Chapter16 valuationofrealoptions

 Centre for Financial Management , Bangalore
CHAPTER 16
VALUATION OF REAL OPTIONS

OUTLINE
. How options work
· Options and their payoffs just before expiration
· Factors determining option values
· Binomial model
· Black and Scholes model
· Types of real options
· Applications of the binomial model
· Applications of the Black and Scholes model
· Revisiting the binomial model
· Judgmental assessment of options
· Managing real options

Strategic NPV
The discounted cash flow (DCF) analysis has been the basic framework for
valuing assets, real and financial, since the 1950s. Hence our discussion of capital
budgeting focused on DCF techniques. In recent years, however, a growing
number of academicians and practitioners have argued that the traditional DCF
model does not fully capture the value of a real project, as it does not reflect the
options embedded therein.
The important types of real options found in capital projects are the option to
delay the project, the option to expand the project, the option to contract or
abandon the project, and the option to change the outputs or inputs of the project.
These options give managers flexibility to amplify gains or to reduce losses.
Given the fact that these options are found in most real projects, particularly
strategic investments, finance theorists and practitioners argue that the strategic
NPV of a project must be defined as:
Strategic NPV = Conventional NPV + Real option value (ROV)

Option Payoffs
Payoff of a Call Option
Payoff of a
Call Option
E (Exercise Price) Stock Price
Payoff of a Put Option
Payoff of a
Put Option
E (Exercise Price) Stock Price

Payoffs to the Seller of Options
Payoff
E
Stock Price
(A) Sell A Call
Payoff
E
Stock Price
(B) Sell a Put

Option Value : Bounds
Upper and Lower Bounds for the Value of Call Option
Value of Upper Lower
Call Option Bound S0) Bound ( S0 – E)
Stock Price
0 E
(

Factors Determining the Option Value
• Exercise Price
• Expiration Date
• Stock Price
• Stock Price Variability
• Interest Rate
C0 = F [S0 , E, 2, T , Rf ]
+ - + + +

Binomial Model
Option Equivalent Method
A single period Binomial (or 2 - State) Model
• S can take two possible values next year, uS or dS (uS > dS)
• B can be borrowed .. or lent at a rate of r, the risk-free rate..
(1 + r) = R
• d < R < u
• E is the exercise price
Cu = Max (u S - E, 0)
Cd = Max (dS - E, 0)

Binomial Model : Option Equivalent Method
Portfolio
 Shares of the stock and B rupees of borrowing
Stock price rises :  uS - RB = Cu
Stock price falls :  dS - RB = Cd
Cu - Cd Spread of possible option price
 = =
S (u- d) Spread of possible share prices
dCu - uCd
B =
(u - d) R
Since the portfolio (consisting of  shares and B debt) has the same
payoff as that of a call option, the value of the call option is
C =  S - B

Illustration
S = 200, u = 1.4, d = 0.9
E = 220, r = 0.10, R = 1.10
Cu = Max (u S - E, 0) = Max (280 - 220, 0) = 60
Cd = Max (dS - E, 0) = Max (180 - 220, 0) = 0
Cu - Cd 60
 = = = 0.6
(u - d) S 0.5 (200)
dCu - uCd 0.9 (60)
B = = = 98.18
(u - d) R 0.5 (1.10)
0.6 of a share + 98.18 borrowing … 98.18 (1.10) = 108 repayt
Portfolio Call Option
When u occurs 1.4 x 200 x 0.6 - 108 = 60 Cu = 60
When d occurs 0.9 x 200 x 0.6 - 108 = 0 Cd = 0
C =  S - B = 0.6 x 200 - 98.18 = 21.82

Binomial model risk-neutral method
We established the equilibrium price of the call option without knowing
anything about the attitude of investors toward risk. This suggests …
alternative method … risk-neutral valuation method
1. Calculate the probability of rise in a risk neutral world
2. Calculate the expected future value ..option
3. Convert .. it into its present value using the risk-free rate

Black - Scholes Model
E
C0 = S0 N (d1) - N (d2)
ert
N (d) = Value of the cumulative normal density function
S0 1
ln E + r + 2 2 t
d1 =
  t
d2 = d1 -   t
r = Continuously compounded risk - free annual interest
rate
 = Standard deviation of the continuously compounded
annual rate of return on the stock

Black - Scholes Model
Illustration
S0 = Rs.60 E = Rs.56  = 0.30
t = 0.5 r = 0.14
Step 1 : Calculate d1 and d2
S0 2
ln E + r + 2 t
d1 =
  t
.068993 + 0.0925
= = 0.7614
0.2121
d2 = d1 -   t
= 0.7614 - 0.2121 = 0.5493
Step 2 : N (d1) = N (0.7614) = 0.7768
N (d2) = N (0.5493) = 0.7086
Step 3 : E 56
= = Rs. 52.21
ert e0.14 x 0.5
Step 4 : C0 = Rs. 60 x 0.7768 - Rs. 52.21 x 0.7086
= 46.61 - 37.00 = 9.61

Replicating Portfolio
Note that the principle of replicating portfolio used in the binomial model
also undergirds the Black-Scholes model. Table shows the replicating
portfolios for calls and puts, in the binomial and the Black-Scholes
models.
Replicating Portfolio
Option Position Binomial Model Black-Scholes Model
Buy Call Option Borrow B Borrow Ee-rt N(d2)
Buy  shares of stock Buy N(d1) shares of stock
Sell Call Option Lend B Lend Ee-rt N(d2)
Sell short  shares Sell short N(d1) shares
Buy Put Option Lend B Lend Ee-rt (1-N(d2))
Sell short  shares Sell short (1-N (d1)) shares
Sell Put Options Borrow B Borrow Ee-rt (1-N(d2))
Buy  shares Buy (1-N(d1)) shares

Adjustment for Dividends
The Black-Scholes model eqn assumes that the stock pays no dividend. When
dividend is paid stock price diminishes. Hence,call options become less
valuable and put options become more valuable. To reflect dividend payments,
two adjustments are commonly made, one for options that have a short life and
the other for options that have a long life.
Short - Term Options
Divt
Adjusted Stock Price = S = S - 
(1 + r)t
E
Value of Call = S N (d1) - N (d2)
ert
S 2
ln E + r + 2 t
d1 =
  t

Adjustment for Dividends - 2
Long - Term Options
C = S e -yt N (d1) - E e -rt N (d2)
S 2
ln E + r - y + 2 t
d1 =
  t
d2 = d1 -   t
The adjustment
• Discounts the value of the stock to the present at the dividend
yield to reflect the expected drop in value on account of the
dividend payments.
• Offsets the interest rate by the dividend yield to reflect the lower
cost of carrying the stock.

Put - Call Parity - Revisited
Just before expiration
C1 = s1 + p1 - e
If there is some time left
C0 = S0 + P0 - E e -rt
The above equation can be used to establish the price of a put option
& determine whether the put - call parity is working

Types of Real Options
• Investment timing options
• Growth options
• Flexibility options
• Abandonment options

Key Differences between Financial and Real Options
 Information about valuing options and making decisions about
exercising them is more readily available for financial options than
for real options.
 While the right to exercise a financial option is unambiguous, the
holder of a real option is often unclear what the precise right is and
how long the same will last.

Value of a Vacant Land - 1
8 - units 3.6 m 12 units 6.2
Price : 0.6
Yearly rental (net) : 0.05M rf = 12%
Wait for a year .. No change constr’n cost
Market condition
Buoyant sluggish
price : 0.75 price : 0.54
8 - unit 0.75 x 8 - 3.6 = 2.4 0.54 x 8 - 3.6 = 0.72
12 - unit 0.75 x 12 - 6.2 = 2.8 0.54 x 12 - 6.2 = 0.28

Value of a Vacant Land
1. Risk - Neutral Prob
p x 0.80 + (1 - p) x 0.59
0.60 =
0.12
p = 0.39
2. Expected Cash Flow
0.39 x 2.8 + 0.61 x 0.72 = 1.531 M
3. Current Value
1.531
= RS 1.367 M
1.12

Value of an Option to Abandon
Payoff (Rs. in million)
SPM (A) GPM (B)
Strong demand 30 28
Weak demand 14 12
Resale value 10 16
Given .. option .. sell .. payoff … GPM
Strong demand 28 Weak demand 16
Suppose the value … B, ignoring the option to abandon, is Rs. 18 M
28 55.6 percent
18
-12 - 33.3%

Year - End Value
Rs. in Million
12 28
Value of Put Option 4 0
Now Year 1
28 (0)
18 (?)
12 (4)
Risk - Free Return = 6%
Risk - Neutral Method
Expected = p x 55.6% + (1 - p) x -33.3%
Return (6%)
p 0.44
Exp. Payoff to Put Option = 0.44 x 0 + 0.56 x 4 = 2.24
2.24
Value of the Abandonment Put = = 2.11
1.06

Value .. Option .. Make A Follow On Investment

• Electriad – II … Double .. Size of
Electriad - I … Investment : Rs. 300 M
• Cash inflows … twice .. pv of Rs. 265 m … year 4
•  = 0.4
S0 = 265 x e - 0.18 x 4 = Rs. 129 Million
E = Rs. 300 M
 = 0.4
t = 4
r = 12 Percent

Step 1 : Calculate d1 and d2
S0  2
ln + r + t
E 2 - 0.843 + (.12 + (.16/2)) 4
d1 = = = -.05375
 t 0.4 4
d2 = d1 -   t = - 0.85375
Step 2 : Find N (d ) and N (d )2
N(d1) = 0.4786
N(d2) = 0.1966
Step 3 : Estimate the Present Value of the Exercise Price
E . e-rt
= 300 / 2.0544 = RS.146.03 Million
Step 4: Plug the numbers obtained in the previous steps in the
Black-Scholes formula
C0
= Rs.129 Million x 0.4786 – Rs.146.03 Million x 0.1966
= Rs.61.74 Million – Rs.28.71 Million = Rs.33.03 Million
1

Value of a Natural Resource Option
• Value of the available reserves of the resource
• Development cost
• Time to expiration of the option
• Variance in value of the underlying asset
• Cost of delay… like dividend yield in a stock loss of
prod’n … each year of delay …
• Development lag

Long - Term Options
C = S e - y t N (d1) - E e - r t N (d2)
S  2
ln + r - y + t
E 2
d1 =
  t
d2 = d1 -   t

Illustration
• Estimated oil reserve : 100 million barrels
• Development cost : $ 1 billion
• Right : 25 years
• Marginal value per barrel : $ 20
•  of ln (Oil Price) = 0.2
• Net prod’n revenue annual = 4% of value of reserve
• r = 8%
• Development lag = 2 years

Illustration
• Estimated oil reserve : 100 million barrel
• Marginal value per barrel : $ 20
• Development lag : 2 years
• Dividend yield : 4%
S0 = Current value of the asset
$ 20 x 100
= = $ 1849.11 million
(1.04)2
E = Development cost = $ 1000 million
 = 0.2
t = Life of the option = 25 yrs
r = 8%
Net Prod’n Revenue
y = = 4%
Value of Reserve

STEP 1: CALCULATE d1 AND d2
S 2
d1 = ln + r – y + t
E 2
 t
= ln (1849.11 / 1000) + [.08 - .04 + (.04/2)] 25 ÷0.2 25
= 0.6147 + 1.5 = 2.1147
d2 = d1 -  t
= 2.1147 – 1.000 = 1.1147
STEP 2: FIND N(d1) AND N(d2)
N(d1) = N (2.1147) = 0.9828
N(d2) = N (1.1147) = 0.8675

STEP 3: ESTIMATE THE PRESENT VALUE OF THE EXERCISE PRICE
E / en = 1000 / e.08 x 25 = $ 135.33 MILLION
STEP 4: PLUG THE NUMBERS OBTAINED IN THE PREVIOUS STEPS
IN THE BLACK-SCHOLES FORMULA
C = $1849.11 MILLION x 0.9828 - $ 135.33 MILLION x 0.8675
= $1699.91 MILLION

Strategy as a Portfolio of Real Options
Leuhrman has proposed an approach to develop strategy using the real options. In this
approach, potential projects are mapped on the basis of two option-value metrics, one
representing a value based on NPV analysis and the other a volatility (risk) measure, as
shown in Exhibit.

Metaphor of a Tomato Garden
He says that managing a portfolio of projects is similar to tending a
garden of tomatoes. On any given day, the tomatoes in a garden (like
projects in a business) have varying prospects for growth and value
creation. An experienced gardener is able to distinguish between ripe
tomatoes, rotten tomatoes, and the rest that fall somewhere “in-between.”
While the ripe tomatoes have to be harvested and the rotten tomatoes
have to be discarded the choice with respect to the “in-between” ones is
not very clear. It is helpful to divide the “in-between” ones into four
groups for determining an appropriate course of action:
 Imperfect, but edible tomatoes: They can be picked now, but would
ripen with more time on the vine. If left on vine, there is some risk
that squirrels will get them first. It makes sense to allow them to
ripen, unless the scavenger risk is high.
Contd…

 Inedible, but very promising tomatoes: It does not make sense to pick
them now, even though they are vulnerable to scavengers. With
enough time left in the season, many of them will ripen beautifully
and eventually be picked.
 Less promising green tomatoes: With sun, water, and a little luck
some of them will ripen before the season is over. Allow them to
remain on the vine and monitor their growth.
 Late blossoms and small green tomatoes: It is unlikely that they will
be harvested in the season. So ignore them or prune them.
Contd…

Metaphor of a Tomato Garden
Using the metaphor of a tomato garden, Luehrman divides the option-value
space of business opportunities into six distinct regions, numbered 1 through 6
clockwise, as shown in Exhibit. Each of these regions calls for a different
strategy.
Region 1: Businesses in this region have a value-to-cost ratio greater than 1
(which means a positive NPV). They have low uncertainty and short time to
expiration. The optimal strategy in this region is to “invest now.”
Region 2: Businesses in this region have positive NPV. They are characterised
by some volatility and / or some time to expiration. So, it may make sense to
wait for a while. However, if there is a possibility of preemption by
competitors, the optimal strategy may be to exercise the option early.
Region 3: Businesses in this region too have a positive NPV. Further, they are
characterised by high levels of uncertainty and / or time to expiration. Hence,
they are likely to become more valuable in future, particularly with active
management. So, the optimal strategy is to defer the exercise of option.

Region 4: Businesses in this region have a value-to-cost ratio of less than 1
(i.e. a negative NPV). They are characterised by high levels of remaining
uncertainty and / or time to expiration. If harvested now, they will destroy
value. However, depending on how uncertainty unravels there may be some
possibility that these businesses may add value with time and active
management. So, a “wait and watch” strategy makes sense.
Region 5: Businesses in this region have a negative NPV. Given their limited
volatility and / or time to expiration, it make sense to ignore them.
Region 6: Businesses in this region have a negative NPV. They are
characterised by little uncertainty and / or will expire soon. The optimal
strategy is never to invest in such businesses.

The Tomato Garden

Judgmental Assessment of Options
Valuing the options embedded in real life projects with the help of Black-
Scholes model often requires heroic assumptions. Yet the insights
provided by this model can be combined with well-informed,
experienced judgment to get a handle over option values. If you can
identify the options and specify the circumstances under which they
would be exercised, you can make an informed estimate of their values.
The procedure for doing this, thus, would broadly involve three steps :
· Identify options
· Analyse environmental uncertainty
· Value options

L
o Cash flows : 75 Cash flows : 75
n Options : 25 Options : 25
g
Duration of
the Project
S
h Cash flows : 95 Cash flows : 75
o Options : 5 Options : 25
r
t
Low High
Environmental Uncertainty
Relative values of Cash Flows and Options

Examples of Real Option Analysis
Here are some examples of real options analysis.
 At Merck, R& D projects are analysed in the real options framework. An R&D
project involves staged investments (which depend on the outcomes in
previous stages) and hence is eminently suitable for real options analysis.
 Shell Oil evaluates capital projects and investment strategies using real options
analysis. When Shell Oil acquires a license to develop an oil field, it has to
decide when to start development, how much to spend on development, often
in stages, and whether to seek an extension of the license.
 Intel uses real options analysis to evaluate its acquisitions. For example, when
Intel purchased Level One Communications it really acquired the option to
produce different types of chips than those it manufactured itself. The demand
for the Chips of Level One Communications is linked to the growth of the
Internet. Thus Intel acquired manufacturing capability in a complementary
business.

Mistakes Made in Real Option Valuation
 Unthinking application of the Black-Scholes model.
 Use of wrong volatility.
 Assumption of a fixed exercise price.
 Overestimation of a the value of flexibility.
 Abuse of real options to justify a project on strategic grounds.
 Failure to consider feedback effects.

Managing Real Options
• Often there is a significant gap between the theoretical and
realised values of real options because of a disconnect between
the way options are valued and the way options are managed.
• What can be done to promote a more timely and rational exercise
of real options? The company’s planning and budgeting system
should reflect the decision trees constructed in developing the
binomial model to value the project.
• More specifically, the planning and budgeting system should:
1. Identify the trigger points
2. Specify the rules governing the exercise decisions
3. Clearly assign responsibilities
4. Motivate people

Managing Real Options Proactively
The key guidelines for managing real options proactively are as follows:
1. Extend the option duration by innovating to hold technological lead,
by signaling the ability to exercise, and by maintaining regulatory
barriers.
2. Increase the uncertainty of expected cash flows by developing
innovative products, by extending opportunity to related markets,
and by encouraging complementary products.
3. Increase the present value of expected cash flows by developing
alliances with low cost suppliers and by developing new marketing
initiatives.
4. Reduce the value lost by waiting to exercise the option by locking
up key resources and by creating implementation hurdles for
customers.
5. Reduce investment by leveraging economies of scale and by
exploiting economies of scope.

SUMMARY
 The standard discounted cash flow analysis calls for evaluating a project on
the basis of its cash flows over its economic life. Such an evaluation, however,
does not capture the value of options embedded in the project.
 The strategic NPV of a project is defined as:
Strategic NPV = Conventional NPV + Real Option Value (ROV)
 An option is a special contract under which the option owner enjoys the right
to buy or sell something without the obligation to do so.
 There are two basic types of options: call options and put options. A call
option gives the option holder the right to buy a stock (or asset) at a fixed
price on or before a certain date. A put option gives the holder the right to sell
stock (or some other asset) at a specified price on or before a certain date.
 The value of a call option depends on five factors: exercise price, expiration
date, stock price, variability, and interest rate. The key insight underlying the
Black and Scholes model may be illustrated through a single-period binomial
(or two-state) model.

 There are two ways of calculating the value of an option in the binomial
world: (i) Option Equivalent Method : Find a portfolio of shares and loan that
imitates the option in its payoff. Since the two alternatives have identical
payoffs in the future, they must command the same price today. (ii) Risk
Neutral Method : Assume that investors are risk-neutral, so that the expected
return on the stock is the same as the interest rates. Calculate the expected
future value of the option and discount it at the risk-free interest rate.
 Fisher Black and Myron Scholes published their celebrated option pricing
model in 1973.
 Black and Scholes developed the following formula (referred to commonly as
the Black-Scholes model) showing how the value of a call option is related to
the basic factors :
E
C0 = S0 N(d1) - ----- N (d2)
ert
 The Black-Scholes model assumes that the stock pays no dividend. To reflect
dividend payments, two adjustments are commonly made, one for options that
have a short life and the other for options that have a long life.

 Options embedded in real life projects may be valued with the help of the
binomial model or the Black-Scholes model, if suitable quantitative estimates
can be defined.
 Where quantitative estimates cannot be defined with a measure of confidence,
Black-Scholes model cannot be applied meaningfully. Yet, the insights
provided by this model can be combined with well-informed, experienced
judgment to get a handle over option values.
 Firms that employ real options for strategic reasons must learn to manage them
well.

Chapter16 valuationofrealoptions

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