Real Options
Valuation of real options in
Corporate Finance
FIN 819: Lecture 8
Today’s plan
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Review what we have learned about options
Real options
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Spot real options
Value real options
• Use the Black-Scholes formula to value real options
• Use the risk-neutral probability to value real options
Some hints about cases to be discussed
What have we learned about
options
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In the last three lectures, we have learned the
concepts about options and option pricing:
Concepts:
• Options: put and call
• Financial options and real options
• Financial options: European and American options
• Position diagram
• No arbitrage argument
• Put-call parity and its application in risky bond valuation
What have you learned about
options?
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Pricing options:
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Replicating portfolios of options
• The binomial tree approach
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•
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to value options (
discrete time case)
Black-Scholes formula (continuous time case)
The basic idea behind the pricing approaches.
The risk-neutral valuation
how to calculate u and d and their meanings
Risk-neutral valuation
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Now we can see that the value of the call
option is just the expected cash flow
discounted by the risk-free rate.
For this reason, p is the risk-neutral probability
for payoff Cu, and (1-p) is the risk-neutral
probability for payoff Cd.
In this way, we just directly calculate the riskneutral probability and payoff in each state.
Then using the risk-free rate as a discount rate
to discount the expected cash flow to get the
value of the call option.
Two-period binomial tree with
risk-neutral valuation
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Suppose that we want to value a call
option with a strike price of $55 and
maturity of six-month. The current stock
price is $55. In each three months, there
is a probability of 0.3 and 0.7,
respectively, that the stock price will go
up by 22.6% and fall by 18.4%. The riskfree rate is 4%.
Do you know how to value this call?
Solution
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First draw the stock price for each period
and option payoff at the expiration 27.67
Stock price
82.67
p
Option
p
67.43
1-p
0
C(K,T)=?
55
p
1-p
55
1-p
0
44.88
36.62
Now
Three
month
Sixth
month
Now
Three
month
Six
month
Solution
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Risk-neutral probability is
•
p=(Rf-d)/(u-d)
=(1.01-0.816)/(1.226-0.816)=0.473
The probability for the payoff of 27.67 is
0.473*0.473, the probability for other two states
are 2*0.473*527, and 0.527*0.527.
 The expected payoff from the option is
0.473*0.473*27.67=
 The present value of this payoff is 6.07
 So the value of the call option is $6.07
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Real options
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Real options
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The options whose underlying assets are real assets.
Examples
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Options to defer investment
Options to shut down temporarily
Options to expand production
Options to be a CEO of big firms after the study at
SFSU
Options to gain investment opportunities in the future
Value Real Options
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Although real options are in all walks of
our life, their valuation is based on the
following two approaches:
• Black-Scholes formula
• Risk-neutral valuation
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In the following, we use two examples to
demonstrate how to use the BlackScholes formula and the risk-neutral
valuation to value real options.
Example 1
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Mark Wang, who got his MBA from SFSU, is asked by his
boss to decide on whether to take the following project.
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The project needs investment of $10 million and will
generate an expected perpetual cash flow of $1.8 million
every year starting next year. The volatility of the return of
the investment is 90%. The cost of capital for the project is
20%. The risk-free rate is 10%. If this project is taken, three
years later, a similarly risky project is available, that is, if you
invest another $10 million in year three and you will receive
another expected perpetual cash flow of $1.8 million every
year starting in year 4. If you don’t invest now, you don’t
have the second investment opportunity.
Simple solution
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I will discuss the full solution in the class. The
following is just a simple solution.
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•
Without considering the second investment
opportunity, NPV= -$1 million
Considering the value of the second investment
opportunity, NPV=-1+C(10,3)=-1+2.53=1.53 >0,
where C(10,3) is the value of a call option with the
strike price of $10 and maturity of 3 years. Here we
use the Black-Sholes formula to calculate
C(10,3)=2.53. (d1=0.5524, d2=-1.0065)
So, when the value of real options is considered, the
project has a positive NPV and should be taken.
Example 2
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Gold is currently trading at $300 per
ounce, and will move up or down as
shown below:
$363
$330
$297
$300
$270
$243
Example 2 (continue)
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Suppose that we can operate a gold mine for three years. We
can only produce 0.1 million ounces of gold per year. Our
extraction cost per ounce is $250, and fixed costs of running
the mine are $4 million. Suppose that the risk-free rate is 5%
per-period.
(a) What is the NPV of running the gold mine for three years?
(b) If we have the option to close the gold mine in the second
period temporarily and reopen it at an extra cost of $500,000 in
the third period, what is the value of this option?
(c) In addition, we have the option to expand production at
period two by 50% ; this expansion will cost $5 million, but will
not altering operating costs. What is the value of this option to
expand and shut down temporarily?
Simple Solution
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I will discuss the full solution in the class. The
following is a simple solution.
Basic idea: calculate the risk-neutral probability
and the cash flow or profit at each node in the
tree
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(a) NPV=$7.08 million
(b) The value of the option to shut down temporarily is
$0.65 million
(c) The value of the option to expand and shut down is
$ 3.22 million