Transcript Chapter 15: Options Pricing

Finance
School of Management
Chapter 15: Options and
Contingent 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
1
School of Management
Finance
Chapter 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 StateContingent Securities
2
School of Management
Finance
Terms

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
3
School of Management
Finance
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
Put
Call
Strike
115
115
115
120
120
120
125
125
125
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
Volume
Last
Open
Interest
Interest
4483
756
1 3/16
9692
2584
10
5
967
15
53
6 3/4
40
8049
873
2 7/8
9849
2561
45
7 1/8
1993
8842 …
…
5259
9764
17
5 3/4
5900
2360 …
…
731
124 …
…
70
4
School of Management
Finance
Table 15.2
List of Index Option Prices
(Source: Wall Street Journal Interactive Edition, June 6, 1998)
S&P500 INDEX -AM
Underlying
S&P500
(SPX)
Jun
Jun
Jul
Jul
Jun
Jun
Jul
Jul
High
1113.88
Strike
1110 call
1110 put
1110 call
1110 put
1120 call
1120 put
1120 call
1120 put
Chicago Exchange
Low
1084.28
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
From
31-Dec
143.43
Net
Change
8 1/2
-11
9 1/2
-12 1/8
7
-11
8 1/4
-11
5
%
Change
14.8
Open
Interest
15,754
17,104
3,712
1,040
16,585
9,947
5,546
4,033
School of Management
Finance
Terminal or Boundary Conditions for Call and Put Options
120
100
Call
Dollars
80
Put
60
40
20
0
0
20
40
60
80
100
120
140
160
-20
Underlying Price
6
180
200
Finance
School of Management
The 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 protectiveput strategy)
– Buying a pure discount bond with a face value equal to
the option’s exercise price and simultaneously buying a
call option
7
School of Management
Finance
Terminal Conditions of a Call and a Put Option with 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
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School of Management
Finance
200
180
Share_Put
160
Call_Bond
Payoffs
140
120
Bond
100
80
Share
60
Call
40
Put
20
0
0
20
40
60
80
100
120
140
160
Stock Price
9
180
200
School of Management
Finance
Payoff Structure for Protective-Put Strategy
Position
Stock
Put
Stock plus put
Value of Position at Maturity Date
If S T < E
ST
E-S T
E
If S T > E
ST
0
ST
10
School of Management
Finance
Payoff Structure for a Pure Discount Bond
Plus a Call
Position
Pure discount bond with face
value of E
Call
Pure discount bond plus call
Value of Position at Maturity Date
If S T < E
If S T > E
E
E
0
E
ST - E
ST
11
School of Management
Finance
Put-Call Parity Equation
Call( Strike, Maturity) 
C
Strike
1  r 
Maturity
E
1  r T
 Put( Strike, Maturity)  Share
 PS
B: bond
12
School of Management
Finance
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
13
School of Management
Finance
Converting a Put into a Call
CS
E
1  r 
T
P
 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
14
School of Management
Finance
Put-Call Arbitrage
C=S+P-B
Immediate Cash
Flow
Sell a call
$18
Buy Replicating Portfolio (Synthetic Call)
Buy a stock
-100.00
Borrow the present value of $100
92.59
Buy a put
-10.00
Net cash flows
0.59
Position
Cash Flow at Maturity Date
If S T < $100 If S T > $100
0
- (S T - $100)
ST
-100.00
ST
-100.00
0.00
0.00
$100 - S T
0.00
15
School of Management
Finance
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
C
or
E
1  r 
T
 P
C  P
F
(1  r )T
F E
(1  r ) T
If the exercise price is equal to the forward price of the
underlying stock, then the put and call have the same price
16
School of Management
Finance
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 exercise price
F is the forward price of underlying share
C is the call price
P is the put price
17
School of Management
Finance
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
18
110
School of Management
Finance
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
120
130
140
-10
100/(1+r)
Share Price
19
150
School of Management
Finance
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
120
Strike
PV Strike
20
School of Management
Finance
Volatility and Option Prices
P0 = $100, Strike Price = $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
The prices of options increase with the volatility of the stock
21
Finance
School of Management
Two-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
22
School of Management
Finance
Binary 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
C  xS  y
23
School of Management
Finance
Binary Model: Creating the Synthetic Call
S = $100, E = $100, T = 1 year, d = 0, r = 0
Position
Call Option
Synthetic Call
Buy x shares of stock
Borrow y
Total replicating portfolio
Immediate
Cash Flow
-$C
Cash Flow at Maturity Date
If S 1 = $120 If S 1 = $80
$20
0
- $100x
$120x
$80x
y
-$100x+y
-y
$120x −y
-y
$80x −y
24
School of Management
Finance
Binary 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:
20  x120  y
0  x80  y
– By inspection, the solution is x=1/2, y = 40.
The Law of One Price
25
School of Management
Finance
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
26
School of Management
Finance
Binary 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
P   xS  y
27
School of Management
Finance
Binary Model: Creating the Synthetic Put
S = $100, E = $100, T = 1 year, d = 0, r = 0
Position
Put option
Synthetic Put
Sell short x shares of stock
Invest y in the risk-free asset
Total replicating portfolio
Immediate
Cash Flow
-$P
Cash Flow at Maturity Date
If S 1 = $120 If S 1 = $80
0
$20
$100x
- $120x
- $80x
-y
$100x-y
y
$0
y
$20
28
School of Management
Finance
Binary 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:
20  x120  y
0  x80  y
– By inspection, the solution is x = 1/2, y = 60
The Law of One Price
29
School of Management
Finance
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
30
School of Management
Finance
Decision Tree for Dynamic Replication
of a Call Option
D
$120
B
$110
E
$100
A
$100
C
$90
F
$80
31
School of Management
Finance
Decision Tree for Dynamic Replication
of a Call Option
$20
D
$120
B
$110
C11
E
$100
0
– 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
32
School of Management
Finance
Decision Tree for Dynamic Replication
of a Call Option
E
$100
C
$90
0
C12
F
$80
0
– The terminal option value for two cases:
90x – y = 0
80x – y = 0
– By inspection, the solution is x=0, y = 0
– Thus, C12 = 0*$90 − $0 = $0
33
School of Management
Finance
Decision Tree for Dynamic Replication
of a Call Option
B
$110
A
$100
$10
C0
C
$90
0
– The terminal option value for two cases:
110x – y = 10
90x – y = 0
– By inspection, the solution is x=1/2, y = $45
– Thus, C0 = (1/2)*$100 − $45 = $5
34
School of Management
Finance
Decision Tree for Dynamic Replication
of a Call Option
Buy another half share of stock
Increase borrowing to $100
D
$120
Sell shares $120
Pay off debt -$100
Total $20
B
$110
E
$100
A
$100
Buy 1/2 share of stock
Borrow $45
Total investment $5
Sell shares $100
Pay off debt -$100
Total 0
C
$90
Sell stock and pay off debt
F
$80
35
School of Management
Finance
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
36
Finance
School of Management
The Black-Scholes Model: The Limiting
Case 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
37
School of Management
Finance
The Black-Scholes Model
1 
S 
ln    r  d   2 T
E 
2 

d1 
 T
1 
S 
ln    r  d   2 T
E 
2 

d2 
 d1   T
 T
Shares of
Stock
C  Se  dT N d1   Ee  rT N d 2 
P   Se  dT N  d1   Ee  rT N  d 2 
38
Bond
School of Management
Finance
The 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
39
School of Management
Finance
The Black-Scholes Model:
Dividend-adjusted Form
 Se r  d T  1 2
ln
  T
E  2
d1  
 T
 Se r  d T  1 2
ln
  T
E  2

d2 
 T
C  e  rT N d1 Se r  d T  N d 2 E 
P  e  rT N  d1 Se r  d T  N  d 2 E 
40
School of Management
Finance
The Black-Scholes Model :
Dividend-adjusted Form (Simplified)
If E  Se r  d T
1
1
d1   T ; d 2    T
2
2
C  Se dT  N d1   N d 2   P
If d  0
C  P  S  N d1   N d 2 
S
 T  0.39886S T
CP
2
41
School of Management
Finance
Determinants of Option Prices
Increases in:
Stock Price, S
Exercise Price, E
Volatility, sigma
Time to Expiration, T
Interest Rate, r
Cash Dividends, d
Call
Put
Increase
Decrease
Increase
Ambiguous
Increase
Decrease
Decrease
Increase
Increase
Ambiguous
Decrease
Increase
42
School of Management
Finance
Value of a Call and Put Options with Strike = Current Stock Price
11
10
call
put
8
7
6
5
4
3
2
1
0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Time-to-Maturity
43
0.0
Call and Put Price
9
School of Management
Finance
Call and Put Prices as a Function of Volatility
6
Call and Put Prices
5
call
put
4
3
2
1
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
Volatility
44
0.20
School of Management
Finance
Implied Volatility


The value of σ that makes the observed market price
of the option equal to its Black-Scholes formula value
S
E
r
T
d
C
100
108.33
0.08
1
0
7.97
σ
0.2
Approximation:

C 2
S T
45
School of Management
Finance
Implied Volatility
46
School of Management
Finance
Computing Implied Volatility
volatility
call
strike
share
rate_dom
rate_for
maturity
0.3154
10.0000
100.0000
105.0000
0.0500
0.0000
0.2500
factor
0.0249
d_1
d_2
0.4675
0.3098
n_d_1
n_d_2
0.6799
0.6217
call_part_1
call_part_2
error
71.3934
-61.3934
Insert any number to
start
Formula for option value
minus the actual call
value
0.0000
47
School of Management
Finance
Computing Implied Volatility
volatility
0.315378127101852
call
strike
share
rate_dom
rate_for
maturity
10
100
105
0.05
0
0.25
factor
=(rate_dom - rate_for + (volatility^2)/2)*maturity
d_1
d_2
=(LN(share/strike)+factor)/(volatility*SQRT(maturity))
=d_1-volatility*SQRT(maturity)
n_d_1
n_d_2
=NORMSDIST(d_1)
=NORMSDIST(d_2)
call_part_1
call_part_2
=n_d_1*share*EXP(-rate_for*maturity)
=- n_d_2*strike*EXP(-rate_dom*maturity)
error
=call_part_1+call_part_2-call
48
Finance
School of Management
Valuation of Uncertain Cash Flows:
CCA / 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
49
School of Management
Finance
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

50
School of Management
Finance
Debtco: 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
51
Finance
School of Management
Two Ways to Think about the Debtco’s
Market 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-equityfinanced “twin” of Debtco is $100 million

52
School of Management
Finance
Payoffs for Bond and Stock Issues
Value of Bond and Stock (Millions)
120
100
80
Bond Value
60
Stock Value
40
20
0
0
20
40
60
80
100
120
140
160
180
Value of Firm (Millions)
53
200
School of Management
Finance
120
100
Value of the Bonds
Value
80
60
40
Value of the Stock
20
0
0
-20
20
40
60
80
100
120
140
160
180
Value of the firm in 1 year
54
200
School of Management
Finance
120
The payoff is identical to a call option in
which the underlying asset is the firm
itself, and the exercise price is the face
value of its debt
100
Value
80
60
40
Value of the Stock
20
0
0
-20
20
40
60
80
100
120
140
160
180
Value of the firm in 1 year
55
200
School of Management
Finance
The
value of the firm’s equity
E  N (d1 )V  N (d 2 ) Be  rT
ln(V / B )  (r   / 2)T
d1 
 T
2
d 2  d1   T
The
value of the debt
D＝V－E
56
School of Management
Finance
Probalility Density of a Firm's Value
0.09
0.08
Probability Density
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
0
20
40
60
80
100
120
140
160
180
Value of a Firm
57
200
School of Management
Finance
Debtco Security Payoff Table ($’000,000)
Security
Payoff State A
Payoff State B
Firm
140
70
Bond
80
70
Stock
60
0
58
School of Management
Finance
Debtco’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
60  140x  1.04Y
0  70 x  1.04Y
6
 x  ; Y  $57,692,308
7
59
School of Management
Finance
Debtco’s Replicating Portfolio ($’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
60
Finance
School of Management
Debtco’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%
61
School of Management
Finance
Another View of Debtco’s
Replicating Portfolio (‘$000)
Security
Total
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
100,000
100,000
0
Bonds + Stock
62
Finance
School of Management
Given 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%
63
School of Management
Finance
Valuing Bonds

We can replicate the firm’s equity using x = 6/7 of the
firm, and about Y = $58 million riskless borrowing
(earlier analysis)
E Y
x
20,000,000  57,692,308

 $90,641,026
67
E  xV  Y  V 

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%
64
Finance
School of Management
Given 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%
65
School of Management
Finance
Replication Portfolio
Position
Purchase x
of firm
Purchase Y
RF Bond
Total
Portfolio
Immediate Cash Flow
Scenario A
V1 = 70
- xV
70 x
140
x
Y (1.04)
Y (1.04)
70
80
-Y
- xV -Y
Scenario B
V1 = 140
66
Finance
School of Management
Determining the Weight of Firm Invested
in Bond, x, and the Value of the R.F.Bond, Y
70  70x  1.04Y 

80  140x  1.04Y 
1
x  ; Y  $57,692,308
7
67
School of Management
Finance
Valuing 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
D Y
D  xV  Y ;  V 
x
72,727,273  57,692,308

 $105,244,753
17
The value of the stock is therefore E = V −D =
$105,244,753 − $72,727,273= $32,517,480
68
School of Management
Finance
Convertible 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
69
Finance
School of Management
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
70
School of Management
Finance
Critical value of Convertident for Conversion

Upon convertion, the total shares of stock will be
2.6 million
1.6 2.6V1  61.5%V1
1.6 2.6V1*  $80 million
*
V1
 $130 million
71
School of Management
Finance
Convertible Bond
140
Value of Stock and Bond Issue
Convertible Bond Value
120
Dilulted Stock Value
100
80
60
40
20
0
0
20
40
60
80
100
120
140
160
Value of the Firm
72
180
200
School of Management
Finance
Payoff for Convertidett’s
Stocks and Bonds
Security
Payoff State A
Payoff State B
140,000,000
70,000,000
Bonds
86,153,846
70,000,000
Stocks
53,846,154
0
Firm
73
School of Management
Finance
Convertidett’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
53.846,154  140x  1.04Y
0  70x  1.04Y
 x  .76,923,077; Y  $51,775,148
74
School of Management
Finance
Values of Convertidett’s Stocks and Bonds
E  $76,923,077  $51,775,148  $25,147,929
D  V  E  $100,000,000 $25,147,929  $74,852,071
Bond Price $935.65
$1,000 $935.65
YTM 
 6.88%
$935.65
75
School of Management
Finance
Decomposition of Convertidett’s
Stocks and Bonds
Security
Total
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
100,000,000
100,000,000
0
Bonds + Stock
76
Finance
School of Management
Pure 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
77
School of Management
Finance
Valuing 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
78
School of Management
Finance
State-Contingent Security #1
70,000,000x  1.04Y  0
1
1
; Y 
 x 
140,000,000x  1.04Y  1 
70,000,000
1.04
P1  1,000,000x  Y
100,000,000
1


 0.467 032 967
70,000,000 1.04
79
School of Management
Finance
State-Contingent Security #2
70,000,000x  1.04Y  1 
2
1
; Y
 x 
140,000,000x  1.04Y  0
1.04
70,000,000
P2  1,000,000x  Y
2
100,000,000
 0.494 505 495


70,000,000 1.04
$1
P1  P2  $0.467 033  $0.494 505  $0.961 538 
1.04
80
School of Management
Finance
Valuing Debtco’s Securities
Security
Firm
Debtco Stock
Debtco Bond


Possible Payoff in 1 Year
$140 million
$70 million
$60
0
$1,000
$875
Price of a Debtco stock = 60P1 = 60*$.4670329 = $28.02
Price of a Debtco bond = 1,000P1 + 875P2
= 1,000*$.4670329 + 875*$.494505 = $899.73
81
School of Management
Finance
Valuing Convertidett’s Securities
Security
Firm


Possible Payoff in 1 Year
$140 million
$70 million
Debtco Stock
$53.85
0
Debtco Bond
$1,076.92
$875
Price of a Convertidett stock = 53.86415P1
= 53.86415*$.4670329 = $25.15
Price of a Convertidett bond = 1,076.923P1 + 875P2
= 1,076.923*$.4670329 + 875*$.494505 = $935.65
82
School of Management
Finance
Payoff for Debtco’s Bond Guarantee
Security
Firm
Bonds
Guarantee
Scenario A
Scenario B
$140,000,000
$70,000,000
$1,000
$875
$0
$125
83
Finance
School of Management
SCS Conformation of Guarantee’s Price

Guarantee’s price = 125P2 = 125* 0.494505 = $61.81
84
School of Management
Finance
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
85
School of Management
Finance
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
86