Session 3: Options III
C15.0008 Corporate Finance
Topics
Summer 2006
Outline
• Risk Neutral Valuation
• Options embedded in projects
• Valuation of real options
– DCF (decision trees)
Risk neutral valuation
What is the expected return on this tree?
0.6
110
100
0.4
90
Recall the definition of “expected” return from the first session
Risk neutral valuation
p
110
100
1-p
90
Why 1-p?
What is the value of p if the expected
return is to be 2%?
Risk-Neutral Valuation
• Given the prices, is it possible to find
implied probabilities such that we can apply
the risk-free rate as our discounting rate.
• Risk-neutral valuation is equivalent to
solving for the replicating strategy, but it is
computationally quicker for large trees.
• Risk-neutral probabilities are not really
probabilities!!
A note on how to get u and d
• If the per period volatility of a stock is σ,
then u and d are given by:
– u = e σ/√2
– d = e -σ/√2
• If we reduce the time period per step of a
tree, so that a very large number of steps
represent a year of time, we get the Black
Scholes formula.
Black Scholes formula
• The Black-Scholes formula for a call option is
• C = S N(d1) – PV(K) N(d2)
– S is the current price of the stock
– PV(K) is the present value of the exercise price (PV(k)
= Ke-rt)
– d1 = (log(S/ PV(K) )/ σ√t) + σ√t/2
– d1 = (log(S/ PV(K) )/ σ√t) - σ√t/2
– σ is the volatility
• Notice that N(d1) is like “H”. PV(K) N(d2) is like
“B”
• Use put-call parity to get the price of a put
Back to Discounted Cash-flows
• Recall that when we compute the NPV of
a project, we discount the expected cash
flows by the cost of capital.
– What exactly are “expected” cash flows?
– Are we missing out on something when we
take these average cash flows?
A hint..
“Discounted cash flow is going to look at
an average scenario," comments Triantis.
"But if you talk to any manager, that's not
how they think. They think about
contingencies — what's going to happen,
how would we react. And even if they don't
think that way, once it's presented to them
that way, they say, 'Yeah, that's the way
we should be thinking.'"
CFO Magazine – “Will Real Options take roots?”, July 2003
Real Options in Projects
Projects have some characteristics
• If things go well, you can expand
• If things go poorly, you can shut down
• Timing options (when to invest, the option
to delay investment)
• These options are related to decisions that
are to be taken when more is known about
the project
Decision Trees
• What does the decision tree for the
decision to take C15.008 look like?
• What are the action points?
• What are the information points?
A Project as an Option
Before the decision is made to undertake a project,
every project is really an option on a project
• NOT accept vs. reject
• Accept today vs. revisit decision tomorrow
Note: generally, the option has no value unless
uncertainty will be resolved over time, i.e.,
information will be revealed about the value of the
underlying asset
Example: Product Introduction
Baldwin Inc. is considering investment in a new technology
to produce colored bowling balls. The primary determinant
of cash flows is demand, i.e., consumer acceptance of
colored bowling balls:
0
Demand
Prob. 1
2
3...
-100 High
1/3
24
24
24...
Low
2/3
1.5
1.5
1.5...
E(CF)
-100
9
9
9…
At a discount rate of 10% the NPV is
-100 + 9/0.1 = -10
The Abandonment Option
The project can be abandoned at the end of year 1 and the
equipment salvaged for 19.5. Should the project be scrapped? If
demand is low…
1
2
3...
PV(10%) in yr 1
Continue
1.5
1.5...
15
Abandon
19.5
19.5
The modified project
Prob.
0
1
1/3
-100
24
2/3
-100
21
E(CF) -100
22
2
24
3...
24...
NPV(10%)
8
8…
-7.27
The Option Value
• -7.27 is the value of the project plus the
option to abandon (using DCF)
• Recall that the value of the project without
the option to abandon was -10
 The value of the abandonment option is
-7.27 - (-10) = 2.73 or
(19.5-15)(2/3)/1.1 = 2.73
The Expansion Option
Baldwin can add a second factory at the end of the first year, which will
also operate at capacity if demand is high.
The modified project
Prob.
0
1
2
3...
NPV(10%)
1/3
-100
24
24
24...
-100
24
24…
2/3
-100
21
E(CF) -100
-11.3
16
16…
35.15
Why not open this second factory immediately?
Why not delay the first investment until time 1?
The Option Value
• 35.15 is the value of the project plus the option to
expand and the option to abandon (using DCF)
• Recall that the value of the project without the option to
expand was -7.27
 The value of the expansion option is
35.15 - (-7.27) = 42.42 or
140(1/3)/1.1 = 42.42
 The total value of the project is
project w/o options + abandonment + expansion =
-10 + 2.73 + 42.42 = 35.15
Valuing Real Options
Why not value real options using a DCF approach (i.e.,
decision trees)?
• In principle, a DCF approach will work
• In practice, the discount rate may be a problem. The
required return on the option is generally not the same as
the required return on the underlying asset (project).
Solution:
 Use a binomial approach
Example: Product Introduction
Underlying project:
0
Demand
Prob. 1
2
3...
-100 High
1/3
24
24
24...
Low
2/3
1.5
1.5
1.5...
Abandonment option: project can be abandoned at time 1
and the equipment salvaged for 19.5
Expansion option: a second factory can be built at time 1
that will also operate at full capacity if demand is high
0
Demand
Prob. 1
2
3...
High
1/3
-100 24
24...
The Option to Abandon
• The abandonment option is a put option where the
underlying asset is an operating bowling ball factory and
the exercise price is the salvage value, i.e., it is the right
to sell (salvage) the factory and receive the salvage
value.
• Conceptually it is a quasi-American option, i.e., it can be
exercised starting in 1 year and at any time thereafter.
If exercised at all, will the option be exercised immediately?
How would you handle a salvage value that varies over
time?
Inputs
• exercise price = salvage value
E = 19.5
• underlying asset = operating factory
value (time 0) = PV(expected cash flows)
0
1
2
3...
9
9
9…
S = 9/0.1 = 90
• t = 1, rf = 2%
The Option to Expand
• The expansion option is a call option where the
underlying asset is a new (second) operating bowling
ball factory and the exercise price is the initial investment
required, i.e., it is the right to buy (build) the factory and
receive the resulting future cash flows.
• Conceptually it is a quasi-American option, i.e., it can be
exercised starting in 1 year and at any time thereafter.
If exercised at all, will the option be exercised immediately?
Inputs
• exercise price = initial investment
E = 100
• underlying asset = operating factory (built at time 1)
value (time 0) = PV(expected cash flows)
0
1
2
3...
9
9…
S = (9/0.1)/1.1 = 90/1.1 = 81.82
Assumption: the second factory looks just like the first
• T = 1, rf = 2%,  = 130%
The Option to Abandon
24 24 24…
90
1.5 1.5 1.5…
E=19.5
264
90
0
240
16.5
15
P
4.5
Option Value
rf = 2%  H = -0.018, B = -4.71, P = 3.07
Note: replicate with the cum-dividend values of the
underlying asset, calculate the put payoff based on
the ex-dividend values
The Option to Expand
0 24 24…
81.82
0 1.5 1.5…
E=100
240
140
81.82
15
C
0
Option Value
rf = 2%  H = 0.622, B = 9.15, C = 41.76
Project Value
DCF
binomial
base abandon expand
-10
2.73
42.42
-10
3.07
41.76
Note:
• DCF uses the wrong discount rate
• B-S uses the wrong distribution
total
35.15
34.83
When is There a Real Option?
• There has to be a clearly defined underlying
asset whose value changes over time in a
predictable way
• The payoffs on this asset have to be contingent
on a specific observable event, i.e, there has to
be a resolution of uncertainty about the value of
the asset
When Does the Option Have
Value?
• For an option to have significant economic
value, there has to be a restriction on
competition in the event of the contingency (in a
perfectly competitive market, no option
generates positive NPVs)
• Real options are most valuable when there is
exclusivity
Assignments
• Reading
– RWJ: Chapters 14, 16.7, 20
– Problems: 14.1, 20.16
– Problem Set 1 due next class