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MS&E247s International Investments Handout #15 Page 1 of 26
Modified on 0806 2008Wednesday July 30, 2008
Course web page: http://Stanford2008.pageout.net
Handout #15 as of 0806 2008
Derivative Security MarketsCurrency and Interest Rate Options
MWF 3:15-4:30 Gates B01
12-2
Levich
Luenberger
Solnik
Blanchard
Chap 12
Chap
Chap 10
Chap
Scan Read
Pages
Pages
Pages 433-483
Pages
Chap Pages
Currency and Interest Rate Options
Wooldridge
Reading Assignments for this Week
12-3
Midterm Exam: Friday Aug 1, 2008
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.
Remote SCPD participants will also take the exam on Friday, August 1, 2008
Please Submit Exam Proctor’s Name, Contact info as SCPD requires, also c.c. to yeetienfu@yahoo.com, preferably a week
before the exam.12-4
Final Exam MS&E 247S Fri Aug 15 2008 12:15PM-3:15PM Gates B03 (alternate arrangement)
Or Saturday Aug 16 2008 7PM-10PM Terman Auditorium (official date/time)Remote SCPD participants will also take the exam on Friday, 8/15
Please Submit Exam Proctor’s Name, Contact info as SCPD requires, also c.c. to yeetienfu@yahoo.com, preferably a week before the exam.Local SCPD students please come to Stanford to take the exam. Light
refreshments will be served.
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
MS&E247s International Investments Handout #15 Page 2 of 26
Modified on 0806 2008Wednesday July 30, 2008
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
The Building Blocks of Contingent Decisions
(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 Building Blocks of Contingent Decisions
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.
Real Options: Amram & Kulatilaka Figure 4.1 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
Real Options: Amram & Kulatilaka Figure 4.2
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).
MS&E247s International Investments Handout #15 Page 3 of 26
Modified on 0806 2008Wednesday July 30, 2008
12-13
Identifying the Building Blocksin Red & White’s Contract
Real Options: Amram & Kulatilaka Figure 4.3
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
Identifying the Building Blocksin Red & White’s Contract
(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
+
+
=
Real Options: Amram & Kulatilaka Figure 4.3
12-15
Identifying the Building Blocksin Red & White’s Contract
Real Options: Amram & Kulatilaka Figure 4.3
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
MS&E247s International Investments Handout #15 Page 4 of 26
Modified on 0806 2008Wednesday July 30, 2008
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
ExamplesAn 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.
MS&E247s International Investments Handout #15 Page 5 of 26
Modified on 0806 2008Wednesday July 30, 2008
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
$
$/£
Profit and Loss Positions
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 $1.50/£
MS&E247s International Investments Handout #15 Page 6 of 26
Modified on 0806 2008Wednesday July 30, 2008
12-31
• Note that the buyer of either a put or call faces limited liability in that her loss is capped at the initial option premium paid.
• The seller of a call faces unlimited liability as the underlying asset could appreciate without limit.
• The seller of a put faces a large liability, which is limited by the fact that the price of the underlying cannot fall below zero.
Spot Currency Options:Prices at Maturity
12-32
Interest Rate Futures Options:Prices at Maturity
Consider the following call and put options on a Eurodollar interest rate futures contract :
• Each option has a strike price (K) of 96.00 (corresponding to a 4.00 % Euro$ 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
Profit and Loss Positions
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
Profit and Loss Positions
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
MS&E247s International Investments Handout #15 Page 7 of 26
Modified on 0806 2008Wednesday July 30, 2008
12-37
A Short Straddle in the $/£
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
$
$/£
Profit and Loss Positions
Sell PutSell Call
Sell CallSell Put
StraddleStraddle
Box 12.1 Figure A 12-38
A Long Straddle in the $/£
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
$
$/£
Profit and Loss Positions
Buy CallBuy Put
Buy PutBuy Call
StraddleStraddle
Box 12.1 Figure B
12-39
Using a Call to Hedge a Short Pound
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
Buy CallShort Pound
Short Pound
Buy Call
CombinedPosition
Box 12.1 Figure C 12-40
Using a Put to Hedge a Long Pound
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
Buy Put
Long Pound
Long Pound
Buy Put
CombinedPosition
Box 12.1 Figure D
12-41
Combining a Floor and Cap : A Collar
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 12-42
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
MS&E247s International Investments Handout #15 Page 8 of 26
Modified on 0806 2008Wednesday July 30, 2008
12-43
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
Value of stock at expiry
-$10
12-44
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
Value of stock at expiry
-$30
$10
12-45
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 $60 on the expiration date.
12-46
Payoff from a Butterfly Spread
Stock price Payoff from Payoff from Payoff from Totalrange first long call second long call short calls payoff *
ST < X1 0 0 0 0
X1 < ST < X2 ST – X1 0 0 ST – X1
X2 < ST < X3 ST – X1 0 –2(ST – X2) X3 – ST
ST > X3 ST – X1 ST – X3 –2(ST – X2) 0
* These payoffs are calculated using the relationship X2 = 0.5(X1 + X3).
12-47
Butterfly Spread Using Call Options
X1 X3
Profit
STX2
12-48
Butterfly Spread Using Put Options
X1 X3
Profit
STX2
MS&E247s International Investments Handout #15 Page 9 of 26
Modified on 0806 2008Wednesday July 30, 2008
12-49
Short Straddle
–30
30 40 60 70
–40
Stock price ($)
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 $10
20
12-50
What are the similarities and differences between the payoffs of the butterfly spread and the short straddle?
12-51
Option Pricing : Formal Models
S1 = 1.40
Period 1
S2,u = 1.5015
S2,d = 1.2293
Spot Rates
Period 2
Option Values
Call2,u = 0.1015Put2,u = 0.0
Call2,d = 0.0Put2,d = 0.1707
Binomial Currency Option Example
Figure 12.8 12-52
Binomial Currency Option Example :Multiple Periods to Expiration
Option Pricing : Formal Models
S5,5 = 1.6594
Period 1 Period 2 Period 3 Period 4 Period 5
S5,4 = 1.5015
S5,3 = 1.3586
S5,2 = 1.2293
S5,1 = 1.1123
Figure 12.9
12-53
Marginal Effect of a Parameter Changeon Option Prices
Option Pricing : Formal Models
Price Effect onCall OptionCall Price ↑Call Price ↓Call Price ↑Call Price ↓Call Price ↑Ambiguous
effect, dependson rd, rf, and σ
Price Effect onPut OptionPut Price ↓Put Price ↑Put Price ↓Put Price ↑Put Price ↑Ambiguous
effect, dependson rd, rf, and σ
VariableSpot Price ↑ (S↑)Exercise price ↑ (K↑)Domestic interest rate ↑ (rd↑)Foreign interest rate ↑ (rf ↑)Spot rate volatility ↑ (σ↑)Time to maturity ↑ (t↑)
Table 12.6 12-54
Chapter 11 Hedging and Insuring
11.9 Options as InsuranceOptions are another ubiquitous form of insurance contract.
Reference: Finance (First Edition) by Zvi Bodie and Robert C. Merton ©1999, ISBN 0-13-310897-X, Prentice Hall
MS&E247s International Investments Handout #15 Page 10 of 26
Modified on 0806 2008Wednesday July 30, 2008
12-55
bond
Put-Call Parity: p0 + S0 = c0 + E/(1+ r)T
25
25
Stock price ($)
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 $25.
Call
Portfolio payoffPortfolio value today = c0 +(1+ r)T
E a+c
a
c
12-56
25
25
Stock price ($)
Opt
ion
payo
ffs (
$)
Consider the payoffs from holding a portfolio consisting of a share of stock and a put with a $25 strike.
Portfolio value today = p0 + S0
Portfolio payoff d + g
Put-Call Parity: p0 + S0 = c0 + E/(1+ r)T
12-57
Since these portfolios have identical payoffs, they must have the same value today: hence
Put-Call Parity: c0 + E/(1+r)T = p0 + S0
25
25
Stock price ($)
Opt
ion
payo
ffs (
$)
25
25
Stock price ($)
Opt
ion
payo
ffs (
$) Portfolio value today = p0 + S0
Portfolio value today
(1+ r)T
E= c0 +
Put-Call Parity: p0 + S0 = c0 + E/(1+ r)T
12-58Buy Call and sell Put, you obtain a € Forward.
12-59
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 “K exp(-rd T) – S1 exp(-rf T)” .
In terms of net cash inflow, buying a call and selling a put nets “–C + P”
12-60
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 “K exp(-rd T) – S1 exp(-rf T)” from “no arbitrage”.Now equate –C + P = “K exp(-rd T) – S1 exp(-rf T)”.
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= S1 exp[(rd-rf)T]
MS&E247s International Investments Handout #15 Page 11 of 26
Modified on 0806 2008Wednesday July 30, 2008
12-61
Now maneuver equate –C + P = “K exp(-rd T) – S1 exp(-rf T)”C – P = S1 exp(-rf T) - K exp(-rd T)={exp(rd T) [S1 exp(-rf T) - K exp(-rd T)]}/exp(rd T)= [S1 exp(rd-rf)T – K]/exp(rd T) = (F1,T – K)/exp(rd T)
12-62
“Leaving No Room for Arbitrage” argument!
Organic: Exchange rate S1 evolves to S2!!!
12-63 12-64
Assignments from Chapter 12Exercises 5, 6, 7, 8.
12-65
PUT-CALL-FORWARD PARITY
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.
SOLUTION:
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.
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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?
SOLUTION: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
MS&E247s International Investments Handout #15 Page 12 of 26
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HEDGING7. 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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SOLUTION:
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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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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SOLUTION:
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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MS&E247s International Investments Handout #15 Page 13 of 26
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12-73
1,000 x 1.2 = 1,200, 1,000 / 1.2 = 833, etc.12-74
Copano’s Decision Tree
12-75 12-76
12-77 12-78
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
MS&E247s International Investments Handout #15 Page 14 of 26
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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.12-80
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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12-83 12-84
Options
Review
Ross Corporate Finance 7E
Chapter 17
MS&E247s International Investments Handout #15 Page 15 of 26
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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
MS&E247s International Investments Handout #15 Page 16 of 26
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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
12-92
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
Exercise price = $50
50
50
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Put Option Profits
–20
20 40 60 80 100
–40
20
40
60
Stock price ($)
Opt
ion
payo
ffs (
$)
Buy a put
Exercise price = $50; option premium = $10
–10
10
50
MS&E247s International Investments Handout #15 Page 17 of 26
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Option Value
• Intrinsic Value¤ Call: Max[ST – E, 0]¤ Put: Max[E – ST , 0]
• Speculative Value¤ The difference between the option premium
and the intrinsic value of the option.
Option Premium = Intrinsic
ValueSpeculative
Value+
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17.4 Selling Options
• The seller (or writer) of an option has an obligation.
• The seller receives the option premium in exchange.
12-99
Call Option Payoffs
–20
12020 40 60 80 100
–40
20
40
60
Stock price ($)
Opt
ion
payo
ffs (
$)
Sell a call
Exercise price = $50
50
12-100
Put Option Payoffs
–20
0 20 40 60 80 100
–40
20
0
40
–50
Stock price ($)
Opt
ion
payo
ffs (
$)
Sell a put
Exercise price = $50
50
12-101
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 p
ut
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--
MS&E247s International Investments Handout #15 Page 18 of 26
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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--
This option has a strike price of $135;
a recent price for the stock is $138.25;July is the expiration month.
12-104
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--
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.
12-105
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--
On this day, 2,365 call options with this exercise price were traded.
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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--
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
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--
On this day, 2,431 put options with this exercise price were traded.
12-108
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--
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 $.8125.
MS&E247s International Investments Handout #15 Page 19 of 26
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17.6 Combinations of Options
• Puts and calls can serve as the building blocks for more complex option contracts.
• If you understand this, you can become a financial engineer, tailoring the risk-return profile to meet your client’s needs.
12-110
Long Straddle
30 40 60 70
30
40
Stock price ($)
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.
$50
–20
12-111
Short Straddle
–30
30 40 60 70
–40
Stock price ($)
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 $10
20
12-112
Put-Call Parity
Since these portfolios have identical payoffs, they must have the same value today: hence
Put-Call Parity: c0 + E/(1+r)T = p0 + S0
25
25
Stock price ($)
Opt
ion
payo
ffs (
$)
25
25
Stock price ($)
Opt
ion
payo
ffs (
$) Portfolio value today = p0 + S0
Portfolio value today
(1+ r)T
E= c0 +
12-113
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.
12-114
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
MS&E247s International Investments Handout #15 Page 20 of 26
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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.
12-116
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.
12-117
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
12-118
Binomial Option Pricing Model
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
12-119
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– $21.2512-120
Binomial Option Pricing Model
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
– $21.25portfolio$7.50
$0
( – ) ==
=
C1$3.75
$0– $21.25
MS&E247s International Investments Handout #15 Page 21 of 26
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Binomial Option Pricing Model
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
– $21.25portfolio$7.50
$0
( – ) ==
=
C1$3.75
$0– $21.2512-122
If the interest rate is 5%, the call is worth:
Binomial Option Pricing Model
( ) 38.2$24.2025$21
)05.1(25.21$25$
21
0 =−=⎟⎟⎠
⎞⎜⎜⎝
⎛−=C
$25
$21.25
$28.75S1S0 debt
– $21.25portfolio$7.50
$0
( – ) ==
=
C1$3.75
$0– $21.25
$2.38
C0
12-123
Binomial Option Pricing Model
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:
12-124
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$075.3$
==−
−=Δ =
Swing of callSwing of stock
12-125
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
12-126
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
+×−+×
=
MS&E247s International Investments Handout #15 Page 22 of 26
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The Risk-Neutral Approach
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 Risk-Neutral Approach
• 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
−
−×+=
12-129
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$25.21$ −×=
12-130
Example of Risk-Neutral Valuation
$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
12-131
Example of Risk-Neutral Valuation
$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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Example of Risk-Neutral Valuation
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
MS&E247s International Investments Handout #15 Page 23 of 26
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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
12-134
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
σ
)2
()/ln(2
1
++=
Tdd σ−= 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.
12-135
The Black-Scholes Model
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.
12-136
The Black-Scholes Model
Let’s try our hand at using the model. If you have a calculator handy, follow along.
Then,
TTσrESd
σ)5.()/ln( 2
1++
=
First calculate d1 and d2
31602.05.30.052815.012 =−=−= Tdd σ
52815.05.30.0
5).)30.0(5.05(.)150/160ln( 2
1 =++
=d
12-137
The Black-Scholes Model
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
12-138
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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Stocks and Bonds as Options
• 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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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
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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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Example: Negative NPV
• 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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Example: Negative NPV
• 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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