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# Table 1 Option Mapping

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Table 1 Option Mapping

REAL OPTIONS MATERIALS FOR CLASSROOM USE

This document contains three parts. The first part is an introduction to real options for students who have studied traditional capital budgeting but have little or no background in derivatives. It is drawn from the article “Getting Up To Speed With Real

Options Analysis: Basics,” by Patrick Larkin, Baeyong Lee, and Abdoul Wane, which appears in Fall 2005 issue of the Midwestern Business and Economic Journal. The

second part contains examples of how to solve real options using binomial spreadsheets and is drawn from the article “Getting Up To Speed With Real Options Analysis: Examples,” by Patrick Larkin, Baeyong Lee, and Abdoul Wane, which appears in Spring 2006 issue of the Midwestern Business and Economic Journal. The third part is drawn

from the article “Hollywood Tycoon: A Real Options Simulation Game” by Patrick Larkin, Baeyong Lee, Abdoul Wane, and Thomas G.E. Williams, which appears in the Summer, 2006 issue of the Journal of Financial Education. It contains the instructions

for using the game in class. Our reason for posting the document in this form is that some instructors may wish to use it as a self-contained unit on real options. This is a teaching document, not an academic paper, but references are included. All of the spreadsheets that are referred to here are available as free downloads at

http://faculty.uncfsu.edu/plarkin/. To get teaching notes for the simulation game, e-mail

Pat Larkin at plarkin@uncfsu.edu. Only instructors are eligible to receive the teaching notes.

INTRODUCTION TO REAL OPTIONS PAGE 3

REAL OPTIONS VALUATION EXAMPLES PAGE 12

DIRECTIONS FOR SIMULATION GAME PAGE 19

TABLE 1.1 PAGE 22

TABLE 2.1 PAGE 23

TABLE 3.1 PAGE 24

TABEL 3.2 PAGE 25

TABLE 3.3 PAGE 26

TABLE 3.4 PAGE 27

EXHIBIT 3.1 PAGE 28

REFERENCES PAGE 29

APPENDIX A PAGE 32

APPENDIX B PAGE 33

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1. AN INTRODUCTION TO REAL OPTIONS

1.1 Calls and Puts

Options are a type of derivative security. They are classified as derivatives because their value can be derived from the value of some other asset, such as a share of stock. The other major types of derivatives are futures and swaps. Call options give investors the right, but not the obligation, to purchase a given asset at some specified price (called the exercise price, or the strike price) over a specified time frame. For example, at the time of this writing General Electric Stock is trading at \$31.70 per share. It is possible to buy a call option on the Chicago Board Option Exchange (CBOE) that conveys the right to buy the stock for \$35.00 at any time before the close of trading on th, 2004 (the time of expiration) for a price of \$.50. The \$.50 Friday, December 17

required to buy the option is called the option premium. Since it would not be profitable to exercise the option now (who wants to buy a stock for \$35.00 when it can be purchased in the market for \$31.70?) it is said to be “out of the money.” If the stock price goes above \$35.00 between now and expiration, then the call will be “in the money.” Since this call option can be exercised at any time before expiration, it is referred to as an American call. In contrast, European options can only be exercised on the date of expiration. In some cases, it is advantageous to exercise the option early. This means that American options are always at least as valuable as European options that are otherwise identical. However, it is not the case that options should always be exercised if they are in the money. Since the call option holder’s downside exposure to the stock price is limited, it is usually better to wait and see if the stock price goes even higher. If the stock price of stGE is at, say, \$37.00 on November 1 and the call holders want to lock in a profit, then

they can usually sell the option for a higher price than the \$2 that they could earn through exercising the option. The exception to this, for call options, occurs when the stock is due to pay a large dividend before expiration. This will cause a predictable fall in the stock price at some point in the future that may make current exercise more attractive than holding the option to expiration and hoping for a large enough increase in the stock price to offset the planned dividend. Besides the current price and expected future dividends on the underlying asset (the GE stock in this example) and the strike price of the option, the other factors that affect option prices are the time to expiration, the expected volatility (standard deviation) of the stock price, and the risk-free rate of interest. The greater the time to expiration and the volatility per unit of time, the greater is the chance that the stock price will go “deep into the money,” and thus the greater the option price. While greater time to expiration and volatility also entail the risk of the stock going deep out of the money, option investors are not as concerned with this possibility. In our example, it doesn’t make any difference to the call investors if GE’s stock price ends up at \$34.99 or \$0 at expiration; in either case the call will expire worthless. The risk-free interest rate also has a positive affect on call prices. Calls allow investors to control part or all of the upside of the underlying asset without tying up the funds required to purchase the asset. Call investors must tie up only the funds necessary to purchase the call, in our example \$.50. Buying a call and investing in risk-free bonds can be thought of as an alternative to buying the stock, and this alternative becomes more attractive at higher interest rates.

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Put options give investors the right, but not the obligation, to sell a given underlying asset at a specified strike price over a specified time frame. As with call options, put options can be either American or European, but most traded options on financial assets, whether calls or puts, are American. As of this writing, it is possible to purchase a put option on GE stock with a strike price of \$35.00 that expires at the close th, 2004, for \$3.90. This put has a much higher price than the of trading on December 17

corresponding call discussed above because it is already well in the money. If the option investors exercised now, they would have a profit of \$3.30. The \$3.30 is sometimes referred to as the “intrinsic value” of the option, while the additional \$.60 is referred to as the “time value.” The put has time value because there is a chance that it may go even

deeper into the money before expiration. As with the call, the greater the volatility of the stock and the time to expiration, the higher the time value of the put. However, in contrast to the call, there is an upper limit on the potential value of the put. If the stock price goes to \$0.00, then the put will have reached its maximum value of \$35.00. It is sometimes optimal to exercise a put option early when the price of the underlying asset is very low (it doesn’t necessarily have to fall to zero) because the gain from investing the funds realized from exercising the put and selling the asset is greater than the potential gain from the price of the underlying asset falling even further. For this reason, the risk-free rate of interest has a negative correlation with the value of the put; higher rates increase the opportunity cost of waiting to realize the proceeds from exercising the put and selling the underlying asset. Of course, since they decrease the value of the underlying asset, expected dividends increase the value of put options.

Options are important in all areas of finance. In addition to listed stocks, call and put options are traded on stock indices, currencies, interest rate futures, and a wide range of other financial assets. Corporate securities such as convertible bonds and convertible preferred stock contain embedded options that are valued using techniques similar to those used to value options on listed stocks. Employee stock options and warrants are call options that are issued by companies on their own stock, often as “sweeteners” to lure capital or skilled labor at more attractive current rates. These securities differ from exchange-listed calls in that their exercise creates more shares of the underlying stock, which results in dilution of the claims of existing shareholders. Equity in a levered firm (a firm that has debt outstanding) can itself be thought of as a call option on the value of the firm, where the strike price is the upcoming debt payment and the underlying asset is the present value of the firm’s future cash flows. Due to limited liability, if the expected present value of the future cash flows is less than the debt service at expiration (the date that the debt payment is due) then the stockholders may simply choose to “walk away”

and leave the firm to the creditors.

1.2 Real Options

Myers (1984) first used the term “real options” to describe corporate investment opportunities that resemble options. He proposed that the value of a firm could be divided into the value of its assets in place and the value of these “future growth options.” Growth options are also frequently referred to as expansion options. For example, the relatively high stock market valuation enjoyed by discount airlines such as Southwest in

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recent years can only be partially attributed to the present value of profits expected from existing routes. The option to invest in additional planes and other assets and expand into new routes accounts for the remainder of that value. Discounted cash flow techniques such as expected net present value (NPV) work well for valuing assets in place, but not so well for valuing growth options. The standard presentation of NPV suggests that investment opportunities must be framed as “now or never”, decisions that require an “all

or nothing” commitment. In reality, management often has considerable flexibility on when to enter or exit a project, and on the scale of the initial and subsequent commitments to make to the project. The value of this flexibility is best captured by “real

options analysis,” the application of option pricing techniques to capital budgeting. Traditional NPV systematically undervalues most investments. Real options analysis allows us to arrive at more accurate valuations of managerial flexibility, or strategic value, that facilitate better investment decisions than we would arrive at using standard NPV analysis. Though real options analysis represents an improvement over standard NPV, it is not productive to think of real options analysis and NPV as competitors. As we will see, standard NPV calculations are usually included in a real options analysis. It is more productive to think of real options analysis as a means of obtaining a more accurate estimate of NPV.

The expansion options possessed by Southwest Airlines can be thought of real call options. The underlying asset in this case is not a share of stock, but the present value of the net cash inflows from opening up a new route. The strike price is the present value of the fixed costs involved in setting up the new route, which may include the purchase or leasing of new airplanes, among other outlays. This is sometimes referred to as the initial investment in capital budgeting. The volatility of the project is the standard deviation of the present value of the project’s net cash inflows. The risk-free interest rate and time to

expiration have the same interpretation with real options that they do with stock options. However, with real options setting the time to expiration is not always a straightforward exercise. It may even be tempting to state that a real option never expires, but this is seldom advisable. Competition often shortens the effective time to expiration of real options. For example, the value of Southwest’s option to expand through introducing new

routes may be diminished by the emergence of other discount carriers who are well positioned to duplicate their business model. Dividends on the real option consist of the present value of any cash outflows from the underlying project that the firm misses out on if they don’t initiate the project immediately. Since the value of the project falls after these cash flows are paid, they decrease the value of the call option on the project in the same way that they reduce the value of a call option on a share of stock. Refer to table 1.1 for an overview of the relationship between real options on capital budgeting projects and listed stock options, and a summary of the determinants of option values discussed above.

While growth options are analogous to financial call options, another important type of real option, known as an abandonment option, is modeled as an American or European put. If the underlying asset is marketable, then the firm may salvage some value from a failed project through liquidation. For example, when a long-haul trucking firm is evaluating the decision to purchase an additional truck, the value of the project is enhanced by the option to sell the truck at the market price. If the value of the future cash

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flows that can be earned from operating the truck falls below the strike price, which is the liquidation value of the truck, then the trucking firm can gain by exercising the put option. Similarly, some projects contain an option to scale down the investment without entirely abandoning it. Another particularly important type of real option that is similar to a call is the option to delay a project. Ingersoll and Ross (1992) show that when the decision to undertake a project can be delayed, it should be delayed unless the project is so far in the money that the gain from immediate exercise outweighs the gain from waiting to see if the value of the project becomes greater over time. Many projects contain more than one type of real option. It many cases, by undertaking a project the firm is giving up the option to delay and purchasing options to expand, to abandon, and to scale down. This implies that many investment opportunities should be valued as a combination of a currently available project using standard NPV and a collection of real options using option-pricing techniques. For example, a retail clothing chain may launch a new store concept in one or two “test markets.” The value of the launch consists of the NPV of the stores in the test markets plus the value of the growth (or expansion) option to extend the chain if the pilot stores succeed plus possibly options to scale down or abandon the stores in the test markets if the pilot stores are not successful. In many cases we can value the options embedded in a project separately and add them together to get the total option value. This is true as long as one option is not dominated by the others. For example, if our analysis determines that the pilot clothing stores have almost no chance of failing so completely over the time frame of the project that we would abandon them, then the abandonment option is dominated by the expansion and scale down options and thus its value should not be included in our valuation of the project. Other types of investment opportunities consist of portfolios of real options that cannot always be valued separately 1 Examples include certain types of switching options, such as the option and aggregated.

to shut down and restart a mine or the option to vary the inputs to a manufacturing process. Two other more complex types of options are rainbow options and compound options. Rainbow options are options with more than one underlying source of uncertainty. An oilfield project may involve uncertainty about both the price of oil and the quantity of oil in the ground. Compound options are essentially options on options. For example, developing a new shaving system may involve a series of irreversible investments that must follow one another in sequence. The first phase may be design, the second testing, and the third making the necessary expenditures to set up a new production line and fund the initial marketing campaign. If design is the first phase, testing the second and production planning and marketing the third, then investing in testing is equivalent to purchasing a call with an exercise price equal to the cost of the initial production and marketing costs, and investing in design is equivalent to purchasing a call on that call.

Of course all of the options described above will not always be present in a meaningful sense in every investment opportunity. Only firms with the potential to exploit an identifiable competitive advantage have real expansion options. For a successful firm with a proven business model like Southwest, the options to expand is likely to be in the money or near the money and to have substantial value. In contrast, for

1 Trigeorgis (1993) provides a more systematic analysis of the affects of the presence of multiple options on the value of an investment opportunity.

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the major U.S. carriers, options to open up new routes are likely to be so far out of the money that their value is likely to be close to zero. For real options analysis to be meaningful, it is also necessary that some degree of uncertainty, flexibility, and irreversibility exist in the underlying project. If there is no uncertainty that can potentially be resolved through time, then we might as well simply apply standard NPV to the project and make an immediate decision to accept or reject. Flexibility is important so that managers can respond to the resolution of uncertainty. If a development firm purchases a tract of land in an urban area, there may be option value in waiting to see if rents on local apartment houses increase before building. However, this value would not be present if the land was purchased from the city with the stipulation that it be developed immediately. The requirement that the investment in the underlying asset be at least partially irreversible may seem to conflict with our assertion that real option analysis is best suited for valuing flexibility. In standard NPV analysis, it is generally assumed that the investment is completely irreversible, while real options analysis allows for the possibility of investment and disinvestment in stages. If the investment is completely reversible, management possesses almost complete flexibility in deciding whether to own the asset or not at any given point in time. This tends to make the calculation of option value trivial. A completely reversible investment carries very little risk, and thus is unlikely to be an investment that allows management to use their expertise to create shareholder value. In practice, most major investments in fixed assets or intangible assets are at least partially irreversible. While there may be an option to abandon, the present value of the strike price or liquidation value is usually less than the present value of the investment required to enter the project. An example of an investment that is not irreversible is an investment in building up working capital.

1.3 Valuing Real Options

We recommend a four step process for finding the value of an investment opportunity using real options analysis. The first step is to frame, or define, the opportunity and identify the embedded options and their parameters. This step can be quite complex if we are dealing with investments that resemble “exotic” options, such as

compound options or rainbow options. Second, we must find the value of the parameters of the options. As we will see, this step will usually also involve finding the NPV of the underlying project. Third, we use an option pricing model to value the options and then determine the total NPV of the project. The fourth step is to evaluate our estimated value using qualitative tools and sensitivity analysis. In part two we provide examples that illustrate the complete valuation process. In the remainder of this section we concentrate on some of the more challenging aspects of the process, including estimating the value of the underlying asset and its volatility, and choosing the best option valuation model.

When attempting to value an asset, it is desirable to base our valuation on market prices. So, if the underlying asset is an oil field project, we may simply multiply the number of barrels of oil in the ground at the project site by the futures market price of a barrel of oil and subtract variable extraction costs per barrel to get the value of the underlying asset. If fixed and variable costs and the quantity of oil are known with certainty, then we may use the standard deviation of oil prices for the volatility. This

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should give us reasonable estimates of project value and volatility because oil is traded in highly liquid and well organized markets. If the value of the project’s net cash inflows does not depend on the value of a traded asset such as oil, then we should attempt to identify a market comparable with equivalent risk and expected future cash flows that are proportional to the expected future cash flows from our project. For example, a Las Vegas casino chain that is contemplating investing in a riverboat casino may determine that 25% of a publicly traded firm that operates four similar riverboat casinos is the project’s comparable. The Las Vegas firm would then estimate the value of the future cash inflows from the riverboat project as 25% of the market capitalization of the riverboat firm, and the volatility of the underlying as the unlevered volatility of the return on the riverboat firm’s stock. Unlevered volatility is calculated as the volatility of equity multiplied by the ratio of equity to total capital. While it is frequently not possible to reliably identify a market comparable, this method can give us valuable information at times. For example, see Nichols (1994) for an interview with Merck CFO Judy Lewent in which she describes how the firm used the volatility of a portfolio of Biotech stocks to estimate the volatility of drug development projects underway at Merck. When it is not possible to explicitly identify a comparable, the usual practice, which you are probably familiar with, is to estimate the cash flows associated with the project and to discount them back to the present using a risk-adjusted discount rate that we believe reflects the return that our investors would require on the project if it were an actively traded asset. Managers approach the valuation this way because they realize that investors compare the value of cash flows from real projects with similar opportunities available in the financial markets. If we accept the validity of this approach, which is of course simply standard NPV analysis, then we have satisfied all of the theoretical conditions necessary to value real options on underlying projects whose values are not derived from traded assets or 2through explicitly identifying comparables. In summary then, we should base the value

and volatility of our underlying asset on market values if possible, but shouldn’t be afraid to estimate them ourselves if necessary.

The issue of estimating volatility is so important in real options analysis that it warrants further discussion. First, we must keep in mind that because options allow us to control the potential upside of an asset with limited exposure to the downside, higher volatility increases option prices. This is the opposite of the case with other assets. Other factors equal, increases in volatility tend to increase systematic risk and thus reduce asset prices. When the underlying asset is traded, or if a comparable can be reliably identified, it is generally not difficult to find published estimates of its volatility or to calculate it ourselves based on published price histories. In the more common case where we have to estimate the volatility of the underlying asset ourselves, things become more complicated. If we have undertaken similar projects many times in the past, then we may be able to use the standard deviation of the outcomes of these projects as our volatility estimate. Some authorities, such as Copeland and Antikarov (2001), recommend running a full simulation of our NPV spreadsheet model using state of the art commercial software, and calculating the standard deviation of the values of the underlying asset based on the

2Many authors, including Mason and Merton (1985), and Copeland and Antikarov (2001), argue that the assumptions necessary to value a project using standard NPV analysis are sufficient to value options on the project using option-pricing techniques.

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simulation output. There is no doubt that in many cases much can be learned about a project from a well-run simulation. However, Michelson and Weaver (2003) point out some possible problems with simulation. First, if the distributions of the variables in the model and the correlations between them are unclear, simulation is difficult to implement with any degree of confidence. Second, the proper implementation of simulation requires a range of skills in the analyst that can be costly to acquire. Michelson and Weaver (2003) propose an alternative method to calculate volatility that is easy to implement in Excel and is based on standard NPV calculations. Another possible alternative to a full scale simulation is a limited simulation using only one or two variables, such as market size and market share, performed with the functions already available in Excel or with an add-in program such as Simtools. See Appendix A for a partial list of resources available for real options analysis. The list includes several free Excel-based valuation tools available for download on the internet. After we have arrived at an estimate of volatility, it can be useful to compare it to some benchmark in the financial markets, even if we were not able to explicitly identify a comparable for the project. For example, if a relatively small semiconductor firm estimates that the volatility of a new product launch is less than the published volatility of Intel common stock after adjusting for leverage, then they may want to revisit their estimation process. We should also perform a sensitivity analysis to determine how much our estimate of the value of the option would be off if our estimate of volatility is incorrect. If the value of the option is extremely sensitive to small changes in volatility and we are not confident in our volatility estimate, then of course we cannot be confident in our valuation. Finally, we should realize that most option pricing models assume that volatility is constant throughout the life of the option. While this may not be a completely realistic assumption, it is also often a difficult one to improve on in practice. The best that we can often do is to actively monitor the project through time and revise our volatility estimate if circumstances dictate.

Once the value and volatility of the underlying asset and the other option parameters have been estimated, we apply an option-pricing model to determine the value of the real option. The first reliable option-pricing model was derived by Black and Scholes (1973). The Black-Scholes formula can be used to obtain the value of European call options on non-dividend paying assets. The value of the European put with identical parameters can be inferred from the call value. Merton (1973) developed an option pricing formula for dividend-paying assets and made other significant contributions to the development of option pricing theory. Merton and Scholes won the Nobel Prize in Economics for their contributions to derivative pricing in 1997. Fischer Black died in 1995. Cox, Ross, and Rubinstein (1979) built on the insights of Black and Scholes (1973) and others to develop the binomial option-pricing model. The binomial model is simpler to understand and explain than the Black-Scholes model, is more versatile, more widely used in practice, and is capable of generating the same results as the Black-Scholes model in situations where either model may be used. Arnold and Crack (2000) extended the binomial model to yield additional probabilistic information about the option that cannot be obtained directly from the Cox, Ross, and Rubinstein (1979) model. Spreadsheets for both binomial models are available at the author’s website listed in Appendix A. We will

leave the decision on whether or not to delve into the mathematics of the binomial model used in the spreadsheets to readers or their instructors. Helpful references on calculating

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option values with the binomial model are also included in Appendix A, and the spreadsheets themselves contain descriptions of the calculations that appear on the individual “sheets.” The mathematics of the Cox, Ross, and Rubinstein (1979) spreadsheet are slightly simpler to understand, but as we are more concerned here with the insights provided by the spreadsheet than the mathematics, we will use the Arnold and Crack (2000) spreadsheet in the example problems that we present in section three. It should be noted that since the binomial trees in the spreadsheets are limited to 52 steps, the spreadsheets do not give precise option values. However, the spreadsheets give option values that are precise enough for most real options applications. For example, the estimated value of the European put option in the first example presented in part 2 is 0.61% greater than the theoretically correct value obtained from the Black Scholes model. While the theoretical value of this option is easy to obtain with the Black Scholes model, this is not the case with most other real options. The binomial spreadsheets could be modified to yield precise estimates of option values by if enough steps were added. This is a less labor-intensive process if an assembly language program is used, but one of our objectives here is to make the valuation process as transparent as possible.

To value an option, it is necessary to specify the process that governs changes in the value of the underlying asset. For stock options, it is usually assumed that the value of the underlying asset moves according to a “random walk” so that small percentage changes in the value of the underlying asset are normally distributed. This is also the most common assumption made when valuing real options and is the assumption that we follow here. There are at least two possible objections to this assumption. First, it implies that the value of the underlying asset can never be negative. This is not a problem with stock options because of limited liability, but if we define the underlying asset in our real options analysis as the net cash inflows from the project, it is possible that this value could be negative. This will be the case for a project when the estimated present value of variable costs exceeds the estimated present value of revenues. The second possible objection is that there may be discrete “jumps” in the value of the underlying asset that are not well represented by a random walk process. Fortunately, the binomial model presented here can be modified to account for either jumps in asset prices or the possibility of the price of the underlying asset being negative. In addition, the unmodified version of the binomial model that we use here will often give a good estimate of option 3 As Friedman (1953) argues, it value even when its assumptions are not satisfied exactly.

isn’t necessary for the assumptions of a model to be completely realistic in order for the model to be useful. A map of New York for example is not a completely realistic representation of the city, but in most cases it will be useful in getting a pedestrian from Lincoln Center to the Empire State building. Another implication of the random walk assumption is that the volatility parameter is actually the estimated standard deviation of the natural logarithm of the value of the underlying asset. For relatively small standard deviations, this value is close to the coefficient of variation of the project (the standard deviation divided by the expected present value) but the difference can be significant as the standard deviation gets larger. The standard deviation of the natural logarithm of a variable is not equal to the natural logarithm of the variable’s standard deviation.

3 Copeland and Antikarov (2001) discuss the versatility of the binomial model with the random walk assumption.

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