1 chapter 15: options & contingent claims copyright © prentice hall inc. 2000. author: nick...
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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
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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
15.6 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
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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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Terninal or Boundary Conditions for Call and Put Options
-20
0
20
40
60
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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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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
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Stock, Call, Put, Bond
0
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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
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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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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),(
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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
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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
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Put and Call as Function of Share Price
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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
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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
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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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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
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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
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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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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
1
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9
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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
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Construction of Pat's Get Rich Portfolio
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-40
-20
0
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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
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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
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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
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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
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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
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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