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1、Real OptionsChapter 331An Alternative to the NPV Rule for Capital InvestmentsDefine stochastic processes for the key underlying variables and use risk-neutral valuationThis approach(known as the real options approach)is likely to do a better job at valuing growth options,abandonment options,etc than
2、 NPV2The Problem with using NPV to Value Options Consider the example from Chapter 11:risk-free rate=12%;strike price=$21 Suppose that the expected return required by investors in the real world on the stock is 16%.What discount rate should we use to value an option with strike price$21?3Stock Price
3、=$22Stock price=$20Stock Price=$18Correct Discount Rates are Counter-IntuitiveCorrect discount rate for a call option is 42.6%Correct discount rate for a put option is 52.5%4General Approach to ValuationWe can value any asset dependent on a variable q byReducing the expected growth rate of q by ls w
4、here l is the market price of q-risk and s is the volatility of q Assuming that all investors are risk-neutral5Extension to Many Underlying VariablesWhen there are several underlying variables qi we reduce the growth rate of each one by its market price of risk times its volatility and then behave a
5、s though the world is risk-neutralNote that the variables do not have to be prices of traded securities6Estimating the Market Price of Risk Using CAPM(equation 33.2,page 740)7rate free-riskterm-short the is and market;the on return expected the is return;smarket the of volatility the is market;the o
6、n returns and variable the in changes percentage between ncorrelatio ousinstantane the is whereby given is variable a of risk of price market Therrmmmml)(Example of Application of Real Options Approach to Valuing A(Business Snapshot 33.1;Schwartz and Moon)Estimate stochastic processes for the compan
7、ys sales revenue and its average growth rate.Estimated the market price of risk and other key parameters(cost of goods sold as a percent of sales,variable expenses as a percent of sales,fixed expenses,etc.)Use Monte Carlo simulation to generate different scenarios in a risk-neutral world.The stock p
8、rice is the average of the present values of the net cash flows discounted at the risk-free rate.8Commodity PricesFutures prices can be used to define the process followed by a commodity price in a risk-neutral world.We can build in mean reversion and use a process for constructing trinomial trees t
9、hat is analogous to that used for interest rates in Chapter 30 9Example(page 746)A company has to decide whether to invest$15 million to obtain 6 million units of a commodity at the rate of 2 million units per year for three years.The fixed operating costs are$6 million per year and the variable cos
10、ts are$17 per unit.The spot price of the commodity is$20 per unit and 1,2,and 3-year futures prices are$22,$23,and$24,respectively.The risk-free rate is 10%per annum for all maturities.10The Process for the Commodity PriceWe assume that this is d ln(S)=q(t)aln(S)dt+dzwhere a=0.1 and =0.211Tree Assum
11、ing q(q(t)=0;Fig 33.1 12EJ0.69280.6928BFK0.34640.34640.3464ACGL0.00000.00000.00000.0000DHM-0.3464-0.3464-0.3464IN-0.6928-0.6928NodeABCDEFGHIpu0.16670.12170.16670.22170.88670.12170.16670.22170.0867pm0.66660.65660.66660.65660.02660.65660.66660.65660.0266pd0.16670.22170.16670.12170.08670.22170.16670.12
12、170.8867Final Tree;Fig 33.2 13EJ44.3545.68BFK30.4931.3732.30ACGL20.0021.5622.1822.85DHM15.2515.6916.16IN11.1011.43NodeABCDEFGHIpu0.16670.12170.16670.22170.88670.12170.16670.22170.0867pm0.66660.65660.66660.65660.02660.65660.66660.65660.0266pd0.16670.22170.16670.12170.08670.22170.16670.12170.8867Valua
13、tion of Base Project;Fig 33.3 14EJ42.240.00BFK38.3221.420.00ACGL14.4610.805.990.00DHM-9.65-5.310.00IN-13.490.00NodeABCDEFGHIpu0.16670.12170.16670.22170.88670.12170.16670.22170.0867pm0.66660.65660.66660.65660.02660.65660.66660.65660.0266pd0.16670.22170.16670.12170.08670.22170.16670.12170.8867Valuatio
14、n of Option to Abandon;Fig 33.4(No Salvage Value;No Further Payments)15EJ0.000.00BFK0.000.000.00ACGL1.940.800.000.00DHM9.655.310.00IN13.490.00NodeABCDEFGHIpu0.16670.12170.16670.22170.88670.12170.16670.22170.0867pm0.66660.65660.66660.65660.02660.65660.66660.65660.0266pd0.16670.22170.16670.12170.08670
15、.22170.16670.12170.8867Value of Expansion Option;Fig 33.5(Company Can Increase Scale of Project by 20%for$2 million)16EJ6.450.00BFK5.662.280.00ACGL1.060.340.000.00DHM0.000.000.00IN0.000.00NodeABCDEFGHIpu0.16670.12170.16670.22170.88670.12170.16670.22170.0867pm0.66660.65660.66660.65660.02660.65660.66660.65660.0266pd0.16670.22170.16670.12170.08670.22170.16670.12170.8867
