Chapter Objective:

Chapter Objective:

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

7Chapter Seven

Futures and Options on Foreign Exchange

7-1

Primary vs. Derivative Products

Primary Financial Products: their values are determined by their own cash flows. E.g., stocks, bonds, currencies, (real, financial, and artificial) commodities, etc

Derivative Products (Derivatives, Contingent Claims): their values are derived from the value of the underlying primary security. E.g., forward, futures, options, swaps, insurance products, etc.

(Currency) Futures Contracts: Preliminaries

A futures contract is like a forward contract: It specifies that a certain currency will be

exchanged for another at a specified time in the future at prices specified today.

A futures contract is different from a forward contract: Futures are standardized contracts trading on

organized exchanges with daily resettlement through a clearinghouse.

7-3

Futures Contracts: Preliminaries

Standardizing Features: CFTC Contract size ~ originally determined around $100k Delivery months ~ 3, 6, 9, 12 Daily settlement (mark to market) Trading costs ~ commissions for a round trip A daily price limit

Initial performance bond (=initial margin, about 2 percent of contract value, cash or T-bills held in a street name at your brokerage).

7-4

Daily Settlement: An Example

Consider a long position in the CME EUR/USD contract.

It is written on €125,000 and quoted in $ per €. The strike price is $1.30 the maturity is 3

months. At initiation of the contract, the long posts an

initial performance bond of $1,350. The maintenance performance bond is $1,000.

7-5


Recall that an investor with a long position gains from increases in the price of the underlying asset.

Our investor has agreed to BUY €125,000 at $1.30 per euro in three months time.

With a forward contract, at the end of three months, if the euro was worth $1.24, he would lose $7,500 = ($1.24 – $1.30) × 125,000.

If instead at maturity the euro was worth $1.35, the counterparty to his forward contract would pay him $6,250 = ($1.35 – $1.30) × 125,000.

7-6

Daily Settlement: An Example With futures, we have daily settlement of gains an

losses rather than one big settlement at maturity. Every trading day:

if the price goes down, the long pays the short if the price goes up, the short pays the long => A zero-sum game!

After the daily settlement, each party has a new contract at the new price with one-day-shorter maturity.

7-7

Performance Bond Money

Each day’s losses are subtracted from the investor’s account.

Each day’s gains are added to the account. In this example, at initiation the long posts an

initial performance bond of $1,350. The maintenance level is $1,000.

If this investor loses more than $350 he has a decision to make: he can maintain his long position only by adding more funds—if he fails to do so, his position will be closed out with an offsetting short position.

7-8


Over the first 3 days, the euro strengthens then depreciates in dollar terms (with $1,500 initial balance):

$1,250

–$1,250

$1.31

$1.30

$1.29 –$1,250

Gain/LossSettle

= ($1.31 – $1.30)×125,000$2,750

$1,500 = $1,750 - $1,250

$250

Account Balance

= $1,500 + $1,250

On third day suppose our investor keeps his long position open by posting an additional $1,100 at minimum to achieve the initial margin requirement of $1,350. Otherwise, his account will be closed out with $250 left for him. A total cost of $1,250 from $1,500 initial balance.

= $1,500 - $1,250

7-9

Toting Up

At the end of the 3rd day, our investor has three ways of computing his gains and losses:Sum of daily gains and losses

– $1,250 = $1,250 – $1,250 – $1,250 Contract size times the difference between initial contract price and last settlement price.

– $1,250 = ($1.29/€ – $1.30/€) × €125,000Ending balance on account minus beginning balance on account, adjusted for deposits or withdrawals.

– $1, 250 = $250 - $1,500

7-10

Currency Futures Markets

The Chicago Mercantile Exchange (CME) is by far the largest.

Others include: The Philadelphia Board of Trade (PBOT) The MidAmerica Commodities Exchange The Tokyo International Financial Futures Exchange The London International Financial Futures Exchange

7-11

The Chicago Mercantile Exchange

Expiry cycle: March, June, September, December.

Delivery date third Wednesday of delivery month.

Last trading day is the second business day preceding the delivery day.

CME hours 7:20 a.m. to 2:00 p.m. CST.

7-12

CME After Hours

Extended-hours trading on GLOBEX runs from 17:00 p.m. to 16:00 p.m CST.

The Singapore Exchange (SIMEX) offers interchangeable contracts.

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

7-13

Reading Currency Futures Quotes, 2/10/11

OPEN HIGH LOW SETTLE CHGOPEN

INT

Euro/US Dollar (CME)—€125,000; $ per €1.4748 1.4830 1.4700 1.4777 .0028Mar 172,396

1.4737 1.4818 1.4693 1.4763 .0025Jun 2,266

Highest price that dayLowest price that day

Closing price?

Daily Change

Number of open contracts

Expiry month

Opening price

7-14

Basic Currency Futures Relationships

Open Interest refers to the number of contracts outstanding for a particular delivery month.

Open interest is a good proxy for demand for a contract.

Some refer to open interest as the depth of the market. The breadth of the market would be how many different contracts (expiry month, currency) are outstanding.

7-15


Notice that open interest is greatest in the nearby contract, in this case March, 2011.

In general, open interest typically decreases with term to maturity of most futures contracts.


INT


1.4737 1.4818 1.4693 1.4763 .0025Jun 2,266

7-16


1 + i€

1 + i$ F

So=

Recall from chapter 6, our interest rate parity condition (arbitrage possibility if there is a sizable disparity):


INT


1.4737 1.4818 1.4693 1.4763 .0025Jun 2,266

7-17

Reading Currency Futures Quotes, 2/10

From March to June 2011, we should expect lower interest rates in dollar denominated accounts: if we find a higher rate in a euro denominated account, we may have found an arbitrage.


INT


1.4737 1.4818 1.4693 1.4763 .0025Jun 2,266

7-18

Eurodollar Interest Rate Futures Contracts

Widely used futures contract for hedging short-term U.S. dollar interest rate risk.

The underlying asset is a hypothetical $1,000,000 90-day Eurodollar deposit—the contract is cash settled.

7-19

Reading Eurodollar Futures Quotes, 2/10/11

Eurodollar futures prices are stated as an index number of three-month LIBOR calculated as F = 100 - LIBOR. The implied yield 3.44% (=100 – 96.56)

If you to secure 3.44% (APR) at minimum for a 3 month investment starting June of this year, you want to take a long position to avoid the F going high (or avoid LIBOR getting low).

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


INTYLD CHG

Eurodollar (CME)—1,000,000; pts of 100%96.56 96.58 96.55 96.56 - 3.44 -Jun 1,398,959

7-20

Trading irregularities

Futures Markets are also a great place to launder money The zero sum nature of futures is the key to

laundering the money.

7-21

James B. Blair, Outside Counsel to Tyson Foods Inc., Arkansas' largest employer, gets Hillary’s discretionary order.

Robert L. "Red" Bone, (Refco broker), allocates trades ex post facto.

Submits identical long and short trades

winners

losers

Money Laundering: Hillary Clinton’s Cattle Futures

7-22

Options Contracts: Preliminaries

An option gives the holder the right, but not the obligation, to buy (Call) or sell (Put) a given quantity of an asset in the future, at prices agreed upon today.

A real-life example of a call option is a rain check. A real-life example of a put option is Ray Lewis contract (Ray is selling his service to the Ravens).

7-23


European vs. American options European options can only be exercised on the

expiration date. American options can be exercised at any time up to

and including the expiration date. Since this option to exercise an option early is more

valuable (greater flexibility), American options are worth more than European options.

7-24


In-the-money It is profitable to exercise the option.

At-the-money Indifferent

Out-of-the-money It is not profitable to exercise.

7-25


Intrinsic Value (price on the rain check vs future spot price => savings!) The difference between the exercise price of the option and the

future spot price of the underlying asset.

Time Value (pay more than the intrinsic value?) The difference between the option premium and the intrinsic value of

the option (this time value is different from TVM).

Option Premium

=Intrinsic Value

Time Value+

7-26

Currency Options Markets

PHLX (Philadelphia Stock Exchange) OTC volume is much bigger than exchange

volume. Trading is in six major currencies against the

U.S. dollar.

7-27

PHLX Currency Option Specifications

Currency Contract Size

Australian dollar AD10,000

British pound £10,000

Canadian dollar CAD10,000

Euro €10,000

Japanese yen ¥1,000,000

Swiss franc SF10,000

http://www.phlx.com/products/xdc_specs.htm

7-28

http://www.phlx.com/products/

Call Option Pricing Relationships at Expiry

(Call) option holder has two prices to choose from, ST or E.

Since a call is used to buy, you look for a lower price to buy. If ST > E (in the money), then Exercise. If ST < E (out of the money), then Do Not Exercise.

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

CT = Max[ST - E, 0]

7-29

Put Option Pricing Relationships at Expiry

(Put) option holder has two prices to choose from, ST or E.

Since a put is used to sell, you look for a higher price to sell. If ST < E (in the money), then Exercise. If ST > E (out of the money), then Do Not Exercise.

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

PT = Max[E – ST, 0]

7-30

Basic Option Profit Profiles

E

ST

Profit

loss

–c0 E + c0

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. In-the-moneyOut-of-the-money

Owner of the call

7-31


E

ST

Profit

loss

c0

E + c0

short 1 call

If the call is in-the-money, the writer loses ST – E.

If the call is out-of-the-money, the writer keeps the option premium.

Seller of the call

7-32


E

ST

Profit

loss

– p0

E – p0long 1 put

E – p0

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.

Out-of-the-moneyIn-the-money

Owner of the put

7-33


E

ST

Profit

loss

p0

E – p0short 1 put

– E + p0

If the put is in-the-money, it is worth E –ST. The maximum loss is – E + p0

If the put is out-of-the-money, it is worthless and the seller of the put keeps the option premium of p0.

Seller of the put

7-34

Example

$1.50

ST

Profit

loss

–$0.25

$1.75

Long 1 call on 1 pound

Consider a call option on €31,250.

The option premium is $0.25 per €

The exercise price is $1.50 per €.

7-35

Example

$1.50

ST

Profit

loss

–$7,812.50

$1.75

Long 1 call on €31,250

Consider a call option on €31,250.


The exercise price is $1.50 per €.

7-36

Example

$1.50

ST

Profit

loss

$42,187.50

$1.35 Long 1 put on €31,250

Consider a put option on €31,250.


The exercise price is $1.50 per euro.

What is the maximum gain on this put option?

At what exchange rate do you break even?

–$4,687.50

$42,187.50 = €31,250×($1.50 – $0.15)/€

$4,687.50 = €31,250×($0.15)/€7-

37

American Option Pricing Relationships

With an American option, you can do everything that you can do with a European option AND you can exercise prior to expiry—this option to exercise early has value, thus:

(note T > t)

Cat > Cet

Pat > Pet

7-38

Market Value, Time Value and Intrinsic Value for an European Call at t

E

ST

Profit

loss

Long 1 callThe 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.

Intrinsic value

Time value

Market Value

In-the-moneyOut-of-the-money

7-39

European Call Option Pricing relationship (determine the call price using the “rep” portfolio & no arbitrage. e.g., $1.1 for $2-B, $1.4 for $3-R, what about $2-B & $6-R? )

Consider two investments in a Call or a “Rep” Portfolio1 Buy a European call option on the British pound futures

contract. The cash flow today is – Ce with CT = Max[ST - E, 0]

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

call in the U.S. at i$ E

(1 + i$)The cash flow today is

2 Lending the present value of One BP (FV=One BP=$ST) at i£

ST

(1 + i£)The cash flow today is –

7-40

European Call Option Pricing Relationships

When the option is in-the-money both strategies have the same payoff at T (i.e., ST – E).When the option is out-of-the-money the call has a higher payoff (0) than the borrowing and lending strategy (ST – E, which is negative). Thus, the present price is:

Ce > MaxST E

(1 + i£) (1 + i$)– , 0

7-41

European Put Option Pricing Relationships

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

Pe > MaxST E

(1 + i£)(1 + i$)– , 0

7-42

A Brief Review of CIRPRecall that if the spot exchange rate is S0 = $1.50/€, and that if i$ = 3% and i€ = 2% then there is only one possible 1-year forward exchange rate that can exist without attracting arbitrage: F1 = $1.5147/€ (note that this diagram is sideway)

1. Borrow $1.5m at i$ = 3%

2. Exchange $1.5m for €1m at spot

3. Invest €1m at i€ = 2%

4. Owe $1.545m

5. Receive €1.02 m

0 1

F1 =$1.5147

€1.00

7-43

Binomial Call Option Pricing Model

Imagine a simple world where the dollar-euro exchange rate is S0 = $1.50/€ today and in the next year, S1 is either $1.875/€ or $1.20/€.

$1.50

$1.20

$1.875

S1S0

7-44

Binomial Option Pricing Model

A call option on the euro with exercise price E = $1.50 (=S0) will have the following payoffs.

C1

$.375

$1.20

$1.875

S1S0

$1.50

By exercising the call option, you can buy €1 for $1.50.If S1 = $1.875/€ the option is in-the-money:

$0

…and if S1 = $1.20/€ the option is out-of-the-money:

7-45


We can replicate the payoffs of the call option. By taking a position in the euro along with some judicious borrowing and lending.

$1.20

$1.875

S1 S0

$1.50

C1

$.375

$07-46

Binomial Option Pricing ModelBorrow 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.

$1.20

$1.875

S1 S0

$1.50

debt

– $1.20

– $1.20

portfolio

= $.675

= $0

C1

$.375

$07-47


The portfolio has 1.8 times the call option’s payoff so the portfolio is worth 1.8 times the option value.

S0 debt portfolio C1 S1

$1.20

$1.875

$1.50

– $1.20

– $1.20

= $.675

= $0

$.375

$0

$.675$.375

1.80 =

7-48

Binomial Option Pricing ModelThe replicating portfolio’s dollar value today (how much it costs to make the portfolio) is the sum of today’s dollar value of the present value of one euro less the present value of a $1.20 debt:

$1.50 $1.20 (1 + i$)€1.00

€1.00(1 + i€)

× –

S0 debt portfolio C1 S1

$1.20

$1.875

$1.50

– $1.20

– $1.20

= $.675

= $0

$.375

$07-49


We can value the call option as 5/9 of the value of the replicating portfolio:

S0

$1.50

debt portfolio C1 S1

$1.20 – $1.20 = $0 $0

$1.875– $1.20 = $.675 $.375

C0 = ×5

9$1.50 $1.20

(1 + i$)€1.00€1.00

(1 + i€)× –

If i$ = 3% and i€ = 2% the call is worth

$0.1697 = ×5

9$1.50 $1.20

(1.03)€1.00€1.00(1.02)

× –

7-50


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.

7-51

The Hedge Ratio In the example just previous, we replicated the

payoffs of the call option with a levered position in the underlying asset. (In this case, borrowing dollars to buy euro at the spot.)

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.

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 – CS1 – S1

downup

downup

7-52

Hedge Ratio This practice of the construction of a riskless

hedge is called delta hedging. The delta of a call option is positive.

Recall from the example:

The delta of a put option is negative.

Deltas change through time.

H = C – CS1 – S1

downup

downup

$0.375 – $0

$1.875 – $1.20

$0.375

$0.675

5

9= ==

7-53

Creating a Riskless HedgeThe standard size of euro options on the PHLX is €10,000.

In our simple world where the dollar-euro exchange rate is S0 = $1.50/€ today and in the next year, S1 is either $1.875/€ or $1.20/€. An at-the-money call on €10,000 has these payoffs:

– $15,000

€10,000

€10,000$1.875

€1.00

$1.20

€1.00×

×

=

=

$12,000

$18,750If the exchange rate at maturity goes up to S1 = $1.875/€ then the option finishes in-the-money.

× €10,000 = $15,000$1.50

€1.00

If the rate goes down, the option finishes out of the money. No one will pay $15,000 for €10,000 worth $12,000

C1 = $3,750up

C1 = $0down7-54

Creating a Riskless HedgeConsider a dealer who has just written 1 at-the-money call on €10,000. He calculates the hedge ratio as 5/9:

He can hedge his position with three trades:

H = C – CS1 – S1

downup

downup=

$3,750 – 0

$18,750 – $12,000=

$3,750

$6,750=

5

9

1. If i$ = 3% then he could borrow $6,472.49 today and owe $6,666.66 in one period.

2. Then buy the present value of €5,555.56 (buy euro at spot exchange rate,

compute PV at i€ = 2%),

3. Invest €5,446.62 at i€ = 2%.

$6,472.49 =$6,666.66

1.03

€5,446.62 =€5,555.56

1.02

$12,000 × = $6,666.665

9

Net cost of hedge = $1,697.44

= €10,000 ×59

7-55

Replicating Portfolio Call on €10,000K = $1.50/€

T = 0 T = 1

– $ 6,666.67

– $6,666

Ser

vice

Loa

n

€10,000 = $15,000

$1.875

S1(

$|€)

€1.00

$1.20

€1.00€5,555.56

€5,555.56

FV

€ in

vest

men

t

×

× = $3,750

= 0Net cost = $1,697.44

Borrow $6,472.49 at i$ = 3% Step 1

Buy €5,446.62 at S0 = $1.50/€

Step 2

Invest €5,446.62 at i€ = 2% Step 3

=

=

$6,666.67

$10,416

FV

€ in

vest

men

t in

$

the replicating portfolio payoffs and the call option payoffs are the same so the call is worth

$1,697.44 = ×59

$1.50 $1.20 (1.03)€1.00

€10,000(1.02)

× –

7-56

Risk Neutral Valuation of Options

Calculating the hedge ratio is vitally important if you are going to use options. The seller needs to know it if he wants to protect his

profits or eliminate his downside risk. The buyer needs to use the hedge ratio to inform his

decision on how many options to buy. Knowing what the hedge ratio is isn’t especially

important if you are trying to value options. Risk Neutral Valuation is a very hand shortcut

to valuation.7-57

Risk Neutral Valuation of OptionsWe can safely assume that CIRP holds:

F1 =$1.5147€1.00

$1.50×(1.03)€1.00×(1.02)=

€10,000 = $15,000

$1.20

€1.00€10,000×$12,000 =

$1.875

€1.00€10,000×$18,750 =

Set the value of €10,000 bought forward at $1.5147/€ equal to the expected value of the two possibilities shown above:

$15,147.06 = p × $18,750 + (1 – p) × $12,000€10,000×$1.5147€1.00

=7-58

Solving for p gives the risk-neutral probability of an “up” move in the exchange rate:

$15,147.06 = p × $18,750 + (1 – p) × $12,000

p = .4662

p =$15,147.06 – $12,000

$18,750 – $12,000


7-59

Now we can value the call option as the present value (discounted at the USD risk-free rate) of the expected value of the option payoffs, calculated using the risk-neutral probabilities.


€10,000 = $15,000

$1.20

€1.00€10,000×$12,000 =

$1.875

€1.00€10,000×$18,750

=←value of €10,000

←value of €10,000

$3,750 = payoff of right to buy €10,000 for $15,000

$0 = payoff of right to buy €10,000 for $15,000

$1,697.44

C0 = $1,697.44 = .4662×$3,750 + (1–.4662)×0

1.037-60

Test Your IntuitionUse risk neutral valuation to find the value of a put option on $15,000 with a strike price of €10,000.

Hint: given that we just found that the value of a call option on €10,000 with a strike price of $15,000 was $1,697.44 this should be easy in the sense that we already know the right answer.

$1.50×1.03€1.00×1.02

F1 =$1.5147€1.00=

$1.50€1.00

S0 =As before, i$ = 3%, i€ = 2%,

7-61

Test Your Intuition (continued)$1.50×1.03€1.00×1.02

F1 =$1.5147€1.00=

€10,000 = $15,000

€1.00

$1.875$15,000×€8,000 =

€1.00

$1.20$15,000×€12,500

=←value of $15,000

←value of $15,000

€9,902.91 = p × €12,500 + (1 – p) × €8,000

$15,000 ×$1.5147

€1.00= €9,902.91

p = .42297-62

Test Your Intuition (continued)

€10,000 = $15,000

€1.00

$1.875$15,000×€8,000 =

€1.00

$1.20$15,000×€12,500

=0 = payoff of right to sell $15,000 for €10,000

€2,000 = payoff of right to sell $15,000 for €10,000

€1,131.63

←value of $15,000

←value of $15,000

€P0 = €1,131.63 = .4229×€0 + (1–.4229)×€2,000

1.02

$P0 = $1,697.44 = €1,131.63 × $1.50€1.00

7-63

Test Your Intuition (continued)

The value of a call option on €10,000 with a strike price of $15,000 is $1,697.44

The value of a put option on $15,000 with a strike price of €10,000 is €1,131.63

At the spot exchange rate these values are the same:

€1,131.63 × €1.00$1.50

= $1,697.44

7-64

Take-Away LessonsConvert future values from one currency to another using forward exchange rates. Convert present values using spot exchange rates.Discount future values to present values using the correct interest rate, e.g. i$ discounts dollar amounts and i€ discounts amounts in euro.To find the risk-neutral probability, set the forward price derived from CIRP equal to the expected value of the payoffs.To find the option value discount the expected value of the option payoffs calculated using the risk neutral probabilities at the correct risk free rate.

7-65

Finding Risk Neutral Probabilities up

F1 = p × S1 + (1 – p) × S1 down

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

$18,750 – $12,000

For a put on $15,000 with a strike price of €10,000 we solved

€9,902.91 = p × €12,500 + (1 – p) × €8,000

p =€9,902.91– €8,000 €12,500 – €8,000

= .4662=$1.5147 – $1.20 $1.875 – $1.20

= .4229 =€0.6602– €.5333 €.8333 – €.5333

7-66

Currency Futures Options

Are an option on a currency futures contract. Exercise of a currency futures option results in a

long futures position for the holder of a call or the writer of a put.

Exercise of a currency futures option results in a short futures position for the seller of a call or the buyer of a put.

If the futures position is not offset prior to its expiration, foreign currency will change hands.

7-67

Currency Futures Options

Why a derivative on a derivative? Transactions costs and liquidity. For some assets, the futures contract can have

lower transactions costs and greater liquidity than the underlying asset.

Tax consequences matter as well, and for some users an option contract on a future is more tax efficient.

The proof is in the fact that they exist.

7-68

Call Option Payoff = $0.3787

Option Payoff = $0

Option Price = ?

Binomial Futures Option PricingA 1-period at-the-money call option on euro futures has a strike price of F1 = $1.5147/€

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

$1.50×1.03€1.00×1.02

F1 =$1.5147

€1.00=

$1.875×1.03

€1.00×1.02F1 =

$1.8934€1.00

=

$1.20×1.03

€1.00×1.02F1($|€) =

$1.2118€1.00

=

7-69

Consider the Portfolio: long futures contractsshort 1 futures call option

Binomial Futures Option Pricing

Portfolio Cash Flow =

H × $0.3603 – $0.3787

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

= 0.5610

Futures Call Payoff = –$0.3787

Option Price = $0.1714

$1.50×1.03€1.00×1.02

F1($|€) =$1.5147

€1.00=

$1.875×1.03

€1.00×1.02F1 =

$1.8934€1.00

=

$1.20×1.03

€1.00×1.02F1($|€) =

$1.2118€1.00

=

Futures Payoff = H × $0.3603


–H×$0.3147

Option Payoff = $0

Futures Payoff = –H×$0.3147

7-70

Binomial Futures Option PricingThe payoffs of the portfolio are –$0.1766 in both the up and down states.

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%


0.5610 × $0.3603 – $0.3787

= –$0.1766


–0.5610×$0.3147 = –$0.1766

Call Option Payoff = –$0.3787

Option Payoff = $0

$1.875×1.03

€1.00×1.02F1($|€) =

$1.8934€1.00

=

$1.20×1.03

€1.00×1.02F1($|€) =

$1.2118€1.00

=

Futures Payoff = H × $0.3603

Futures Payoff = –0.5610×$0.3147

$1.50×1.03€1.00×1.02

F1($|€) =$1.5147

€1.00=

$0.17661.03

C0 = $0.1714 =

7-71

Option Pricing

Find the value of an at-the-money call and a put on €1 with

Strike Price = $1.50

i$ = 3%

i€ = 2%

u = 1.25

d = .8

$1.50

$1.875 = 1.25 × $1.50

$1.20 = 0.8 × $1.50

$0.375 = Call payoff

$0 =Call payoff

$0 = Put payoff

$0.30 = Put payoff

.4662× $0.375C0 = 1.03

= $.169744.5338 × $0.30

P0 = 1.03= $0.15555

– .80

p =1.25 – 0.80

= .4662

1.031.02

P0 = $0.15555

C0 = $.169744

7-72

Hedging a Call Using the Spot MarketWe want to sell call options. How many units of the underlying asset should we hold to form a riskless portfolio?

$1.50

$1.875 = 1.25 × $1.50

$1.20 = 0.8 × $1.50

$0.375 = Call payoff

$0 = Call payoff

$0.375 – $0 =$1.875 – $1.20

= 5/9

Sell 1 call option; buy 5/9 of the underlying asset to form a riskless portfolio.

If the underlying is indivisible, buy 5 units of the underlying and sell 9 calls.

7-73

Hedging a Call Using the Spot Market

Call finishes in-the-money, so we must buy an additional €4 at $1.875.

Cost = 4 × $1.875 = $7.50Cash inflow call exercise = 9 × $1.50 = $13.50

Portfolio cash flow = $6.00

S0 = $1.50/€

T = 0 Cash Flows T = 1S1 = $1.875

S1 = $1.20

Call finishes out-of-the-money, so we can sell our now-surplus €5 at $1.20.

Cash inflow = 5 × $1.20 = $6.00

Handy thing to notice: $5.8252 × 1.03 = $6.00

Write 9 calls: Cash inflow = 9 × $0.169744 = $1.5277

Portfolio cash flow today = –$5.8252

$0.375 – $0 =

$1.875 – $1.20= 5/9

Go long PV of €5.

Cost today =€5

1.02× = $7.3529

$1.50€1.00

C1= $.375

C1= $0

7-74

Hedging a Put Using the Spot MarketWe want to sell put options. How many units of the underlying asset should we hold to form a riskless portfolio?

S0 = $1.50/€

Put payoff = $0.0

Put payoff = $0.30

$0 – $0.30 =

$1.875 – $1.20= – 4/9

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.

S1 = $1.875

S1 = $1.20

7-75

Hedging a Put Using the Spot Market

Put finishes out-of-the-money.To repay loan buy €4 at $1.875.

Cost = 4 × $1.875 = $7.50Option cash inflow = 0

Portfolio cash flow = $7.50

T = 0 Cash Flows T = 1

put finishes in-the-money, so we must buy 9 units of underlying at

$1.50 each = 9×1.50 = $13.50 use 4 units to cover short sale, sell remaining 5 units at $1.20 = $6.00

Portfolio cash flow = $7.50 Handy thing to notice: $7.2816 × 1.03 = $7.50

Write 9 puts: Cash inflow = 9 × $0.15555 = $1.3992

Portfolio Inflow today = $7.2816

Borrow the PV of €4 at i€ = 2%.

Inflow =

$0 – $0.30 =

$1.875 – $1.20= – 4/9

€41.02

× = $5.8824$1.50€1.00

S1 = $1.20

S1 = $1.875

S0 = $1.50/€

7-76

Hedging a Call Using Futures

Futures contracts matures: buy 5 units at forward price. Cost = 5× $1.5147 = $7.5735

Call finishes out-of-the-money, so we sell our 5 units of underlying at $1.20.

Cash inflow = 5 × $1.20 = $6.00Portfolio cash flow = –$1.5735


Write 9 calls: Cash inflow = 9 × $0.169744 = $1.5277

Portfolio cash flow today = $1.5277

Go long 5 futures contracts. Cost today = 0Forward Price = × = $1.5147

1.031.02

$1.50€1.00

Call finishes in-the-money, we must buy 4 additional units of underlying at S1($/€) = $1.875. Cost = 4 × $1.875 = $7.50

Option cash inflow = 9 × $1.50 = $13.50Portfolio cash flow = –$1.5735

Futures contracts matures: buy 5 units at forward price. Cost = 5× $1.5147 = $7.5735

S1 = $1.875

S0 = $1.50/€

S1 = $1.20

7-77

Hedging a Put Using Futures

Put finishes out-of-the-money. Option cash flow = 0Portfolio cash flow = –$1.4412

Put finishes in-the-money, we must buy €9 at $1.50/€ = 9×1.50 = $13.50 Futures contracts matures: sell €4 at

forward price $1.5147/€4× $1.5147 = $6.0588

sell remaining €5 at $1.20 = $6.00Portfolio cash flow = –$1.4412


Write 9 puts: Cash inflow = 9 × $0.15555 = $1.3992

Portfolio Inflow today = $1.3992

Go short 4 futures contracts. Cost today = 0Forward Price = × = $1.5147

1.031.02

$1.50€1.00

Futures contracts matures: sell €5 at forward price. Loss = 4× [$1.875 – $1.5147] = $1.4412

S1 = $1.875

S0 = $1.50/€

S1 = $1.20

7-78

2-Period OptionsValue a 2-period call option on €1 with a strike price = $1.50/€

i$ = 3%; i€ = 2%u = 1.25; d = .8

– .80

p =1.25 – 0.80

= .4662

1.031.02

S0 = $1.50/€

S1 = $1.875

S1 = $1.20down

up

S2 = $2.3438

S2 = $1.50up-down

up-up

S2 = $0.96down-down

1.03

.4662× $1.0609C0 = = $0.4802

.4662× $0.8468C1 = 1.03

= $1.06up

C0 = $0.4802

C2 = $0down-down

C2 = $0up-down

C2 = $0.8468up-up

C1 = $1.0609up

C1 = $0down

7-79

Chapter Objective:

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