day 2($1=¥146)

-$1,723.82$1,050.23

(3424.66)$1,723.82 $5,616.43


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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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The Chicago Mercantile Exchange

Expiry cycle: March, June, September, December. Delivery date 3rd 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.


9-21

CME After Hours

Extended-hours trading on GLOBEX runs from 2:30 p.m. to 4:00 p.m dinner break and then back at it from 6:00 p.m. to 6:00 a.m. CST.

Singapore International Monetary Exchange (SIMEX) offer interchangeable contracts.

There’s other markets, but none are close to CME and SIMEX trading volume.


9-22

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.


9-23

Reading a Futures Quote Open Hi Lo Settle Change Lifetime

High Lifetime

Low Open

Interest

Sept .9282 .9325 .9276 .9309 +.0027 1.2085 .8636 74,639

Expiry monthOpening price

Highest price that day

Lowest price that dayClosing price

Daily ChangeHighest and lowest

prices over the lifetime of the

contract.

Number of open contracts


9-24

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 $1,000,000 90-day 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.


9-25

Reading Eurodollar Futures Quotes

EURODOLLAR (CME)—$1 million; pts of 100%

Open High Low Settle Chg YieldSettle Change

Open Interest

July 94.69 94.69 94.68 94.68 -.01 5.32 +.01 47,417

Eurodollar futures prices are stated as an index number of three-month LIBOR calculated as F = 100-LIBOR.

The closing price for the July contract is 94.68 thus the implied yield is 5.32 percent = 100 – 94.68

The change was .01 percent of $1 million representing $100 on an annual basis. Since it is a 3-month contract one basis point corresponds to a $25 price change.


9-26

Daily Resettlements: Long Position*

Suppose that, the July contract closes at 94.50 today. Thus, the implied yield is 5.50% (=100-94.5). This means that if you buy (long) one 90-day Eurodollar interest rate futures (contract size=$1 million), you are guaranteed to receive $13,750= $1,000,000×5.5%×(90/360) on your July investment.

On the next day, the July contract closes at 94.51 (one basis point more than yesterday), what happens to your margin account?

The implied yield now is 5.49% (100-94.51). This means that the buyer of the contract is guaranteed to receive $13,725= $1,000,000×5.49%×(90/36) on the same July investment.

You earn $25=$13,750-$13,725 more than the new investor. This “opportunity profit” (because you are guaranteed with higher interest rate) will be added into your margin account.


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Your profit is calculated by [5.5%-5.49%] ×$1,000,000×(90/360)=$25.

Note that this is can also be calculated by [(100-94.50)-(100-94.51)] ×1%×$1,000,000×(90/360).That is , your profit can also be calculated by[94.51-94.5] ×1%×$1,000,000×(90/360)=$25, as the usual formula calculating futures contracts’ payoff.

Then you can see why the interest rate futures contracts are quoted this way.

Note also that from this example, you shall see why one basis point change in quotation corresponds to a $25 price change.

Daily Resettlements: Long Position*


9-28

Daily Resettlements: Short Position*

Suppose that, the July contract closes at 94.50 today. Thus, the implied yield is 5.50% (=100-94.5). This means that if you sell (short) one 90-day Eurodollar interest rate futures (contract size=$1 million), you are guaranteed that you will have to pay $13,750= $1,000,000×5.5%×(90/360) on your July borrowing.

On the next day, the July contract closes at 94.51 (one basis point more than yesterday), what happens to your margin account?

The implied yield now is 5.49% (100-94.51). This means that the seller of the contract is guaranteed with interest cost of $13,725= $1,000,000×5.49%×(90/36) on the same July borrowing.

You loss $25=$13,750-$13,725 because you have to pay more interest payment on the same borrowing. This “opportunity cost” will be deducted from your margin account.


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Your profit is calculated by [5.49%-5.5%] ×$1,000,000×(90/360)= -$25.

Note that this is can also be calculated by [(100-94.51)-(100-94.50)] ×1%×$1,000,000×(90/360).That is , your profit can also be calculated by[94.50-94.51] ×1%×$1,000,000×(90/360)= -$25, as the usual formula calculating futures contracts’ payoff.

Now, it’s your term to think which position you will take if you expect that the 90-day LIBOR will be rising, and wish to speculate from interest rate futures.

Also, think what if you expect that the 90-day LIBOR is declining?

Daily Resettlements: Short Position*


9-30

Eurodollar Futures Hedge if You are an Investor *

You are a treasurer of a MNC, and expect to receive $2,000,000 on Sept. 17, 2003, because of your merchandise sale. You wish to invest these $2,000,000 in Eurodollar deposit because you know that the money will not be needed for a period of 90 days.

Today, the settlement price of the Sept03 interest rate futures is 94.00.

If you are uncertain about what interest rate you are going to receive, and wish to lock the return of your investment. How will you hedge against this interest rate risk? (You are worrying that the prevailing interest rate on that day will be very low.)


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You will buy (long) 2 Euro dollar interest rate futures contracts at 94.00. This means that you can lock your return from investment at 6% p.a. (that is, you lock your return from 90-day euro dollar deposit at $2,000,000×6%×(90/360)=$30,000.)

We will demonstrate that no matter what the realized interest rate at Sept. 17 2003 is, your return is not altered since you’ve hedged your interest rate risk.



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1. What happens if the 3-month LIBOR is 5% p.a. on Sept. 2003?a. You deposit your $2,000,000 at that realized interest rate 5% p.a., which gives you $2,000,000×5%×(1/4)=$25,000 interest payment.

b. You profit from your futures investment.The settlement price of your futures contract is 95 (=100-5). Your profit earned on the futures position is calculated as [95-94] ×(1/100) ×(1/4) ×$1,000,000×2=$5,000.

Your total return is then $25,000+$5,000=$30,000.



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2. What happens if the 3-month LIBOR is 7% p.a. on Sept. 2003?a. You deposit your $2,000,000 at that realized interest rate 7% p.a., which gives you $2,000,000×7%×(1/4)=$35,000 interest payment.

b. You lose from your futures investment.The settlement price of your futures contract is 93 (=100-7). Your profit earned on the futures position is calculated as [93-94] ×(1/100) ×(1/4) ×$1,000,000×2=-$5,000.

Your total return is then $35,000-$5,000=$30,000.



9-34

Eurodollar Futures Hedge if You are a Borrower *

You are a treasurer of a MNC, and expect to borrow $2,000,000 for 90 days on Sept. 17, 2003, because of your imports on that day..

Today, the settlement price of the Sept03 interest rate futures is 94.00.

If you are uncertain about what your interest cost will be, and wish to lock the cost of your loan. How will you hedge against this interest rate risk? (You are worrying that the prevailing interest rate on that day will be very high.)


9-35

You will sell (short) 2 Euro dollar interest rate futures contracts at 94.00. This means that you can lock your cost of borrowing at 6% p.a. (that is, you lock your interest cost at $2,000,000×6%×(90/360)=$30,000.)

We will demonstrate that no matter what the realized interest rate at Sept. 17 2003 is, your interest cost is not altered since you’ve hedged your interest rate risk.



9-36

1. What happens if the 3-month LIBOR is 5% p.a. on Sept. 2003?a. You borrow $2,000,000 at that realized interest rate 5% p.a., which yields $2,000,000×5%×(1/4)=$25,000 interest cost.

b. You lose from your futures investment.The settlement price of your futures contract is 95 (=100-5). Your profit earned on the futures position is calculated as [94-95] ×(1/100) ×(1/4) ×$1,000,000×2=-$5,000.

Your total cost is then $25,000+$5,000=$30,000.



9-37

2. What happens if the 3-month LIBOR is 7% p.a. on Sept. 2003?a. You borrow $2,000,000 at that realized interest rate 7% p.a., which yields $2,000,000×7%×(1/4)=$35,000 interest cost.

b. You profit from your futures investment.The settlement price of your futures contract is 93 (=100-7). Your profit earned on the futures position is calculated as [94-93] ×(1/100) ×(1/4) ×$1,000,000×2=$5,000.

Your total cost is then $35,000-$5,000=$30,000.



9-38

Options Contracts: Preliminaries

An option gives the holder the right, but not the obligation, to buy (call) or sell (put) a given quantity (contract size) of an asset (underlying asset) in the future (expiration date), at prices (strike price) agreed upon today.

The price specified in the contract is called the strike or exercise price. The specified maturity date is called the expiration date. The price paid for the option is called the option premium.


9-39


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 early generally has value,

American options are usually worth more than European options, other things equal.


9-40

Types of Contracts

Examples An American call option on spot € :

The right to buy € 1 million for $1.10 per € from today until expiration on Dec 15, 2001.

This “call on € ” is also a “put on US$”.

A European put option on Swiss franc futures : The right to sell SFr 10 million March 2002 futures for

$0.65 per SFr on (and only on) Mar 15, 2002. This “put on SFr” is also a “call on US$”.


9-41


In-the-money (it’s worthwhile exercising your option) Call: ST>E

Put: ST<E

At-the-money (you’re indifferent whether you exercise the option) Call: ST=E

Put: ST=E

Out-of-the-money (you will not exercise your option) Call: ST<E

Put: ST>E


9-42

Currency Options

Currency options began trading on the Philadelphia Stock Exchange (PHLX) in 1982.

Since then, the markets have expanded : more option exchanges around the world, more currencies and debt instruments on which options

are traded, option contracts with longer maturities, more “styles” of option contracts, and greater volume of trading activity.


9-43

PHLX Currency Option Specifications

Currency Contract SizeAustralian dollar AD50,000British pound £31,250Canadian dollar CD50,000Deutsche mark DM62,500French franc FF250,000Japanese yen ¥6,250,000Swiss franc SF62,500Euro 62,500


9-44

Location and Scale of Trading

Currency options are traded by banks on an over-the-counter (OTC) basis and on organized futures and options exchanges.

According to surveys conducted by the Bank for International Settlements, the volume of trading in terms of billions per day is:

OTC Organized Exchanges 1995 1998 1995 1998

Currency Options $41.0 $87.1 $3.8 $1.8


9-45

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.


9-46

Basic Option Pricing Relationships at Expiry

At expiry, an American call option is worth the same as a European option with the same characteristics.

If the call is in-the-money, it is worth ST – E. If the call is out-of-the-money, it is worthless.

CaT = CeT = Max[ST - E, 0]


9-47

Basic Option Pricing Relationships at Expiry

At expiry, an American put option is worth the same as a European option with the same characteristics.

If the put is in-the-money, it is worth E - ST. If the put is out-of-the-money, it is worthless.

PaT = PeT = Max[E - ST, 0]


9-48

Basic Option Profit Profiles


profit

loss

E E+CST

Long 1 call


9-49



profit

loss

EE+C

STshort 1 call


9-50


PaT = PeT = Max[E - ST, 0]

profit

loss

EE - p

ST

long 1 put


9-51



profit

loss

EST

Short 1 put

E - p


9-52

American Option Pricing Relationships

With an American option, you can do everything that you can do with a European option—this option to exercise early has value.

CaT > CeT = Max[ST - E, 0]

PaT > PeT = Max[E - ST, 0]


9-53

Market Value, Time Value and Intrinsic Value for an American Call

CaT > Max[ST - E, 0]

Profit

loss

E ST

Market Value

Intrinsic value

S T - E

Time value

Out-of-the-money In-the-money


9-54

European Option Pricing Relationships

Consider two investments

1 Buy a call option on the British pound futures contract. The cash flow today is -Ce

2 Replicate the upside payoff of the call by 1 Borrowing the present value of the exercise price of

the call in the U.S. at i$ The cash flow today is E /(1 + i$)

2 Lending the present value of ST at i£ The cash flow is - ST /(1 + i£)


9-55


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 the borrowing and lending strategy.

Thus:

0,)1()1(

max$£

i

E

i

SC T

e


9-56


Using a similar portfolio to replicate the upside potential of a put, we can show that:

0,)1()1(

max£$

i

S

i

EP T

e


9-57

Option Pricing with Risk Neutrality: Call*

Notations:Ct: the price of the call at time tCT: the call option’s payoff at expiration date TST: the spot exchange rate of the underlying asset at TX: the strike price of the underlying assetrf: the risk-free rate

The value of the option at time t (assuming risk neutral) is : Ct=E(CT)/(1+rf)


9-58

Example: Assume the underlying asset is $/ ￡ exchange rate, and the spot exchange rate at T has a uniform distribution between $1/ ￡ and $2/ ￡ . Also assume X=$1.5/ ￡ ; rf between t and T is 5%.Then, what’s the value of this call option at time t?


1.5 21

Prob.

ST


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1.5 21

Prob.

ST

AB

If ST falls in A, ST>XCT=ST-1.5

If ST falls in B, ST<XCT=0

E(CT)= {the expected payoff given ST>X} ×Prob. {ST>X} + {the expected payoff given ST<X} ×Prob. {ST<X} =E[ST-1.5|given ST>1.5] ×0.5+0×0.5

=0.25*0.5=0.125


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Then we know that Ct=E(CT)/(1+rf)=0.125/1.05=0.1190

Other things being equal, what the value of the option will be if there is another call with higher strike price (say, X=$1.8/ ￡ )?

1.8 21

Prob.

ST

AB


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If ST falls in A, ST>XCT=ST-1.8

If ST falls in B, ST<XCT=0

E(CT)= {the expected payoff given ST>X} ×Prob. {ST>X} + {the expected payoff given ST<X} ×Prob. {ST<X} =E[ST-1.8|given ST>1.8] ×0.2+0×0.8 =0.1*0.2=0.02

Ct=E(CT)/(1+rf)=0.02/1.05=0.0190<0.1190 Then we know that a call with higher exercise price would be less

valuable. Why?1. It has lower possibility to be “in-the-money”.2. Even if it is in-the-money, its expected payoff is lower.


9-62


What if the expected volatility of the ST becomes larger? (say, if ST is distributed between 0.5 and 2.5?)

ST1.5 21

Prob.1

0.5

0.5 2.5


9-63


If ST falls in A, ST>XCT=ST-1.5 If ST falls in B, ST<XCT=0 E(CT)= {the expected payoff given ST>X} ×Prob. {ST>X}

+ {the expected payoff given ST<X} ×Prob. {ST<X} =E[ST-1.5|given ST>1.5] ×0.5+0×0.5

=0.5*0.5=0.25

1.5

Prob.

0.5

0.5 2.5

AB


9-64


Then we know that Ct=E(CT)/(1+rf)=0.25/1.05=0.2381>0.1190 When the variance of the expected spot exchange rate at T becomes

larger, the call is more valuable. Why?

* the increase in variance leads to the possibility of both higher and lower ST.

* the higher price causes expected payoff to be higher, but the lower price does not drag down the expected payoff.

* The increase in the variance of the spot exchange rate increases the upside potential for the investment, while leaving the downside potential unaffected.

* Or, loosely speaking, when the volatility is higher, people need the option to hedge their risk more eagerly. Thus led to a higher call value.


9-65

Option Pricing with Risk Neutrality: Put*

Notations:Pt: the price of the call at time tPT: the call option’s payoff at expiration date TPT: the spot exchange rate of the underlying asset at TX: the strike price of the underlying assetrf: the risk-free rate

The value of the option at time t (assuming risk neutral) is : Pt=E(PT)/(1+rf)


9-66

Example: Assume the underlying asset is $/ ￡ exchange rate, and the spot exchange rate at T has a uniform distribution between $1/ ￡ and $2/ ￡ . Also assume X=$1.5/ ￡ ; rf between t and T is 5%.Then, what’s the value of this put option at time t?


1.5 21

Prob.

ST


9-67


1.5 21

Prob.

ST

AB

If ST falls in A, ST>XPT=0

If ST falls in B, ST<XPT=1.5-ST

E(PT)= {the expected payoff given ST>X} ×Prob. {ST>X} + {the expected payoff given ST<X} ×Prob. {ST<X} =0×0.5+E[1.5-ST|given ST<1.5] ×0.5

=0.25*0.5=0.125


9-68


Then we know that Pt=E(PT)/(1+rf)=0.125/1.05=0.1190

Other things being equal, what the value of the option will be if there is another put with higher strike price (say, X=$1.8/ ￡ )?

1.8 21

Prob.

ST

AB


9-69


If ST falls in A, ST>XCT=0

If ST falls in B, ST<XCT=1.8-ST

E(PT)= {the expected payoff given ST>X} ×Prob. {ST>X} + {the expected payoff given ST<X} ×Prob. {ST<X} =0×0.2+E[1.8-ST|given ST<1.8] ×0.8 =0.4×0.8=0.32Pt=E(PT)/(1+rf)=0.32/1.05=0.3048>0.1190Then we know that a call with higher exercise price would be more valuable. Why?1. It has higher possibility to be “in-the-money”.2. When it is in-the-money, its expected payoff is greater.


9-70


What if the expected volatility of the ST becomes larger? (say, if ST is distributed between 0.5 and 2.5?)

ST1.5 21

Prob.1

0.5

0.5 2.5


9-71


If ST falls in A, PT>XCT=0If ST falls in B, ST<XCT=1.5-STE(PT)= {the expected payoff given ST>X} ×Prob. {ST>X} + {the expected payoff given ST<X} ×Prob. {ST<X} =0×0.5+E[1.5-ST|given ST<1.5] ×0.5 =0.5*0.5=0.25

1.5

Prob.

0.5

0.5 2.5

AB


9-72


Then we know that Pt=E(PT)/(1+rf)=0.25/1.05=0.2381>0.1190 When the variance of the expected spot exchange rate at T becomes

larger, the put is more valuable. Why?

* the increase in variance leads to the possibility of both higher and lower ST.

* the lower price causes expected payoff to be higher, but the higher price does not drag down the expected payoff.

* The increase in the variance of the spot exchange rate increases the upside potential for the investment, while leaving the downside potential unaffected.

* Or, loosely speaking, when the volatility is higher, people need the option to hedge their risk more eagerly. Thus led to a higher option value ( either call or put).


9-73

Binomial Option Pricing Model

Imagine a simple world where the dollar-euro exchange rate is S0($/ ) = $1 today and in the next year, S1($/ ) is either $1.1 or $.90.

$1

$.90

$1.10

S0($/ ) S1($/ )


9-74


$1

$.90

$1.10

S0($/ ) S1($/ )

$.10

$0

C1($/ )

A call option on the euro with exercise price S0($/ ) = $1 will have the following payoffs.


9-75

$1

$.90

$1.10

S0($/ ) S1($/ )

$.10

$0

C1($/ )


We can replicate the payoffs of the call option. With a levered position in the euro.


9-76

$1

$.90

$1.10

S0($/ ) S1($/ )

$.10

$0

C1($/ )


debt

-$.90

-$.90

portfolio

$.20

$.00

Borrow the present value of $.90 today and buy 1. Your net payoff in one period is either $.2 or $0.


9-77


$1

$.90

$1.10

S0($/ ) S1($/ )

$.10

$0

C1($/ )debt

-$.90

-$.90

portfolio

$.20

$.00

The portfolio has twice the option’s payoff so the portfolio is worth twice the call option value.


9-78

$1

$.90

$1.10

S0($/ ) S1($/ )

$.10

$0

C1($/ )debt

-$.90

-$.90

portfolio

$.20

$.00


The portfolio value today is today’s value of one euro less the present value of a $.90 debt:

)1(

90$.1$

$i


9-79


$1

$.90

$1.10

S0($/ ) S1($/ )

$.10

$0

C1($/ )debt

-$.90

-$.90

portfolio

$.20

$.00

We can value the option as half of the value of the portfolio:

)1(

90$.1$

2

1

$0 i

C


9-80


The most important lesson from the binomial option pricing model is:

the replicating portfolio intuition.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.


9-81

Binominal Option Pricing Model*

We have 3 ways to price an option, and they should yield the same pricing formula. These 3 methods are:1. riskless hedge2. replication method3. risk-neutral pricing


9-82

Binominal Option Pricing Model-Assumptions and Notations*

The spot rate begins at S0 and either rises to u S0 or falls to d S0 in the next period. (u>d)

S0

Period 0

S1,u = u S0

S1,d = d S0

Period 1


9-83


Also assume that the forward rate (expires at period 1) of the underlying asset is F. If there is a call with strike price E, its value can be represented as:

C0

Period 0

CuT = Max{u S0-E,0}

CdT = Max{d S0-E,0}

Period 1


9-84


Also assume that the interest rate of the domestic currency is r.

Our main goal is to determine what C0 should be, using the 3 methods we are going to introduce.


9-85

Binominal Option Pricing Model- riskless hedge*

The main idea is to construct a “riskless portfolio” that yields a constant payoff, regardless of which value S1 is going to be.

We can long a call, and short h futures (or forward) contracts. (why short? Because thus your futures position’s payoff is opposite to that of the call!)

h is called the hedge ratio, it is the size of the short (or long) position the investor must have in the underlying asset per option to yield a risk-free portfolio.


9-86


If we form a portfolio that longs one call and shorts h futures, its payoff at period 1 will be:

Period 0CuT +(F-u S0) h=A

CdT +(F-d S0) h=B

Period 1

•Since it is “riskless”, A must be equal to B.

•Thus we can determine what h should be to make A=B.


9-87


)(

)()(

0

00

duS

CCh

BhdSFChuSFCA

dTuT

dTuT

•Substituting it into the period 1 payoff, we are sure that this portfolio’s payoff will be:

)(

)()(

)()(

0

00

00

duS

CuSFCdSF

duS

CCuSFCBA

dTuT

dTuTuT


9-88


Since it is a certain profit, to rule out arbitrage opportunity, it should yield the same return as the riskless bond. That is:

)()1/()(

)()(

)1/(

1

0

00

0

0

arduS

CuSFCdSF

rAC

rC

A

dTuT


9-89

Digression: What will happen if (a) does not hold?

A riskless arbitrage opportunity exists. We will demonstrate how the arbitrage activities would

drive prices back to their equilibrium levels. That is, we’ll show that arbitrage makes (a) hold with equality.

Case I: if A/C0>(1+r)The return (cost) of your riskless portflolio is greater than that of buying (selling) the riskless bond.

How will you profit from this? (i.e. what is your arbitrage strategy?)


9-90


In period 0, you can borrow C0 and use the very money you borrowed to buy a call and short h futures simultaneously.(Note that on doing this, your period-0 cash flow is zero! )

In period 1, you can earn a certain profit of A from your riskless portfolio. And, at the same time, you have to repay your loan by paying the bond holder C0(1+r).

Since A>C0(1+r), your period-1 cash flow is A-C0(1+r)>0. You earn a certain, riskless profit with zero period 0 investment!


9-91


What happens then if all the astute traders like you try to arbitrage from this?

Look at the period-0 strategy:1. You first borrow from the bond market r goes up.2. You then use the money you borrowed to long a call call price C0 goes up.3. You also short h futures simultaneously F goes down and thus A goes down.

Now you see 3 forces that drive prices back to the levels that make (a) holds with equality!


9-92


Case II: if A/(C0)<(1+r)The return (cost) of the riskless portfolio is smaller than that of buying (selling) the riskless bond.

How will you profit from this? (i.e. what is your arbitrage strategy?)


9-93


In period 0, you can short a call and long h futures to form your riskless portfolio, and invest C0 from writing the call.(Note that on doing this, your period-0 cash flow is zero! )

In period 1, you will receive C0(1+r) from your bond holding. And, at the same time, your riskless portfolio is making “profit” of -A.

Since A<C0(1+r), your period-1 cash flow is C0(1+r)-A>0. You earn a certain, riskless profit with zero period 0 investment!


9-94


What happens then if all the astute traders like you try to arbitrage from this?

Look at the period-0 strategy:1. You first short a call call price C0 goes down.2. You also long h futures simultaneously F goes up and thus A goes up.3. You then deposit C0 in the bond market r goes down.

Now you see 3 forces that drive prices back to the levels that make (a) holds with equality!


9-95

Binominal Option Pricing Model- replication method*

The main idea is to construct a portfolio that will yield exactly the same period 1 payoff as a call. That is, we construct a portfolio that replicates a call’s payoff.

Since they have the same payoff, to rule out arbitrage opportunity, their cost at period 0 should also be equal.

Thus, we can use this portfolio with known period 0 cost to get the value of the call, C0.


9-96


If our goal is to replicate the payoff of a long position of call. We can invest B in bond holding, and long forward contracts.

The first problem facing us is to decide what B and should be. Then we can use this replicating portfolio to price the call.


9-97


This portfolio have the following pattern of payoff :

B

Period 0

B(1+r) + (u S0-F)=G

B(1+r)+ (d S0-F)=H

Period 1

•If this portfolio replicates the call’s payoff, we must have G=CuT, and H=CdT.

•Using these two relationships, we can solve for B and .


9-98


0

0

00

)(

)1/()(

)()(

)()1(

)()1(

Sdu

CC

rSdu

CuSFCdSFB

CFdSrB

CFuSrB

dTuT

dTuT

dTo

uTo


9-99


We can summerize the cash flows as what follows:

Domestic deposit (B) -B B(1+r) B(1+r)

long forwards 0 (u S0-F) (d S0-

F)

B + F -B G H

Call option purchase -C0 CuT CdT

TransactionCash Flows

Period 0Cash Flows in Period 1

if S1 = uS0 if S1 = dS0

• Since we’ve selected B and so that G=CuT, H=CdT, the period one cash flows is exactly the same. The period 0 cash flow should also be the same (or you will have a riskless arbitrage opportunity).C0=B


9-100


Now we’ve priced the call using replication method:

)1/()(

)()(

0

000 r

Sdu

CuSFCdSFBC dTuT

•Note that this is the very pricing formula we’ve derived using riskless hedge method!


9-101

Binominal Option Pricing Model- risk-neutral pricing*

The main idea is to calculate the “risk-neutral probability” that makes the expected payoff of the period 1 payoff equal to its riskless price ~F.

Then we can use this risk-neutral probability to calculate the expected payoff of the call.

Discounting this expected payoff of the call, we than get what C0 should be.

To start, assume that there is a artificial probability q, named risk-neutral probability.


9-102


Assume that S1 goes to uS0 with probability q, dS0 with prob. 1-q.

S0

Period 0S1,u = u S0

S1,d = d S0

Period 1q

1-q

• Our first step is to find q that makes S1 behave as riskless. That is, Eq(S1)=F.


9-103


0

0

00

)(

*)1(*

Sdu

dSFq

FdSquSq

•Since this probability makes S behaves as riskless (its expected payoff is the forward exchange rate that we’ve known today.), we can use this probability to make the call “riskless” .Then we only have to discount this expected payoff to period 0, and get the price of the call, C0.


9-104


r

CqCqr

CEC

dTuT

Tq

1

*)1(*1

)(0

)1/()(

1)( 0

0

0

00 rC

Sdu

dSFC

Sdu

dSFC dTuT

•Substitute the q we’ve derived into it, we than get exactly the same pricing formula as the previous two methods!

)1/()(

)()(

0

00 rSdu

CuSFCdSF dTuT


9-105

An Example

Now, you should be able to calculate C0 using the example in the textbook.

Suppose there is a September SF call with strike price E=$0.67/SF. The spot exchange is $63.86/SF. The interest rate from now to expiry date is 0.00908.

Also assume the September futures price now is F=$0.6433/SF.


9-106

An Example

The spot exchange rate behaves in the following way:

63.86

Period 0S1,u = u S0=67.78

S1,d = d S0=60.17

Period 1

•The call’s payoffs are thus:

C0

Period 0CuT = 67.78-67=0.78

CdT = 0

Period 1


9-107

An Example-riskless hedge*

To form a riskless portfolio that longs one call and short h futures. This portfolio has the following payoff pattern:

Making A=B, we must long h=0.1025 forward contracts: h=(.78-0)/(67.78-60.17)=0.1025

Thus A=B=0.78+(64.33-67.78)0.1025=0.4264 C0=A/(1+r)=0.4264/1.00908=0.42 cents per SF.

Period 0CuT +(F-u S0) h=A=0.78+(64.33-67.78)h

CdT +(F-d S0) h=B=0+(64.33-60.17)h

Period 1


9-108

An Example-replication method

If our goal is to replicate the payoff of a long position of call. We can invest B in bond holding, and long forward contracts.

This portfolio has the following payoff pattern:

B

Period 0

B(1+r) + (u S0-F)=G=1.00908B+ 3.45

B(1+r)+ (d S0-F)=H=1.00908B-4.16

Period 1


9-109


To replicate the call’s payoff, we must have G=CuT=0.78, H=CdT=0.

Solve for B and , we know that we have to deposit B=0.42 cents, and long 0.1025 futures contracts.

Domestic deposit (B) -0.42 0.43 0.43

long forwards 0 0.35 -0.43

B + F -0.42 0.78 0

Call option purchase -C0 0.78 0

TransactionCash Flows

Period 0Cash Flows in Period 1

if S1 = uS0 if S1 = dS0


9-110


• Since we’ve selected B and so that the period one cash flows is exactly the same. The period 0 cash flow should also be the same (or you will have a riskless arbitrage opportunity).C0=B=0.4222


9-111

An Example- risk-neutral pricing

63.86

Period 0S1,u = u S0=67.78

S1,d = d S0=60.17

Period 1q

1-q

• Risk-neutral probability makes q*67.88+(1-q)60.17=F=64.33.

•Solve for q, we have the risk-neutral probability q=0.5466


9-112

An Example- risk-neutral pricing

Using this probability, we can calculate the expected payoff of the call, and discount it to get the call price:

C0=[q*CuT+(1-q)CdT]/(1+r) =[.5466(0.78)+(1-.5466)(0)]/1.00908 =0.42 cents per SF


9-113

Pricing Spot Currency OptionsThe Discrete Time Binomial Model

A put option can be priced similarly. The two-period model can be extended to a multiperiod

setting:


9-114

European Option Pricing Formula

We can use the replicating portfolio intuition developed in the binomial option pricing formula to generate a faster-to-use model that addresses a much more realistic world.

Using the above-mentioned idea, you generalize it to the standard Black-Scholes options pricing formula.


9-115


The model isTredNEdNFC $)]()([ 210

Where

C0 = the value of a European option at time t = 0Trr

teSF )( £$ r$ = the interest rate available in the U.S.

r£ = the interest rate available in the foreign country—in this case the U.K.

,5.)/ln( 2

1T

TEFd

Tdd 12


9-116

European Option Pricing FormulaFind the value of a six-month call option on the British pound with an exercise price of $1.50 = £1The current value of a pound is $1.60

The interest rate available in the U.S. is r$ = 5%.

The interest rate in the U.K. is r£ = 7%.The option maturity is 6 months (half of a year).The volatility of the $/£ exchange rate is 30% p.a.Before we start, note that the intrinsic value of the option is $.10—our answer must be at least that.


9-117


Let’s try our hand at using the model. If you have a calculator handy, follow along.

Then, calculate d1 and d2

106066.05.4.

5.)4.0(5.)50.1/485075.1ln(5.)/ln( 22

1

T

TEFd

First calculate

485075.150.1 50.0)07.05(.)( £$ eeSF Trrt

176878.05.4.106066.012 Tdd


9-118


N(d1) = N(0.106066) = .5422

N(d2) = N(-0.1768) = 0.4298TredNEdNFC $)]()([ 210

157.0$]4298.50.15422.485075.1[ 5.*05.0 eC

485075.1F

106066.01 d

176878.02 d


9-119

Option Value Determinants

Call Put1. Exchange rate + –2. Exercise price – +3. Interest rate in U.S. + –4. Interest rate in other country + –5. Variability in exchange rate + +6. Expiration date + +

The value of a call option C0 must fall within

max (S0 – E, 0) < C0 < S0.

The precise position will depend on the above factors.


9-120

Combining put and call to beat market volatility

Portfolios of put and call options are called “strategies”.

Straddle: a particular kind of “strategies” that combines a put and a call at the same strike price.

Practically, we can use straddle to beat market volatility.


9-121


+C+P

profit

loss

E

ST

+C+P

+C+P

UD


9-122


If ST>U: gain profit from the callIf ST<D: gain profit from the putIf D<ST<U: loss money

So, if you think the market volatility will be very large, you can invest in this kind of straddle to profit from market volatility.


9-123


-C-P

profit

loss

E ST

-C-P

-C-P

UD


9-124


If ST>U: loss from the callIf ST<D: loss from the putIf D<ST<U: gain profit

So, if you think the market volatility will be very small, you can invest in this kind of straddle.


9-125

Empirical Tests

The European option pricing model works fairly well in pricing American currency options.

It works best for out-of-the-money and at-the-money options.

When options are in-the-money, the European option pricing model tends to underprice American options.


9-126

End Chapter Nine

international financial management

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