chapter 15: options and contingent claims

1

FinanceFinance School of Management School of Management

Chapter 15: Options and Chapter 15: Options and Contingent ClaimsContingent Claims

Objective• To show how the law of one price may

be used to derive prices of options• To explore the range of financial decisions

that can be fruitfully analyzed in terms of options

2


Chapter 15 ContentsChapter 15 Contents

How Options Work Investing with Options The Put-Call Parity

Relationship Volatility & Option Prices Two-State Option Pricing Dynamic Replication &

the Binomial Model

The Black-Scholes Model Implied Volatility Contingent Claims

Analysis of Corporate Debt and Equity

Convertible Bonds Valuing Pure State-

Contingent Securities

3


TermsTerms

A option is the right (not the obligation) to purchase or sell something at a specified price (the exercise price) in the future

– Underlying Asset, Call, Put, Strike (Exercise) Price, Expiration (Maturity) Date, American / European Option

– Out-of-the-money, In-the-money, At-the-money

– Tangible (Intrinsic) value, Time Value

4


Table 15.1 List of IBM Option Prices

(Source: Wall Street Journal Interactive Edition, May 29, 1998)

IBM (IBM) Underlying stock price 120 1/16

Call PutStrike Expiration Volume Last Open Volume Last Open

Interest Interest115 Jun 1372 7 4483 756 1 3/16 9692115 Oct … … 2584 10 5 967115 Jan … … 15 53 6 3/4 40120 Jun 2377 3 1/2 8049 873 2 7/8 9849120 Oct 121 9 5/16 2561 45 7 1/8 1993120 Jan 91 12 1/2 8842 … … 5259125 Jun 1564 1 1/2 9764 17 5 3/4 5900125 Oct 91 7 1/2 2360 … … 731125 Jan 87 10 1/2 124 … … 70

5


Table 15.2 List of Index Option Prices (Source: Wall Street Journal Interactive Edition, June 6, 1998)

S&P500 INDEX -AM Chicago ExchangeUnderlying High Low Close Net From %

Change 31-Dec ChangeS&P500 1113.88 1084.28 1113.86 19.03 143.43 14.8

(SPX) Net Open Strike Volume Last Change Interest

Jun 1110 call 2,081 17 1/4 8 1/2 15,754Jun 1110 put 1,077 10 -11 17,104Jul 1110 call 1,278 33 1/2 9 1/2 3,712Jul 1110 put 152 23 3/8 -12 1/8 1,040Jun 1120 call 80 12 7 16,585Jun 1120 put 211 17 -11 9,947Jul 1120 call 67 27 1/4 8 1/4 5,546Jul 1120 put 10 27 1/2 -11 4,033

6


Terminal or Boundary Conditions for Call and Put Options

-20

0

20

40

60

80

100

120

0 20 40 60 80 100 120 140 160 180 200

Underlying Price

Do

llar

s

Put

Call

7


The Put-Call Parity RelationThe Put-Call Parity Relation

Two ways of creating a stock investment that is insured against downside price risk:– Buying a share of stock and a put option (a protective-

put strategy)

– Buying a pure discount bond with a face value equal to the option’s exercise price and simultaneously buying a call option

8


Terminal Conditions of a Call and a Put Option with Strike = 100

Share Call Put Share_Put Bond Call_Bond0 0 100 100 100 100

10 0 90 100 100 10020 0 80 100 100 10030 0 70 100 100 10040 0 60 100 100 10050 0 50 100 100 10060 0 40 100 100 10070 0 30 100 100 10080 0 20 100 100 10090 0 10 100 100 100

100 0 0 100 100 100110 10 0 110 100 110120 20 0 120 100 120130 30 0 130 100 130140 40 0 140 100 140150 50 0 150 100 150160 60 0 160 100 160170 70 0 170 100 170180 80 0 180 100 180190 90 0 190 100 190200 100 0 200 100 200

9


0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100 120 140 160 180 200

Stock Price

Pay

offs

Put

Share

Share_Put

Bond

Call

Call_Bond

10


Payoff Structure for Protective-Put StrategyPayoff Structure for Protective-Put Strategy

If S T < E If S T > E

Stock S T S T

Put E-S T 0

Stock plus put E S T

Value of Position at Maturity DatePosition

11


Payoff Structure for a Pure Discount Bond Payoff Structure for a Pure Discount Bond Plus a CallPlus a Call

If S T < E If S T > E

Call 0 S T - EPure discount bond plus call E S T

Value of Position at Maturity DatePosition

E EPure discount bond with facevalue of E

12


Put-Call Parity EquationPut-Call Parity Equation

ShareMaturityStrikePut

r

StrikeMaturityStrikeCall

Maturity

),(

1),(

SP

r

EC

T

1

13


Synthetic SecuritiesSynthetic Securities

The put-call parity relationship may be solved for any of the four security variables to create synthetic securities C=S+P-B

S=C-P+B

P=C-S+B

B=S+P-C

14


Converting a Put into a CallConverting a Put into a Call

S = $100, E = $100, T = 1 year, r = 8%, P = $10:

C = 100 – 100/1.08 + 10 = $17.41

If C = $18, the arbitrageur would sell calls at a price of $18, and synthesize a synthetic call at a cost of $17.41, and pocket the $0.59 difference between the proceed and the cost

P

r

ESC

T

1

15


Put-Call ArbitragePut-Call Arbitrage

Sell a call $18 0 - (S T - $100)

Buy a stock -100.00 S T S T

Borrow the present value of $100 92.59 -100.00 -100.00

Buy a put -10.00 $100 - S T 0.00Net cash flows 0.59 0.00 0.00

Cash Flow at Maturity Date

If S T < $100 If S T > $100

Buy Replicating Portfolio (Synthetic Call)

PositionImmediate Cash

Flow

16


Options and ForwardsOptions and Forwards We saw in the last chapter that the discounted value of

the forward was equal to the current spot The relationship becomes

TT r

FP

r

EC

)1(1

Tr

EFPC

)1(

or

If the exercise price is equal to the forward price of the underlying stock, then the put and call have the same price

17


Implications for European OptionsImplications for European Options

If (F > E) then (C > P) If (F = E) then (C = P) If (F < E) then (C < P)

− E is the common exercise price− F is the forward price of underlying share− C is the call price− P is the put price

18


Call and Put as a Function of Forward

0

2

4

6

8

10

12

14

16

90 92 94 96 98 100 102 104 106 108 110

Forward

Put

, Cal

l Val

ues

callput

asy_call_1asy_put_1

Strike = Forward

Call = Put

19


Put and Call as Function of Share Price

-10

0

10

20

30

40

50

60

50 60 70 80 90 100 110 120 130 140 150

Share Price

Pu

t an

d C

all P

rice

s

call

put

asy_call_1

asy_call_2

asy_put_1

asy_put_2

20


Put and Call as Function of Share Price

0

5

10

15

20

80 85 90 95 100 105 110 115 120

Share Price

Pu

t an

d C

all

Pri

ces

call

put

asy_call_1

asy_call_2

asy_put_1

asy_put_2

PV Strike

Strike

21


Volatility and Option PricesP0 = $100, Strike Price = $100

Stock Price Call Payoff Put Payoff

Low Volatility Case

Rise 120 20 0Fall 80 0 2

0Expectation 100 10 10

High Volatility Case

Rise 140 40 0

Fall 60 0 40Expectation 100 20 20

The prices of options increase with the volatility of the stock

22


Two-State Option Pricing: SimplificationTwo-State Option Pricing: Simplification

The stock price can take only one of two possible values at the expiration date of the option: either rise or fall by 20% during the year

The option’s price depends only on the volatility and the time to maturity

The interest rate is assumed to be zero

23


Binary Model: CallBinary Model: Call

The synthetic call, C, is created by– buying a fraction x (which is called the hedge

ratio) of shares, of the stock, S– selling short risk-free bonds with a market

value y

yxSC

24


Binary Model: Creating the Synthetic CallBinary Model: Creating the Synthetic Call

Call Option -$C $20 0

Buy x shares of stock - $100x $120x $80x

Borrow y y - y - y Total replicating portfolio -$100x+y $120x −y $80x −y


If S 1 = $120 If S 1 = $80

Synthetic Call

PositionImmediateCash Flow

S = $100, E = $100, T = 1 year, d = 0, r = 0

25


Binary Model: CallBinary Model: Call

Specification:

– We have an equation, and give the value of the terminal share price, we know the terminal option value for two cases:

– By inspection, the solution is x=1/2, y = 40.

yx

yx

800

12020

The Law of One Price

26


Binary Model: CallBinary Model: Call Solution:

– We now substitute the value of the parameters x=1/2, y = 40 into the equation

– to obtainyxSC

10$401002

1C

27


Binary Model: PutBinary Model: Put

The synthetic put, P, is created by– selling short a fraction x of shares, of the

stock, S– buying risk free bonds with a market value y

yxSP

28


Binary Model: Creating the Synthetic PutBinary Model: Creating the Synthetic Put

Put option -$P 0 $20

Sell short x shares of stock $100x - $120x - $80x

Invest y in the risk-free asset - y y y Total replicating portfolio $100x-y $0 $20


If S 1 = $120 If S 1 = $80

Synthetic Put

PositionImmediateCash Flow

S = $100, E = $100, T = 1 year, d = 0, r = 0

29


Binary Model: PutBinary Model: Put Specification:

– We have an equation, and give the value of the terminal share price, we know the terminal option value for two cases:

– By inspection, the solution is x = 1/2, y = 60

yx

yx

800

12020

The Law of One Price

30


Binary Model: PutBinary Model: Put

Solution:– We now substitute the value of the parameters

x=1/2, y = 60 into the equation

– to obtain:

yxSP

10$601002

1P

31


Decision Tree for Dynamic Replication Decision Tree for Dynamic Replication of a Call Optionof a Call Option

F$80

E$100

D$120

C$90

A$100

B$110

32



E$100

D$120

B$110

0

C11

$20

– The terminal option value for two cases:

120x – y = 20

100x – y = 0

– By inspection, the solution is x=1, y = $100

– Thus, C11 = 1*$110 − $100 = $10

33



F$80

E$100

C$90

0

C12

0


90x – y = 0

80x – y = 0

– By inspection, the solution is x=0, y = 0

– Thus, C12 = 0*$90 − $0 = $0

34



C$90

B$110

A$100

0

C0

$10


110x – y = 10

90x – y = 0

– By inspection, the solution is x=1/2, y = $45

– Thus, C0 = (1/2)*$100 − $45 = $5

35



F$80

E$100

D$120

C$90

A$100

B$110

Buy 1/2 share of stockBorrow $45Total investment $5

Buy another half share of stockIncrease borrowing to $100

Sell stock and pay off debt

Sell shares $120Pay off debt -$100

Total $20

Sell shares $100Pay off debt -$100

Total 0

36



<---------0 Months----------> <------------------6 Months----------------> 12 MonthsStockPrice x y CallPrice x y CallPrice

$120.00 $20.00$110.00 $10.00 100.00% -$100.00$100.00 50.00% -$45.00 $0.00$90.00 $0.00 0.00% $0.00$80.00 $0.00

($120*100%) + (-$100) = $20

37


The Black-Scholes Model: The Limiting The Black-Scholes Model: The Limiting Case of Binomial ModelCase of Binomial Model

One can continuously and costlessly adjust the replicating portfolio over time

As the decision intervals in the binomial model become shorter, the resulting option price from the binomial model approaches the Black-Scholes option price

38


The Black-Scholes ModelThe Black-Scholes Model

21

21

1

2

2

2

1

21

ln

21

ln

dNEedNSeP

dNEedNSeC

TdT

TdrES

d

T

TdrES

d

rTdT

rTdT

Shares of Stock

Bond

39


The Black-Scholes Model: NotationThe Black-Scholes Model: Notation

C = price of call P = price of put S = price of stock E = exercise price T = time to maturity ln(·) = natural logarithm e = 2.71828... N(·) = cum. norm. dist’n

The following are annual, compounded continuously:– r = domestic risk free rate

of interest – d = foreign risk free rate

or constant dividend yield– σ = volatility

40


The Black-Scholes Model: The Black-Scholes Model: Dividend-adjusted FormDividend-adjusted Form

EdNSedNeP

EdNSedNeC

T

TE

Se

d

T

TE

Se

d

TdrrT

TdrrT

Tdr

Tdr

21

21

2

2

2

1

21

ln

21

ln

41


TSTS

PC

dNdNSPC

d

PdNdNSeC

TdTd

SeE

dT

Tdr

39886.02

0 If

21

;21

If

21

21

21

The Black-Scholes Model :The Black-Scholes Model :Dividend-adjusted Form (Simplified)Dividend-adjusted Form (Simplified)

42


Determinants of Option PricesDeterminants of Option Prices

Increases in: Call Put Stock Price, S Increase Decrease Exercise Price, E Decrease Increase Volatility, sigma Increase Increase Time to Expiration, T Ambiguous Ambiguous Interest Rate, r Increase Decrease Cash Dividends, d

Decrease Increase

43


Value of a Call and Put Options with Strike = Current Stock Price

0

1

2

3

4

5

6

7

8

9

10

11

0.00.10.20.30.40.50.60.70.80.91.0

Time-to-Maturity

Cal

l an

d P

ut

Pri

ce

call put

44


Call and Put Prices as a Function of Volatility

0

1

2

3

4

5

6

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Volatility

Cal

l an

d P

ut

Pri

ces

call put

45


Implied VolatilityImplied Volatility

The value of σ that makes the observed market price of the option equal to its Black-Scholes formula value

Approximation:

TS

C 2

S E r T d C σ

100 108.33 0.08 1 0 7.97 0.2

46


Implied Volatility

47


Computing Implied Volatility

volatility 0.3154

call 10.0000strike 100.0000share 105.0000rate_dom 0.0500rate_for 0.0000maturity 0.2500

factor 0.0249

d_1 0.4675d_2 0.3098

n_d_1 0.6799n_d_2 0.6217

call_part_1 71.3934call_part_2 -61.3934

error 0.0000

Insert any number to start

Formula for option value minus the actual

call value

48


Computing Implied Volatility

volatility 0.315378127101852

call 10strike 100share 105rate_dom 0.05rate_for 0maturity 0.25

factor =(rate_dom - rate_for + (volatility^2)/2)*maturity

d_1 =(LN(share/strike)+factor)/(volatility*SQRT(maturity))d_2 =d_1-volatility*SQRT(maturity)

n_d_1 =NORMSDIST(d_1)n_d_2 =NORMSDIST(d_2)

call_part_1 =n_d_1*share*EXP(-rate_for*maturity)call_part_2 =- n_d_2*strike*EXP(-rate_dom*maturity)

error =call_part_1+call_part_2-call

49


Valuation of Uncertain Cash Flows: Valuation of Uncertain Cash Flows: CCA / DCFCCA / DCF

The DCF approach discounts the expected cash flows using a risk-adjusted discount rate

The Contingent-Claims Analysis (CCA) uses knowledge of the prices of one or more related assets and their volatilities

50


An Example: Debtco Corp.An Example: Debtco Corp. Debtco is in the real-estate business It issues two types of securities:

– common stock (1 million shares)

– corporate bonds with an aggregate face value of $80 million (80,000 bonds, each with a face value of $1,000) and maturity of 1 year

– risk-free interest rate is 4% The total market value of Debtco is $100 million

51


Debtco: NotationDebtco: Notation

– V be the current market value of Debtco’s assets ($100 million)

– V1 be the market value of Debtco’s assets a year from now

– E be the market value of Debtco’s stocks

– D be the market value of Debtco’s bonds

52


Two Ways to Think about the Debtco’s Two Ways to Think about the Debtco’s Market ValueMarket Value

To think of the assets of the firm, real estates in Debtco’s case, as having a market value of $100 million

To imagine another firm that has the same assets as Debtco but is financed entirely with equity, and the market value of this all-equity-financed “twin” of Debtco is $100 million

53


Payoffs for Bond and Stock Issues

0

20

40

60

80

100

120

0 20 40 60 80 100 120 140 160 180 200

Value of Firm (Millions)

Val

ue

of

Bo

nd

an

d S

tock

(M

illio

ns)

Bond Value

Stock Value

54


-20

0

20

40

60

80

100

120

0 20 40 60 80 100 120 140 160 180 200

Value of the firm in 1 year

Va

lue

Value of the Bonds

Value of the Stock

55


-20

0

20

40

60

80

100

120

0 20 40 60 80 100 120 140 160 180 200

Value of the firm in 1 year

Va

lue

Value of the Stock

The payoff is identical to a call option in which the underlying asset is the firm itself, and the exercise price is the face value of its debt

56


The value of the firm’s equity

The value of the debt

D ＝ V ＝ E

1 2

2

1

2 1

( ) ( )

ln( / ) ( / 2)

rTE N d V N d Be

V B r Td

T

d d T

57


Probalility Density of a Firm's Value

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 20 40 60 80 100 120 140 160 180 200

Value of a Firm

Pro

bab

ility

Den

sity

58


Debtco Security Payoff TableDebtco Security Payoff Table ($’000,000)($’000,000)

Security Payoff State A Payoff State B

Firm 140 70

Bond 80 70

Stock 60 0

59


Debtco’s Replicating PortfolioDebtco’s Replicating Portfolio Let

– x be the fraction of the firm in the replication

– Y be the borrowings at the risk-free rate in the replication

– The following equations must be satisfied

308,692,57$;7

6

04.1700

04.114060

Yx

Yx

Yx

60


Debtco’s Replicating Portfolio Debtco’s Replicating Portfolio ($’000)($’000)

Position Immediate Case A Case B

6/7 assets -85,714 120,000 60,000

Bond (RF) 57,692 -60,000 -60,000

Total 28,022 60,000 0

61


Debtco’s Replicating PortfolioDebtco’s Replicating Portfolio

We know the value of the firm is $1,000,000, and the value of the total equity is $28,021,978, so the market value of the debt with a face of 80,000,000 is $71,978,022

The yield on this debt is (80…/71…) -1=11.14%

62


Another View of Debtco’s Another View of Debtco’s Replicating Portfolio (‘$000)Replicating Portfolio (‘$000)

SecurityTotal

Market Value Equivalent

Amount of Firm Equivalent

Amount of Rf Debt

Bonds 71,978 14,286(1/7) 57,692

Stock 28,022 85,714(6/7) -57,692

Bonds + Stock 100,000 100,000 0

63


Given the Price of the StockGiven the Price of the Stock Suppose:

– 1 million shares of Debtco’s stock outstanding, and the market price is $20 per share

– two possible future value of for Debtco, $70 million and $140 million

– the face value of Debtco’ bonds is $80 million– risk-free interest rate is 4%

64


Valuing BondsValuing Bonds

We can replicate the firm’s equity using x = 6/7 of the firm, and about Y = $58 million riskless borrowing (earlier analysis)

The implied value of the bonds is then $90,641,026 − $20,000,000 = $70,641,026 & the yield is (80.00 − 70.64)/70.64 = 13.25%

026,641,90$76

308,692,57000,000,20

x

YEVYxVE

65


Given the Price of the BondsGiven the Price of the Bonds

Suppose: – the face value of Debtco’ bonds is $80 million,

the yield-to-maturity on the bonds is 10% (i.e., the price of Debtco bonds is $909.09)

– two possible future value of for Debtco, $70 million and $140 million

– risk-free interest rate is 4%

66


Replication PortfolioReplication Portfolio

Position Immediate Cash Flow

Scenario A V1= 70

Scenario B

Purchase x

of firm

- x V

70 x

Purchase Y

RF Bond - Y Y (1.04) Y (1.04)

Total Portfolio

- x V - Y 70 80

V1= 140

140 x

67


Determining the Weight of Firm Determining the Weight of Firm Invested in Bond, Invested in Bond, xx, and the Value of the , and the Value of the

R.F.-Bond, R.F.-Bond, YY

308,692,57$;7

1

04.114080

04.17070

Yx

Yx

Yx

68


Valuing StockValuing Stock We can replicate the bond by purchasing 1/7 of the

company, and $57,692,308 of default-free 1-year bonds The market value of the bonds is $909.0909 * 80,000 =

$72,727,273

The value of the stock is therefore E = V −D = $105,244,753 − $72,727,273= $32,517,480

753,244,105$71

308,692,57273,727,72

;

x

YDVYxVD

69


Convertible BondsConvertible Bonds

A convertible bond obligates the issuing firm either to redeem the bond at par value upon maturity or to allow the bondholder to convert the bond into a prespecified number of shares of common stock

70


An Example: Convertidett Corp.An Example: Convertidett Corp. Convertidett has assets identical to those of Debtco Its capital structure consists of

– 1 million shares of common stock

– one-year zero-coupon bonds with a face value of $80 million (80,000 bonds, each with a face value of $1,000), that are convertible into 20 shares of Convertidett stock at maturity

– risk-free interest rate is 4% The total market value of Debtco is $100 million

71


Critical value of Convertident for ConversionCritical value of Convertident for Conversion

Upon convertion, the total shares of stock will be 2.6 million

11 %5.616.26.1 VV

millionV

millionV

130$

80$6.26.1*

1

*1

72


Convertible Bond

0

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140 160 180 200

Value of the Firm

Val

ue

of

Sto

ck a

nd

Bo

nd

Issu

e

Convertible Bond Value

Dilulted Stock Value

73


Payoff for Convertidett’s Payoff for Convertidett’s Stocks and BondsStocks and Bonds

Security Payoff State A Payoff State B

Firm 140,000,000 70,000,000

Bonds 86,153,846 70,000,000

Stocks 53,846,154

0

74


Convertidett’s Replicating PortfolioConvertidett’s Replicating Portfolio Let

– x be the fraction of the firm in the replication– Y be the borrowings at the risk-free rate in the

replication– The following equations must be satisfied

148,775,51$;077,923,76.

04.1700

04.1140154,846.53

Yx

Yx

Yx

75


Values of Convertidett’s Stocks and BondsValues of Convertidett’s Stocks and Bonds

929,147,25$148,775,51$077,923,76$ E

071,852,74$929,147,25$000,000,100$ EVD

%88.665.935$

65.935$000,1$

YTM

65.935$PriceBond

76


Decomposition of Convertidett’s Decomposition of Convertidett’s Stocks and BondsStocks and Bonds

SecurityTotal

Market Value Equivalent

Amount of Firm Equivalent

Amount of Rf Debt

Bonds 74,852,071 23,076,923(0.23) 51,775,148

Stock 25,147,929 76,923,077(0.77) -51,775,148

Bonds + Stock 100,000,000 100,000,000 0

77


Pure State-Contingent SecuritiesPure State-Contingent Securities

Securities that pay $1 in one of the states and nothing in the others

For Debtco and Convertidett, if we know the prices of the two pure state-contingent securities, then we are able to price any securities issued by the firms—stocks, bonds, convertible bonds, or other securities

78


Valuing Pure State-Contingent SecuritiesValuing Pure State-Contingent Securities

Security Payoff Scenario a Payoff Scenario b

Firm $70,000,000 $140,000,000

Contingent Security #1

$0

$1

Contingent Security #2

$1 $0

79


State-Contingent Security #1State-Contingent Security #1

967032467.004.1

1

000,000,70

000,000,100

000,000,1

04.1

1;

000,000,70

1

104.1000,000,140

004.1000,000,70

1

YxP

YxYx

Yx

80


State-Contingent Security #2State-Contingent Security #2

04.1

1$538961.0$505494.0$033467.0$

495505494.004.1

2

000,000,70

000,000,100

000,000,1

04.1

2;

000,000,70

1

004.1000,000,140

104.1000,000,70

21

2

PP

YxP

YxYx

Yx

81


Valuing Debtco’s SecuritiesValuing Debtco’s Securities

Price of a Debtco stock = 60P1 = 60*$.4670329 = $28.02

Price of a Debtco bond = 1,000P1 + 875P2

= 1,000*$.4670329 + 875*$.494505 = $899.73

Security Possible Payoff in 1 Year

Firm $140 million $70 million

Debtco Stock $60 0

Debtco Bond $1,000 $875

82


Valuing Convertidett’s SecuritiesValuing Convertidett’s Securities

Price of a Convertidett stock = 53.86415P1

= 53.86415*$.4670329 = $25.15 Price of a Convertidett bond = 1,076.923P1 + 875P2

= 1,076.923*$.4670329 + 875*$.494505 = $935.65

Security Possible Payoff in 1 Year

Firm $140 million $70 million

Debtco Stock $53.85 0

Debtco Bond $1,076.92 $875

83


Payoff for Debtco’s Bond GuaranteePayoff for Debtco’s Bond Guarantee

Security Scenario A Scenario B

Firm $140,000,000 $70,000,000

Bonds $1,000 $875

Guarantee $0 $125

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SCS Conformation of Guarantee’s PriceSCS Conformation of Guarantee’s Price

Guarantee’s price = 125P2 = 125* 0.494505 = $61.81

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Credit Guarantees Guarantees against credit risk pervade the financial

system and play an important role in corporate and public finance– Parent corporations routinely guarantee the debt obligations of

their subsidiaries

– Commercial banks and insurance companies offer guarantees in return for fees on a broad spectrum of financial instruments ranging from traditional letters of credit to interest rate and currency swaps

– The largest providers of financial guarantees are almost surely governments and governmental agencies

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Credit Guarantees Fundamental identity:

– Risky loan+ loan guarantee=default-free loan

– Risky loan= default-free loan-loan guarantee

The credit guarantee is equivalent to writing a put option– on the firm's assets

– with a strike price equal to the face value of the debt. The guarantee's value can, therefore, be computed using the adjusted put-option-pricing formula

chapter 15: options and contingent claims

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