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**Valuation of real options in Corporate Finance**

FIN 819: Lecture 8

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**Today’s plan Some hints about cases to be discussed**

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

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**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

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**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

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**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.

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**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?

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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 C(K,T)=? 55 p 1-p 55 1-p 44.88 36.62 Three month Six month Now Three month Sixth month Now

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**Solution Risk-neutral probability is**

p=(Rf-d)/(u-d) =( )/( )=0.473 The probability for the payoff of 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 Real options Examples**

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

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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.

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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.

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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)= =1.53 >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= ) So, when the value of real options is considered, the project has a positive NPV and should be taken.

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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

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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?

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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

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