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Lecture 24Real OptionsOptions, Futures, Derivatives / April 28, 2008 1Real OptionsWe can try to use our knowledge of derivatives to help value real objects, such as buildings, land,and equipment.Often there are options e mbedded in thes e investment opportunities, and often difficult to pricewell.Capital Investment AppraisalTraditionally one values a potential capital investment project is known as net present value(NPV). The NPV of a project is the present value of its expected future incremental cash flows.The discount rate used to calculate the present value is a risk-adjusted discount rate, chosen toreflect the risk of the project.As the riskiness of the project increases, the discount rate also increases.Example Consider an investment that costs $100 million and will last 5 years. The expected cashinflow in each year is estimated to be $25 million. If the risk-adjusted discount rate is 12% (withcont. compounding),Options, Futures, Derivatives / April 28, 2008 2the net present value of the investment is−100 + 25e−0.12×1+ 25e−0.12×2+ 25e−0.12×3+ 25e−0.12×4+ 25e−0.12×5= −11.53• A negative NPV indicates that the project will reduce the value of the company to itsshareholders and should not be undertaken.• A positive NPV indicates that the project should be undertaken because it will incre ase thevalue of the company.The risk-adjusted discount rate should be the return required by the company. This can becalculated in a number of ways.One approach often used involves the capital asse t pricing model:1. Take a sample of c ompanies whose main line of business is the same as that of the projectbeing contemplated.2. Calculate the betas of the companies and average them to obtain a proxy beta for the project3. Set the required rate of return equal to t he risk-free rate plus the proxy beta times t he excessreturn of the market portfolio over the risk-free rate.There are difficulties are using this approach, since there are usually embedded options in eachproject. In particular a company may consider abandoning a certain plant upgrade, etc. This willbe discussed further next week.Options, Futures, Derivatives / April 28, 2008 3Extension of the risk-neutral valuation frameworkRecall the market price of risk for variable θ:λ =µ − rσwhere r is the risk-free rate, µ is the return on a traded security dependent only on θ, and σ is itsvolatility. We get the same market price of risk λ regardless of the traded security chosen.Suppose a real asset depends on several variables θi. Let miand sibe the expected growth rateand volatility of θiso thatdθiθi= midt + sidziwhere ziis the Wiener process. Define λias the market price of risk of θi. We can extendrisk-neutral valuation to show that any asset dependent on the θican be valued by:1. Reducing the expected growth rate of each θifrom mito mi− λisi2. Discounting cash flows at the risk-free rate.Options, Futures, Derivatives / April 28, 2008 4ExampleThe cost of renting commercial real estate in a certain city is quoted as the amount that would bepaid per square foot per year in a new 5-year rental agreement.The current c ost is $30 per square foot. The expected growth rate of the cos t is 12% per annum,its volatility is 20% per annum, and its market price of risk is 0.3.A company has the opportunity to pay $1 million now for the option to rent 100,000 square feet at$35 per square foot for a 5-year period starting in 2 years.The risk-free rate is 5% per annum (assumed constant). Define V as the quoted cost per squarefoot of office space in 2 years. We make the simplifying assumption that rent is paid annually inadvance. The payoff from the option is100, 000A max{V − 35, 0}where A is an annuity factor given byA = 1 + 1 × e−0.05×1+ 1 × e−0.05×2+ 1 × e−0.05×3+ 1 × e−0.05×4= 4.5355The expected payoff in a risk-neutral world is therefore100, 000 × 4.5355 ׈E [max{V − 35, 0}] = 453, 550 ׈E [max{V − 35, 0}]Options, Futures, Derivatives / April 28, 2008 5whereˆE denotes expectations in a risk-neutral world. Then453, 550hˆE(V )N(d1) − 35N(d2)iwhered1=lnˆE(V )35+0.22×220.2√2d2= d1− 0.2√2The expected growth rate in the cost of commercial real estate in a risk-neutral world is m − λs,where m is the real-world growth rate, s is the volatility, and λ is the market price of risk.In this c ase m = 0.12, s = 0 .2 , and λ = 0.3, so that the expected risk-neutral growth rate is0.06 or 6% per year. It follows thatˆE(V ) = 30e0.06×2= 33.82. Substituting this in theexpression above gives the expected payoff in a risk-neutral world as $1.5015 million. Discountingat the risk-free rate the value of the option is 1.5015e−0.05×2= $1.3586 million. This showsthat it is worth paying $1 million for the option.Options, Futures, Derivatives / April 28, 2008 6Estimating the Market Price of RiskThe real-option approach to evaluating an investment avoids the need to estimate risk-adjusteddiscount rates, but it does require market price of risk parameters for all stochastic variables.When historical data are available for a particular variable, its market price of risk can be estimatedusing the capital asset pricing model.To show how this is done, we consider an investment asset dependent solely on the variable anddefineµ = Expected return of the investment assetσ = Volatility of the return of the investment assetλ = Market price of risk of the variableρ = Instantaneous correlation between the percentage changesin the variable and returns on a broad index of stock market pricesµm= Expected return on broad index of stock market pricesσm= Volatility of return on the broad index of stock market pricesr = Short-term risk-free rateOptions, Futures, Derivatives / April 28, 2008 7Because the investment asset is dependent solely on the market variable, the instantaneouscorrelation between its return and the broad index of stock market prices is also ρ.From the continuous-time version of the capital asset pricing model, we haveµ − r =ρσσm(µm− r)since λ =µ−rσfor all securities derived from the variable and σm= ρσ.Therefore, since µ − r = λσ thenλ =ρσm(µm− r) (1)We can use (1) to estimate the value of λ.Example: A historical analysis of company’s sales, quarter by quarter, show that percentagechanges in s ales have a correlation of 0.3 with returns on the S&P 500 is 20% per annum andbased on historical data the expected excess return of the S&P 500 over the


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U of M MATH 5076 - Real Options

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