Transcript Chapter 15

Chapter 15: Option Pricing
Objective
Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley
•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
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
from option prices
3
Introduction
• This chapter explores how option prices
are affected by the volatility of the
underlying security
• Exchange traded options appeared in
1973, enabling us to determine the
market’s estimate of future volatility,
rather than relying on historical values
4
Definition of an Option
• Recall that an American {European} call
(put) option is the right, but not the
obligation to buy (sell) an asset at a
specified price any time before its
expiration date {on its expiration date}
5
Ubiquitous Options
• This chapter focuses on traded options,
but it would be a mistake to believe that
the tools we will be developing are
restricted to traded options
• Some examples of options are given on
the next few slides
6
Government Price Supports
• Governments sometimes provide
assistance to farmers by offering to
purchase agricultural products at a
specified support price
• If the market price is lower than the
support, then a farmer will exercise her
right to ‘put’ her crop to the government
at the higher price
7
Old Mortgage
• Traditional US mortgages give the
householder the right to call the
mortgage at a strike equal to the
outstanding principle
• If interest rates have fallen below the
note’s rate, then the home owner will
consider refinancing the mortgage
8
New Mortgage
• You pay some ‘points’ to lock-in an
interest rate on a mortgage
– If rates fall, you may renegotiate the
mortgage, and then pay more points to lock
in the new lower rate
– If rates rise, then you will go to the
settlement table with a lower-than-market
interest rate
9
Tenure and Seniority
• In a company that has a policy of last-in
first-out, a worker with seniority may
forego a higher salary in another
company because of the loss of job
security
• The worker has been given the right, but
not the obligation, to have work under a
set of adverse economic conditions
10
Copper Pennies, Silver Coin
• Silver and copper coinage has been
replaced by zinc and cooper alloys/
composites to reduce their minting costs
• The old coins are often legal currency,
and so contain an option feature:
– If the price of the underlying metal falls
below its legal value, I have the right to
return the coin into circulation
11
Insurance
• Insurance policies often give you the
right, but not the obligation to do
something, it is therefore option-like
– The renewable rider on a term life policy is
an option
– If somebody:
• is terminally ill, then the rider is very valuable
• remains in good health, then it is not valuable
12
Supply Contracts
• A nuclear power plant supplier once got
into serious trouble by guaranteeing to
supply enriched uranium at a fixed price
• The market price of enriched uranium
rose precipitously
13
Technological Leases
• A computer leasing company had a
clause in its lease stating that the
customer had the right to cancel
• The computer manufacturer introduced
next generation of computers, and the
leasing company’s customer’s canceled
their leases, resulting in a massive
inventory of obsolete computers
14
Limited Liability
• The owners of a limited liability
corporation have the right, but not the
obligation, to ‘put’ the company to the
corporation’s creditors and bondholders
• Limited liability is, in effect, a put option
15
Trading on Commission
• You are a trader with a contract giving
you a commission of 20% of each
months trading profits
• If you make a loss, then you walk away,
but if you make a profit, you stay
– You may be tempted to increase your
volatility to boost the value of your option
16
15.1 How Options Work
• The Language of Options
– Contingent Claim: Any asset whose future pay-off depends
upon the outcome of an uncertain event
– Call: an option to buy
– Put: an option to sell
– Strike or Exercise Price: the fixed price specified in an
option contract
– Expiration or Maturity Date: The date after which an
option can’t be exercised
– American Option: an option that can be exercised at any
time up to and including maturity date
17
– European Option: an option that can only be exercised on
the maturity date
– Tangible Value: The hypothetical value of an option if it
were exercised immediately
– At-the-Money: an option with a strike price equal to the
value of the underlying asset
– Out-of-the-Money: an option that’s not at-the-money, but
has no tangible value
– In-the-Money: an option with a tangible value
– Time Value: the difference between an option’s market
value and its tangible value
– Exchange-Traded Option: A standardized option that an
exchange stands behind in the case of a default
– Over the Counter Option: An option on a security that is
not an exchange-traded option
18
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
19
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
20
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
15.2 Investing with Options
• The payoff diagram (terminal conditions,
boundary conditions) for a call and a put
option, each with a strike (exercise price)
of $100, is derived next
21
Option Payoff Diagrams
• The value of an option at expiration
follows immediately from its definition
– In the case of a call option with strike of
$100, if the stock price is $90 ($110), then
exercising the option results purchasing the
share for $100, which is $10 more expensive
($10 less expensive) than buying it, so you
wouldn't (would) exercise your right
22
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
23
160
180
200
15.3 The Put-Call Parity
Relationship
• Consider the following two strategies
– Purchase a put with a strike price of $100,
and the underlying share
– Purchase a call with a strike price of $100
and a bond that matures at the same date
with a face of $100
• The maturity values are tabulated and
plotted against share price:
24
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
25
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
Stock Price
26
140
160
180
200
Observation
• The most important point to observe is
that the value of the “call + bond”
strategy, is identical (at maturity) with
the protective-put strategy “put + share”
• So, if the put and the call have the same
strike price, we obtain the put-call parity
relationship: put + share = call + bond
27
Technical Note
– The above relationship is true, in general, for
dividend-less European options, but the
actual proof requires taking expectations of
the option and security boundary conditions
– The argument is therefore a heuristic for
remembering and ‘seeing’ the relationship
– The full proof is left for your investment class
28
Put-Call Parity for American
and European Options
• A European option that pays no dividend
during its life fully satisfies the
requirements of put-call parity
• In the case of American options, the
relationship is fully accurate only at
maturity, because American puts are
sometimes exercised early
29
Put-Call Parity Equation
Call ( Strike , Maturity ) 
Strike
 Put( Strike , Maturity )  Share
Maturity
1  rf 
30
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
31
Synthetic Securities
C=S+P-B and P=C-S+B may be used by
floor traders to flip between a call and a put
S=C-P+B may be used by short-term traders
wishing to take advantage of lower
transaction costs
B=S+P-C may be used to create a synthetic
bond said to pay a slightly higher return than
the physical bond
32
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
33
Implications for European
Options
• If the forward price of the underlying
stock is equal to the strike price, then the
value of the call is equal to the value of
the put
• This relationship is so important, that
some option traders define ‘at-themoney’ in terms of the forward rather
than the spot
34
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
35
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
36
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
37
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
Share Price
PV
Strike
38
110
115
Strik
e
120
15.4 Volatility and Option
Prices
• We next explore what happens to the
value of an option when the volatility of
the underlying stock increases
– We assume a world in which the stock price
moves during the year from $100 to one of
two new values at the end of the year when
the option matures
– Assume risk neutrality
39
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
40
Illustration Explained
• The stock volatility in the second scenario
is higher, and the expected payoffs for
both the put and the call are also higher
– This is the result of truncation, and holds in
all empirically reasonable cases
• Conclusion: Volatility increases all option
prices
41
15.5 Two-State (Binomial)
Option-Pricing
– We are now going to derive a relatively
simple model for evaluating options
• The assumptions will at first appear totally
unrealistic, but using some underhand
mathematics, the model may be made to
price options to any desired level of accuracy
• The advantage of the method is that it does
not require learning stochastic calculus, and
yet it illustrates all the key steps necessary to
derive any option evaluation model
42
Binary Model Assumptions
• Assume:
– the exercise price is equal to the forward
price of the underlying stock
• option prices then depend only on the
volatility and time to maturity, and do not
depend on interest rates
• the put and call have the same price
43
Binary Model Assumptions
• More specifically we assume:
– share price = strike price = $100
– time to maturity = 1 year
– dividend rate = interest rate = 0
– stock prices either rise or fall by 20% in the
year, and so are either $80 or $120 at
yearend
44
Binary Model: Call
• Strategy:
– replicate the call using a portfolio of
• the underlying stock
• the riskless bond
– by the law of one price, the price of the
actual call must equal the price of the
synthetic call
45
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
46
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
47
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
48
Binary Model: Put
• Strategy:
– replicate the put using a portfolio of the
underlying stock and riskless bond
– by the law of one price, the price of the
actual put must equal the price of the
synthetic put replicated above
• Minor changes to the call argument are
made in the next few slides for the put
49
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
50
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
51
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
52
15.6 Dynamic Replication and
the Binomial Model
– We now take the next step towards greater
realism by dividing the year into 2 subperiods of half a year each. This gives 3
possible outcomes
– Our first task is to find a self-financing
investment strategy that does not require
injection or withdrawal of new funds during
the life of the option
– We first create a decision tree:
53
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
54
Reading the Decision Tree
• The tree is constructed backwards
because we know only the future
contingent call prices
– For Example, when constructing the weights
for time 6-months, the option prices for 12months are used
• For consistency with the next model, the
discrete stock prices are usually fixed ratios,
i.e. 121, 110, 100, 90.91, 82.64
55
The Power of Lattice Models
– Lattice models, of which the binary model is
the simplest, are very important to traders
because they may be modified to handle
different distributions, the possibility of early
exercise, and discrete dividend payments
– To see how easy it is to change the
distributional assumption, the above
illustration results in stock prices being
normally distributed, and the modification
results in a lognormal distribution
of prices
56
15.7 The Black-Scholes Model
• The most widely used model for pricing
options is the Black-Scholes model
– This model is completely consistent with the
binary model as the interval between stock
prices decreases to zero
– The model provides theoretical insights into
option behavior
– The assumptions are elegant, simple, and
quite realistic
57
The Black-Scholes Model
• We will work with the generalized form of
the model because the small additional
complexity results in considerable
additional power and flexibility
• First, notation:
58
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
59
The Normal Problem
• It is not unusual for a student to have a
problem computing the cumulative
normal distribution using tables
– table structures vary, so be careful
– using standard-issue normal tables degrades
computed option values because of errors
caused by catastrophic subtraction
– {Many professionals use Hasting’s formula as reported in
Abramowitz and Stegun as equation 26.2.19 (never, never
use 26.2.18). Its certificate valid in 0<=x<Inf, so use
symmetry to get -Inf<x<0}
60
The Normal Problem
• The functions that come with Excel have
adequate accuracy, so consider using
‘Normsdist()’ in the statistical functions
(note the s in Normsdist)
61
The Black-Scholes Model:
What’s missing
• There are no expectations about future
returns in the model
• The model is preference-free
• σ-risk, not b-risk, is the relevant risk
62
The Black-Scholes Model:
Equations
1 2
S 
ln    r  d   T
E 
2 

d1 
 T
1 2
S 
ln    r  d   T
E 
2 

d2 
 d1   T
 T
C  Se  dT N d1   Ee  rT N d 2 
P   Se  dT N  d1   Ee  rT N  d 2 
63
The Black-Scholes Model:
Equations (Forward 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 
64
The Black-Scholes Model:
Equations (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
CP
 T  0.39886S T
2
65
So What Does it Mean?
• You can now obtain the value of nondividend paying European options
• With a little skill, you can widen this to
obtain approximate values of European
options on shares paying a dividend, and
to some American options
66
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
67
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
Time-to-Maturity
68
0.2
0.1
0.0
Call and Put Price
9
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
Volatility
69
0.14
0.16
0.18
0.20
Observable Variables
• All the variables are directly observable,
excepting the volatility, σ, and possibly,
the next cash dividend, d
• We do not have to delve into the psyche
of investors to evaluate options
• We do not forecast future prices to
obtain option values
70
15.8 Implied Volatility
• The following slides show how to
estimate volatility using Excel
– The option most commonly used to estimate
volatility is the one closest to the present
value of the strike price: That is, the option
that is has a strike closest to the forward
price
– This option has the most “oomph”
71
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
Insert any number to
start
Formula for option value
minus the actual call
value
71.3934
-61.3934
0.0000
72
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
73
Pat’s Plan
• Pat has a plan to get rich with no risk:
– Set up special portfolio, (Pat calls this a “self
financing, delta neutral portfolio with positive
curvature,” but Pat has this thing with words)
74
Pat, the Strategist
• Short some shares, and off-set small
price changes about the current price
with some call options, then invest the
difference in bonds
75
Construction of Pat's Get Rich Portfolio
80
60
Portfolio Values
40
20
0
-20
50
60
70
80
90
100
110
120
-40
-60
-80
-100
call
P_Share
P_bond
Portfolio
Tangent
-120
76
Share Price
130
140
150
Pat, the Cartographer
• Apparently, what Pat has done is to find
the tangent (at today’s share price) of
the call value curve, using bonds and
stock in the right proportions
• This is what we did earlier when we
constructed the binary pricing model
• At the current price of $103, the tangent
mimics the call curve
77
Pat (Continued)
• Pat then went short the tangency
portfolio, and long the call to create the
thick black portfolio
• Observe that the minimum value of the
portfolio is zero, and this occurs at the
current price, so it is self-financing
• Pat makes money if share prices move up
or down
78
Pat Triumphant
• This clearly defeats the law of one price:
There is no downside risk, no
construction costs, and yet will yield a
positive profit almost all the time
• What’s wrong with Pat’s analysis?
79
Pat Dejected
• The answer is that it takes time for a
price to move, and during that time, all
other things being equal, the value of the
option will decay
• Think of a downwards sloping, very slick,
rain-gutter containing a critter:
– The critter may climb the walls of the gutter,
but it is constantly sliding down the gutter
80
Pat in Despair
• The next diagram shows the value of the
portfolio today and one week hence
• The construction lines have been
removed, and the graph has been rescaled
81
Strategy 1-Week Later
2.0
Strategy Value
1.5
Portfolio
PortfolioLater
1.0
0.5
0.0
90
95
100
105
110
-0.5
Share Price
82
115
120
Pat Condemned to Poverty
• The diagram shows that, if next week’s
share prices fall between about $97 and
$105.5, Pat will enjoy a loss
• As time passes, decay will make this
strategy a very risky one
– Another factor Pat did not take into account
is that volatility is itself volatile, so the hedge
may disintegrate
83
15.9 Contingent Claims
Analysis (CCA)of Corporate
Debt and Equity
• The CCA approach uses a different set of
informational assumptions than the
discounted cash flow (DCF) method:
– it uses the risk-free rate rather than a riskadjusted discount rate
– it uses knowledge of the prices of one or
more related assets and their volatilities
84
Contingent-Claims Analysis of
Stock and Bonds: Debtco
• Debtco is a real-estate holding company
and has issued
– 1,000,000 common shares
– 80,000 pure discount bonds, face $,1000,
maturity 1-year
85
Debtco, Continued
– The total market value of Debtco is
$100,000,000
– The risk-free rate, (and therefore, by the law
of one price, Debtco’s bond rate,) is 4%
86
Debtco, Notation
• E the market value of the stock issue
• D the market value of the debt issue
• V the total current market value; V = E + D
• V1 the total market value one year hence
• (The law of one price ensures that V = E + D
must be true, otherwise there will be an
arbitrage opportunity)
87
Debtco, Security Valuation
• Value of the bonds
– By the rule of one price, the value of the
bonds must equal their face value discounted
at the risk-free rate for a year
• D = 80,000 * $1,000 / 1.04 = $76,923,077
– By the total value of the firm, V = E + D, the
value of the stock is
• E = V - D = $100,000,000 - $76,923,077 =
$23,076,923
88
Debtco, Payoff
– A consequence of Debtco’s having bonds
with a risk-free rate is that the company has
either purchased bond default insurance
from a third party, or that the firm’s assets
have no (downside) risk
– For many companies, a more realistic
assumption is that the assets do have risk,
and to evaluate such securities requires a
payoff function for the bonds or stock:
89
Payoffs for Bond and Stock Issues
Value of Bond and Stock (Millions)
120
100
80
60
40
BondValue
StockValue
20
0
0
20
40
60
80
100
120
140
90
Value of Firm (Millions)
160
180
200
Negative Firm Values
– We have assumed that the value of the firm
never falls below zero, but while unusual, it
is possible for the market value of a firm’s
assets to be less than zero
– Consider Enviromess Inc., a firm that for
years polluted the Hudson River with a
byproduct of Lifecide®
• The cost of cleaning up the river may well
greatly exceed the firm’s financial resources
91
Negative Firm Values
– Negative values of limited liability companies
is irrelevant to the construction of the payoff
diagrams
– It does influence the value of the firm’s debt
and equity through the (truncated)
distribution of the firms future values
92
Debtco, Probabilities
– In addition to the payoff diagrams, we need
information about the probabilities of the
future values of the firm
93
Probability Density of a Firm's Value
•0.09
•Probability Density
•0.08
•0.07
•0.06
•0.05
•0.04
•0.03
•0.02
•0.01
•0.00
•0
•20
•40
•60
•80
•100
•120
•140
•Value of a Firm
94
•160
•180
•200
Debtco, Probabilities
– Considerable effort is employed when
estimating the probabilities used in CCA, but,
to obtain a basic understanding of CCA it is
enough to assume a very simple distribution
– We will assume that the firm may take on
only one of two possible values a year from
now, when the bond matures, namely $70 or
$140 million
– (The two-state assumption can be generalized into a nstate lattice model with any specified degree of accuracy)
95
Debtco Security Payoff Table
($’000,000)
Security
Payoff State a Payoff State b
Firm
140
70
Bond
80
70
Stock
60
0
96
Debtco’s Replicating Portfolio
• Let
– x be the fraction of the firm in replicator
– Y be the borrowings at the risk-free rate in
the replicator
– In $’000,000 the following equations must
be satisfied
60  140 x  1.04Y ; 0  70 x  1.04Y 
6
x  ; Y  $57,692,308
7
97
Debtco’s Replicating Portfolio
($’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
98
Debtco’s Replicating Portfolio
• Note that the above slide shows that
(with the weights on the last-but-one
slide) the value of the replicating
portfolio and the stock are identical now
and one-year hence
• By the law of one price, the value of
Debtco’s Stock is $28.02 each (One
million issued)
99
Debtco’s Replicating Portfolio
• We know 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%
100
Another View of Debtco’s
Replicating Portfolio (‘$000)
Security
Bonds
Stock
Bonds +
Stock
Total
Equivalent Equivalent
market
Amount
Amount
Value
of Firm of Rf Debt
71,978
14,286
57,692
28,022
85,714
-57,692
100,000
100,000
0
101
Interpretation
– The market value of the firm’s risky debt
consists of about $58 million riskless debt
and about $14 million of the risky firm
• There is a sense that holders of risky debt
accept some of the firm-as-a-whole’s risky
cash flow, just as shareholders do
– Shareholders accept the remaining $85
million of the firm-as-a-whole, and finance it
with the equivalent of about $58 million in
default-free debt
102
Valuing the Bonds given the
Price of the Stock
• There are three values that are in
equilibrium with each other:
– the value of the firm ( done)
– the value of the bond (to be done next)
– the value of the stock
• If we know one, the others may be
deduced
103
Valuing Bonds Given the
Stock Price ($’000,000’s)
• Assume that, as before we have:
–
–
–
–
Scenario a: value of firm in 1-year = $70
Scenario b: value of firm in 1-year = $140
Risk-free 1-year bonds produce a 4% yield
Total face value of Debtco’s bonds is $80
• Assume also:
– Debtco has a million shares outstanding,
total market value $20
104
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  xV  Y ;  V 
E  Y 20,000 ,000  57 ,692 ,308

 $90,641,026
6
x
7
– 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%
105
Valuing Stock Given the Bond
Yield
• Assume that, as before we have:
– Scenario a: value of firm in 1-year = $70
– Scenario b: value of firm in 1-year = $140
– Risk-free 1-year bonds produce a 4% yield
• Assume also:
– Debtco’s bonds yield 10% (current value
$909.09)
106
Bond Replication
• In order to replicate the bond, we will
purchase a fraction, x, of the firm, and
purchase the value Y of risk-free bonds
• At maturity, the value of the bonds is
– Scenario a, V = $70 million: $70 million
– Scenario b, V = $140million: $80 million
107
Replication Portfolio
Position
Purchase x
of firm
Purchase
Y RF Bond
Total
Portfolio
Immediate Scenario a Scenario b
Cash Flow V1 = 70
V1 = 140
- x* V
70 x
140 x
-Y
Y (1.04)
Y (1.04)
-x*V-Y
70
80
108
Determining the Weight of
Firm Invested in Bond, x, and
the Value of the R.F.-Bond, Y
70  70 x  1.04Y 

80  140 x  1.04Y 
1
x  ; Y  $57,692,308
7
109
Valuing Stock
– We can replicate the bond by purchasing 1/7
of the company, and $57,692,308 of defaultfree 1-year bonds
– The market value of the bonds is $909.0909
* 80,000 = $72,727,273
D  xV  Y ;  V 
E  Y 72,727 ,273  57 ,692 ,308

 $105 ,244 ,753
1
x
7
– The value of the stock is therefore E=V -D =
$105,244,753 - $72,727,273= $32,517,480
110
Convertible Bonds
• A convertible bond obligates the issuing
firm to redeem the bond at par value
upon maturity, or to allow the bond
holder to convert the bond into a prespecified number of share of common
stock
111
Convertible Bonds: The
Convertidebt Corporation
• Assume that Convertidebt is in every way
like Debtco, but each bond is convertible
to 20 common stock at maturity
– If all the debt is converted, then the number
of common stock will rise from 1,000,000 to
1,000,000 + 80,000 * 20 = 2,600,000
shares
112
Convertible Bond
140
Value of Stock and Bond Issue
ConvertibleBondValue
120
DilultedStockValue
100
80
60
40
20
0
0
20
40
60
80
100
120
Value of the Firm
113
140
160
180
200
Bondholder Entitlements
• Given that a conversion occurs, the value
of each common stock will be
– Value of firm / 2,600,000
• The bond holders will receive 1,600,000 of
these shares, so the bondholders will own
1.6/2.6 of the firm, leaving the shareholders
with 1/2.6 of the firm
• The critical value for conversion is firm’s value
= 80 million*2.6/1.6 = $130 million
114
Payoffs from Convertible
Bonds
• Scenario a: The value of the firm is 70
million: The bondholders will own the
company, so the value is 70 million
• Scenario b: The value of the firm is 140
million: The bondholders will own
1.6/2.6 of the company, or $86,153,846
– The remaining analysis follows exactly the
same recipe as the conventional bond
115
Dynamic Replication
• If you refer back to the convertible bond
example, you will observe that only two
points on the doubly-kinked payoff curve
were sampled (clearly unrealistic)
• As in the case of binary option
evaluation, by increasing the number of
sample points, you may achieve any
desired level of accuracy
116
Outline of Method
• Bifurcate the year into two six-month
intervals
• Starting at the present time at Node-A
(value of the firm is $100 million) there
are two scenarios
– the price rises to $115 MM (Node-B)
– the price falls to $90 MM (Node-C)
117
Setup Assumptions (See Next
Diagram)
– Given Node-B occurred at month 6 ($115
MM), then after a further 6-months, there
are two more scenarios
• the price rises to $140 MM (Node-D)
• the price falls all the way to $90 MM (Node-E)
– Given Node-C occurred at month 6 ($90
MM), then after a further 6-months, there
are two more scenarios
• The price rises to $110 MM (Node-F)
• The price falls further to $70 MM (Node-G)
118
Month 0
Month 6
Month 12
Outline Decision Tree
Node-D
$140MM
Node-B
$115MM
Node-F
$110MM
Node-A
$100MM
Node-E
$90MM
Node-C
$90MM
119
Node-G
$70MM
Outline of Method
• There are three decision nodes A, B, C
– At each node, a replicating decision is made
• First B (from D & E), and independently C
(from F & G)
• Then, using the backwards induction, A (from
B & C)
– The steps are exactly as outlined in detail for
a non-composite decisions
• The example is featured in the textbook
120
Outline of Method
• The only detail that requires your
attention is that the portfolio is
completely self-financing at each node
121
Summary
– The fundamental principle of CCA is that one
may replicate the securities issued by a firm
through the purchase and sale of the firm
(as a whole) and the risk-free asset, and that
this dynamic replication strategy is selffinancing
– The CCA is a direct consequence of the law
of one price
– Note that informational inputs are relatively
few
122
Probabilities
– The selection of the prices at which to
replicate the portfolios is equivalent to
selecting a probability distribution
• It is usual to construct lattices that reassociate at given values, forming a fishnet
• The basis of the spacing, and the ratio of
temporal spacing to spatial spacing,
determine the distribution
– As mentioned in the chapter, this step requires
considerable skill
123
Valuing Pure State-Contingent
Securities
• Recall: In Debtco and Convertidebt,
there were only two possible values that
the firm could take at maturity
• Define a pure state-contingent security to
be a security that pays $1 in one of these
states and $0 in the other
124
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
125
State-Contingent Security #1
S. C. S. #1
70,000,000 x  1.04Y  0
1
1

x

;
Y



140,000,000 x  1.04Y  1 
70,000,000
1.04
100,000,000
1
P1  1,000,000 x  Y 

 0.467 032 967
70,000,000 1.04
S. C. S. #2
70,000,000 x  1.04Y  1 
1
2
; Y
 x  
140,000,000 x  1.04Y  0
70,000,000
1.04
100,000,000
2

 0.494 505 495
70,000,000 1.04
$1
P1  P2  $0.467 033  $0.494 505  $0.961 538 
1.04
126
P2  1,000,000 x  Y  
State Contingent Securities to
Evaluate Real Securities
• Price of Debtco stock is equivalent to 60
type 1 state contingent securities and no
type 2, so stock price = 60 * 0.467033 =
$28.02
• The price of a Debtco bond is equivalent
to 1000 type 1 SCS and 875 type 2, so
bond price = 1000 * $0.467033 + 875 *
$0.494505 = $899.73
127
Advantages of Pure State
Contingent Securities
• SCS enable us to evaluate any security
that is dependent upon the value of the
firm and the risk-free bond
– For example, Convertidebt securities are
priced quick using SCS prices (already
computed) and the payoff schedule
– Think of a SCS as the conditional probability
of an event, weighted by the riskless time
value of money
128
15.10 Pricing a Bond
Guarantee
• Guarantees against credit risk common
– Parent corporations guarantee the debt of subsidiaries
– Commercial banks and insurance companies offer
guarantees for a fee on a spectrum of financial
instruments including swaps & letters of credit
– U.S. Government guarantees bank deposits, SBA loans,
pensions, farm & student loans, mortgages, the debt of
other sovereign countries, and huge strategic corporations
– They occur implicitly every time a risky loan is made
129
Example
• Earlier, we computed the value of the
Debtco bond and found it to be $899.73,
but the value of a risk-free bond was
$961.54
• A third-party might be willing to insure
Debtco’s bonds against default, and the
equilibrium price for this is $961.54 $899.73 = $61.81
130
• (We assume that the insuring company
itself has no credit risk)
– We may compute the cost of this kind of
insurance using our computed SCS values
and a Bond Guarantee payoff table (next
slide)
– The insurance pays off only when the firm’s
price is $70,000,000
131
Payoff for Debtco’s Bond
Guarantee
Security
Scenario a
Firm
$70,000,000 $140,000,000
Bonds
Guarantee
Scenario b
$1,000
$875
$0
$125
132
SCS Conformation of
Guarantee’s Price
• Guarantee’s price is 125 * $0.494505 =
$61.81
133
15.11 Other Applications of
Option-Pricing Methodology
– This slide presentation started with a range
of options that are embedded in products
and contracts
– Options not associated with financial
instruments are called real options
– The future is uncertain, so having flexibility
to decide what to do after some of the
uncertainty has been removed has value
134
Options in Project-Investment
Valuations:
– Option to initiate
– Option to expand
– Option to abandon
– Option to reduce scale
– Option to adjust timing
– Option to exploit a future technology
135
Examples:
– Choice of oil or gas to generate electricity
– Product development of pharmaceuticals
– Making a sequel to a movie
– Vocational education
– Litigation decisions
– strategic decisions
136