Real Options

1
Real Options

• Valuation of real options in Corporate Finance

2
Todays plan

• Review what we have learned about options
• Real options
• 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

3
What have we learned about options

• 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

4
What have you learned about options?

• Pricing options
• Replicating portfolios of options
• The binomial tree approach 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

5
Risk-neutral valuation

• 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
  risk-neutral 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.

6
Two-period binomial tree with risk-neutral
valuation

• 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 risk-free rate is
  4.
• Do you know how to value this call?

7
Solution

• First draw the stock price for each period and
  option payoff at the expiration

27.67
p
Stock price
Option
82.67
p
67.43
0
1-p
C(K,T)?
55
p
1-p
55
1-p
0
44.88
36.62
Three month
Six month
Now
Three month
Sixth month
Now
8
Solution

• 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
• 0.4730.473, the probability for other two
  states are 20.473527, and 0.5270.527.
• The expected payoff from the option is
• 0.4730.47327.67
• The present value of this payoff is 6.07
• So the value of the call option is 6.07

9
Real options

• Real options
• The options whose underlying assets are real
  assets.
• Examples
• 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

10
Value Real Options

• 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
• In the following, we use two examples to
  demonstrate how to use the Black-Scholes formula
  and the risk-neutral valuation to value real
  options.

11
Example 1

• Mark Wang, who got his MBA from SFSU, is asked by
  his boss to decide on whether to take the
  following project.
• 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 dont invest now, you dont have the second
  investment opportunity.

12
Simple solution

• I will discuss the full solution in the class.
  The following is just a simple solution.
• Without considering the second investment
  opportunity, NPV -1 million
• Considering the value of the second investment
  opportunity, NPV-1C(10,3)-12.531.53 gt0,
• 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. (d10.5524, d2-1.0065)
• So, when the value of real options is
  considered, the project has a positive NPV and
  should be taken.
  

13
Example 2

• Gold is currently trading at 300 per ounce, and
  will move up or down as shown below

363
330
297
300
270
243
14
Example 2 (continue)

• 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?

15
Simple Solution

• 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
• (a) NPV7.08 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