Corporate finance chapter 015 option and contingent claims

Chapter 15: Options &
Contingent Claims

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 option
1
prices

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


Chapter 15 Contents
15.1 How Options Work

15.7 The Black-Scholes Model

15.2 Investing with Options

15.8 Implied Volatility

15.3 The Put-Call Parity
Relationship

15.9 Contingent Claims Analysis of
Corporate Debt and Equity

15.4 Volatility & Option Prices

15.10 Credit Guarantees

15.5 Two-State Option Pricing

15.11 Other Applications of
Option-Pricing Methodology

15.6 Dynamic Replication & the
Binomial Model

2


Objectives
•

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

•

To show how to infer implied volatility form option prices

3


Table 15.1 List of IBM Option Prices
(Source: Wall Street Journal Interactive Edition, May 29, 1998)

IBM (IBM)

Strike
115
115
115
120
120
120
125
125
125

Underlying stock price 120 1/16
Call .
Put .

Expiration Volume
Jun
Oct
Jan
Jun
Oct
Jan
Jun
Oct
Jan

Last

1372
…

…

7
…
…

2377
121
91
1564
91
87

3 1/2
9 5/16
12 1/2
1 1/2
7 1/2
10 1/2

Open
Interest
4483
2584
15
8049
2561
8842
9764
2360

124

4

Volume

Last

756
10
53
873
45
…

1 3/16
5
6 3/4
2 7/8
7 1/8
…

17
…
…

5 3/4
…
…

Open
Interest
9692
967
40
9849
1993
5259
5900
731
70


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

S & P 500 INDEX -AM
Underlying
S&P500
(SPX)
Jun
Jun
Jul
Jul
Jun
Jun
Jul
Jul

High

Low

1113.88

1084.28

Strike
1110 call
1110 put
1110 call
1110 put
1120 call
1120 put
1120 call
1120 put

Chicago Exchange
Close
1113.86

Net
Change
19.03

Volume
2,081
1,077
1,278
152

80
211
67
10

Last
17 1/4
10
33 1/2
23 3/8
12
17
27 1/4
27 1/2

5

From
31-Dec
143.43
Net
Change
8 1/2
-11
9 1/2
-12 1/8
7
-11
8 1/4
-11

%
Change
14.8
Open
Interest
15,754
17,104
3,712
1,040
16,585
9,947
5,546
4,033


Terninal or Boundary Conditions for Call and Put Options
120

100

Dollars

80

Call

Put

60

40

20

0
0

20

40

60

80

100

120

140

-20

Underlying Price

6

160

180

200


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

100

Share
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180

190
200

Call
0
0
0
0
0
0
0
0
0
0
0
10
20
30
40
50
60
70
80
90
100

Put
Share_Put
100
100

90
100
80
100
70
100
60
100
50
100
40
100
30
100
20
100
10
100
0
100
0
110
0
120
0
130
0
140
0
150

0
160
0
170
0
180
0
190
0
200

Bond Call_Bond
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100

100
100
100
100
100
110
100
120
100
130
100
140
100
150
100
160
100
170
100
180
100
190
100
200

7


Stock, Call, Put, Bond
200

Call
Put

160

Share_Put

140

Bond

Stock, Call, Put, Bond, Put+Stock,
Call+Bond

180

Call_Bond

120

Share

100
80
60
40
20
0
0

20

40

60

80

100

120

140

Stock Price

8

160

180

200


Put-Call Parity Equation
Call ( Strike, Maturity ) +

Strike

= Put ( Strike, Maturity ) + Share
Maturity
(1 + rf )

9


Synthetic 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

10


Options and Forwards
•

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

•

The relationship becomes

Call ( Strike, Maturity ) +

Strike
Forward
=
Put
(
Strike
,
Maturity
)
+
(1 + rf ) Maturity
(1 + rf ) Maturity

11


Implications 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 strike price
• F is the forward price of underlying share
• C is the call price
• P is the put price
12


Call and Put as a Function of Forward
Call = Put
16
call
put
asy_call_1
asy_put_1

14
Put, Call Values

12
10
8
6
4
2
0
90

92

94

96

98

100

102

104

106

108

Forward

Strike = Forward

13

110


Put and Call as Function of Share Price
60
call

Put and Call Prices

50

put
asy_call_1

40

asy_call_2
asy_put_1

30

asy_put_2

20
10
0
50

60

70

80

90

100

110

-10
Share Price

14

120

130

140

150


Put and Call as Function of Share Price
20

call

Put and Call Prices

put
asy_call_1

15

asy_call_2
asy_put_1
asy_put_2

10

5

0
80

85

90

95

100

105

110

115

Share Price

PV
Strike

15

Strik
e

120


Volatility and Option Prices, P0 = $100, Strike = $100
Stock Price Call Payoff Put Payoff
Low Volatility Case
Rise
Fall
Expectation

120
80
100

20
0
10

0
20
10

140
60
100

40
0
20

0
40
20

High Volatility Case
Rise
Fall
Expectation

16


Binary Model: Call
•

Implementation:

– the synthetic call, C, is created by
• buying a fraction x of shares, of the stock, S,
and simultaneously selling short risk free
bonds with a market value y
• the fraction x is called the hedge ratio

C = xS − y
17

Binary Model: Call
•

Specification:

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

20 = x120 − y
0 = x80 − y

– By inspection, the solution is x=1/2, y = 40
18


Binary Model: Call
•

Solution:

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

C = xS − y

– to obtain:

1
C = 100 − 40 = $10
2
19


Binary Model: Put
•

Implementation:

– the synthetic put, P, is created by
• sell short a fraction x of shares, of the stock,
S, and simultaneously buy risk free bonds
with a market value y
• the fraction x is called the hedge ratio

P = − xS + y
20


Binary Model: Put
•

Specification:

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

20 = x120 − y
0 = x80 − y

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

21


Binary Model: Put
•

Solution:

– We now substitute the value of the
parameters x=1/2, y = 60 into the equation

P = − xS + y

– to obtain:

1
P = − 100 + 60 = $10
2
22


Decision Tree for Dynamic
Replication of a Call Option
<---------0 Months----------> <------------------6 Months----------------> 12 Months
StockPrice

x
y
CallPrice
x
y
CallPrice
$120.00
$110.00
$100.00
$90.00
$80.00

$20.00
$10.00
50.00%

100.00%

-$100.00

-$45.00

$0.00
$0.00

0.00%

$0.00
$0.00

($120*100%) + (-$100) = $20
23


The Black-Scholes Model:
Notation
• C = price of call

• N(.) = cum. norm. dist’n

• P = price of put

• The following are annual,
compounded continuously:

• S = price of stock
• E = exercise price
• T = time to maturity

• r = domestic risk free
rate of interest

• ln(.) = natural logarithm

• d = foreign risk free rate
or constant dividend yield

• e = 2.71828...

• σ = volatility

24


The Black-Scholes Model:
Equations
1 2
S 
ln  +  r − d + σ T
E 
2 

d1 =
σ T
1 2
S 
ln  +  r − d − σ T
E 
2 

d2 =
= d1 − σ T
σ T
C = Se − dT N ( d1 ) − Ee − rT N ( d 2 )

P = − Se − dT N ( − d1 ) + Ee − rT N ( − d 2 )
25