Option Pricing
1. The following are prices of options
traded on Microsoft Corporation, which pays no dividends.
Call Put
K=85 K=90 K=85 K=90
1 month 2.75 1.00 4.50 7.50
3 month 4.00 2.75 5.75 9.00
6 month 7.75 6.00 8.00 12.00
The stock is trading at $83, and the
annualized riskless rate is 3.8%. The standard deviation in ln stock prices
(based upon historical data) is 30%.
a. Estimate
the value of a threemonth call, with a strike price of 85.
b. Using the
inputs from the BlackScholes model, specify how you would replicate this call.
c. What is
the implied standard deviation in this call?
d. Assume
now that you buy a call with a strike price of 85 and sell a call with a strike
price of 90. Draw the payoff diagram on this position.
e. Using putcall
parity, estimate the value of a threemonth put with a strike price of 85.
2. You are trying to value threemonth
call and put options on Merck, with a strike price of 30. The stock is trading
at $28.75, and expects to pay a quarterly dividend per share of $0.28 in two
months. The annualized riskless interest rate is 3.6%, and the standard
deviation in ln stock prices is 20%.
a. Estimate
the value of the call and put options, using the BlackScholes.
b. What
effect does the expected dividend payment have on call values? on put values?
Why?
3. There is the possibility that the
options on Merck, described above, could be exercised early.
a. Use the
pseudoAmerican call option technique to determine whether this will affect the
value of the call.
b. Why does
the possibility of early exercise exist? What types of options are most likely
to be exercised early?
4. You have been provided the following
information on a threemonth call:
S
= 95 K=90 t=0.25 r=0.04
N(d1)
= 0.5750 N(d2)
= 0.4500
a. If you
wanted to replicate buying this call, how much money would you need to borrow?
b. If you wanted to
replicate buying this call, how many shares of stock would you need to buy?
5. Go Video, a manufacturer of video
recorders, was trading at $4 per share in May 1994. There were 11 million
shares outstanding. At the same time, it had 550,000 oneyear warrants
outstanding, with a strike price of $4.25. The stock has had a standard
deviation (in ln stock prices) of 60%. The stock does not pay a dividend. The
riskless rate is 5%.
a. Estimate
the value of the warrants, ignoring dilution.
b. Estimate
the value of the warrants, allowing for dilution.
c. Why does
dilution reduce the value of the warrants.
6. You are trying to value a long term
call option on the NYSE Composite Index, expiring in five years, with a strike
price of 275. The index is currently at 250, and the annualized standard
deviation in stock prices is 15%. The average dividend yield on the index is
3%, and is expected to remain unchanged over the next five years. The fiveyear
treasury bond rate is 5%.
a. Estimate
the value of the long term call option.
b. Estimate
the value of a put option, with the same parameters.
c. What are
the implicit assumptions you are making when you use the BlackScholes model to
value this option? Which of these assumptions are likely to be violated? What
are the consequences for your valuation?
7.
A new security on AT&T will entitle the investor to all dividends on
AT&T over the next three years, limit upside potential to 20%, but also
provide downside protection below 10%. AT&T stock is trading at $50, and
threeyear call and put options are traded on the exchange at the following
prices
Call
Options Put
Options
K 1
year 3 year 1
year 3
year
45 $8.69 $13.34 $1.99 $3.55
50 $5.86 $10.89 $3.92 $5.40
55 $3.78 $8.82 $6.59 $7.63
60 $2.35 $7.11 $9.92 $10.23
How much would you be
willing to pay for this security?
a. Estimate the value of
the mine using traditional capital budgeting techniques.
b. Estimate the value of
the mine based upon an option pricing model.
c. How would you explain
the difference between the two values?
a. What is the value of
project, based upon traditional NPV?
b. What is the value of the
project as an option?
c. Why are the two values
different? What factor or factors determine the magnitude of this difference?
a. Estimate the value of
this company.
b. How sensitive is this
value estimate to the variance in project cash flows? What broader lessons
would you draw from this analysis?
The
Option to Expand and Abandon
1.
NBC has the
rights to televise the Winter Olympics in 2 years and is trying to estimate the
value of these rights for possible sale to another network. NBC expects it to
cost $40 million (in present value terms) to televise the Olympics and based
upon current assessments expects to have a Nielsen rating[1] of 15 for the games. Each rating point
is expected to yield net revenue of $2 million to NBC (in present value terms).
There is substantial variability in this estimate and the standard deviation in
the expected net revenues is 30%. The riskless rate is 5%.
a. What is the net present value of these
rights, based upon current assessments?
b. Estimate the value of these rights for
sale to another network.
2.
You are
analyzing Skates Inc., a firm that manufactures skateboards. The firm is
currently unlevered and has a cost of equity of 12%. You estimate that Skates
would have a cost of capital of 11% at its optimal debt ratio of 40%. The
management, however, insists that it will not borrow the money because of the
value of maintaining financial flexibility and they have provided you with the
following information.
·
Over the
last 10 years, reinvestments (net capital expenditures + working capital
investments) have amounted to 10% of firm value, on an annual basis. The
standard deviation in this reinvestment has been 0.30.
·
The firm
has traditionally used only internal funding (net income + depreciation) to
meet these needs and these have amounted to 6% of firm value.
·
In the most
recent year, the firm earned $180 million in net income on a book value of
equity of $1 billion and it expects to earn these excess returns on new
investments in the future.
·
The
riskless rate is 5%.
a.
Estimate
the value of financial flexibility as a percent of firm value, on an annual
basis.
b.
Based upon
part a, would you recommend that Skates use its excess debt capacity?
3.
Disney is
considering entering into a joint venture to build condominiums in Vail,
Colorado, with a local real estate developer. The development is expected to
cost $1 billion overall and, based on Disney’s estimate of the cashflows,
generate $900 million in present value cash flows. Disney will have a 40% share
of the joint venture (requiring it to put up $400 million of the initial
investment and entitling it to 40% of the cashflows) but it will have the right
to sell its share of the venture back to the developer for $300 million anytime
over the next 5 years. (The project life is 25 years)
a.
If the
standard deviation in real estate values in Vail is 30% and the riskless rate
is 5%, estimate the value of the abandonment option to Disney.
b.
Would you
advice Disney to enter into the joint venture?
c.
If you were
advising the developer, how much would he need to generate in present value
cashflows from the investment to make this a good investment?
4.
Quality
Wireless is considering making an investment in China. While it knows that the
investment will cost $1 billion and generate only $800 million in cashflows (in
present value terms), the proponents of expansion are arguing that the
potential market is huge and that Quality should go ahead with its investment.
a.
Under what
conditions will the expansion potential have option value?
b.
Assume now
that there is an option value to expansion that exactly offsets the negative
net present value on the initial investment. If the cost of the subsequent
expansion in 5 years is $2.5 billion, what is your current estimate of the
present value of the cash flows from expansion? (You can assume that the
standard deviation in the present value of the cashflows is 25% and that the
riskless rate is 6%.)
5.
Reliable
Machinery Inc. is considering expanding its operations in Thailand. The initial
analysis of the projects yields the following results.
·
The project
is expected to generate $85 million in aftertax cash flows every year for the
next 10 years.
·
The initial
investment in the project is expected to be $750 million.
·
The cost of
capital for the project is 12%.
If the project generates much higher cash
flows than anticipated, you will have the exclusive right for the next 10 years
(from a manufacturing license) to expand operations into the rest of South East
Asia. A current analysis suggests the following about the expansion
opportunity.
·
The
expansion will cost $2 billion (in current dollars).
·
The
expansion is expected to generate $150 million in after tax cash flows each
year for 15 years. There is substantial uncertainty about these cash flows and
the standard deviation in the present value is 40%.
·
The cost of
capital for this investment is expected to be 12% as well. The riskfree rate is
6.5%.
a. Estimate the net present value of the
initial investment.
b. Estimate the value of the expansion
option.
1. Designate the following statements as
true or false.
a. Equity can be viewed as an option
because equity investors have limited liability (limited to their equity
investment in the firm).
b. Equity investors will sometimes take
bad projects (with negative net present value) because they can add to the
value of the firm.
c. Investing in a good project (with
positive NPV)  which is less risky than the average risk of the firm  can
negatively impact equity investors.
d. The value of equity in a firm is an
increasing function of the duration of the debt in the firm (i.e., equity will
be more valuable in a firm with longer term debt than an otherwise similar firm
with short term debt).
e. In a merger in which two risky firms
merge and do not borrow more money, equity can become less valuable because
existing debt will become less risky.
2. XYZ Corporation has $500 million in
zerocoupon debt outstanding, due in five years. The firm had earnings before
interest and taxes of $40 million in the most recent year (the tax rate is
40%). These earnings are expected to grow 5% a year in perpetuity and the firm
paid no dividends. The firm had a return on capital of 12% and a cost of
capital of 10%. The annualized standard deviation in firm values of comparable
firms is 12.5%. The fiveyear bond rate is 5%.
a. Estimate
the value of the firm.
b. Estimate
the value of equity, using an option pricing model.
c. Estimate
the market value of debt and the appropriate interest rate on the debt.
3. McCaw Cellular Communications reported
earnings before interest and taxes of $850 million in 1993, with a depreciation
allowance of $400 million and capital expenditures of $ 550 million in that
year; the working capital requirements were negligible. The earnings before
interest and taxes and net cap ex are expected to grow 20% a year for the next
five years. The cost of capital is 10% and the return on capital is expected to
15% in perpetuity after year 5; the growth rate in perpetuity is 5%. The firm
has $10 billion in debt outstanding with the following characteristics.
Duration Debt
1
year $2
billion
2
years $4
billion
5
years $4
billion
The annualized standard deviation in the
firm's stock price is 35%, while the annualized standard deviation in the
traded bonds is 15%. The correlation between stock and bond prices has been 0.5
and the average debt ratio over the last few years has been 60%. The threeyear
bond rate is 5% and the tax rate is 40%.
a. Estimate the value of
the firm.
b. Estimate the value of
the equity.
c. The stock was trading
at $60 and there were 210 million shares outstanding in January 1994. Estimate
the implied standard deviation in firm value.
d. Estimate the market
value of the debt.
4. You have been asked to analyze the
value of equity in a company that has the following features.
·
The
earnings before interest and taxes is $25 million and the corporate tax rate is
40%.
·
The
earnings are expected to grow 4% a year in perpetuity and the return on capital
is 10%. The cost of capital of comparable firms is 9%.
·
The firm
has two types of debt outstanding  2year zerocoupon bonds with a face value
of $250 million and bank debt with ten years to maturity with a face value of
$250 million (The duration of this debt is 4 years.).
·
The firm is
in two businesses  food processing and auto repair – of equal size. The
average standard deviation in firm value for firms in food processing is 25%,
whereas the standard deviation for firms in auto repair is 40%. The correlation
between the businesses is 0.5.
·
The
riskless rate is 7%.
Use the option pricing model to value
equity as an option.
5. You are valuing the equity in a firm
with $800 million (face value) in debt with an average duration of 6 years and
assets with an estimated value of $400 million. The standard deviation in asset
value is 30%. With these inputs (and a riskless rate of 6%) we obtain the
following values (approximately) for d1 and d2.
d1
=  0.15 d2
=  0.90
Estimate the default spread (over and
above the riskfree rate) that you would charge for the debt in this firm.
Solutions
to Option Pricing Problems
A. The values of the option parameters are as follows:
S = $83
K = $85
t = 0.25
r = 3.80%
Variance = 0.09
Value of call = $4.42
B.
To replicate this call, you would have to:
Buy 0.4919 Shares of Stock (this is
N(d1) from the model)
and
Borrow K ert N(d2) =
85 exp(0.038)(0.25) (0.4324) = $36.40
C.
At an implied variance of 0.075, the call has a
value of approximately $4.00 (the market price).
Implied Standard Deviation =
√0.075 = 0.2739
D.
E.
Value of Threemonth Put = C  S + Kert = $4.42  $83 + 85 exp(0.038)(0.25) = $5.62
A. S
= $28.75
K
= $30
t
= 0.25
r
= 3.60%
s2 = 0.04
PV
of Expected Dividends = $0.28/(1.036)2/12 = $0.28
Value
of Call = $0.64
B. The payment of a dividend reduces the expected stock
price, and hence reduces the value of calls and increases the value of puts.
A. First value the threemonth call, as above:
Value
of Call = $0.64
Then, value a call to the first (and only) dividend
payment,
S
= $28.75
K
= $30
t
= 2/12
r
= 3.60%
s2 = 0.04
y
= 0 (since it assumes exercise before the dividend payment)
Value
of Call = $0.51
Since the value of the threemonth call is higher,
there is no anticipated exercise.
B. If the dividend payment is large enough, it may pay to
exercise the call just before the exdividend day (before the stock price
drops) rather than wait until expiration. This early exercise is more likely
for call options:
(a)
the larger the dividend on the stock, and
(b)
the closer the option is to expiration.
A. You would need to borrow Kert N(d2) = 90 exp(0.04)(0.25) (0.4500) = $40.10
B. You would need to buy 0.575 shares of stock.
A. S= $4.00
K
= $4.25
r
= 5%
t
= 1
Variance
= 0.36
Value
of Warrant = $0.93
B. Adjusted Stock Price = (Stock Price * Number of Shares
Outstanding) + (Warrant price * Number of Warrants Outstanding)/(Number of
Shares+Number of Warrants)
=
($4.00 * 11,000,000 + $0.93 * 550,000)/(11,550,000) = $3.85
(To avoid the circular reasoning problem, the price
from the nodilution case is used.)
Adjusted Exercise Price = $4.25
r = 5%
t = 1
Variance = 0.36
Value of Warrant = $0.80
(If you are using a spreadsheet with iterations turned
on, and are feeding the option prices back to calculate the adjusted stock
price, the value of the warrants is still $0.80.)
C. Dilution increases the number of shares outstanding.
For any given value of equity, each share is worth less.
A. S = 250
K
= 275
t
= 5
r
= 5%
s2 = (0.15)2
y
= 0.03
Value
of call = $29.09
B. Value of put with same parameters = $28.09
C.
(1) The variance will be unchanged for the life of the
option. This is likely to be violated because stock price variances do change
substantially over time.
(2) There will be no early exercise. This is
reasonable and is unlikely to be violated.
(3) Any deviations from the option value will be
arbitraged away.
While there are plenty of arbitrageurs eager to exploit deviations from true value, arbitraging an index is clearly more difficult to do than arbitraging an individual stock.
New Security =
AT & T stock  Call (K=60) + Put (K=45)
=
$50  $7.11 + $3.55 = $46.44
The
call with a strike price of $60 is sold, eliminating upside potential above
$60.
The put with a strike
price of $45 is bought, providing downside protection.
Solutions: The Option to Delay
Question 1
S = PV of $25 million a year for 20 years at 16% = $148.22 million
K = Cost of Taking Project = $300 million
t = 10 years
Standard Deviation = 20%
r = 12%
y = Dividend Yield = 1/ Project Life = 10%
a. PV of Inflows = 400,000 * 0.85 * (1  1.04^25/1.07^25)/(.07  .04) 400,000 * 0.40 * (1  1.03^25/1.07^25)/(.07  .03) = $3,309,756
Fixed Costs associated with opening
= 3,000,000
NPV = 3,309,756 3,000,000 = $309,756
b. S = 3,309,756
K = 3,000,000
t = 25
r = 7%
s = 0.25
y = 1/25 = 4%
Value of the Call Option = $828,674
c. The latter considers the option characteristics of owning the mine, i.e., that copper prices may go up, and that the mineowner will be more likely to develop the mine at higher copper prices.
Question 3
Current Value of Developed Reserve = 10,000,000 * ($20  $6) = $140,000,000
Exercise Price = Cost of Developing Reserve = $120,000,000
t = 20 years
r = 7%
s = 20%
y = 4% (Alternatively, you can use 1/20 or 5% as your cost of delay)
Value of Call (Natural Resource Reserve) = $37,360,435
Question 4
a. NPV of Project = $250  $200 = $50 million
b. The option has the following characteristics:
S = 250
K = 200
r = 8%
t = 5
Variance = 0.04
Dividend Yield = 12.5/250 = 5%
Value of Call (Project Rights) = $68.68
c. The latter captures the value of delaying the project. The difference between the two values will increase as the variance in the project cash flows increases.
Question 5
a. S = PV of Cash Inflows on Project = 250
K = Cost of Taking Project = 500
t = 10 years
r = 6%
s = 0.6
y = 10/250 = 4%
Value of Call (Product Patent) = $95 million
b. It is an increasing function of the variance in project cash flows. This analysis suggests that the rights to products in technologically volatile areas are likely to be worth a great deal, even though the products may not be viable now.
Solutions to
the Option to Expand
Question 1
a. Net present value of the project = $ 30  $ 40 =
 $ 10 million
b. Inputs
S = Present Value of Net
Revenues = $
30 million
K = Cost of televising the
Olympics = $ 40
million
t = Time until Olympics = 2
years
r = Riskless rate = 5%
Variance in value = 0.09
y = Cost of delay = 0
d1= 0.2302 N(d1)
= 0.4090
d2 = 0.6545 N(d2)
= 0.2564
Value of the Rights = 30 (0.409)  40 exp
(0.05)(2) (.2564) = 2.99
c. Probability that rights will be profitable =
0.2564  0.4090
a.
S = Expected
reinvestment needs as percent of firm value = 10%
K = Reinvestment
needs that can be met without excess debt capacity = 6%
T = 1 year
Standard deviation
in reinvestment needs = 0.30
The option pricing
value with these inputs is 4.32%. If we assume that the current excess returns
(18%  12%) continue in perpetuity, the value of flexibility is
Value of
flexibility (on an annual basis) = 4.32% * .06/.12 = 2.16%
b.
Based upon part a,
would you recommend that Skates use its excess debt capacity?
The value of
flexibility exceeds what the firm would save by moving to its optimal (only
1%). The firm should not use its excess debt capacity.
Question 3
a. Value of abandonment option
S = PV of cashflows from development = $ 900 million* 0.4 =
$ 360 million
K = Abandonment value = $ 300 million
T = 5 years
Riskless rate = 5%
Standard deviation = 40%
Value of abandonment option = $ 63.51 million
b. The net present value of this project to
Disney is $ 40 million.
Net present value = 400 + 360 = 40 million
The value of the abandonment option is greater than the
negative net present value. I would advice Disney to make the investment.
c. If you were the developer, you would need
to make a net present value equal to at least $63.51 million to cover the cost
of the abandonment option.
PV of cash flows to developer = (63.51) + .6 (1000) = $
663.51 million
Question 4
a. For the expansion potential to have
option value, Quality Wireless has to have exclusive rights to expand.
b. Net present value of initial investment =
 $ 200 million
S = PV of cashflows from expansion (currently) = ?
K = $2500 million
T = 5 years
Standard deviation in firm value = 25%
Riskless rate = 5%
Setting up the option value = $ 200 million and solving for
S, we get
S = $ 1511 million
(Sorry. The only way to get there is by trial and error. An
approximate answer would have been sufficient)
Question 5
a. Net present value of initial investment =
750 + 85 (PV of annuity, 10 years, 12%)
=
 $269.73 million
b. Value of expansion option
S = 150 (PV of annuity, 12%, 15 years) = $1,021.63 million
K = Cost of expansion = $ 2,000 million
Riskless rate = 6.5%
Standard deviation in value = 40%
Life of the option = 10 years
Value of expansion option = $ 477.28 million
Solutions to Equity as an option in a deeply
troubled firm
a. True. Equity investors cannot lose more than their equity investment.
b. False. They can make equity more valuable, not the firm.
c. True. It transfers wealth to the bondholders.
d. True. This is the equivalent of the life of the option.
e. True. There is a transfer of wealth to bondholders.
Problem 2
a. Reinvestment rate = g/ROC = 5%/12% = 41.67%
Value of the firm = 40(1.05)(1.5)(10.4)/(.10.05) = $294 million
b.
The value of the equity is computed as a call option on the value of the firm, using the call option pricing formula, , where , d_{2} = d_{1}  sÖt.
S = $294
K = $500
t = 5 years
r = 5%
s = 0.125
The equity or call option value can be written as 294 N(0.8657) 500 e^^{0.25 }N(1.1452). Since N(d_{1}) = 0.1933; N(d_{2}) = 0.1261, the option value is $7.75 million.
Value of Call (Equity) = $7.75 million
c. Value of Debt = $294  $7.75 = $286.25 million
Appropriate Interest Rate = (500/286.25)^{1/5}  1 = 11.80%
Problem 3
a. Value of firm
Current free cashflow to firm = $ 850* (1.4) – (550 – 400) = $ 700 million
Year 
EBIT (1t) 
Net cap ex 
FCFF 
PV 
1 
$612.00 
$180.00 
$432.00 
$392.73 
2 
$734.40 
$216.00 
$518.40 
$428.43 
3 
$881.28 
$259.20 
$622.08 
$467.38 
4 
$1,057.54 
$311.04 
$746.50 
$509.87 
5 
$1,269.04 
$373.25 
$895.80 
$556.22 
Terminal 
$1,332.50 
$444.17 
$888.33 

I used a reinvestment rate of 33.33% (5/15) in the terminal year.
Terminal value = 888.33/(.10.05) = $ 17,766
Value of firm = 392.73 + 428.43 + 467.38 + 509.87 + 556.22 + 17766.60/1.1^{5} = $13,386.28 million
b. Value of equity as an option
S = 13386.28
K = 10000.00
T = Weighted duration of debt = 3 years
Riskless rate = 5%
Variance in firm value = (.35)(.4)^^{2}+(.15)(.6)^^{2}+ 2 (.35)(.15)(.5)(.4)(.6) = .20 = 0.0403
Value of equity = $ 4958 million
c. If the market value of equity = 30 * 210 = $ 6300 million
Trial and error yields an implied standard deviation of 46.53%.
d. Value of debt = Firm value – Value of equity
= 13386 – 4958 = $8,428 million
Problem 4
Value of firm = EBIT (1t) (1 Reinvestment rate) (1+g)/(r – g)
= 25 (1.4) (1 – 4/10) (1.04)/(.09.04) = $ 187.20 million
Face value of debt = $ 250 + $ 250 = $ 500 million
Average duration of debt = (2+4)/2 = 3 years
Standard deviation in firm value = 0.25^{2}(.5)^^{2}+0.4^{2}(.5)^^{2}+ 2*.25*.4*.5*(.5)^^{2} = 28.39%
Riskless rate = 7%
Value of equity as an option = $ 3.30 million
Problem 5
d1 = 0.15 N(d1) =0.4404
Value of Equity = 400 (.4404)  800 exp (.06*6) (.1841) =$ 73.41
Value of Debt = 400  73.41= $ 326.59
Interest rate on debt = (800/326.59)^(1/6)  1 = 16.08%
Default spread on debt = 16.08%  6% = 10.08%
[1] There are 99.4 million households in the United States. Each rating point represents 1% of roughly 994,000 households.