Real Options Valuation of real options in Corporate Finance FIN 819: Lecture 8
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 • 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? • 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 • 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 • 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 • 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
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 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 • 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 • 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 • 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 • 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) • 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 • 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
