Presentation on theme: "Valuation of real options in Corporate Finance"— Presentation transcript:
1Valuation of real options in Corporate Finance FIN 819: Lecture 8
2Today’s plan Some hints about cases to be discussed Review what we have learned about optionsReal optionsSpot real optionsValue real optionsUse the Black-Scholes formula to value real optionsUse the risk-neutral probability to value real optionsSome hints about cases to be discussed
3What have we learned about options In the last three lectures, we have learned the concepts about options and option pricing:Concepts:Options: put and callFinancial options and real optionsFinancial options: European and American optionsPosition diagramNo arbitrage argumentPut-call parity and its application in risky bond valuation
4What have you learned about options? Pricing options:Replicating portfolios of optionsThe 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 valuationhow to calculate u and d and their meanings
5Risk-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.
6Two-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?
7SolutionFirst draw the stock price for each period and option payoff at the expiration27.67pStock priceOption82.67p67.431-pC(K,T)=?55p1-p551-p44.8836.62ThreemonthSixmonthNowThreemonthSixthmonthNow
8Solution Risk-neutral probability is p=(Rf-d)/(u-d)=( )/( )=0.473The probability for the payoff of is0.473*0.473, the probability for other two states are 2*0.473*527, and 0.527*0.527.The expected payoff from the option is0.473*0.473*27.67=The present value of this payoff is 6.07So the value of the call option is $6.07
9Real options Real options Examples The options whose underlying assets are real assets.ExamplesOptions to defer investmentOptions to shut down temporarilyOptions to expand productionOptions to be a CEO of big firms after the study at SFSUOptions to gain investment opportunities in the future
10Value Real OptionsAlthough real options are in all walks of our life, their valuation is based on the following two approaches:Black-Scholes formulaRisk-neutral valuationIn the following, we use two examples to demonstrate how to use the Black-Scholes formula and the risk-neutral valuation to value real options.
11Example 1Mark 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.
12Simple solutionI will discuss the full solution in the class. The following is just a simple solution.Without considering the second investment opportunity, NPV= -$1 millionConsidering 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.
13Example 2Gold is currently trading at $300 per ounce, and will move up or down as shown below:$363$330$297$300$270$243
14Example 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?
15Simple SolutionI 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