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Real Options. Valuation of real options in Corporate Finance. Today’s 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

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real options

Real Options

Valuation of real options in Corporate Finance

FIN 819: Lecture 8

today s plan
Today’s 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
what have we learned about options
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
what have you learned about options
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
risk neutral valuation
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.
two period binomial tree with risk neutral valuation
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?
solution
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

1-p

0

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

solution8
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

0.473*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
real options9
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
value real options
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.
example 1
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 don’t invest now, you don’t have the second investment opportunity.
simple solution
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=-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
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

example 2 continue
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?
simple solution15
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) 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