1

Chapter 15: Options & Chapter 15: Options & Contingent ClaimsContingent Claims

Copyright © Prentice Hall Inc. 2000. Author: Nick Bagley, bdellaSoft, Inc.

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

2

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

15.6 Dynamic Replication & 15.6 Dynamic Replication & the Binomial Modelthe Binomial Model

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 form option prices volatility form option prices

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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

5

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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Terninal 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

lla

rs

Call Put

7

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

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

8

Stock, Call, Put, Bond

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100 120 140 160 180 200

Stock Price

Sto

ck,

Cal

l, P

ut,

Bo

nd

, P

ut+

Sto

ck,

Cal

l+B

on

d

Call

Put

Share_Put

Bond

Call_Bond

Share

9

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

11

Options and ForwardsOptions and Forwards

• We saw in the last chapter that the We saw in the last chapter that the discounted value of the forward was discounted value of the forward was equal to the current spotequal to the current spot

• The relationship becomesThe relationship becomes

MaturityMaturity rf

ForwardMaturityStrikePut

rf

StrikeMaturityStrikeCall

1),(

1),(

12

Implications for European Implications for European OptionsOptions

• If (F > E) then (C > P)If (F > E) then (C > P)

• If (F = E) then (C = P)If (F = E) then (C = P)

• If (F < E) then (C < P)If (F < E) then (C < P)• E is the common strike priceE is the common strike price

• F is the forward price of underlying shareF is the forward price of underlying share

• C is the call priceC is the call price

• P is the put price P is the put price

13

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

14

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

Pri

ces

call

put

asy_call_1

asy_call_2

asy_put_1

asy_put_2

15

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

16

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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Binary Model: CallBinary Model: Call

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

• buying a fraction x of shares, of the buying a fraction x of shares, of the stock, S, and simultaneously selling stock, S, and simultaneously selling short risk free bonds with a market short risk free bonds with a market value yvalue y

• the fraction x is called the the fraction x is called the hedge ratiohedge ratioyxSC

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Binary Model: CallBinary Model: Call

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

of the terminal share price, we know the of the terminal share price, we know the terminal option value for two cases:terminal option value for two cases:

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

yx

800

12020

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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=1/2, y = 40 into the parameters x=1/2, y = 40 into the equationequation

– to obtain:to obtain:

yxSC

10$401002

1C

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Binary Model: PutBinary Model: Put

• Implementation:Implementation:– the synthetic put, P, is created bythe synthetic put, P, is created by

• sell short a fraction x of shares, of the sell short a fraction x of shares, of the stock, S, and simultaneously buy risk stock, S, and simultaneously buy risk free bonds with a market value yfree bonds with a market value y

• the fraction x is called the the fraction x is called the hedge ratiohedge ratio

yxSP

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Binary Model: PutBinary Model: Put

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

of the terminal share price, we know the of the terminal share price, we know the terminal option value for two cases:terminal option value for two cases:

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

yx

800

12020

22

Binary Model: PutBinary Model: Put

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

parameters x=1/2, y = 60 into the parameters x=1/2, y = 60 into the equationequation

– to obtain:to obtain:

yxSP

10$601002

1P

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Decision Tree for Dynamic Decision Tree for Dynamic Replication of a Call Replication of a Call OptionOption

<---------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

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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 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

26

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

27

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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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

29

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

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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

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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

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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

33

Construction of Pat's Get Rich Portfolio

-120

-100

-80

-60

-40

-20

0

20

40

60

80

50 60 70 80 90 100 110 120 130 140 150

Share Price

Po

rtfo

lio

Val

ues

callP_ShareP_bondPortfolioTangent

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Pat in DespairPat in Despair

• The next diagram shows the the The next diagram shows the the value of the portfolio today and one value of the portfolio today and one week henceweek hence

• The construction lines have been The construction lines have been removed, and the graph has been removed, and the graph has been re-scaledre-scaled

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Strategy 1-Week Later

-0.5

0.0

0.5

1.0

1.5

2.0

90 95 100 105 110 115 120

Share Price

Str

ateg

y V

alu

e

Portfolio

PortfolioLater

36

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

illi

on

s)

BondValue

StockValue

37

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

ilit

y D

ensi

ty

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

Security Payoff State a Payoff State b

Firm 140 70

Bond 80 70

Stock 60 0

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Debtco’s Replicating Debtco’s Replicating PortfolioPortfolio• LetLet

– x be the fraction of the firm in x be the fraction of the firm in replicatorreplicator

– Y be the borrowings at the risk-free Y be the borrowings at the risk-free rate in the replicatorrate in the replicator

– In $’000,000 the following equations In $’000,000 the following equations must be satisfiedmust be satisfied

308,692,57$;7

6

04.1700;04.114060

Yx

YxYx

40

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

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Debtco’s Replicating Debtco’s Replicating PortfolioPortfolio

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

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

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

Security TotalmarketValue

EquivalentAmountof Firm

EquivalentAmount

of Rf DebtBonds 71,978 14,286 57,692

Stock 28,022 85,714 -57,692

Bonds +Stock

100,000 100,000 0

43

Valuing BondsValuing Bonds

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

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

026,641,90$

76

308,692,57000,000,20;

xYE

VYxVE

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Replication PortfolioReplication Portfolio

Position ImmediateCash Flow

Scenario aV1 = 70

Scenario bV1 = 140

Purchase xof firm

- x* V 70 x 140 x

PurchaseY RF Bond

- Y Y (1.04) Y (1.04)

TotalPortfolio

- x * V - Y 70 80

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Determining the Weight of Determining the Weight of Firm Invested in Bond, x, Firm Invested in Bond, x, and the Value of the R.F.-and the Value of the R.F.-Bond, YBond, Y

308,692,57$;7

1

04.114080

04.17070

Yx

Yx

Yx

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Valuing StockValuing Stock

– We can replicate the bond by purchasing We can replicate the bond by purchasing 1/7 of the company, and $57,692,308 of 1/7 of the company, and $57,692,308 of default-free 1-year bondsdefault-free 1-year bonds

– The market value of the bonds is $909.0909 The market value of the bonds is $909.0909 * 80,000 = $72,727,273* 80,000 = $72,727,273

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

753,244,105$

71

308,692,57273,727,72;

x

YEVYxVD

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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

Iss

ue

ConvertibleBondValue

DilultedStockValue

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Outline Decision TreeOutline Decision Tree

Node-B$115MM

Node-C$90MM

Node-D$140MM

Node-F$110MM

Node-E$90MM

Node-G$70MM

Node-A$100MM

Month 0 Month 6 Month 12

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Valuing Pure State-Valuing Pure State-Contingent SecuritiesContingent Securities

Security PayoffScenario a

PayoffScenario b

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

ContingentSecurity #1

$0 $1

ContingentSecurity #2

$1 $0

50

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

04.1

1$538961.0$505494.0$033467.0$

495505494.004.1

2

000,000,70

000,000,100000,000,1

04.1

2;

000,000,70

1

004.1000,000,140

104.1000,000,70

#2 S. C. S.

967032467.004.1

1

000,000,70

000,000,100000,000,1

04.1

1;

000,000,70

1

104.1000,000,140

004.1000,000,70

#1 S. C. S.

21

2

1

PP

YxP

YxYx

Yx

YxP

YxYx

Yx

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Payoff for Debtco’s Bond Payoff for Debtco’s Bond GuaranteeGuarantee

Security Scenario a Scenario b

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

Bonds $1,000 $875

Guarantee $0 $125

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SCS Conformation of SCS Conformation of Guarantee’s PriceGuarantee’s Price

• Guarantee’s price is 125 * Guarantee’s price is 125 * $0.494505 = $0.494505 = $61.81$61.81