Fin 575 Special Topics in FinanceClass# 4355

Andras Fekete

Real Option Analysis

Case: Gas Station Investment

2007 Spring

Prepared for Dr. James Haskins

May 11th, 2007

Table of Contents

Introduction..........................................................................................................................2About the resources.............................................................................................................3What is Real option Analysis?.............................................................................................3Real Option Case:................................................................................................................4Valuing a gas station with the options of expand or abandon.............................................4

The methodoly of the valuation process build up from the following steps...................4Computing Pro Forma Income Statement...........................................................................5Computing Free Cash Flows and Net Present Value...........................................................7Calculating Volatility...........................................................................................................8

Calculating the up and down factors.............................................................................10Construct Binominal Lattice..........................................................................................10

Identifying Real Options....................................................................................................11Conclusion.........................................................................................................................15References..........................................................................................................................16APPENDIX A....................................................................................................................17APPENDIX B....................................................................................................................19

List of Figures

Table 1 Expense and revenue estimates..............................................................................6Table 2 Gas price.................................................................................................................6Table 3 Pro Forma Financial Statement for the gas station (in thousand HUF)..................7Table 4 Cash Flow Schedule...............................................................................................8Table 5 Underlying Asset Lattice......................................................................................11Table 6 Inputs for SSLS....................................................................................................15

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Introduction

This study is about real options. Why they are important and how to use them. The main sources of the paper were two books. The first called Real Options from Tom Copeland and Vladimir Antikarov (2000). The other one was Real Option Analysis from Johnathan Mun (2006).

The structure of the paper wallows a logical flow to learn about real options in a short introduction then the demonstration of a case will follow. The case based on an a fictive investment but it includes most of the aspects of the analysis one would do to evaluate any investment or make a financial decision. This case is also useful to demonstrate the use of Super Lattice Solver and Risk Simulator software for Real Option Analysis.

About the resources

As I mentioned above, the primary resources were the two books. The first book was very useful to learn about real options and why we use it and how. It gives a good overall idea about the history of the real options, what companies used it first and why was it necessary to develop this new evaluation method. It was also useful to learn about Crystal Ball and practice many useful tools available within the software. But there was a couple of reasons to switch to the other book.

First, the software accompanying the book was not available in the mean time. The Real Option book from Copeland and Antikarov closely follows the methodology of using the accompanying Crystal Ball, Optquest, and other Decisioneering products, but not closely enough to give an easy to follow, step-by-step guide.

Second, The first book has been published seven years ago, when Real option analysis and methods were not clearly defined and not complete. Therefore the wording of the book and the calculations it contains has a little reflect on the commonly used financial analysis methods (cash flows, NPV, probabilities). The second book was published only a year ago and it is a revised version of the previous one, with a new software. This new software package contains an updated version of the previously existing tools with a user friendly engine. Previous problems and limitations were also eliminated.

Finally, it is a good idea to examine the topic from a few different sources to gain more information and see different approaches.

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What is Real option Analysis?

A real option is the right but not the obligation, to undertake some business decision, typically the option to make a capital investment. For example, the opportunity to invest in the expansion of a firm's factory is a real option. In contrast to financial options a real option is not tradeable - e.g. the factory owner cannot sell the right to extend his factory to another party, only he can make this decision. The terminology "real option" is relatively new, whereas business operators have been making capital investment decisions for centuries. However the description of such opportunities as real options has occurred at the same time as thinking about such decisions in new, more analytically-based, ways. As such the terminology "real option" is closely tied to these new methods. The term "real option" was coined by Professor Stewart Myers at the MIT Sloan School of Management. (Wikipedia “Real Option Analysis”)

This kind of option is not a derivative instrument, but an actual tangible option (in the sense of "choice") that a business may gain by undertaking certain endeavors. For example, by investing in a project or property, a company may have the real option of expanding, contracting, or abandoning other projects in the future. Other examples of real options may be opportunities for research and development, mergers and acquisitions, licensing or franchising.

These are called "real options" because they pertain to physical or tangible assets, such as equipment, rather than financial instruments. Taking into account real options can greatly affect the valuation of potential investments. However, valuation methods, such as Net present value (NPV), do not include the benefits that real options might provide.

Generally, the most widely used methods are: Closed form solutions, partial differential equations and the binomial lattices. In business strategy, Real Options have been advanced by the construction of option space, where volatility is compared with value-to-cost.

I our case we use the binominal lattice apporach bacause this method can be used widely, easy to understand, and return the same result than the other two methods. A detailed comparison with explonation can be found in the book of Johnathan Mun: Real Option Analysis.

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Real Option Case:

Valuing a gas station with the options of expand or abandon

The idea of valuing a gas station came from the research I recently made on the world’s largest campany, Exxon Mobil. I have chosen the operation of a gas station for several reasons.

Determining the value of a project with Real Option Analysis (ROA) is not new for Exxon Mobil. Exxon and Mobil were one of the first companies to use ROA in the 1990s for valuing development of a natural gas field and oil exploration and production.

The oil industry is the most profitable in these days. Demand for energy consistently rising. Gas stations are one of the possible last links to deliver the gas to the end users. In my experince a well run station can yield a good profit. There is 2 gas stations in my home town and both percieved to be quite profitable. I was curious how profitable they really are and if there is market for a third gas station and what the value is for such operation. Also I am familiar with my home town and able to make fairly accurate estimates about demand, prices, and costs.

The methodoly of the valuation process build up from the following steps:

1. Compute base case present value without flexibility using DCF valuation model:

- Use the expected values of price, quantify, cost investments, terminal value, and WACC to compute free cash flow

- Compute NPV of FCFs- Obtain expected PV evolution over time. (i.e. compute the PV of all future FCF

for each point in time).

2. Calculate the standard deviation of the expected cash flows:- use the after tax operating income to make sure the model is robust- calculate the relative return of the NOPAT- take the natural log of the returns- calculate volatility as the geometric average of the natural logs

3. Construct a basic Lattice:- use Lattice maker to construct the lattice with the following inputs:

S: the present value of the underlying assetX: the present value of implementation cost of the real optionσ: volatility in percentageT: time to expiration in yearsrf: risk free rate of returnb: continuous dividend outflows in percentageOption style: American

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- Lattice Maker will calculate:u: up factor (the multiplier in case of favorable event)d: down factor (the multiplier in case of unfavorable event)p: risk natural probability measure (used for intermediate calculations)

Using these factors Lattice Maker will then build a basic event tree for the desired period.

4. Identify and incorporate managerial flexibilities creating a decision tree:- Identify possible real options (what kind: American, European, compound,

Bermuda, custom, and the effect of the option on the underlying asset)- If the options used are similar to abandonment, expansion or contraction options,

make a new event tree with the options incorporated.- If the options can not be entered into Lattice Maker, go to step 5.

5. Calculate real option value (ROA)- Use Single Super Lattice Solver (SSLS) to get the value of the options.- Use the same inputs as you used for the Lattice Maker- Chose custom options and add the equations for the options- Read the total value of the options

Computing Pro Forma Income Statement

A Pro Forma Income Statement is necessary to compute cash flows (CF), free cash flows (FCF), present values (PV), and net present value (NPV).

To calculate expenses and revenues we use estimates. The input data can be found in Table 1 below.

Table 1 Expense and revenue estimatesSales volume annually Fixed costs(000)

personal weekly consumption(l) 10780,000 labor

3,600

# of people uses the gas station 1,500 utilities 1,200

daily commercial use(l) 80 29,200 maintenance 2,000

TOTAL 809,200 total 6,800

The data above is based on the population of the town (13,000), bypassing traffic, commercialization of the area and the habits of the people (weekly consumption). Fixed cost based on current salaries and other estimated expenses.

After the sales volume we calculate the expected price of the gas. Since there is three different types of fuel selling at the station we calculate an average selling price

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per liter to simplify further calculations (assuming that the sales mix remains constant over the next 5 years (Table 2).

Table 2 Gas priceGas Type 95 98 Dieselprice 265 275 249ratio of sales 65% 15% 20%average P/l 263.3

2008 2009 2010 2011 2012Gas Price 303 333 360 381 400Units sold 809,200 890,120 961,330 1,019,009 1,059,770change in price 15% 10% 8% 6% 5%sales growth 10% 8% 6% 4%

Current prices were obtained from http://benzin.totalcar.hu/ but since the will lunch in 2008, the estimated price and future changes are adjusted from Hungarian energy price forecasts.

Having the sales volume, price, fixed cost, and the dynamics of those we can create the Pro Forma Income statement in a way that it will be useful for financial analysis.The Statement can be found in Table 3.

Table 3 Pro Forma Financial Statement for the gas station (in thousand HUF)2008 2009 2010 2011 2012

Revenue 245,022 296,476 345,810 388,552 424,299Variable cost 220,520 266,829 311,229 349,697 381,869Fixed cost 6,800 7,480 8,228 9,051 9,956EBITDA 17,702 22,168 26,353 29,804 32,474Depreciation 8,000 8,000 8,000 8,000 8,000EBIT 9,702 14,168 18,353 21,804 24,474taxes 4,851 7,084 9,176 10,902 12,237NOPAT 4,851 7,084 9,176 10,902 12,237

Assumptions:- The COG remains 90% of the price over the 5 year period- Fixed cost will increase 10% each year- The depreciation is 20% of the initial investments each year

Depreciate the investment during the five year period with straight line method.- Tax is 50%

Variable cost is basically the cost directly associated with the production and transportation of the gas.

Earnings before interests and taxes, depreciation and amortization (EBITDA) are the revenues less fixed and variable costs.

Net operating income after tax (NOPAT) is the income available to investors.

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Computing Free Cash Flows and Net Present Value

A measure of financial performance calculated as operating cash flow minus capital expenditures. In other words, free cash flow (FCF) represents the cash that a company is able to generate after laying out the money required to maintain or expand its asset base. Free cash flow is important because it allows a company to pursue opportunities that enhance shareholder value. Without cash, it's tough to develop new products, make acquisitions, pay dividends and reduce debt.

 It can also be calculated by taking operating cash flow and subtracting capital

expenditures. (Investopedia “Free Cash Flows”)

NPV is the sum of all future cash flows discounted with the weighted average cost of capital (WACC). This is the current value of the investment without considering any options.

See the schedule of cash flows in Table 4.

Table 4 Cash Flow Scheduleperiod 0 1 2 3 4 5Δcapital Spending (40,000)NOPAT 4,851 7,084 9,176 10,902 12,237 DEPRECIATION 8,000 8,000 8,000 8,000 8,000 FCF assuming cont ops (40,000) 12,851 15,084 17,176 18,902 154,710 NPV 111,111 PV each period (40,000) 11,779 12,672 13,227 13,342 100,091

The WACC is 9.10% (obtained from http://www.247wallst.com/2006/12/analyzing_exxon.html)Period “0” represents the time when the investment made (for the gas station it should be January 1st , 2008).The terminal value of HUF 134,473,000 is added the last year cash flows. It is the last year NOPAT discounted with 9.10% as perpetuity.

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Calculating Volatility

If we calculate the volatility of a period which is less then a year, the periodic volatility estimate used in a real option or financial option analysis has to be an annualized volatility.

Notice that the number of returns in Table 5 is one less than the total number of periods. That is because when calculating the relative return we need to have prior cash flows so the relative return can be calculated from year two. The approach is valid and correct when estimating the volatilities of liquid and highly traded assets-historical stock prices, historical prices of oil and electricity-and less valid for computing volatilities in a real options word, where the underlying asset generates cash flows. This is because to obtain valid results, many data points are required, and in modeling real options, the cash flows generated using a DCF model may only be for 5 to 10 periods. In contrast, a large number of historical stock prices or oil prices are available for analysis. With smaller data sets, this approach typically overestimates the volatility.

The DCF cash flows may very well take on negative values, returning an error in our computation (i.e., log of negative value does not exist). However we can take certain approaches to avoid this error. The first is to move up our DCF model, from FCFs to net income, to operating income (EBITDA), and even all the way up to revenues and prices, where all the values are positive. If doing it this way, then care must be taken such that all other options and projects are modeled this way for comparability’s sake. Also this approach is justified in situations where the volatility, risk, and uncertainty stem from a certain variable above the line is used. For instance the only critical success factor for a gas station is the price of gas and the sales volume (quantity), where both are multiplied to obtain revenues. In addition if all other items in the DCF are proportional ratios (e.g. operating expenses are 25 percent of revenues or EBITDA values are 10 percent of revenues and so forth), then we are only interested in the volatility of revenues. In fact, if the proportions remain constant the volatilities computed are identical. Taking our example a step further, computing the volatility of revenues, assuming no other market risks exist below this revenue line in the DCF, is justified because the firm may have global operations with different tax conditions and financial leverages. For these reasons we will use NOPAT to estimate volatility. NOPAT is a good measure of the returns and safe enough to eliminate the problems mentioned above.

Step 1: Collect the relevant data and determine the periodicity and time frame. Our data set is the projected NOPAT in years 2008 – 2012.

Step 2: Compute relative returns. Relative returns are used in geometric averages while absolute returns are used in arithmetic averages. The arithmetic average tends to over inflate the average when fluctuations occur. Fluctuations do occur in the stock market or for many real option project, otherwise the volatility is very low and there is no option value, and hence, no point in doing an option analysis. For these reasons the geometric average is a better way to compute the return. The computation is seen below (as part of the geometric average calculation relative returns are computed).

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Step 3: Compute natural logarithm of the relative returns. The natural log is used for two reasons. The first is to be comparable to the exponential Brownian Motion stochastic process.

To compute the volatility (σ) used in an equivalent computation (regardless of weather it is used in simulation, lattices, or closed-form models because these three approaches require the Brownian Motion as a fundamental assumption), a natural log is used. In the equation the exponential of natural log cancels each other out. The second reason is that in computing the geometric average, relative returns were used than multiplied and taken to the root of the number of periods. By taking a natural log of a root (n), we reduce the root (n) in the geometric average equation. This is why natural logs are used in this step.

Step 4: Compute the sample standard deviation to obtain the periodic volatility. A sample standard deviation is used instead of a population standard deviation because your dataset might be small. For larger datasets, the sample standard deviation converges to the population standard deviation, so it is always safer to use the sample standard deviation. The sample stander deviation is the average of the deviation of each point of a dataset from its mean, adjusted for a degree of freedom for small datasets, where a higher standard deviation implies a wider distributional width and, thus, carries a higher risk. The variation of each point around the mean is squared to capture its absolute distances, and the entire result is taken to the square root to bring the value back to its original unit. Finally, the denominator (n-1) adjusts for a degree of freedom in small sample sizes.

Volatility = sqrt (1/(1-n)*∑(Xi-Xaverage)^2)

Step 5: Since we used yearly data our volatility is annual. This is the volatility of the expected revenues for the gas station.

Calculating the up and down factors

Inputs:

S: the present value of the underlying assetX: the present value of implementation cost of the real optionσ: volatility in percentageT: time to expiration in yearsrf: risk free rate of returnb: continuous dividend outflows in percentage

Using Lattice Maker we don’t have to worry about the calculation because it will automatically calculate the values. We just enter the inputs mentioned above and run the simulation. But I describe briefly the mechanism of the software and the way it calculates the values.

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With the above inputs we can calculate the up and down factors (u and d) as well as the risk natural probability measure (p).

The up factor is the exponential function of the cash flow returns volatility multiplied by the square root of time-steps. That is, if an option has a one-year maturity and the binominal lattice that is constructed has 10 steps, each time-steps has a stepping time of 0.1 years. The volatility measure is an annualized value; multiplying it by the square root of time-steps breaks it down into the time-step’s equivalent volatility. The down factor is simply the reciprocal of the up factor. In addition the higher the volatility measure, the higher the up and down factors. This reciprocal magnitude ensures that the lattices are recombining because the up and down steps have the same magnitude but different signs; at places along the future path these binominal bifurcations must meet.

The second required calculation is that of the risk-natural probability, defined as the ratio of the exponential function of the difference between risk-free rate and dividend, multiplied by the stepping time less the down factor, to the difference between the up and down factors. This risk natural probability value is a mathematical intermediate and by itself has no particular meaning.

Construct Binominal Lattice

Now we can create a binominal lattice of the underlying asset value, shown in Table 5.

Table 5 Underlying Asset LatticeASSET($) VOLATILITY RISKFREE DIVIDEND MATURITY(Y) STEPS 111111.00 11.45% 5.00% 0.00% 5.00 4

UP SIZE DOWN SIZE UP PROBDOWN PROB

DISC FACTOR

OPTION STYLE

1.136569928 0.879840277 0.719255377 0.280744623 0.939413063 AMERICAN

111111.00 126285.42 143532.21 163134.40 185413.6597759.93 111111.00 126285.42 143532.21

86013.13 97759.93 111111.0075677.81 86013.13

66584.39

The inputs defined earlier are in the upper rows colored orange.In the middle cells in blue, the calculated factors.

I would like to point out that similar results can be obtained by using Crystal Ball. There are many useful tools concerning distributions, confidence intervals, and correlations of the variables. The process for calculating the same outputs are the following:

Monte Carlo process for building a value based event tree

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1. Use expected free cash flows to estimate PV:- build PV spreadsheet- Discount at WACC

2. Model variable uncertainties:- Capture autocorrelation of each variable with itself (includes mean reversion).- Capture cross-sectional correlations among variables.- Decide on how confidence band changes through time.

3. Use Monte Carlo Simulation to generate distribution of PVs.- Show distribution of PVs.- Volatility to be used in lattice is based on: ln(Vt/Vo)

4. Construct PV lattice (event tree)Present Value (with cash flows reinvested follows geometric Brownian motion)

Identifying Real Options

As we mentioned real option can be any possible action that can add value to a project by exercising it. First let’s see some of the most common options.

The real-options analysis gives management the flexibility to address uncertainties as they're resolved through the following decision actions: defer, abandon, shut down and restart, expand, contract, or switch use.

The real-options methodology more closely matches the manner in which businesses operate. It allows evaluation of the company's flexibility to abandon, contract, switch, expand, or otherwise modify its actions after the situation--or "state of nature"--has revealed itself.

The resulting management decision can have a variety of outcomes on the financial impact of the project. For example, if the project is deferred, management must wait to determine if a "good" state of nature will return. The deferral option is the one that's generally exhibited and is treated as analogous to a call option.

If the climate is such that a project investment is going to be abandoned, management still obtains salvage value or opportunity cost from the asset. In a shutdown-and-restart scenario, management waits for a good state of nature to return before reprising the project. In a time-to-build situation, the optimal decision is to delay or default on the project, which presents a compound option. In contraction, management reduces operations if the state of nature is worse than expected. In a switch scenario, management considers alternative technologies, depending on input price.

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In a temporary expansion period, management puts more into the project if the state of nature is better than expected. And in a full-fledged growth cycle, management takes advantage of future interrelated opportunities through collaborative, interdepartmental efforts.

Defer Wait to determine if a "good" state of nature returns

Abandon Salvage value of the existing assets

Choose Choose among real options to execute

Mixed Combination of real options

Contract Reduce operations if state of nature is worse than expected

Switch Use alternative technologies depending on input prices

Expand Scale up if state of nature is better than expected

GrowthTake advantage of future interrelated opportunities

American and European Options: An option which can be exercised at any time between the purchase date and the expiration date. Most options in the U.S. are of this type. This is the opposite of a European-style option, which can only be exercised on the date of expiration. Since an American option provides an investor with a greater degree of flexibility than a European style option, the premium for an American style option is at least equal to or higher than the premium for a European-style option which otherwise has all the same features. (Optionshouse “American Option”)

Bermudan Option: A Bermudan option is a call or put option which can be exercised on specified days during the life of the option. It is reasonable to say that Bermudan options are a hybrid of European options, which can only be exercised on the option expiry date, and American options, which can be exercised at any time during the option life time. As a consequence, under same conditions, the value of a Bermudan option is greater than (or equal to) a European option but less than (or equal to) an American option. (Fincad “Bermudan Options”)

Custom option: is an option than can be exercised at times and in circumstances specified by the contracting parties.

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To keep it simple we use 3 of these options in our model; the option to expand, the option the contract, and the option the abandon. The Super Lattice Solver can calculate the value of these options considering all three styles: American, European, and Bermudan.

The Option to Expand: Firms sometimes invest in projects because the investments allow them either to make further investments or to enter other markets in the future. In such cases, we can view the initial projects as options allowing the firm to invest in other projects and we should therefore be willing to pay a price for such options. Put another way, a firm may accept a negative net present value on the initial project because of the possibility of high positive net present values on the future project.

The Option to Contract: This is the reverse option of the option to expand. When a firm’s assets and available capacity is much higher than the demand and there is no profitable way to sell the excess production the firm might want to downsize. Sell out some of the assets and continue to operate in an efficient, profitable level. This is possible only when the nature of the asset makes possible to continue to operate after selling out part if it. A common for of the option to contract is the downsizing of the labor.

The Option to Abandon: When investing in new projects, firms worry about the risk that the investment will not pay off and that actual cash flows will not measure up to expectations. Having the option to abandon a project that does not pay off can be valuable, especially on projects with a significant potential for losses.

The Contraction, Expansion, and Abandonment Option applies when a firm has three competing and mutually exclusive options on a single project to choose from at different times up to the time of expiration. Be aware that these options are mutually wxclusive. That is, we cannot execute any combinations of expansion, contraction or abandonment at the same time. Only one option can be executed at any time. In this case we use a single model to compute the option value. However if the options were non mutually exclusive, we would have to calculete the value of them individually in different models and add up the values for the total value of the strategy.

We calculate the value of these options with both American style and European style to see the difference, but the relevant value for our project is the American style option value. We don’t use Bermudan style option because there is no restriction on when to exercise these options.

First let’s run the Super Lattice Solver without any specified option to see how it works. Type the following inputs:

PV Underlying assets: 111,111 (NPV of the project)Implementation cost: 0 (the NPV already containes the initioal investments)Maturity (Years): 5Lattice Steps: 4

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Risk-Free Rate: 5Dividend Rate: 0Volatility: 11.45

Running the solver returns the initial value of the project: 111,111. That is because there is no option added.

Now define and add the options we selected.

Consider theses options:

Expansion: if the sales in gas station reaches above a certain limit, we can build a convinience store at the station for the cost of 10 million HUF. This store would further increase the revenues that would raise the value of the operation buy 15%.

Contraction: Suppose the station operates with 6 pumps on 3 driveways. In the case of low traffic we can sell out 2 pumps and eliminate one driveway. By doing that we decrease our revenues buy 10% (the third driveway were used very rarely only in summer peaks) and we save 3 million on maintenance and other regular expences associated with the 2 pumps and driveway.

Abandonmend: say we build the gas station in a center location and that is the only property Wal-Mart would need to build its Super Center in the city. Wal-Mart can wait 5 years but would buy the gas station -as is- any time during the 5 year period for 20 million HUF.

Now translate these options to factors that we can work with.

Open Super Lattice Solver and add the variables into the “Custom Variables” section (Table 6), defining their effect on the project. Our variables are the following:

Table 6 Inputs for SSLSCustom Variables    Variable Name Value Starting StepExpansion 1.15 0ExpansCost 10000 0Contraction 0.9 0ContractSavings 3000 0Salvage 20000 0

Starting Steps are all 0 because we can exercises these options any time from the beginning.

Then define the Terminal Node Equation and the Intermediate Node Equation. These equations are to tell the Solver how to calculate the value from the given options. We calculate the value of the project at each node as we would exercise the options and see how it would change the value in that node. Then we plug the highest value in the node.

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The equation looks like this:Max(Asset, Asset*Expansion-ExpandCost, Asset*Contraction+ContractSavings, Salvage,@@)

We type this equation into both the terminal and intermediate equation fields.Then run the Solver.

The results are the same for both the American style and European style options:119990 HUF.That reads as the value combined value of these options are 119990-

111111=8880 thousand HUF. Because the net present value of the project increased by that much.The value of the options is relatively low (compare to the value of the underlying asset) because of two reasons:

1. The volatility of the project is relatively low, which means we can determinate to outcome of the investment with high certainty, thus, there is not much extra benefit of eliminating some more of these uncertainties.

2. The options defined cover such an extreme cases that rarely occur. Therefore the probability to execute an option is quite low.

Conclusion

Real option valuation is a power tool for financial managers. Identifying ways they can add value to the operations is really useful. Knowing the value of those options… priceless. Especially when negotiating these contracts. As we see in our example adding real options increased the value of the operation by 8 percent. This increase is basically the result of eliminating some risk associated with future performance. If these options are given by nature, they greatly increase the value of the project. If they are not given, then the management should consider pursuing the right to exercises the options even if it cost some money. As long as the cost of this right to exercises or to have an option is less than the value of that option.

In my opinion Johnathan Mun’s book of Real Option Analysis is a really good guide for anybody interest in real options. The software accompanying the book is a power tool for real option analysis and one can save a lot of time by using them. Although we need to be familiar with the underlying assumptions that the software built on and possibly customize the model to fit for our case. There is no 2 case identical, and although these models cover the most frequently used calculation, sometimes we have to build our own model.

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References

Tom Copeland and Vladimir Antikarov: Real Options (2000)

Johnathan Mun: Real Option Analysis (2006)

(Wikipedia “Real Option Analysis”) (2006) accessed from http://en.wikipedia.org/wiki/Real_option

(Wikipedia “Brownian Motion”) (2006) accessed from http://en.wikipedia.org/wiki/Brownian_motion

(Wikipedia “Volatility”) (2006) accessed fromhttp://en.wikipedia.org/wiki/Volatility

(Investopedia “Free Cash Flows”) (2006) accessed from http://www.investopedia.com/terms/f/freecashflow.asp

(Optionshouse “American Option”) (2006) accessed from http://www.investorwords.com/196/American_option.html

(Fincad “Bermudan Options”) (2006) accessed from http://www.fincad.com/support/developerfunc/mathref/Bermud.htm

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APPENDIX A(tables and figures)

Table 1 Expense and revenue estimatesSales volume annually Fixed costs(000)

personal weekly consumption(l) 10780,000 labor

3,600

# of people uses the gas station 1,500 utilities 1,200

daily commercial use(l) 80 29,200 maintenance 2,000

TOTAL 809,200 total 6,800

Table 2 Gas priceGas Type 95 98 Dieselprice 265 275 249ratio of sales 65% 15% 20%average P/l 263.3

2008 2009 2010 2011 2012Gas Price 303 333 360 381 400Units sold 809,200 890,120 961,330 1,019,009 1,059,770change in price 15% 10% 8% 6% 5%sales growth 10% 8% 6% 4%

Table 3 Pro Forma Financial Statement for the gas station (in thousand HUF)2008 2009 2010 2011 2012

Revenue 245,022 296,476 345,810 388,552 424,299Variable cost 220,520 266,829 311,229 349,697 381,869Fixed cost 6,800 7,480 8,228 9,051 9,956EBITDA 17,702 22,168 26,353 29,804 32,474Depreciation 8,000 8,000 8,000 8,000 8,000EBIT 9,702 14,168 18,353 21,804 24,474taxes 4,851 7,084 9,176 10,902 12,237NOPAT 4,851 7,084 9,176 10,902 12,237

Table 4 Cash Flow Scheduleperiod 0 1 2 3 4 5Δcapital Spending (40,000)NOPAT 4,851 7,084 9,176 10,902 12,237 DEPRECIATION 8,000 8,000 8,000 8,000 8,000 FCF assuming cont (40,000) 12,851 15,084 17,176 18,902 154,710

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opsNPV 111,111 PV each period (40,000) 11,779 12,672 13,227 13,342 100,091

Table 5 Underlying Asset LatticeASSET($) VOLATILITY RISKFREE DIVIDEND MATURITY(Y) STEPS 111111.00 11.45% 5.00% 0.00% 5.00 4

UP SIZE DOWN SIZE UP PROBDOWN PROB

DISC FACTOR

OPTION STYLE

1.136569928 0.879840277 0.719255377 0.280744623 0.939413063 AMERICAN

111111.00 126285.42 143532.21 163134.40 185413.6597759.93 111111.00 126285.42 143532.21

86013.13 97759.93 111111.0075677.81 86013.13

66584.39

Table 6 Inputs for SSLSCustom Variables    Variable Name Value Starting StepExpansion 1.15 0ExpansCost 10000 0Contraction 0.9 0ContractSavings 3000 0Salvage 20000 0

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APPENDIX B(definitions)

Volatility most frequently refers to the standard deviation of the change in value of a financial instrument with a specific time horizon. It is often used to quantify the risk of the instrument over that time period. Volatility is typically expressed in annualized terms, and it may either be an absolute number ($5) or a fraction of the initial value (5%).

For a financial instrument whose price follows a Gaussian random walk, or Wiener process, the volatility increases by the square-root of time as time increases. Conceptually, this is because there is an increasing probability that the instrument's price will be farther away from the initial price as time increases.

More broadly, volatility refers to the degree of (typically short-term) unpredictable change over time of a certain variable. It may be measured via the standard deviation of a sample, as mentioned above. However, price changes actually do not follow Gaussian distributions. Better distributions used to describe them actually have "fat tails" although their variance remains finite. Therefore, other metrics may be used to describe the degree of spread of the variable. As such, volatility reflects the degree of risk faced by someone with exposure to that variable.

Historical volatility (or ex-post volatility) is the volatility of a financial instrument based on historical returns. This phrase is used particularly when it is wished to distinguish between the actual volatility of an instrument in the past, and the current (ex-ante, or forward-looking) volatility implied by the market. (Wikipedia “Volatility”)

Brownian motion (named in honor of the botanist Robert Brown) is either the random movement of particles suspended in a fluid or the mathematical model used to describe such random movements, often called a Wiener process.

The mathematical model of Brownian motion has several real-world applications. An often quoted example is stock market fluctuations. Another example is the evolution of physical characteristics in the fossil record.

Brownian motion is among the simplest continuous-time stochastic processes, and it is a limit of both simpler and more complicated stochastic processes (see random walk and Donsker's theorem). This universality is closely related to the universality of the normal distribution. In both cases, it is often mathematical convenience rather than the accuracy

of the models that motivates their use. (Wikipedia “Brownian Motion”)

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