Currency and Interest Rate Options - Stanford Universityweb.stanford.edu/class/msande247s/2009/1110 2009 po… · · 2009-11-10Currency and Interest Rate Options M. Brandt Reading

Handout #15Derivative Security Markets

Currency and Interest Rate Options

Tuesdays 6:10-9:00 p.m.Commerce 260306

Wednesdays 9:10 a.m.-12 noonCommerce 260508

Course web pages:http://finance2010.pageout.net

ID: California2010 Password: blueskyID: Oregon2010 Password: greenland

12-2

Levich

Luenberger

Solnik

Fabozzi

Chap 12

Chap

Chap 10

Interest RateModels

Scan Read

Pages

Pages

Pages 433-483

Pages

Chap 27Interest Rate Options

Pages

Currency and Interest Rate Options

M. Brandt

Reading Assignments for this Week

12-3

Midterm Exam: See University Calendar (November 16-20, 2009)

Coverage: Chapters 3, 4, 5, 6, 7, 8, 9, 10 + Ben Bernanke’s semi-annual testimony

It’s a closed-book exam. However,a two-sided formula sheet (11 x 8.5) is required;

calculator/dictionary is okay; notebook is NOT okay.

75 minutes, 7 questions, 100 points total; five questions require calculation and

two questions require (short) essay writing.

12-4

Final Exam See University Calendar (January 8-14, 2010) A Three-hour Exam

Open-Book, Open Notes

Derivative Security MarketsCurrency and Interest Rate Options

MS&E 247S International InvestmentsYee-Tien Fu

12-6

A First Example of No Arbitrage

Consider a forward contract that obliges us to hand over an amount $F at time T to receive the underlying asset.

• Today’s date is t and the price of the asset is currently $S(t) (this is the spot price).

• When we get to maturity, we will hand over the amount $F and receive the asset, then worth $S(T).

• Is there any relationship among F, S(t), tand T?

1

Derivatives: The Theory and Practice of Financial Engineering . Paul Wilmott . John Wiley & Sons 1998

12-7

A First Example of No Arbitrage• Enter into the forward contract, and sell the

asset simultaneously. Selling something one does not own is called going short. Put the cash (S(t)) in the bank to get interest.

• At maturity, we hand over F to receive the asset, hence canceling our short asset position. Our net position is therefore

S(t)er (T - t ) – F or S(t) (1+r)T-t – F

F = S(t)er(T-t) or F = S(t) (1+r)T-t

1

12-8

A First Example of No Arbitrage1

Cashflows in a hedged portfolio of asset and forward.

Worthtoday (t )

0- S(t )S(t )

0

HoldingForward- StockCash

Total

Worth atmaturity (T )

S(T ) - F- S(T )

S(t)er (T-t )

S(t )er (T-t ) - F

12-9

The Building Blocks of Contingent Decisions

(a) Long a bond (invest in a zero-coupon bond)

0

0

(b) Short a bond (issue a zero-coupon bond)

Payo

ff at

mat

urity

Payo

ffat

mat

urity

+

−

12-10


(c) Purchase of Right toBuy at a Fixed Price

(d) Purchase of Right toSell at a Fixed Price

Opt

ion

Payo

ff

Value of Underlying Asset at Decision Date

(e) Sell Right toBuy at a Fixed Price

(f) Sell Right toSell at a Fixed Price

$0 $0

Real Options: Amram & Kulatilaka Figure 4.1

12-11


The top row shows the payoffs from holding an asset that confers the right (but not the obligation) to buy or sell at a fixed price. The payoffs in the bottom row are the mirror images and show the position of the party on the other side of the transaction.


12-12

The Building Blocks of Noncontingent Decisions

(g) Forward Purchase(long position)

(h) Forward Sale(short position)

Payo

ff

Value of Underlying Asset at Decision Date

$0 $0


A forward contract is the right to buy or sell an asset at a specified date in the future at a specified price. The payoffs to a forward are not contingent on a future decision (hence there is no kink) but do depend on the realized value of an uncertain asset (the line is sloped).

12-13

Identifying the Building Blocksin Red & White’s Contract


Suppose Red & White, a soft drink manufacturer, was offered a new sugar contract with a price floor of 90 cents per pound and a price cap of $1.10 per pound. Using the building-block approach, the contract can be seen as the sum of three parts:

1. Buy sugar on the world spot market.2. Sell a put option with a strike price of $1.10.3. Buy a call option with a strike price of 90 cents.

12-14


(a) The Payoff to a Contract witha Price Cap and a Price Floor

Mon

thly

Con

trac

tPa

yoffs

$0.90 $1.10

Price of Sugar in World Spot Market (per pound)

(b) Buy on Spot Market

(c) Sell Put Option

$1.10

(d) Buy Call Option

$0.90

+

+

=


12-15



a This contract requires Red & White to pay a minimum of 90 cents per pound and a maximum of $1.10 per pound. The building-block approach shows that the contract is equivalent to

b purchases in the world spot market, plusc a put option with an exercise price at $1.10 per

pound, plusd a call option with an exercise price at 90 cents

per pound.

12-16

• Currency options began trading on the Philadelphia Stock Exchange in 1982, while interest rate options began trading on the Chicago Mercantile Exchange in 1985.

• Since then, there has been expansion in many directions :¤ 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.

Introduction to Options

12-17

Types of Contracts

• An option is the right, but not the obligation, to buy (or to sell) a fixed quantity of an underlying financial asset or commodity at a given price on (or on or before) a specified date.

• A call option bestows on the owner the right, but not the obligation, to buy the underlying financial asset or commodity.

• A put option, conveys to the owner the right, but not the obligation, to sell the underlying financial asset or commodity.

12-18

With currency options, there are two choices of the underlying financial asset :

• An option on spot takes spot foreign exchange as the underlying asset. Spot foreign exchange is transferred if such an option is exercised.

• If an option on futures is exercised, positions in the underlying currency futures contract are created: a long position (at the strike price) for the holder of a call or seller of a put, and a short position (at the strike price) for the holder of a put or seller of a call.

Types of Contracts

12-19

• The owner of a futures call has the right to buy a futures, so the seller of a futures call must deliver a long position if the call is exercised (i.e., the seller takes a short position).

• The owner of a put has the right to sell a futures, so the seller of a futures put must deliver a short position (i.e., the seller takes a long position).

• Because futures contracts have zero-net supply, the underlying positions must be “created” before they can be delivered.

Types of Contracts

12-20

• With interest rate options, the underlying contract is typically an interest rate futures. The interest rate can be taken from a 3-month Eurodollar deposit, a long-term Treasury bond, or some other interest-bearing instrument.

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

Types of Contracts

12-21

Examples An American call option on spot DM :

The right to buy DM 1 million for $0.63 per DM from today until expiration on Dec 15, 1999.

A European put option on Swiss franc futures :The right to sell SFr 10 million March 1999 futures for $0.76 per SFr on (and only on) Mar 15, 1999.

Note that US$ is used as the second currency or numeraire currency in both the above examples.

Types of Contracts

12-22

• With the rise of significant trading volume in cross-exchange rates - notably the £/¥, DM/¥, DM/£, and DM against other European currency rates - options written in currency pairs that do not involve the US$ have been introduced.

• These cross-rate options gained further popularity after 1991 as exchange rate volatility among European currencies increased.

Types of Contracts

12-23

Location of Trading

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

Over-the-counter (OTC) market -- An informal network of brokers and dealers who negotiate sales of securities.

12-24

Contract Specifications

Calls PutsVol. Last Vol. Last

German Mark 63.1962,500 German Marks EOM-cents per unit.54 Aug 6350 0.47 …… ……61 Jul 26 2.52 …… ……61 Aug …… …… 6350 0.3863½ Jul 2 0.88 4 1.03… … … … … …

62,500 German Marks-European Style.61½ Jul …… …… 1350 0.1961½ Sep …… …… 340 0.7063 Jul 57 0.97 …… ……63 Aug 10 1.25 10 0.95… … … … … …

62,500 German Marks-cents per unit.57 Sep …… …… 10 0.0658 Sep 20 5.56 …… ……61½ Aug 80 2.10 25 0.4163 Aug 65 1.17 …… ……… … … … … …

OPTIONSPHILADELPHIA EXCHANGE

This newspaper extract reports prices of Philadelphia Stock Exchange options on spot currency.For the German mark, there are three sections of figures.

Prices on end-of-month (EOM) options that mature on the last Friday of the given month.

Prices of European-style German mark options.

Prices of American-style German mark options.

12-25

Calls PutsVol. Last Vol. Last

German Mark 63.1962,500 German Marks EOM-cents per unit.54 Aug 6350 0.47 …… ……61 Jul 26 2.52 …… ……61 Aug …… …… 6350 0.3863½ Jul 2 0.88 4 1.03… … … … … …

62,500 German Marks-European Style.61½ Jul …… …… 1350 0.1961½ Sep …… …… 340 0.7063 Jul 57 0.97 …… ……63 Aug 10 1.25 10 0.95… … … … … …

62,500 German Marks-cents per unit.57 Sep …… …… 10 0.0658 Sep 20 5.56 …… ……61½ Aug 80 2.10 25 0.4163 Aug 65 1.17 …… ……… … … … … …

OPTIONSPHILADELPHIA EXCHANGE

Contract Specifications

The German mark contract size (62,500 DM) is written at the top of each section.

The strike prices (in US cents per DM).

The closing spot price of the DM ($0.6319/DM) is listed across from the word “German Mark”.

The closing or settlement price was reported as 1.17 cents per DM. Since the contract size is 62,500 DM, the buyer of this call option would expect to pay 62,500x$0.0117=$731.25 plus commission charges.

Consider this call option.

12-26

Spot Currency Options: Prices at Maturity

Consider the following call and put options on the £ :

• Each option has a strike price (K) of $1.50/£and an option premium of about $0.10. If we designate C and P as the values of the call and put prices, then at maturity:C = Max [0, S-K ] and P = Max [0, K-S ]

where S is the spot price of the £ at the maturity of the option.

• It is clear from the above equations that the value of an option can never be negative.

(12.1)

12-27

Buyer and Seller of a Spot Currency Call on £with Strike at $1.50/£

0.15

0.10

0.05

0.00

-0.05

-0.10

-0.151.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75

Pro

fit o

r Los

s in

$

$/£

Profit and Loss Positions

Seller of a Call

Buyer of a Call

Figure 12.4A

12-28

• For spot prices $1.50, the expiration value of the call is 0. The buyer suffers a loss - the $0.10 premium he originally paid for the call.

• At spot prices $1.50, the expiration value of the option is positive (= S - $1.50), which reduces the call buyer’s losses. The break-even exchange rate is $1.60.

• The seller keeps the premium when the spot rate $1.50, because the call is not exercised.

• At spot rates $1.50, the opportunity cost for the seller is .

Buyer and Seller of a Spot Currency Call on £with Strike at $1.50/£

12-29

0.15

0.10

0.05

0.00

-0.05

-0.10

-0.151.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75

Pro

fit o

r Los

s in

$

$/£


Seller of a Put

Buyer of a Put

Buyer and Seller of a Spot Currency Put on £with Strike at $1.50/£

Figure 12.4B

12-30

• For spot prices $1.50, the expiration value of the put is 0. The buyer suffers a loss of $0.10.

• At spot prices $1.50, the expiration value of the put is positive (= $1.50 - S) which reduces the put buyer’s losses. The break-even exchange rate is $1.40.

• The seller earns the option premium when the spot rate $1.50, because the put is not exercised.

• At spot rates $1.50, the seller’s opportunity cost becomes .

Buyer and Seller of a Spot Currency Put on £with Strike at $1.50/£

12-31

• Note that the buyer of either a put or call faces limited liability in that her loss is capped at the initial option premium paid.

• The seller of a call faces unlimited liability as the underlying asset could appreciate without limit.

• The seller of a put faces a large liability, which is limited by the fact that the price of the underlying cannot fall below zero.

Spot Currency Options:Prices at Maturity

12-32

Interest Rate Futures Options:Prices at Maturity

Consider the following call and put options on a Eurodollar interest rate futures contract :

• Each option has a strike price (K) of 96.00 (corresponding to a 4.00 % Euro$ interest rate) and an option premium of about 0.25. If we designate Cf and Pf as the values of the call and put prices, then at maturity:

Cf = Max [0, F-K ] and Pf = Max [0, K-F ]where F is the price of the underlying Eurodollar futures contract on the maturity of the option.

(12.2)

12-33

Euro-$ Interest Rate Call with Strike at 96.00

94.00 94.50 95.00 95.50 96.00 96.50 97.00 97.50 98.00

Pro

fit o

r Los

s in

%

Interest Rate Futures Price at Maturity


Seller of a Call

Buyer of a Call-1.0

-1.5

-2.0

-0.5

0.0

0.5

1.0

1.5

2.0

Figure 12.5A

12-34

• Note the inverse relationship between interest rates and futures prices.

• For futures prices 96.00 (interest rates 4.00%), the expiration value of the call is 0. The buyer suffers a loss equal to the original premium paid for the option (25 basis points x $25/b.p. = $625).

• If the price at maturity 96.00, the expiration value is positive (= F - 96.00 basis points), which reduces the call buyer’s losses. The break-even Euro$ futures rate is 96.25 (3.75% interest rate).

Euro-$ Interest Rate Call with Strike at 96.00

12-35

94.00 94.50 95.00 95.50 96.00 96.50 97.00 97.50 98.00

Pro

fit o

r Los

s in

%

Interest Rate Futures Price at Maturity


Seller of a Put

Buyer of a Put-1.0

-1.5

-2.0

-0.5

0.0

0.5

1.0

1.5

2.0

Euro-$ Interest Rate Put with Strike at 96.00

Figure 12.5B

12-36

• For futures prices 96.00 (interest rates 4.00%), the expiration value of the put is zero. The buyer incurs a loss equal to the original premium paid for the option.

• If the Euro$ futures price at maturity 96.00, the expiration value of the put is positive (equal to 96.00 - F basis points), which reduces the put buyer’s losses.

• The break-even Euro$ futures rate is 95.75 (4.25% interest rate ). If interest rates rise above 4.25%, the put buyer earns a profit.

Euro-$ Interest Rate Put with Strike at 96.00

12-37

12-38

12-39

12-40

Buy putVery bearish

Write callSlightly bearish

Write putSlightly bullish

Buy callVery bullish

StrategyCircumstance

12-41

Google Earnings Gave Options Traders a 17,530% Gain (Update1)By Michael Patterson and Jeff KearnsApril 18, 2008 (Bloomberg) -- Options traders who predicted Google Inc. would beat estimates earned as much as 17,530 percent on their investments today, the most-profitable bet among all U.S. equity derivatives.Contracts giving the right to buy Google shares for $530 before the close of trading today jumped as high as $17.63 from their 10-cent closing price yesterday.That gain almost matched the 18,760 percent advance in the Dow Jones Industrial Average since the beginning of 1900, according to Bloomberg data.

12-42

``Today in Google you see the power of leverage in options, especially going into earnings,'' said Peter Bottini, executive vice president of trading at OptionsXpress Holdings Inc., a Chicago-based online brokerage. ``We were swamped with customers who were calling in at the open of the market.''Google shares climbed 20 percent, the most since its initial public offering in 2004, to $539.41 after the owner of the most popular Internet search engine beat the average analyst profit estimate by 7.1 percent.Google had dropped 35 percent in 2008 on concern the U.S. economic slump would hurt spending on online advertising. Google call-option volume jumped to 311,139 contracts, the most since January 2006. Those contracts outnumbered trading in bearish bets, or puts, by 1.6-to-1. Call options give the right to buy a security for a certain amount, called the strike price, by a given date. Puts convey the right to sell.

12-43

A Short Straddle in the $/£ : betting on convergence

0.10

0.05

0.00

-0.05

-0.10

-0.15

-0.201.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75

Pro

fit o

r Los

s in

$

$/£


Sell PutSell Call

Sell CallSell Put

StraddleStraddle

Box 12.1 Figure A

12-44

A Long Straddle in the $/£ : betting on divergence

0.20

0.15

0.10

0.05

0.00

-0.05

-0.101.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75

Pro

fit o

r Los

s in

$

$/£


Buy CallBuy Put

Buy PutBuy Call

StraddleStraddle

Box 12.1 Figure B

12-45

Using a Call to Hedge a Short Pound: hedge an A/P

0.3

0.2

0.1

0

-0.1

-0.2

-0.31.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75

Pro

fit o

r Los

s in

$

$/£


Buy CallShort Pound

Short Pound

Buy Call

CombinedPosition

Box 12.1 Figure C

12-46

Using a Put to Hedge a Long Pound: hedge an A/R

0.3

0.2

0.1

0

-0.1

-0.2

-0.31.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75

Pro

fit o

r Los

s in

$

$/£


Buy Put

Long Pound

Long Pound

Buy Put

CombinedPosition

Box 12.1 Figure D

12-47

Combining a Floor and Cap : A CollarBeginning cash outlay is minimized, customized hedge of an A/R

0.3

0.2

0.1

0

-0.1

-0.2

-0.31.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75

Pro

fit o

r Los

s in

$

$/£

Profit and Loss Positions at Maturity

Sell a Call

Long Pound

Buy a Put

A Collar :Sell Call + Buy Put

Box 12.1 Figure E

12-48

Protective Put Strategy (Payoffs)

Buy a put with an exercise price of $50

Buy the stock

Protective Put payoffs

$50

$0

$50

Value at expiry

Value of stock at expiry

Portfolio value today = p0 + S0

12-49

Protective Put Strategy (Profits)

Buy a put with exercise price of $50 for $10

Buy the stock at $40

$40

Protective Put strategy has

downside protection and upside potential

$40

$0

-$40

$50

Value at expiry


-$10

12-50

Covered Call Strategy

Sell a call with exercise price of $50 for $10

Buy the stock at $40

$40

Covered Call strategy

$0

-$40

$50

Value at expiry


-$30

$10

12-51

Use of a Butterfly SpreadFrom the Trader’s DeskA stock is currently selling for $61. The prices of call options expiring in 6 months are quoted as follows:

Strike price = $55, call price = $10Strike price = $60, call price = $7Strike price = $65, call price = $5

An investor feels it is unlikely that the stock price will move significantly in the next 6 months.

The StrategyThe investor sets up a butterfly spread:

1. Buy one call with a $55 strike.2. Buy one call with a $65 strike.3. Sell two calls with a $60 strike.

The cost is $10 + $5 - (2 x $7) = $1. The strategy leads to a net loss (maximum $1) if the stock price moves outside the $56-to-$64 range but leads to a profit if it stays within this range. The maximum profit of $4 is realized if the stock price is $60 on the expiration date.

12-52

Payoff from a Butterfly Spread

Stock price Payoff from Payoff from Payoff from Totalrange first long call second long call short calls payoff *

ST < X1 0 0 0 0

X1 < ST < X2 ST – X1 0 0 ST – X1

X2 < ST < X3 ST – X1 0 –2(ST – X2) X3 – ST

ST > X3 ST – X1 ST – X3 –2(ST – X2) 0

* These payoffs are calculated using the relationship X2 = 0.5(X1 + X3).

12-53

Butterfly Spread Using Call Options

X1 X3

Profit

STX2

12-54

Butterfly Spread Using Put Options

X1 X3

Profit

STX2

12-55

Short Straddle

–30

30 40 60 70

–40

Stock price ($)

Opt

ion

payo

ffs (

$)

$50

This Short Straddle only loses money if the stock price moves $20 away from $50 (diverges).

Sell a put with exercise price of$50 for $10

Sell a call with an exercise price of $50 for $10

20

12-56

What are the similarities and differences between the payoffs of the butterfly spread and the short straddle?

12-57

Option Pricing : Formal Models

S1 = 1.40

Period 1

S2,u = 1.5015

S2,d = 1.2293

Spot Rates

Period 2

Option Values

Call2,u = 0.1015Put2,u = 0.0

Call2,d = 0.0Put2,d = 0.1707

Binomial Currency Option Example

Figure 12.8

12-58

Binomial Currency Option Example :Multiple Periods to Expiration


S5,5 = 1.6594

Period 1 Period 2 Period 3 Period 4 Period 5

S5,4 = 1.5015

S5,3 = 1.3586

S5,2 = 1.2293

S5,1 = 1.1123

Figure 12.9

12-59

12-60

Marginal Effect of a Parameter Changeon Option Prices


Price Effect onCall OptionCall Price Call Price Call Price Call Price Call Price Ambiguous

effect, dependson rd, rf, and

Price Effect onPut OptionPut Price Put Price Put Price Put Price Put Price Ambiguous

effect, dependson rd, rf, and

VariableSpot Price (S)Exercise price (K)Domestic interest rate (rd)Foreign interest rate (rf )Spot rate volatility ()Time to maturity (t)

Table 12.6

12-61

Chapter 11 Hedging and Insuring

11.9 Options as InsuranceOptions are another ubiquitous form of insurance contract.

Reference: Finance (First Edition) by Zvi Bodie and Robert C. Merton ©1999, ISBN 0-13-310897-X, Prentice Hall

12-62

bond

Put-Call Parity: p0 + S0 = c0 + E/(1+ r)T

25

25

Stock price ($)

Opt

ion

payo

ffs (

$)

Consider the payoffs from holding a portfolio consisting of a call with a strike price of $25 and a bond with a future value of $25.

Call

Portfolio payoffPortfolio value today = c0 +(1+ r)T

E a+c

a

c

12-63

25

25

Stock price ($)

Opt

ion

payo

ffs (

$)

Consider the payoffs from holding a portfolio consisting of a share of stock and a put with a $25 strike.

Portfolio value today = p0 + S0

Portfolio payoff d + g


12-64

Since these portfolios have identical payoffs, they must have the same value today: hence

Put-Call Parity: c0 + E/(1+r)T = p0 + S0

25

25

Stock price ($)

Opt

ion

payo

ffs (

$)

25

25

Stock price ($)O

ptio

n pa

yoff

s ($) Portfolio value today

= p0 + S0

Portfolio value today

(1+ r)T

E= c0 +


12-65

Stocks and Bonds as Options

• It all comes down to put-call parity.

Value of a call on the

firm

Value of a put on the

firm

Value of a risk-free

bond

Value of the firm= + –

Stockholder’s position in terms of call options

Stockholder’s position in terms of put options

c0 = S0 + p0 – (1+ r)TE

12-66Observe: Buy Call and sell Put, you obtain a € Forward.

12-67

What did IRP say about Forward???Yes! (in terms of cost concept) F1,T= borrow in local & pay rd – invest in foreign currency & receive rf= S1exp[(rd-rf)T]

12-68

A synthetic foreign currency forward is borrowing local currency,Exchange for foreign currency, and lend foreign currency till maturity.Or, in terms of net cash inflow, “K(1+rd)T – S(1+rf)T”, or in continuous term “Kexp(-rdT) – S1exp(-rfT)” .

In terms of net cash inflow, buying a call and selling a put nets “–C + P”

12-69

A synthetic foreign currency forward is borrowing local currency,Exchange for foreign currency, and lend foreign currency till maturity.Or, in terms of net cash inflow, “K(1+rd)T – S(1+rf)T”, or in continuous term “Kexp(-rdT) – S1exp(-rfT)” from “leaving no room for arbitrage”.Now equate –C + P = “Kexp(-rdT) – S1exp(-rfT)”.

What did IRP say about Forward???Yes! (in terms of cost concept) F1,T= borrow in local & pay rd – invest in foreign currency & receive rf= S1exp[(rd-rf)T]

12-70

Now maneuver to equate –C + P = “Kexp(-rdT) – S1exp(-rfT)”C – P = S1exp(-rfT) - Kexp(-rdT)={exp(rdT)[S1exp(-rfT) - Kexp(-rdT)]}/exp(rdT)= [S1exp(rd-rf)T – K]/exp(rdT) = (F1,T – K)/exp(rdT)

12-71

“Leaving No Room for Arbitrage” argument!

Organic: Exchange rate S1 evolves to S2!!!

12-72

12-73

Assignments from Chapter 12Exercises 5, 6, 7, 8.

12-74

PUT-CALL-FORWARD PARITY

12.5. Using the Put-Call-Forward Parity, demonstrate that the price of a call with the strike price equal to the futures price is equal to the price of a put with the same strike price and the same maturity.

HINTS:

Put-Call-Forward Parity tells us that:C - P = (F0,T - X) / exp(rdT)If the strike price equals the futures price, or X = F0,T, then C - P = 0, or C = P. Therefore, for a strike price equals to the futures price, the price of the call and the price of the put are equal.

12-75

12.6. Suppose call options on the DM with a strike price of $0.63/DM and maturity of one month are traded at $0.01/DM. One-month futures on the DM are traded at $0.624/DM. One-month US Treasury Bills yield 5.5%. One-month German government securities yield 7.5%. The spot $/DM exchange rate is $0.625/DM.

a. Using the Put-Call-Forward Parity, determine the value of the put option with a strike price of $0.63 and one month maturity.

b. How would you take advantage of arbitrage opportunities if you find that the actual price of the put is below the theoretical price determined using the Parity condition?HINTS: P = C - (F0,T - X) / exp(rdT)

C = .010X = .63F0,T = .624rd = 5.5%T = 1/12P = .010 - (.624 - .63) / exp(.055* 1 / 12) P = .016

b. I believe in my model price (i.e., theoretical price), so I’ll buy from the market if the market price is lower than what my model suggest and I will “sell C, buy F, and sell bond” to capitalize the difference in model price and market price.

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b. How would you take advantage of arbitrage opportunities if you find that the actual price of the put is below the theoretical price determined using the Parity condition?HINTS: P = C - (F0,T - X) / exp(rdT)

C = .010X = .63F0,T = .624rd = 5.5%T = 1/12P = .010 - (.624 - .63) / exp(.055* 1 / 12) P = .016

I believe in my model price (i.e., theoretical price), so I’ll buy from the market if the market price is lower than what my model suggest and I will “sell C, buy F, and sell bond” to capitalize the difference in model price and market price.

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HEDGING12.7. The treasurer of the XYZ company is expecting a dividend payment of

DM 10,000,000 from a German subsidiary in two months. His/her expectations of the future DM spot rate are mixed: The DM could strengthen or stay flat over the next two months. The current exchange rate is $0.63/DM. The two-month futures rate is at $0.6279/DM. The two-month German interest rate is 7.5%. The two-month US T-Bill yields 5.5%. Puts on the DM with maturity of two months and strike price of $0.63 are traded on the CME at $0.0128/DM. Compare the following choices offered to the Treasurer:

• Sell a futures on the DM for delivery in two months for a total amount of DM 10 million.

• Buy 80 put options on the CME with expiration in two months and strike price equal to the current price.

• Set up a forward contract with the firm's bank XYZ.a. What is the respective cost of each strategy?b. Which strategy would best fit the treasurer's mixed forecast for the future

spot rate of the DM?

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HINTS:

a. Strategy One: Sell futures contracts at the current price of .6279. In this case, the US firm is assured to get $ 6,279,000 (DM 10,000,000 * $ .6279/DM). However, it has to deposit the margin requirements and risks having to make payments to maintain the maintenance margin, introducing cash-flow issues in the future.

b. Strategy Two: Buy 80 put options on the DM at a strike price of 63. Total cost: 80 * 125,000 * $0.0128 = $128,000. The firm is assured to get at least $6,172,000 but it reserves itself the right to sell the DM on the spot market and not to exercise its options. Break-even rate is $.6428/DM. If the DM falls below that rate, the firm will be able to sell on the spot market and get more for its DM.

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12.8. Refer to the previous question. Suppose the DM actually rose in value to $0.67/DM when the dividend payment is made.

a. Which of the three strategies enables the treasurer to take advantage of the rise in the DM against the dollar?

b. What is the final gain (loss) incurred in each case?

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HINTS:

a. Strategy Three: Set up a forward contract with bank XYZ. No cash-flows are involved until maturity. However, the firm needs a credit line with its bank.

The only strategy that allows for a potential future gain from a falling DM is the second strategy:

b. If the DM rises to 67, the firm can forego its put option at 63 and actually sell its DM on the spot market. The firm loses the initial $128,000 spent on the put option. It gets $6,700,000 for its DM. Total cash proceeds from the dividend payment: $6,700,000 -128,000 = $6,572,000.

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A Real-World Way to Manage Real OptionsBy Tom Copeland and Peter Tufano

Harvard Business ReviewMarch 2004

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1,000 x 1.2 = 1,200, 1,000 / 1.2 = 833, etc.

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Copano’s Decision Tree

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12-86

12-87

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Replicating a call option by borrowing some money from the bank and buy some units of the underlying asset, i.e., by “M (1,440) - B”:

Contingent claims of a synthesized / replicated call option: M (1,728) – (1 + .08) (B) = 928M (1,200) – (1 + .08) (B) = 400M=1, B=741Asset value at current time:1 * 1,440 – 741 = 699

Two Years

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Contingent claims of a synthesized / replicated call option: M (1,200) – (1 + .08) (B) = 400M (833) – (1 + .08) (B) = 33M=1, B=741Asset value at current time:1 * 1,000 – 741 = 259

Two Years

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Replicating a call option by borrowing some money from the bank and buy some units of the underlying asset, i.e., by “M (694) - B”:

Contingent claims of a synthesized / replicated call option: M (833) – (1 + .08) (B) = 33M (579) – (1 + .08) (B) = 0M=0.13, B=70Asset value at current time:0.13 * 694 – 70 = 20

Two Years

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Contingent claims of a synthesized / replicated call option: M (1,400) – (1 + .08) (B) = 699M (1,000) – (1 + .08) (B) = 259M=1, B=686Asset value at current time:1 * 1,200 – 686 = 514

One Year

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Replicating a call option by borrowing some money from the bank and buy some units of the underlying asset, i.e., by “M (833) - B”:

Contingent claims of a synthesized / replicated call option: M (1,000) – (1 + .08) (B) = 259M (694) – (1 + .08) (B) = 21M=.777, B=479Asset value at current time:.777 * 833 – 479 = 169

One Year

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Contingent claims of a synthesized / replicated call option: M (1,200) – (1 + .08) (B) = 114M (833) – (1 + .08) (B) = 0M=.31, B=239Asset value at current time:.31 * 1,000 – 239 = 71

Today

12-94

Real Option CalculationBy YTF

M (1,728) – (1 + .08) (B) = 928M (1,200) – (1 + .08) (B) = 400• M=1, B=741• 1 * 1,440 – 741 = 699M (1,200) – (1 + .08) (B) = 400M (833) – (1 + .08) (B) = 33• M = 1, B=741• 1 * 1,000 – 741 = 259M (833) – (1 + .08) (B) = 33M (579) M (579) –– (1 + .08) (B) = 0(1 + .08) (B) = 0• M = 0.13, B = 70• 0.13 * 694 – 70 = 20

M (1,440) – (1 + .08) (B) = 699M (1,000) – (1 + .08) (B) = 259• M = 1, B = 686• 1 * 1,200 – 686 = 514M (1,000) – (1 + .08) (B) = 259M (694) – (1 + .08) (B) = 21• M = .777, B = 479• .777 * 833 – 479 = 169M (1,200) – (1 + .08) (B) = 114M (833) M (833) –– (1 + .08) (B) = 0(1 + .08) (B) = 0• M = .31, B = 239• .31 * 1,000 – 239 = 71

From Event Tree Strategic Decision with flexibilityStrategic Decision with flexibility

12-95

Cases forMS&E 247S International Investments

Summer 2006

Topic 8: International Financial Innovations

Case—Deutsche Bank: Discussing the Equity Risk Premium.

Case—Swedish Lottery Bonds. Case—Bank Leu’s Prima Cat Bond Fond. Case—Catastrophe Bonds at Swiss Re. Case—Mortgage Backs at Ticonderoga. Case—KAMCO and the Cross-Border Securitization of

Korean Non-Performing Loans. Case—Nexgen: Structuring Collateralized Debt

Obligations (CDOs). Case—The Enron Odyssey (A):The Special Purpose of

SPEs.

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Learning Objective:To introduce students to the use of equity derivatives in the context of money management, particularly the use of delta-hedging using options. Also, to learn about how and why risk management decisions are made in a simple levered hedge fund.

Description of Pine Street Capital:A technology hedge fund is trying to decide whether and/or how to hedge equity market risk. Its hedging choices are short-selling and options. The fund has just gone through one of the most volatile periods in NASDAQ's history, it is trying to decide whether it should continue its risk management program of short-selling the NASDAQ index or switch to a hedging program utilizing put options on the index.

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Setting: San Francisco, CA; Financial services; $1.3 million revenues; 4 employees; 2000

Subjects Covered:Derivatives, Financial strategy, Hedging, Investment management, Options, Risk management.

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Google Earnings Gave Options Traders a 17,530% GainBy Michael Patterson and Jeff KearnsApril 18 (Bloomberg) -- Options traders who predicted Google Inc. would beat estimates earned as much as 17,530 percent on their investments today, the most-profitable bet among all U.S. equity derivatives.Contracts giving the right to buy Google shares for $530 before the close of trading today jumped as high as $17.63 from their 10-cent closing price yesterday. That gain almost matched the 18,760 percent advance in the Dow Jones Industrial Average since the beginning of 1900, according to Bloomberg data.Google call-option volume jumped to 311,139 contracts, the most since January 2006. Those contracts outnumbered trading in bearish bets, or puts, by 1.6-to-1. Call options give the right to buy a security for a certain amount, called the strike price, by a given date. Puts convey the right to sell.

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12-101

Options

Review

Ross Corporate Finance 7E

Chapter 17

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Key Concepts and Skills

• Understand option terminology• Be able to determine option payoffs and

profits• Understand the major determinants of option

prices• Understand and apply put-call parity• Be able to determine option prices using the

binomial and Black-Scholes models

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Chapter Outline

17.1 Options17.2 Call Options17.3 Put Options17.4 Selling Options17.5 Option Quotes17.6 Combinations of Options17.7 Valuing Options17.8 An Option Pricing Formula17.9 Stocks and Bonds as Options17.10 Options and Corporate Decisions: Some Applications17.11 Investment in Real Projects and Options

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17.1 Options

• An option gives the holder the right, but not the obligation, to buy or sell a given quantity of an asset on (or before) a given date, at prices agreed upon today.

• Exercising the Option¤ The act of buying or selling the underlying asset

• Strike Price or Exercise Price¤ Refers to the fixed price in the option contract at which

the holder can buy or sell the underlying asset.• Expiry (Expiration Date)

¤ The maturity date of the option

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Options

• European versus American options¤ European options can be exercised only at expiry.¤ American options can be exercised at any time up to

expiry.• In-the-Money

¤ Exercising the option would result in a positive payoff. • At-the-Money

¤ Exercising the option would result in a zero payoff (i.e., exercise price equal to spot price).

• Out-of-the-Money¤ Exercising the option would result in a negative payoff.

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17.2 Call Options

• Call options gives the holder the right, but not the obligation, to buy a given quantity of some asset on or before some time in the future, at prices agreed upon today.

• When exercising a call option, you “call in” the asset.

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Call Option Pricing 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:

C = Max[ST – E, 0]Where

ST is the value of the stock at expiry (time T)E is the exercise price.C is the value of the call option at expiry

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Call Option Payoffs

–20

12020 40 60 80 100

–40

20

40

60

Stock price ($)

Opt

ion

payo

ffs (

$) Buy a

call

Exercise price = $50

50

12-109

Call Option Profits

Exercise price = $50; option premium = $10

Buy a call

–20

12020 40 60 80 100

–40

20

40

60

Stock price ($)

Opt

ion

payo

ffs (

$)

50–10

10

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17.3 Put Options

• Put options gives the holder the right, but not the obligation, to sell a given quantity of an asset on or before some time in the future, at prices agreed upon today.

• When exercising a put, you “put” the asset to someone.

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Put Option Pricing 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.

P = Max[E – ST, 0]

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Put Option Payoffs

–20

0 20 40 60 80 100

–40

20

0

40

60

Stock price ($)

Opt

ion

payo

ffs (

$)

Buy a put


50

50

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Put Option Profits

–20

20 40 60 80 100

–40

20

40

60

Stock price ($)

Opt

ion

payo

ffs (

$)

Buy a put

Exercise price = $50; option premium = $10

–10

10

50

12-114

Option Value

• Intrinsic Value¤ Call: Max[ST – E, 0]¤ Put: Max[E – ST , 0]

• Speculative Value¤ The difference between the option premium

and the intrinsic value of the option.

Option Premium = Intrinsic

ValueSpeculative

Value+

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17.4 Selling Options

• The seller (or writer) of an option has an obligation.

• The seller receives the option premium in exchange.

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Call Option Payoffs

–20

12020 40 60 80 100

–40

20

40

60

Stock price ($)

Opt

ion

payo

ffs (

$)

Sell a call


50

12-117

Put Option Payoffs

–20

0 20 40 60 80 100

–40

20

0

40

–50

Stock price ($)

Opt

ion

payo

ffs (

$)

Sell a put


50

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Option Diagrams Revisited

Exercise price = $50; option premium = $10 Sell a call

Buy a call

50 6040 100

–40

40

Stock price ($)

Opt

ion

payo

ffs (

$)

Buy a put

Sell a put

–10

10

Buy a call

Sell a

put

Buy a put

Sell a call

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17.5 Option Quotes

Option/Strike Exp. Vol. Last Vol. LastIBM 130 Oct 364 15¼ 107 5¼138¼ 130 Jan 112 19½ 420 9¼138¼ 135 Jul 2365 4¾ 2431 13/16138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

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Option Quotes


--Put----Call--

This option has a strike price of $135;

a recent price for the stock is $138.25;July is the expiration month.

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Option Quotes


--Put----Call--

This makes a call option with this exercise price in-the-money by $3.25 = $138¼ – $135.

Puts with this exercise price are out-of-the-money.

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Option Quotes


--Put----Call--

On this day, 2,365 call options with this exercise price were traded.

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Option Quotes


--Put----Call--

The CALL option with a strike price of $135 is trading for $4.75.

Since the option is on 100 shares of stock, buying this option would cost $475 plus commissions.

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Option Quotes


--Put----Call--

On this day, 2,431 put options with this exercise price were traded.

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Option Quotes


--Put----Call--

Since the option is on 100 shares of stock, buying this option would cost $81.25 plus commissions.

The PUT option with a strike price of $135 is trading for $.8125.

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17.6 Combinations of Options

• Puts and calls can serve as the building blocks for more complex option contracts.

• If you understand this, you can become a financial engineer, tailoring the risk-return profile to meet your client’s needs.

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Long Straddle

30 40 60 70

30

40

Stock price ($)

Opt

ion

payo

ffs (

$)

Buy a put with exercise price of $50 for $10

Buy a call with exercise price of $50 for $10

A Long Straddle only makes money if the stock price moves $20 away from $50.

$50

–20

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Short Straddle

–30

30 40 60 70

–40

Stock price ($)

Opt

ion

payo

ffs (

$)

$50

This Short Straddle only loses money if the stock price moves $20 away from $50.

Sell a put with exercise price of$50 for $10

Sell a call with an exercise price of $50 for $10

20

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Put-Call Parity

Since these portfolios have identical payoffs, they must have the same value today: hence

Put-Call Parity: c0 + E/(1+r)T = p0 + S0

25

25

Stock price ($)

Opt

ion

payo

ffs (

$)

25

25

Stock price ($)O

ptio

n pa

yoff

s ($) Portfolio value today

= p0 + S0

Portfolio value today

(1+ r)T

E= c0 +

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17.7 Valuing Options

• The last section concerned itself with the value of an option at expiry.

• This section considers the value of an option prior to the expiration date.¤ A much more

interesting question.

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American Call

C0 must fall within max (S0 – E, 0) < C0 < S0.

25

Opt

ion

payo

ffs (

$) Call ST

loss

E

Profit

ST

Time valueIntrinsic value

Market Value

In-the-moneyOut-of-the-money

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Option Value Determinants

Call Put1. Stock price + –2. Exercise price – +3. Interest rate + –4. Volatility in the stock price + +5. Expiration date + +The value of a call option C0 must fall within

max (S0 – E, 0) < C0 < S0.The precise position will depend on these

factors.

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17.8 An Option Pricing Formula

• We will start with a binomial option pricing formula to build our intuition.

• Then we will graduate to the normal approximation to the binomial for some real-world option valuation.

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Binomial Option Pricing Model

Suppose a stock is worth $25 today and in one period will eitherbe worth 15% more or 15% less. S0= $25 today and in one year S1is either $28.75 or $21.25. The risk-free rate is 5%. What is the value of an at-the-money call option?

$25

$21.25 = $25×(1 –.15)

$28.75 = $25×(1.15)S1S0

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1. A call option on this stock with exercise price of $25 will have the following payoffs.

2. We can replicate the payoffs of the call option with a levered position in the stock.

$25

$21.25

$28.75S1S0 C1

$3.75

$0

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Binomial Option Pricing ModelBorrow the present value of $21.25 today and buy 1 share. The net payoff for this levered equity portfolio in one period is either $7.50 or $0. The levered equity portfolio has twice the option’s payoff, so the portfolio is worth twice the call option value.

$25

$21.25

$28.75S1S0 debt

– $21.25portfolio$7.50

$0

( – ) ==

=

C1$3.75

$0– $21.25

12-137


The value today of the levered equity portfolio is today’s value of one share less the present value of a $21.25 debt: )1(

25.21$25$fr

$25

$21.25

$28.75S1S0 debt


$0

( – ) ==

=

C1$3.75

$0– $21.25

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We can value the call option today as half of the value of the levered equity portfolio:

)1(25.21$25$

21

0fr

C

$25

$21.25

$28.75S1S0 debt


$0

( – ) ==

=

C1$3.75

$0– $21.25

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If the interest rate is 5%, the call is worth:


38.2$24.2025$21

)05.1(25.21$25$

21

0

C

$25

$21.25

$28.75S1S0 debt


$0

( – ) ==

=

C1$3.75

$0– $21.25

$2.38

C0

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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.

The most important lesson (so far) from the binomial option pricing model is:

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Delta

• 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.

21

5.7$75.3$

25.21$75.28$075.3$

Swing of callSwing of stock

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Delta

• Determining the Amount of Borrowing:

Value of a call = Stock price × Delta – Amount borrowed

$2.38 = $25 × ½ – Amount borrowedAmount borrowed = $10.12

38.2$24.20$25$21

)05.1(25.21$25$

21

0

C

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The Risk-Neutral Approach

We could value the option, V(0), as the value of the replicating portfolio. An equivalent method is risk-neutral valuation:

S(0), V(0)

S(U), V(U)

S(D), V(D)

q

1- q

)1()()1()()0(

frDVqUVqV

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S(0) is the value of the underlying asset today.

S(0), V(0)

S(U), V(U)

S(D), V(D)

S(U) and S(D) are the values of the asset in the next period following an up move and a down move, respectively.

q

1- q

V(U) and V(D) are the values of the option in the next period following an up move and a down move, respectively.

q is the risk-neutral probability of an “up” move.

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• The key to finding q is to note that it is already impounded into an observable security price: the value of S(0):

S(0), V(0)

S(U), V(U)

S(D), V(D)

q

1- q

)1()()1()()0(

frDVqUVqV

)1()()1()()0(

frDSqUSqS

A minor bit of algebra yields:)()(

)()0()1(DSUS

DSSrq f

12-146

Example of Risk-Neutral Valuation

$21.25,C(D)

q

1- q

Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. The risk-free rate is 5%. What is the value of an at-the-money call option?The binomial tree would look like this:

$25,C(0)

$28.75,C(U)

)15.1(25$75.28$

)15.1(25$25.21$

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$21.25,C(D)

2/3

1/3

The next step would be to compute the risk neutral probabilities

$25,C(0)

$28.75,C(U)

)()()()0()1(

DSUSDSSr

q f

3250.7$5$

25.21$75.28$25.21$25$)05.1(

q

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$21.25, $0

2/3

1/3

After that, find the value of the call in the up state and down state.

$25,C(0)

$28.75, $3.75

25$75.28$)( UC

]0,75.28$25max[$)( DC

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Finally, find the value of the call at time 0:

$21.25, $0

2/3

1/3

$25,C(0)

$28.75,$3.75

)1()()1()()0(

frDCqUCqC

)05.1(0$)31(75.3$32)0(

C

38.2$)05.1(

50.2$)0( C

$25,$2.38

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This risk-neutral result is consistent with valuing the call using a replicating portfolio.

Risk-Neutral Valuation and the Replicating Portfolio

38.2$24.2025$21

)05.1(25.21$25$

21

0

C

38.2$05.150.2$

)05.1(0$)31(75.3$32

0

C

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The Black-Scholes Model

)N()N( 210 dEedSC rT

WhereC0 = the value of a European option at time t = 0r = the risk-free interest rate.

T

TσrESd

)2

()/ln(2

1

Tdd 12

N(d) = Probability that a standardized, normally distributed, random variable will be less than or equal to d.

The Black-Scholes Model allows us to value options in the real world just as we have done in the 2-state world.

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Find the value of a six-month call option on Microsoft with an exercise price of $150.

The current value of a share of Microsoft is $160.The interest rate available in the U.S. is r = 5%.The option maturity is 6 months (half of a year).The volatility of the underlying asset is 30% per annum.Before we start, note that the intrinsic value of the

option is $10—our answer must be at least that amount.

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

Then,

TTσrESd

)5.()/ln( 2

1

First calculate d1 and d2

31602.05.30.052815.012 Tdd

52815.05.30.0

5).)30.0(5.05(.)150/160ln( 2

1

d

12-154


N(d1) = N(0.52815) = 0.7013N(d2) = N(0.31602) = 0.62401

52815.01 d31602.02 d

)N()N( 210 dEedSC rT

92.20$62401.01507013.0160$

0

5.05.0

CeC

12-155

17.9 Stocks and Bonds as Options

• Levered equity is a call option.¤ The underlying asset comprises the assets of the

firm.¤ The strike price is the payoff of the bond.

• If at the maturity of their debt, the assets of the firm are greater in value than the debt, the shareholders have an in-the-money call. They will pay the bondholders and “call in” the assets of the firm.

• If at the maturity of the debt the shareholders have an out-of-the-money call, they will not pay the bondholders (i.e. the shareholders will declare bankruptcy) and let the call expire.

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• Levered equity is a put option.¤ The underlying asset comprises the assets of the firm.¤ The strike price is the payoff of the bond.

• If at the maturity of their debt, the assets of the firm are less in value than the debt, shareholders have an in-the-money put.

• They will put the firm to the bondholders.• If at the maturity of the debt the shareholders have an

out-of-the-money put, they will not exercise the option (i.e. NOT declare bankruptcy) and let the put expire.

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• It all comes down to put-call parity.

Value of a call on the

firm

Value of a put on the

firm

Value of a risk-free

bond

Value of the firm= + –

Stockholder’s position in terms of call options

Stockholder’s position in terms of put options

c0 = S0 + p0 – (1+ r)TE

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Mergers and Diversification

• Diversification is a frequently mentioned reason for mergers.

• Diversification reduces risk and, therefore, volatility.• Decreasing volatility decreases the value of an option.• Assume diversification is the only benefit to a merger:

¤ Since equity can be viewed as a call option, should the merger increase or decrease the value of the equity?

¤ Since risky debt can be viewed as risk-free debt minus a put option, what happens to the value of the risky debt?

¤ Overall, what has happened with the merger and is it a good decision in view of the goal of stockholder wealth maximization?

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Example

• Consider the following two merger candidates.• The merger is for diversification purposes only with no

synergies involved.• Risk-free rate is 4%.

50%40%Asset return standard deviation

4 years4 yearsDebt maturity

$7 million$18 millionFace value of zero coupon debt

$15 million$40 millionMarket value of assets

Company BCompany A

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Example

• Use the Black and Scholes OPM (or an options calculator) to compute the value of the equity.

• Value of the debt = value of assets – value of equity

5.1214.28Market Value of Debt

9.8825.72Market Value of EquityCompany BCompany A

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Example

• The asset return standard deviation for the combined firm is 30%

• Market value assets (combined) = 40 + 15 = 55• Face value debt (combined) = 18 + 7 = 25

20.82Market value of debt

34.18Market value of equity

Combined Firm

Total MV of equity of separate firms = 25.72 + 9.88 = 35.60

Wealth transfer from stockholders to bondholders = 35.60 – 34.18 = 1.42 (exact increase in MV of debt)

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M&A Conclusions

• Mergers for diversification only transfer wealth from the stockholders to the bondholders.

• The standard deviation of returns on the assets is reduced, thereby reducing the option value of the equity.

• If management’s goal is to maximize stockholder wealth, then mergers for reasons of diversification should not occur.

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Options and Capital Budgeting

• Stockholders may prefer low NPV projects to high NPV projects if the firm is highly leveraged and the low NPV project increases volatility.

• Consider a company with the following characteristics:¤ MV assets = 40 million¤ Face Value debt = 25 million¤ Debt maturity = 5 years¤ Asset return standard deviation = 40%¤ Risk-free rate = 4%

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Example: Low NPV

• Current market value of equity = $22.706 million• Current market value of debt = $17.294 million

$15.169$19.169MV of debt$25.381$23.831MV of equity

50%30%Asset return standard deviation

$41$43MV of assets$1$3NPV

Project IIProject I

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Example: Low NPV

• Which project should management take?• Even though project B has a lower NPV, it is

better for stockholders.• The firm has a relatively high amount of

leverage:¤ With project A, the bondholders share in the

NPV because it reduces the risk of bankruptcy.¤ With project B, the stockholders actually

appropriate additional wealth from the bondholders for a larger gain in value.

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Example: Negative NPV

• We’ve seen that stockholders might prefer a low NPV to a high one, but would they ever prefer a negative NPV?

• Under certain circumstances, they might.• If the firm is highly leveraged, stockholders

have nothing to lose if a project fails, and everything to gain if it succeeds.

• Consequently, they may prefer a very risky project with a negative NPV but high potential rewards.

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• Consider the previous firm.• They have one additional project they are

considering with the following characteristics¤ Project NPV = -$2 million¤ MV of assets = $38 million¤ Asset return standard deviation = 65%

• Estimate the value of the debt and equity¤ MV equity = $25.453 million¤ MV debt = $12.547 million

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• In this case, stockholders would actually prefer the negative NPV project to either of the positive NPV projects.

• The stockholders benefit from the increased volatility associated with the project even if the expected NPV is negative.

• This happens because of the large levels of leverage.

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Options and Capital Budgeting

• As a general rule, managers should not accept low or negative NPV projects and pass up high NPV projects.

• Under certain circumstances, however, this may benefit stockholders:¤ The firm is highly leveraged¤ The low or negative NPV project causes a

substantial increase in the standard deviation of asset returns

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17.12 Investment in Real Projects and Options

• Classic NPV calculations generally ignore the flexibility that real-world firms typically have.

• Quick Quiz for Chapter 17 of Ross Corporate Finance

• What is the difference between call and put options? • What are the major determinants of option prices?• What is put-call parity? What would happen if it doesn’t hold?• What is the Black-Scholes option pricing model?• How can equity be viewed as a call option?• Should a firm do a merger for diversification purposes only?

Why or why not?• Should management ever accept a negative NPV project? If

yes, under what circumstances?

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We will now study Duke Professor Michael Brandt’s Interest Rate Models. Please see the notes posted to out class website.

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Reverse mortgage (Bank of America site):http://reversemortgage.bankofamerica.com/?cm_mmc=CRE-SEMAX-_-Google-PS-_-reverse%20mortgage-_-SECN_Mortgage

Currency and Interest Rate Options - Stanford Universityweb.stanford.edu/class/msande247s/2009/1110 2009 po… · · 2009-11-10Currency and Interest Rate Options M. Brandt Reading

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