Chapter
TwentyFour
Option Valuation
© 2003 The McGraw-Hill Companies, Inc. All rights reserved.
24.1
Key Concepts and Skills
• Understand and be able to use Put-Call Parity
• Be able to use the Black-Scholes Option
Pricing Model
• Understand the relationships between option
premiums and stock price, time to expiration,
standard deviation and the risk-free rate
• Understand how the OPM can be used to
evaluate corporate decisions
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24.2
Chapter Outline
•
•
•
•
Put-Call Parity
The Black-Scholes Option Pricing Model
More on Black-Scholes
Valuation of Equity and Debt in a Leveraged
Firm
• Options and Corporate Decisions: Some
Applications
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24.3
Protective Put
• Buying the underlying asset and a put option
to protect against a decline in the value of the
underlying asset
• Pay the put premium to limit the downside risk
• Similar to paying an insurance premium to
protect against potential loss
• Trade-off between the amount of protection
and the price that you pay for the option
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24.4
An Alternative Strategy
• You could buy a call option and invest the
present value of the exercise price in a riskfree asset
• If the value of the asset increases, you can buy
it using the call option and your investment
• If the value of the asset decreases, you let your
option expire and you still have your
investment in the risk-free asset
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24.5
Comparing the Stratgies
Value at Expiration
S<E
S≥E
Stock + Put
E
S
Call + PV(E)
E
S
Initial Position
• Stock + Put
– If S < E, exercise put and receive E
– If S ≥ E, let put expire and have S
• Call + PV(E)
– PV(E) will be worth E at expiration of the option
– If S < E, let call expire and have investment, E
– If S ≥ E, exercise call using the investment and have S
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24.6
Put-Call Parity
• If the two positions are worth the same at the
end, they must cost the same at the beginning
• This leads to the put-call parity condition
– S + P = C + PV(E)
• If this condition does not hold, there is an
arbitrage opportunity
– Buy the “low” side and sell the “high” side
• You can also use this condition to find any of
the variables
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24.7
Example: Finding the Call Price
• You have looked in the financial press and
found the following information:
–
–
–
–
–
Current stock price = $50
Put price = $1.15
Exercise price = $45
Risk-free rate = 5%
Expiration in 1 year
• What is the call price?
– 50 + 1.15 = C + 45 / (1.05)
– C = 8.29
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24.8
Continuous Compounding
• Continuous compounding is generally used
when working with option valuation
• Time value of money equations with
continuous compounding
– EAR = eqt
– PV = FVeRt
– FV = PVe-Rt
• Put-call parity with continuous compounding
– S + P = C + Ee-Rt
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24.9
Example: Continuous Compounding
• What is the present value of $100 to be
received in three months if the required return
is 8%, with continuous compounding?
– PV = 100e-.08(3/12) = 98.02
• What is the future value of $500 to be received
in nine months if the required return is 4%,
with continuous compounding?
– FV = 500e.04(9/12) = 515.23
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24.10
PCP Example: PCP with Continuous Compounding
• You have found the following information;
–
–
–
–
–
Stock price = $60
Exercise price = $65
Call price = $3
Put price = $7
Expiration is in 6 months
• What is the risk-free rate implied by these prices?
–
–
–
–
S + P = C + Ee-Rt
60 + 7 = 3 + 65e-R(6/12)
.9846 = e-.5R
R = -(1/.5)ln(.9846) = .031 or 3.1%
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24.11
Black-Scholes Option Pricing Model
• The Black-Scholes
model was originally
developed to price call
options
• N(d1) and N(d2) are
found using the
cumulative normal
distribution tables
McGraw-Hill/Irwin
C  SN (d1 )  Ee Rt N (d 2 )
d1 
2


S

 
ln     R  t
2 
E 
 t
d 2  d1   t
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24.12
Example: OPM
• You are looking at a call
.2 2 
 45  
.5
ln     .04 
option with 6 months to
2 
 35  
d1 
 1.99
expiration and an
.2 .5
exercise price of $35.
d 2  1.99  .2 .5  1.85
The current stock price
is $45 and the risk-free •Look up N(d1) and N(d2) in Table 24.3
rate is 4%.
•N(d1) = (.9761+.9772)/2 = .9767
The standard deviation •N(d2) = (.9671+.9686)/2 = .9679
of underlying asset
C = 45(.9767) – 35e-.04(.5)(.9679)
returns is 20%. What
is the value of the call C = $10.75
option?
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24.13
Example: OPM in a Spreadsheet
• Consider the previous example
• Click on the excel icon to see how this
problem can be worked in a spreadsheet
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24.14
Put Values
• The value of a put can be found by finding the
value of the call and then using put-call parity
– What is the value of the put in the previous
example?
• P = C + Ee-Rt – S
• P = 10.75 + 35e-.04(.5) – 45 = .06
• Note that a put may be worth more if
exercised than if sold, while a call is worth
more “alive than dead” unless there is a large
expected cash flow from the underlying asset
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24.15
European vs. American Put Options
• The Black-Scholes model is strictly for
European options
• It does not capture the early exercise value that
sometimes occurs with a put
• If the stock price falls low enough, we would
be better off exercising now rather than later
• A European option will not allow for early
exercise and therefore, the price computed
using the model will be too low relative to an
American option that does allow for early
exercise
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24.16
Table 24.4
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24.17
Work the Web Example
• There are several good options calculators on
the Internet
• The Chicago Board Options Exchange has one
such calculator
• Click on the web surfer to go to the calculator
under trading tools and price the call option
from the earlier example
– S = $45; E = $35; R = 4%; t = .5;  = .2
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24.18
Varying Stock Price and Delta
• What happens to the value of a call (put)
option if the stock price changes, all else
equal?
• Take the first derivative of the OPM with
respect to the stock price and you get delta.
– For calls: Delta = N(d1)
– For puts: Delta = N(d1) - 1
– Delta is often used as the hedge ratio to determine
how many options we need to hedge a portfolio
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24.19
Figure 24.1
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24.20
Example: Delta
• Consider the previous example:
– What is the delta for the call option? What does it
tell us?
• N(d1) = .9767
• Change in option value is approximately equal to delta
times change in stock price
– What is the delta for the put option?
• N(d1) – 1 = .9767 – 1 = -.0233
– Which option is more sensitive to changes in the
stock price? Why?
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24.21
Varying Time to Expiration and Theta
• What happens to the value of a call (put) as we
change the time to expiration, all else equal?
• Take the first derivative of the OPM with
respect to time and you get theta
• Options are often called “wasting” assets,
because the value decreases as expiration
approaches, even if all else remains the same
• Option value = intrinsic value + time premium
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24.22
Figure 24.2
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24.23
Example: Time Premiums
• What was the time premium for the call and
the put in the previous example?
– Call
• C = 10.75; S = 45; E = 35
• Intrinsic value = max(0, 45 – 35) = 10
• Time premium = 10.75 – 10 = $0.75
– Put
• P = .06; S = 45; E = 35
• Intrinsic value = max(0, 35 – 45) = 0
• Time premium = .06 – 0 = $0.06
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24.24
Varying Standard Deviation and Vega
• What happens to the value of a call (put) when
you vary the standard deviation of returns, all
else equal?
• Take the first derivative of the OPM with
respect to sigma and you get vega
• Option values are very sensitive to changes in
the standard deviation of return
• The greater the standard deviation, the more
the call and the put are worth
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24.25
Figure 24.3
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24.26
Varying the Risk-Free Rate and Rho
• What happens to the value of a call (put) as
you vary the risk-free rate, all else equal?
• Take the first derivative of the OPM with
respect to the risk-free rate and you get rho
• Changes in the risk-free rate have very little
impact on options values over any normal
range of interest rates
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24.27
Figure 24.4
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24.28
Implied Standard Deviations
• All of the inputs into the OPM are directly
observable, except for the expected standard
deviation of returns
• The OPM can be used to compute the market’s
estimate of future volatility by solving for the
standard deviation
• This is called the implied standard deviation
• Online options calculators are useful for this
computation since there is not a closed form
solution
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24.29
Equity as a Call Option
• Equity can be viewed as a call option on the firm’s
assets whenever the firm carries debt
• The strike price is the cost of making the debt
payments
• The underlying asset price is the market value of the
firm’s assets
• If the intrinsic value is positive, the firm can exercise
the option by paying off the debt
• If the intrinsic value is negative, the firm can let the
option expire and turn the firm over to the
bondholder
• This concept is useful in valuing certain types of
corporate decisions
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24.30
Valuing Equity and Changes in Assets
• Consider a firm that has a zero coupon bond
that matures in 4 years. The face value is $30
million and the risk-free rate is 6%. The
current market value of the firm’s assets is $40
million and the firm’s equity is currently worth
$18 million. Suppose the firm is considering a
project with an NPV = $500,000.
– What is the implied standard deviation of returns?
– What is the delta?
– What is the change in stockholder value?
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24.31
PCP and the Balance Sheet Identity
• Risky debt can be viewed as a risk-free bond
minus the cost of a put option
– Value of risky bond = Ee-Rt – P
• Consider the put-call parity equation and
rearrange
– S = C + Ee-Rt – P
– Value of assets = value of equity + value of a risky
bond
• This is just the same as the traditional balance
sheet identity
– Assets = liabilities + equity
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24.32
Mergers and Diversification
• Diversification is a frequently mentioned reason for
mergers
• Is this a good reason from a stockholder perspective?
• Diversification reduces risk and therefore volatility
• Decreasing volatility decreases the value of an option
• Since equity can be viewed as a call option, should a
merger increase or decrease the value of the equity
assuming diversification is the only benefit to the
merger?
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24.33
Extended Example – Part I
• Consider the following two merger candidates
• The merger is for diversification purposes only with no
synergies involved
• Risk-free rate is 4%
Market value of assets
Face value of zero coupon
debt
Debt maturity
Asset return standard
deviation
McGraw-Hill/Irwin
Company A Company B
$40 million $15 million
$18 million $7 million
4 years
40%
4 years
50%
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24.34
Extended Example – Part II
• Use the OPM (or an options calculator) to compute
the value of the equity
• Value of the debt = value of assets – value of equity
Company A Company B
Market Value of Equity
25.681
9.867
Market Value of Debt
14.319
5.133
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24.35
Extended Example – Part III
• The asset return standard deviation for the combined
firm is 30%
• Market value assets (combined) = 40 + 15 = 55
• Face value debt (combined) = 18 + 7 =25
Combined Firm
Market value of equity
34.120
Market value of debt
20.880
Total MV of equity of separate firms = 25.681 + 9.867 = 35.548
Wealth transfer from stockholders to bondholders = 35.548 – 34.120 = 1.428
(exact increase in MV of debt)
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24.36
M&A Conclusions
• Mergers for diversification only transfer
wealth from the stockholders to the
bondholders
• The standard deviation of returns on the assets
is reduced, thereby reducing the option value
of the equity
• If management’s goal is to maximize
stockholder wealth, then mergers for reasons
of diversification should not occur
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24.37
Extended Example: Low NPV – Part I
• Stockholders may prefer low NPV projects to
high NPV projects if the firm is highly
leveraged and the low NPV project increases
volatility
• Consider a company with the following
characteristics
–
–
–
–
–
MV assets = 40 million
FV debt = 25 million
Debt maturity = 5 years
Asset return standard deviation = 40%
Risk-free rate = 4%
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24.38
Extended Example: Low NPV – Part II
• Current market value of equity = $22.657 million
• Current market value of debt = $17.343 million
NPV
MV of assets
Asset return standard
deviation
MV of equity
MV of debt
McGraw-Hill/Irwin
Project I
$3
$43
30%
Project II
$1
$41
50%
$23.769
$19.231
$25.339
$15.661
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24.39
Extended Example: Low NPV – Part III
• Which project should management take?
• Even though project B has a lower NPV, it is
better for stockholders
• The firm has a relatively high amount of
leverage
– With project A, the bondholders share in the NPV
because it reduces the risk of bankruptcy
– With project B, the stockholders actually
appropriate additional wealth from the
bondholders for a larger gain in value
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24.40
Extended Example: Negative NPV – Part I
• We’ve seen that stockholders might prefer a
low NPV to a high one, but would they ever
prefer a negative NPV?
• Under certain circumstances they might
• If the firm is highly leveraged, stockholders
have nothing to lose if a project fails and
everything to gain if it succeeds
• Consequently, they may prefer a very risky
project with a negative NPV but high potential
rewards
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24.41
Extended Example: Negative NPV – Part II
• Consider the previous firm
• They have one additional project they are
considering with the following characteristics
– Project NPV = -$2 million
– MV of assets = $38 million
– Asset return standard deviation = 65%
• Estimate the value of the debt and equity
– MV equity = $25.423 million
– MV debt = $12.577 million
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24.42
Extended Example: Negative NPV – Part III
• In this case, stockholders would actually
prefer the negative NPV project over either of
the positive NPV projects
• The stockholders benefit from the increased
volatility associated with the project even if
the expected NPV is negative
• This happens because of the large levels of
leverage
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24.43
Conclusions
• As a general rule, managers should not go
around accepting low or negative NPV
projects and passing up high NPV projects
• Under certain circumstances, however, this
may benefit stockholders
– The firm is highly leveraged
– The low or negative NPV project causes a
substantial increase in the standard deviation of
asset returns
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24.44
Quick Quiz
• What is put-call parity? What would happen if
it doesn’t hold?
• What is the Black-Scholes option pricing
model?
• How can equity be viewed as a call option?
• Should a firm do a merger for diversification
purposes only? Why or why not?
• Should management ever accept a negative
NPV project? If yes, under what
circumstances?
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