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Chapter 15: Option Chapter 15: Option PricingPricing

Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley

Objective•To show how the law of one price may

be used to derive prices of options•To show how to infer implied

volatility from optionprices

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Chapter 15 ContentsChapter 15 Contents

15.1 How Options Work15.1 How Options Work

15.2 Investing with Options15.2 Investing with Options

15.3 The Put-Call Parity 15.3 The Put-Call Parity RelationshipRelationship

15.4 Volatility & Option Prices15.4 Volatility & Option Prices

15.5 Two-State Option Pricing15.5 Two-State Option Pricing

15.7 The Black-Scholes Model15.7 The Black-Scholes Model

15.8 Implied Volatility15.8 Implied Volatility

15.9 Contingent Claims 15.9 Contingent Claims Analysis of Corporate Debt Analysis of Corporate Debt and Equityand Equity

15.10 Credit Guarantees15.10 Credit Guarantees

15.11 Other Applications of 15.11 Other Applications of Option-Pricing Option-Pricing MethodologyMethodology

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ObjectivesObjectives

• To show how the Law of One Price To show how the Law of One Price can be used to derive prices of can be used to derive prices of optionsoptions

• To show how to infer implied To show how to infer implied volatility from option prices volatility from option prices

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IntroductionIntroduction

• This chapter explores how option This chapter explores how option prices are affected by the volatility of prices are affected by the volatility of the underlying securitythe underlying security

• Exchange traded options appeared in Exchange traded options appeared in 1973, enabling us to determine the 1973, enabling us to determine the market’s estimate of future volatility, market’s estimate of future volatility, rather than relying on historical valuesrather than relying on historical values

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Definition of an OptionDefinition of an Option

• Recall that an American {European} Recall that an American {European} call (put) option is the right, but not call (put) option is the right, but not the obligation to buy (sell) an asset the obligation to buy (sell) an asset at a specified price any time before at a specified price any time before its expiration date {on its expiration its expiration date {on its expiration date}date}

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Ubiquitous OptionsUbiquitous Options

• This chapter focuses on traded This chapter focuses on traded options, but it would be a mistake to options, but it would be a mistake to believe that the tools we will be believe that the tools we will be developing are restricted to traded developing are restricted to traded optionsoptions

• Some examples of options are given Some examples of options are given on the next few slideson the next few slides

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Government Price Government Price SupportsSupports

• Governments sometimes provide Governments sometimes provide assistance to farmers by offering to assistance to farmers by offering to purchase agricultural products at a purchase agricultural products at a specified support pricespecified support price

• If the market price is lower than the If the market price is lower than the support, then a farmer will exercise her support, then a farmer will exercise her right to ‘put’ her crop to the right to ‘put’ her crop to the government at the higher pricegovernment at the higher price

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Old MortgageOld Mortgage

• Traditional US mortgages give the Traditional US mortgages give the householder the right to call the householder the right to call the mortgage at a strike equal to the mortgage at a strike equal to the outstanding principleoutstanding principle

• If interest rates have fallen below the If interest rates have fallen below the note’s rate, then the home owner will note’s rate, then the home owner will consider refinancing the mortgageconsider refinancing the mortgage

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InsuranceInsurance

• Insurance policies often give you the Insurance policies often give you the right, but not the obligation to do right, but not the obligation to do something, it is therefore option-likesomething, it is therefore option-like– The renewable rider on a term life policy is The renewable rider on a term life policy is

an option an option

– If somebody:If somebody:• is terminally ill, then the rider is very valuableis terminally ill, then the rider is very valuable

• remains in good health, then it is not valuableremains in good health, then it is not valuable

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Limited LiabilityLimited Liability

• The owners of a limited liability The owners of a limited liability corporation have the right, but not corporation have the right, but not the obligation, to ‘put’ the company the obligation, to ‘put’ the company to the corporation’s creditors and to the corporation’s creditors and bondholdersbondholders

• Limited liability is, in effect, a put Limited liability is, in effect, a put optionoption

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15.1 How Options Work15.1 How Options Work• The Language of OptionsThe Language of Options

– Contingent Claim: Any asset whose future pay-off depends Contingent Claim: Any asset whose future pay-off depends upon the outcome of an uncertain eventupon the outcome of an uncertain event

– Call: an option to buyCall: an option to buy

– Put: an option to sellPut: an option to sell

– Strike or Exercise Price: the fixed price specified in an option Strike or Exercise Price: the fixed price specified in an option contractcontract

– Expiration or Maturity Date: The date after which an option Expiration or Maturity Date: The date after which an option can’t be exercisedcan’t be exercised

– American Option: an option that can be exercised at any American Option: an option that can be exercised at any time up to and including maturity datetime up to and including maturity date

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– European Option: an option that can only be exercised European Option: an option that can only be exercised on the maturity dateon the maturity date

– Tangible Value: The hypothetical value of an option if Tangible Value: The hypothetical value of an option if it were exercised immediatelyit were exercised immediately

– At-the-Money: an option with a strike price equal to At-the-Money: an option with a strike price equal to the value of the underlying assetthe value of the underlying asset

– Out-of-the-Money: an option that’s not at-the-money, Out-of-the-Money: an option that’s not at-the-money, but has no tangible valuebut has no tangible value

– In-the-Money: an option with a tangible valueIn-the-Money: an option with a tangible value

– Time Value: the difference between an option’s Time Value: the difference between an option’s market value and its tangible valuemarket value and its tangible value

– Exchange-Traded Option: A standardized option that Exchange-Traded Option: A standardized option that an exchange stands behind in the case of a defaultan exchange stands behind in the case of a default

– Over the Counter Option: An option on a security that Over the Counter Option: An option on a security that is not an exchange-traded optionis not an exchange-traded option

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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 . Put .

Strike Expiration Volume Last Open Volume Last OpenInterest Interest

115 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

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

S & P 500 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

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15.2 Investing with 15.2 Investing with OptionsOptions

• The payoff diagram (terminal The payoff diagram (terminal conditions, boundary conditions) for conditions, boundary conditions) for a call and a put option, each with a a call and a put option, each with a strike (exercise price) of $100, is strike (exercise price) of $100, is derived nextderived next

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Option Payoff DiagramsOption Payoff Diagrams

• The value of an option at expiration The value of an option at expiration follows immediately from its definitionfollows immediately from its definition– In the case of a call option with strike of In the case of a call option with strike of

$100, if the stock price is $90 $100, if the stock price is $90 ($110)($110), then , then exercising the option results purchasing exercising the option results purchasing the share for $100, which is $10 more the share for $100, which is $10 more expensive expensive ($10 less expensive)($10 less expensive) than than buying it, so you wouldn't buying it, so you wouldn't (would)(would) exercise exercise your rightyour right

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Terninal or Boundary Conditions for Call and Put Options

-20

0

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100

120

0 20 40 60 80 100 120 140 160 180 200

Underlying Price

Do

lla

rs

Call Put

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15.3 The Put-Call Parity 15.3 The Put-Call Parity RelationshipRelationship

• Consider the following two Consider the following two strategiesstrategies– Purchase a put with a strike price of Purchase a put with a strike price of

$100, and the underlying share$100, and the underlying share

– Purchase a call with a strike price of Purchase a call with a strike price of $100 and a bond that matures at the $100 and a bond that matures at the same date with a face of $100same date with a face of $100

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ObservationObservation

• The most important point to observe is The most important point to observe is that the value of the “call + bond” that the value of the “call + bond” strategy, is identical (at maturity) with strategy, is identical (at maturity) with the protective-put strategy “put + the protective-put strategy “put + share”share”

• So, if the put and the call have the same So, if the put and the call have the same strike price, we obtain the put-call parity strike price, we obtain the put-call parity relationship: put + share = call + bondrelationship: put + share = call + bond

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Put-Call Parity for Put-Call Parity for American and European American and European OptionsOptions

• A European option that pays no A European option that pays no dividend during its life fully satisfies dividend during its life fully satisfies the requirements of put-call parity the requirements of put-call parity

• In the case of American options, the In the case of American options, the relationship is fully accurate only at relationship is fully accurate only at maturity, because American puts maturity, because American puts are sometimes exercised earlyare sometimes exercised early

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Put-Call Parity EquationPut-Call Parity Equation

ShareMaturityStrikePut

rf

StrikeMaturityStrikeCall Maturity

),(

1),(

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Synthetic SecuritiesSynthetic Securities

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

S=C-P+BS=C-P+B

P=C-S+BP=C-S+B

B=S+P-CB=S+P-C

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Synthetic SecuritiesSynthetic Securities

C=S+P-B and P=C-S+B may be used by C=S+P-B and P=C-S+B may be used by floor traders to flip between a call and a floor traders to flip between a call and a putput

S=C-P+B may be used by short-term S=C-P+B may be used by short-term traders wishing to take advantage of traders wishing to take advantage of lower transaction costslower transaction costs

B=S+P-C may be used to create a B=S+P-C may be used to create a synthetic bond said to pay a slightly synthetic bond said to pay a slightly higher return than the physical bondhigher return than the physical bond

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15.4 Volatility and Option 15.4 Volatility and Option PricesPrices

• We next explore what happens to the We next explore what happens to the value of an option when the volatility value of an option when the volatility of the underlying stock increasesof the underlying stock increases– We assume a world in which the stock We assume a world in which the stock

price moves during the year from $100 price moves during the year from $100 to one of two new values at the end of to one of two new values at the end of the year when the option maturesthe year when the option matures

– Assume risk neutralityAssume risk neutrality

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Volatility and Option Prices, P0 = $100, Strike = $100

Stock Price Call Payoff Put Payoff

Low Volatility Case

Rise 120 20 0Fall 80 0 20Expectation 100 10 10

High Volatility Case

Rise 140 40 0Fall 60 0 40Expectation 100 20 20

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Illustration ExplainedIllustration Explained

• The stock volatility in the second The stock volatility in the second scenario is higher, and the expected scenario is higher, and the expected payoffs for both the put and the call are payoffs for both the put and the call are also higheralso higher

– This is the result of truncation, and holds in This is the result of truncation, and holds in all empirically reasonable casesall empirically reasonable cases

• Conclusion: Volatility increases all Conclusion: Volatility increases all option pricesoption prices

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15.5 Two-State (Binomial) 15.5 Two-State (Binomial) Option-PricingOption-Pricing

– We are now going to derive a relatively We are now going to derive a relatively simple model for evaluating optionssimple model for evaluating options• The assumptions will at first appear totally The assumptions will at first appear totally

unrealistic, but using some underhand unrealistic, but using some underhand mathematics, the model may be made to mathematics, the model may be made to price options to any desired level of accuracyprice options to any desired level of accuracy

• The advantage of the method is that it does The advantage of the method is that it does not require learning stochastic calculus, and not require learning stochastic calculus, and yet it illustrates all the key steps necessary yet it illustrates all the key steps necessary to derive any option evaluation modelto derive any option evaluation model

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Binary Model Assumptions Binary Model Assumptions

• We assume:We assume:– share price = strike price = $100share price = strike price = $100

– time to maturity = 1 yeartime to maturity = 1 year

– interest rate = 5%interest rate = 5%

– stock prices either rise or fall by 20% in stock prices either rise or fall by 20% in the year, and so are either $80 or $120 the year, and so are either $80 or $120 at yearendat yearend

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Binary Model: CallBinary Model: Call

• Strategy:Strategy:– replicate the call using a portfolio of replicate the call using a portfolio of

• the underlying stock the underlying stock

• a zero coupon riskless bond with a face a zero coupon riskless bond with a face value of $100value of $100

– by the law of one price, the price of the by the law of one price, the price of the actual call must equal the price of the actual call must equal the price of the synthetic callsynthetic call

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Binary Model: CallBinary Model: Call

• Implementation:Implementation:– the synthetic call, C, is created bythe synthetic call, C, is created by

• holding x number of shares of the stock, holding x number of shares of the stock, S, and y number of risk free bonds with S, and y number of risk free bonds with a market value B a market value B

• C = xS + yBC = xS + yB

• C = x *100 + y * 95.2381C = x *100 + y * 95.2381

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Binary Model: CallBinary Model: Call

• Specification:Specification:– We have an equation, and given the We have an equation, and given the

value of the terminal share price, we value of the terminal share price, we know the terminal option value for two know the terminal option value for two cases: cases: 20 = x * 120 + y * 100 20 = x * 120 + y * 100 0 = x * 80 + y * 100 0 = x * 80 + y * 100 By inspection, the solution is x= 0.5, By inspection, the solution is x= 0.5, y = - 0.4 y = - 0.4

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Binary Model: CallBinary Model: Call

• Solution:Solution:– We now substitute the value of the We now substitute the value of the

parameters x= 0.5 , y = - 0.4 into the parameters x= 0.5 , y = - 0.4 into the equation in slide 30 to obtain: equation in slide 30 to obtain:

C = 0.5 *100 - 0.4 * C = 0.5 *100 - 0.4 * 95.2381 95.2381 C = $11.905C = $11.905

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15.7 The Black-Scholes 15.7 The Black-Scholes ModelModel

• The most widely used model for pricing The most widely used model for pricing options is the Black-Scholes modeloptions is the Black-Scholes model– This model is completely consistent with the This model is completely consistent with the

binary model as the interval between stock binary model as the interval between stock prices decreases to zeroprices decreases to zero

– The model provides theoretical insights into The model provides theoretical insights into option behavioroption behavior

– The assumptions are elegant, simple, and The assumptions are elegant, simple, and quite realisticquite realistic

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The Black-Scholes ModelThe Black-Scholes Model

• We will work with the generalized We will work with the generalized form of the model because the form of the model because the small additional complexity results small additional complexity results in considerable additional power in considerable additional power and flexibilityand flexibility

• First, notation:First, notation:

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The Black-Scholes Model: The Black-Scholes Model: NotationNotation

• C = price of callC = price of call

• P = price of putP = price of put

• S = price of stockS = price of stock

• E = exercise priceE = exercise price

• T = time to maturityT = time to maturity

• ln(.) = natural logarithmln(.) = natural logarithm

• e = 2.71828...e = 2.71828...

• N(.) = cum. norm. dist’nN(.) = cum. norm. dist’n

• The following are annual, The following are annual, compounded compounded continuously:continuously:

• r = domestic risk free r = domestic risk free rate of interest rate of interest

• d = foreign risk free rate d = foreign risk free rate or constant dividend or constant dividend yieldyield

• σ = volatilityσ = volatility

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The Normal ProblemThe Normal Problem• It is not unusual for a student to have It is not unusual for a student to have

a problem computing the cumulative a problem computing the cumulative normal distribution using tablesnormal distribution using tables

– table structures vary, so be carefultable structures vary, so be careful

– using standard-issue normal tables using standard-issue normal tables degrades computed option values degrades computed option values because of errors caused by catastrophic because of errors caused by catastrophic subtractionsubtraction

– {Many professionals use Hasting’s formula as reported {Many professionals use Hasting’s formula as reported in Abramowitz and Stegun as equation 26.2.19 (never, in Abramowitz and Stegun as equation 26.2.19 (never, never use 26.2.18). Its certificate valid in 0<=x<Inf, never use 26.2.18). Its certificate valid in 0<=x<Inf, so use symmetry to get -Inf<x<0}so use symmetry to get -Inf<x<0}

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The Normal ProblemThe Normal Problem

• The functions that come with Excel The functions that come with Excel have adequate accuracy, so have adequate accuracy, so consider using ‘Normsdist()’ in the consider using ‘Normsdist()’ in the statistical functions (note the statistical functions (note the ss in in NormNormssdist)dist)

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The Black-Scholes Model: The Black-Scholes Model: What’s missingWhat’s missing

• There are no expectations about There are no expectations about future returns in the modelfuture returns in the model

• The model is preference-free The model is preference-free

• σ-risk, not σ-risk, not -risk, is the relevant risk-risk, is the relevant risk

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The Black-Scholes Model: The Black-Scholes Model: EquationsEquations

21

21

1

2

2

2

1

21

ln

21

ln

dNEedNSeP

dNEedNSeC

TdT

TdrES

d

T

TdrES

d

rTdT

rTdT

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The Black-Scholes Model: The Black-Scholes Model: Equations (Forward Form)Equations (Forward 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

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The Black-Scholes Model: The Black-Scholes Model: Equations (Simplified)Equations (Simplified)

TSTS

PC

dNdNSPC

d

PdNdNSeC

TdTd

SeE

dT

Tdr

39886.02

0 If

21

;21

If

21

21

21

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So What Does it Mean?So What Does it Mean?

• You can now obtain the value of non-You can now obtain the value of non-dividend paying European options dividend paying European options

• With a little skill, you can widen this With a little skill, you can widen this to obtain approximate values of to obtain approximate values of European options on shares paying a European options on shares paying a dividend, and to some American dividend, and to some American optionsoptions

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Determinants 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

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

0

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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

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Call and Put Prices as a Function of Volatility

0

1

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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

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Observable VariablesObservable Variables

• All the variables are directly All the variables are directly observable, excepting the volatility, σ, observable, excepting the volatility, σ, and possibly, the next cash dividend, dand possibly, the next cash dividend, d

• We do not have to delve into the We do not have to delve into the psyche of investors to evaluate optionspsyche of investors to evaluate options

• We do not forecast future prices to We do not forecast future prices to obtain option valuesobtain option values

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15.8 Implied Volatility15.8 Implied Volatility

• Implied volatility is defined as the value Implied volatility is defined as the value of the volatility that makes the observed of the volatility that makes the observed market price of the option equal to the market price of the option equal to the value computed using the Black-Scholes value computed using the Black-Scholes option formula. option formula.

• Solver or Goal Seek function in Excel Solver or Goal Seek function in Excel can be used to compute the implied can be used to compute the implied volatility volatility

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15.9 Contingent Claims 15.9 Contingent Claims Analysis (CCA)of Analysis (CCA)of Corporate Debt and Corporate Debt and EquityEquity• The CCA approach uses a different set The CCA approach uses a different set

of informational assumptions than the of informational assumptions than the discounted cash flow (DCF) method:discounted cash flow (DCF) method:– it uses the risk-free rate rather than a risk-it uses the risk-free rate rather than a risk-

adjusted discount rate adjusted discount rate

– it uses knowledge of the prices of one or it uses knowledge of the prices of one or more related assets and their volatilitiesmore related assets and their volatilities

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Contingent-Claims Contingent-Claims Analysis of Stock and Analysis of Stock and Bonds: DebtcoBonds: Debtco

• Debtco is a real-estate holding Debtco is a real-estate holding company and has issuedcompany and has issued– 1,000,000 common shares1,000,000 common shares

– 80,000 pure discount bonds, face 80,000 pure discount bonds, face $,1000, maturity 1-year$,1000, maturity 1-year

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Debtco, ContinuedDebtco, Continued

– The total market value of Debtco is The total market value of Debtco is $100,000,000$100,000,000

– The risk-free rate, (and therefore, by The risk-free rate, (and therefore, by the law of one price, Debtco’s bond the law of one price, Debtco’s bond rate,) is 4%rate,) is 4%

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Debtco, NotationDebtco, Notation

• E the market value of the stock issueE the market value of the stock issue

• D the market value of the debt issueD the market value of the debt issue

• V the total current market value; V = E + DV the total current market value; V = E + D

• VV11 the total market value one year hence the total market value one year hence

• (The law of one price ensures that V = E + D (The law of one price ensures that V = E + D must be true, otherwise there will be an must be true, otherwise there will be an arbitrage opportunity)arbitrage opportunity)

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Debtco, Security ValuationDebtco, Security Valuation

• Value of the bondsValue of the bonds– By the rule of one price, the value of the By the rule of one price, the value of the

bonds must equal their face value discounted bonds must equal their face value discounted at the risk-free rate for a yearat the risk-free rate for a year• D = 80,000 * $1,000 / 1.04 = $76,923,077D = 80,000 * $1,000 / 1.04 = $76,923,077

– By the total value of the firm, V = E + D, the By the total value of the firm, V = E + D, the value of the stock isvalue of the stock is• E = V - D = $100,000,000 - $76,923,077 = E = V - D = $100,000,000 - $76,923,077 =

$23,076,923$23,076,923

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Debtco, PayoffDebtco, Payoff

– A consequence of Debtco’s having bonds A consequence of Debtco’s having bonds with a risk-free rate is that the company with a risk-free rate is that the company has either purchased bond default has either purchased bond default insurance from a third party, or that the insurance from a third party, or that the firm’s assets have no (downside) riskfirm’s assets have no (downside) risk

– For many companies, a more realistic For many companies, a more realistic assumption is that the assets do have risk, assumption is that the assets do have risk, and to evaluate such securities requires a and to evaluate such securities requires a payoff function for the bonds or stock:payoff function for the bonds or stock:

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Payoffs for Bond and Stock Issues

0

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Value of Firm (Millions)

Val

ue

of

Bo

nd

an

d S

tock

(M

illi

on

s)

BondValue

StockValue

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Convertible BondsConvertible Bonds

• A convertible bond obligates the A convertible bond obligates the issuing firm to redeem the bond at issuing firm to redeem the bond at par value upon maturity, or to allow par value upon maturity, or to allow the bond holder to convert the bond the bond holder to convert the bond into a pre-specified number of share into a pre-specified number of share of common stockof common stock

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Convertible Bonds: The Convertible Bonds: The Convertidebt CorporationConvertidebt Corporation

• Assume that Convertidebt is in Assume that Convertidebt is in every way like Debtco, but each every way like Debtco, but each bond is convertible to 20 common bond is convertible to 20 common stock at maturitystock at maturity– If all the debt is converted, then the If all the debt is converted, then the

number of common stock will rise from number of common stock will rise from 1,000,000 to 1,000,000 + 80,000 * 20 1,000,000 to 1,000,000 + 80,000 * 20 = 2,600,000 shares= 2,600,000 shares

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Convertible Bond

0

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0 20 40 60 80 100 120 140 160 180 200

Value of the Firm

Val

ue

of

Sto

ck a

nd

Bo

nd

Iss

ue

ConvertibleBondValue

DilultedStockValue

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Bondholder EntitlementsBondholder Entitlements

• Given that a conversion occurs, the Given that a conversion occurs, the value of each common stock will bevalue of each common stock will be

– Value of firm / 2,600,000Value of firm / 2,600,000• The bond holders will receive 1,600,000 of The bond holders will receive 1,600,000 of

these shares, so the bondholders will own these shares, so the bondholders will own 1.6/2.6 of the firm, leaving the shareholders 1.6/2.6 of the firm, leaving the shareholders with 1/2.6 of the firmwith 1/2.6 of the firm

• The critical value for conversion is firm’s The critical value for conversion is firm’s value = 80 million*2.6/1.6 = $130 millionvalue = 80 million*2.6/1.6 = $130 million

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15.10 Pricing a Bond 15.10 Pricing a Bond Guarantee Guarantee

• Guarantees against credit risk commonGuarantees against credit risk common– Parent corporations guarantee the debt of subsidiariesParent corporations guarantee the debt of subsidiaries

– Commercial banks and insurance companies offer Commercial banks and insurance companies offer guarantees for a fee on a spectrum of financial guarantees for a fee on a spectrum of financial instruments including swaps & letters of creditinstruments including swaps & letters of credit

– U.S. Government guarantees bank deposits, SBA loans, U.S. Government guarantees bank deposits, SBA loans, pensions, farm & student loans, mortgages, the debt of pensions, farm & student loans, mortgages, the debt of other sovereign countries, and huge strategic other sovereign countries, and huge strategic corporationscorporations

– They occur implicitly every time a risky loan is madeThey occur implicitly every time a risky loan is made

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15.11 Other Applications 15.11 Other Applications of Option-Pricing of Option-Pricing MethodologyMethodology

– This slide presentation started with a This slide presentation started with a range of options that are embedded in range of options that are embedded in products and contractsproducts and contracts

– Options not associated with financial Options not associated with financial instruments are called instruments are called real optionsreal options

– The future is uncertain, so having The future is uncertain, so having flexibility to decide what to do after some flexibility to decide what to do after some of the uncertainty has been removed has of the uncertainty has been removed has valuevalue

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Options in Project-Options in Project-Investment Valuations:Investment Valuations:

– Option to initiateOption to initiate

– Option to expandOption to expand

– Option to abandonOption to abandon

– Option to reduce scaleOption to reduce scale

– Option to adjust timingOption to adjust timing

– Option to exploit a future technologyOption to exploit a future technology

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Examples:Examples:

– Choice of oil or gas to generate electricityChoice of oil or gas to generate electricity

– Product development of pharmaceuticalsProduct development of pharmaceuticals

– Making a sequel to a movieMaking a sequel to a movie

– Vocational educationVocational education

– Litigation decisionsLitigation decisions

– strategic decisionsstrategic decisions