Second Order Moment Approach to Real Options Analysis
Submitted as a Component of Required Courses for the Award of Bachelor of Engineering (Civil) Honours School of Civil Engineering University of New South Wales Author: Ariel Hersh October 2010

Supervisor: Professor David G. Carmichael

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 ORIGINALITY STATEMENT ‘I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project's design and conception or in style, presentation and linguistic expression is acknowledged.’ Signed …………………………………………….............. Date ……………………………………………..............

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1. ABSTRACT
Real options analysis can be used by investors to determine the value of potential investments that offer an owner the right but not the obligation to exercise a strategic decision at a predetermined time and price. Tools which are popular for valuing financial options, such as Black Scholes analysis, can be used to determine the value of real options. However, Black Scholes analysis has been criticized for its unintuitive approach to real options and also for its difficulty in determining the volatility of an investment. The purpose of this thesis is to investigate whether or not a second order moment approach can be used as a substitute for Black Scholes analysis. Comparison between Black Scholes analysis and the second order moment approach is done in three ways. Firstly, the manner in which each approach captures the upside value of a real option is compared mathematically including an analysis of input parameters; specifically, cash flows, rate of return, strike price, standard deviation, volatility and time until expiration. Secondly, an analysis of each method’s valuation of various examples of real option cases including sensitivity studies for each input parameter is carried out. Finally, each method’s valuation of a case, based on an actual real option, is analysed. A mathematical comparison of the two approaches illustrates that each method prices an option by calculating the present worth of the truncated mean of the investment when the investment’s value is above zero. In each of the various examples of real option cases, and the case based on an actual real option, the values of these options, using each method, are within 10% of each other. The comparison suggests that an intuitive second order moment approach can be used with confidence by investors to determine the value of real options.

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2. ACKNOWLEDGMENTS
First and foremost I would like to thank Professor Carmichael for his generosity throughout this year. Professor Carmichael was abundantly generous with both his patience and knowledge. His door was always open to my many questions and the entire experience was a pleasure due to his positive attitude. I would like to thank Dr Gareth Peters for sharing his time and mathematical expertise with me. As part of my research I contacted many people involved in strategic decisions that contain real options. To all those people that answered their phones and gave me their time I am very thankful. Very special thanks are due to the person who gave generously of his time and understanding so that I could analyse an actual industry based real option. I would like to thank all my friends that gave their time to listen to my ideas and offer suggestions. Finally, without my family and fiancée Alana this thesis would not have been possible. They have allowed me the environment and support in which I could apply myself to the best of my ability.

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LIST OF CONTENTS
1. 2. 3. 4. 5. ABSTRACT ACKNOWLEDGMENTS ACRONYMS AND NOTATION INTRODUCTION BACKGROUND AND LITERATURE REVIEW Introduction Background to Real Options Financial Options Black Scholes Analysis Financial Option Characteristics Black Scholes Analysis of Real Options Criticisms of Real Options Analysis Real Options Difficulties in the Industry Suggested Solutions to Real Options Analysis Difficulties A New Approach to Valuing Real Options Input Parameters for the Second Order Moment Approach iii iv viii 9 11 11 11 14 17 19 19 24 25 26 28 32

5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10. 5.11.

5.12. Equivalence between Second Order Moment Approach and Black Scholes A Analysis 34 5.13. 6. Summary 36 36 37 38 45 46 47 47 48 67 143 157 v 35

METHODOLOGY Introduction Mathematical Equivalence Example Real Option Values Determined by Both Approaches Case Based on Actual Industry Real Option Conclusion

6.1. 6.2. 6.3. 6.4. 6.5. 7.

COMPARISON Introduction Mathematical Equivalence Example Real Option Values Determined by Both Approaches Case Based on Actual Industry Real Option Conclusion of Comparison

7.1. 7.2. 7.3. 7.4. 7.5.

8.

DISCUSSION Introduction Summary of results Advantages of the Second Order Moment Approach Further Research Conclusion

158 158 158 161 162 162 163 164 166 170 172 174 178 180 182 186 188 190 194 196 198 200 202 204

8.1. 8.2. 8.3. 8.4. 8.5. 9.

CONCLUSION

10. REFERENCES APPENDIX A – Case 1 – Black Scholes Analysis Calculations APPENDIX C – Case 1 – Upside Value Representation APPENDIX D – Case 1 – Comparison of Approaches APPENDIX E – Case 2 – Black Scholes Analysis Calculations APPENDIX G – Case 2 – Upside Value Representation APPENDIX H – Case 2 – Comparison of Approaches APPENDIX I – Case 3 – Black Scholes Analysis Calculations APPENDIX K – Case 3 – Upside Value Representation APPENDIX L – Case 3 – Comparison of Approaches APPENDIX M – Case 4 – Black Scholes Analysis Calculations APPENDIX O – Case 4 – Upside Value Representation APPENDIX P – Case 4 – Comparison of Approaches APPENDIX Q – Case Based on Actual Industry Real Option – B Black Scholes Analysis Calculations APPENDIX R – Case Based on Actual Industry Real Option – S Second Order Moment Approach Calculations APPENDIX S – Case Based on Actual Industry Real Option – U Upside Value Representation APPENDIX T – Case Based on Actual Industry Real Option – C Comparison of Approaches

APPENDIX B – Case 1 – Second Order Moment Approach Calculations 168

APPENDIX F – Case 2 – Second Order Moment Approach Calculations 176

APPENDIX J – Case 3 – Second Order Moment Approach Calculations 184

APPENDIX N – Case 4 – Second Order Moment Approach Calculations 192

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LIST OF FIGURES
Figure 1 – “Call Option”, Wikipedia, Accessed 30/3/2010 15 Figure 2 – “Put Option”, Wikipedia, Accessed 30/3/2010 15 Figure 3 – Example Cash Flows 21 Figure 4 – Representation of Various Parameters for Black Scholes Analysis 22 Figure 5 – Normal Distribution of the Present Worth of the Investment 30 Figure 6 – Upside of the Present Worth of the Investment 31 Figure 7 – Example Cash Flows 32 Figure 8 – Representation of Parameters of the Second Order Moment Approach 33 Figure 9 – Example Cash Flows 48 Figure 10 – Representation of Parameters from Black Scholes Analysis 49 Figure 11 – Probability Distribution of Standard Normal Variable 51 Figure 12 – Second Order Moment Representation of the Cash Flows 54 Figure 13 – Probability Distribution of the Present Worth of the Investment 57 Figure 14 – Upside of the Present Worth of the Investment 57 Figure 15 – Probability Distribution of a Standard Normal Variable Shaded to the Left of / 59 Figure 16‐ Cash Flows for Real Option to Expand which is Close to the Money 69 Figure 17 – Representation of the Parameters Used by Black Scholes Analysis 71 Figure 18 ‐ Representation of Parameters Used by the Second Order Moment Approach 72 Figure 19 – Cash Flows for a Real Option to Expand, Which is Far to the Money 89 Figure 20 ‐ Representation of the Parameters Used by Black Scholes Analysis 90 Figure 21 ‐ Representation of Parameters Used by the Second Order Moment Approach 91 Figure 22– Cash Flows for a Real Option to Expand, Which has both Positive and Negative Cash Flows 105 Figure 23 ‐ Representation of the Parameters Used by Black Scholes Analysis 106 Figure 24 ‐ Representation of Parameters Used by the Second Order Moment Approach 108 Figure 25 – Cash Flows for a Real Option to Abandon 125 Figure 26 ‐ Representation of the Parameters Used by Black Scholes Analysis 126 Figure 27 ‐ Representation of Parameters Used by the Second Order Moment Approach 127 Figure 28 – Cash Flows for Case Based on Industry Real Option 146 Figure 29 ‐ Representation of the Parameters Used by Black Scholes Analysis 149 Figure 30 ‐ Representation of Parameters Used by the Second Order Moment Approach 151

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3. ACRONYMS AND NOTATION
DCF – Discounted Cash Flow NPV – Net Present Value PERT – Program Evaluation and Review Technique BSA – Black Scholes Analysis SOMA – Second Order Moment Analysis PW – Present Worth CBA – Commonwealth Bank of Australia CPI – Consumer Price Index LIBOR – London Interbank Offered Rate BBSW – Bank Bill Swap Reference Rate c – Value of a call option p – Value of a put option SD – Standard Deviation Var ‐ Variance σ – Volatility µ – Mean value

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4. INTRODUCTION
When a company is faced with the strategic decision of whether or not to invest in a project many tools are available to determine the value of the project. Investors usually use net present value analysis to make strategic investment decisions. Net present value analysis discounts future cash flows according to the time value and risk associated with each cash flow. This approach ignores the ability of the investor to make profitable decisions during the lifespan of the investment and as such calculates a lower value of the overall investment when a real option is present. A real option offers an investor the right but not the obligation to make a strategic decision at a particular time, at a particular price. This strategic decision may be to delay, expand or abandon an investment. Real options analysis uses a risk free rate of discount to capture only the upside potential of the investment. Research done to determine the value of a real option compares its value to the value of a corresponding financial option (Damodaran, 2008). Financial options can be valued using Black Scholes analysis, Binomial Lattices or Monte Carlo Simulations (Hull, 1997). One of the difficulties in comparing the value of a real option with the value of a financial option is the difficulty of determining volatility for a real option, a requirement of Black Scholes analysis. In addition to this, the engineering industry is hesitant to use real options analysis because of its unintuitive nature (Block, 2007). While solutions have been suggested to resolve the various challenges of real options analysis there is no unanimous position on how to determine the value of volatility for a real option (Borison, 2003). Carmichael and Balatbat (2009) suggest using a second order moment approach to determine the value of real options, requiring only the mean and variance of the cash flows of the investment. This intuitive method would not require the investor to determine the volatility of the asset. The aim of this thesis is to determine whether or not a second order moment approach can be used as a substitute for Black Scholes analysis in determining the value of real options. In addition to showing equivalence between the second order moment approach and Black

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Scholes analysis the thesis explores whether a second order moment approach is appropriate for real options, and discusses its advantages. In Section 5, the background and literature review introduces the concept of real options and the current tools that are used to value them. Existing challenges to the use of Black Scholes analysis to determine the value of real options are discussed, and a number of suggested solutions are presented. Finally, the second order moment approach is introduced as a possible substitute for Black Scholes analysis. The methodology, in Section 6, outlines the way in which the thesis shows equivalence between Black Scholes analysis and the second order moment approach. The methodology is split into three parts; the first is a mathematical comparison of the way in which each approach captures the upside value of real options. Secondly, the values of four basic real option cases, as calculated by each approach, are compared. For each basic case sensitivity analyses are carried out on various input parameters to ensure that equivalence exists over a large range of real options. The variance, volatility, cash flows, risk free rate of return and strike price are each changed. Finally, a case based on an actual industry real option is valued by each approach and the results are compared. The mathematical comparison of Black Scholes analysis and the second order moment approach, in Section 7, shows that each approach values real options in a similar way. The basic real option cases and the case based on an industry real option are all valued equivalently by each approach. The sensitivity analyses of the input parameters also show that each approach calculates an equivalent value for a large number of real options. Section 8 discusses the similarities and differences of the two approaches that are found through the comparison of the two approaches. Also the advantages of the second order moment approach are discussed. The conclusion, in section 9, suggests that the second order moment approach can be used as a substitute for Black Scholes analysis in determining the value of real options.

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5. BACKGROUND AND LITERATURE REVIEW
5.1. Introduction This review introduces the concept of real options and describes the existing tools for determining their values. Various challenges to the use of real options analysis are discussed and a number of suggested solutions are examined. A new method of real options analysis, a second order moment approach, is proposed as a possible substitute for the existing methods. Finally, equivalence between the second order moment approach and Black Scholes analysis is questioned and an existing measure of equivalence is discussed. 5.2. Background to Real Options Currently, strategic decisions are made using net present value analysis (Damodaran, 2008). The net present value of an asset is equivalent to its present worth .The net present value of an asset’s future cash flows are determined by discounting the future cash flows to today’s value with a discount rate, the rate of return. Cash flows are further discounted according to their likelihood of occurring. The less likely a cash flow is to occur the more it is discounted. The rate at which it is discounted is called the ‘risk adjusted rate of discount’. The risk adjusted rate of discount increases as the perceived risk increases. Investors with a large number of investment opportunities should choose the investment choice with the highest net present value. A real option is an investment in which an investor is offered the exclusive right, but not the obligation, to make a strategic decision at a predetermined time at a predetermined price. This strategic decision may be to delay, abandon or expand an investment. The investor is required to pay a premium in order to gain this right. The three main types of real options, according to Damodaran (2008), offer investors the following three different options: 1) Option to wait: This is when a company pays a premium to have the option to

delay further investment to a time when it is profitable to do so. Example: A company buys land to drill for oil and has the option to wait for the price of oil to go up before drilling.

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2)

Option to expand: This is when a company pays a premium to have the

option to expand a particular investment at a later time. Example: A company builds a building with extra strength foundations to allow for further storeys to be built at a later date. 3) Option to abandon: This is when a company pays a premium to be able to

leave an investment at a particular time. Example: A company pays a premium for an escape clause in a contract. Kodukula and Papudesu (2006) mention a number of other real options which are based on the three real options above. 1) Option to contract: This is when a company pays a premium to have the option to contract another company to take over a portion of their business. This is similar to the option to abandon. 2) Option to sell: This is when a company pays a premium to have the option to sell an investment. This is similar to the option to abandon. 3) Option to choose: This is when a company pays a premium to choose between a number of different real options cases. This real option is based on the different options that the investor can choose from, whether they are options to expand, abandon, sell, contract or delay. 4) Parallel compound option: This is when a company pays a premium to have the option on an option. This option is based on the relevant options in question, whether they are options to expand, abandon, sell, contract or delay. 5) Sequential compound option: This is when a company pays a premium to have the option whose value is dependent on a subsequent option. This option is based on the relevant options in question, whether they are options to expand, abandon, sell, contract or delay. A predetermined time exists at which the owner of the real option can decide whether or not to exercise their right. The owner will exercise their right only if it is profitable to do so. In the case of the real option to expand, should the owner decide not to expand, only the

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premium has been forfeited; whereas, when it is profitable to do so and the owner does exercise their right and expand the investment the profits are limitless. Since the owner of the real option is not obligated to exercise their right, the value of the real option only takes into account the upside potential of the investment (Howell et al., 2001). The potential downside of exercising the right is not considered because the investor will not exercise their right if it is not profitable to do so. The theory of real options was first discussed by Myers (1977) who suggested that the value of a firm’s discretionary investment opportunity, or real option, can be determined by relating it to a financial option on real assets (Reuer and Tong, 2007). By comparing a real option to a financial option the upside value of the real option can be calculated. The value of a real option, when determined by relating its value to a financial option, is higher than its value if it were determined by net present value analysis. This is because net present value analysis discounts future outcomes with a risk adjusted discount rate, ignoring the ability of the investor to make profitable decisions during the lifetime of the investment. (Amram and Kulatilaka, 1999). Since the investors’ strategic decisions are not taken into account in a net present value analysis, it is referred to as a ‘static discounted cash flow analysis’ This is in contrast to real options analysis, which does incorporate the investor’s potential discretionary decisions, and is referred to as ‘dynamic discounted cash flow analysis’. The value that the real option adds to the net present value of the investment may influence overall strategic investment decisions. As such the value of this real option should be included when determining the viability of a project. For example, in the case of the oil rig that contains the option to wait, it may be determined that the risk adjusted present worth of the oil that will be drilled is worth less than the cost of building the rig. However, when the option to wait is considered, the investment may in fact be profitable. As with financial options, the longer the time before the real option expires and the higher the uncertainty of the cash flows, the higher the value of the option. This is because both

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higher uncertainty and longer duration until expiration cause the potential profits of the option to rise but not the potential losses, as the downside of the option is not considered. 5.3. Financial Options Real options can be valued using the same mathematical approaches used to determine financial options (Damodaran, 2008). A financial option is a derivative of an underlying asset where an investor pays a premium for the right but not the obligation to purchase the underlying asset at a predetermined time and at a predetermined price (Hull, 1997). A real option is similar to a financial option because both offer an investor the right but not the obligation to make a strategic decision at a predetermined time and at a predetermined price. An option, either real or financial, that can only be exercised at a predetermined time is referred to as a European option, whereas an option that can be exercised at anytime during the life of the option is referred to as an American option. This thesis focuses only on European options because the purpose of this thesis is to show equivalence between Black Scholes analysis and the second order moment approach and Black Scholes analysis can only determine the value of European options. There are two main types of financial options, call options and put options. A call option is when an investor pays a premium for the right but not the obligation to buy an underlying asset at a predetermined time and at a predetermined price. A put option is when an investor pays a premium for the right but not the obligation to sell an underlying asset at a predetermined time and at a predetermined price.

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 Figure 1 – “Call Option”, Wikipedia, Accessed 30/3/2010

Figure 2 – “Put Option”, Wikipedia, Accessed 30/3/2010 Figures 1 and 2 illustrate an advantage of a call option over buying actual stock. For example, Commonwealth Bank of Australia (CBA) shares could be bought for $20 per share or a call option of CBA shares could be bought where a premium of $2 is paid for the right but not the obligation to buy CBA stock in 2 months time at $20. If the price of the underlying stock goes up, both investments will return profits, the option returns the same profit less $2. However if the price of the stock goes down by $10, the holder of the call option will not exercise their right and will lose only $2, much less than the holder of the actual stock. As an option protects the holder against all losses, no matter how large, and rewards the holder for gains, the more volatile the stock the more the option is worth.
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A call option is considered ‘out of the money’ if the strike price is below the current value of the stock and ‘in the money’ if the strike price is above the current value of the stock. The opposite is true for a put option. An option is considered ‘close to the money’ if it is either just ‘in the money’ or just ‘out of the money’ and considered ‘far from the money’ if it is either ‘deep in the money’ or ‘deep out of the money’. When an option is close to the money the uncertainty of the asset has a large influence on the price of the option. The three main methods of determining the value of a financial option are binomial trees, Monte Carlo simulation and Black‐Scholes analysis (Hull, 1997). In order to show that a second order moment approach can be utilised as a substitute for Black Scholes analysis a detailed explanation of Black Scholes analysis is presented in Section 5.4. Following, is a brief explanation of Binomial trees and Monte Carlo simulation as a background to calculation option values. The binomial trees approach first splits the life of the option into many small time intervals. At each time interval the value of the asset may go up or down. The probability of the asset going up or down and the amount by which it goes up and down is calculated. A binomial tree is generated representing all of the possibilities over the life of the option and the value of the option for each of the possibilities is determined. The value of the option at each possibility is calculated working back from the time of expiration until the value of the option at time zero is determined. Monte Carlo simulation simulates a large number of stock prices and calculates the option payoff for each stock price. The average of all the payoffs is calculated and the value of the option is equal to the present worth of this average.

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5.4. Black Scholes Analysis Fischer Black and Myron Scholes derived a differential equation, which could be used to determine the value of derivatives of stocks such as options (Hull, 1997). The Black‐Scholes equation is based on the following assumptions (Hull, 1997): 1. Short‐selling (selling an asset that one does not own at the time of sale) is

permitted. 2. 3. There is no cost of transactions and the stocks are perfectly divisible. The derivatives (financial instruments whose price is based on another asset)

have no dividends. 4. There is no possibility for riskless arbitrage (an opportunity for a risk free

profit at zero cost). 5. 6. 7. Stock trading is continuous. The risk free rate stays constant. The log of stock prices follows Geometric Brownian motion with constant

drift and volatility. A riskless portfolio is created consisting of a position in an option and a position in the stock. The return on the portfolio is equal to the return from a risk free investment at the risk free rate. The portfolio is riskless because the uncertainty of the option and the underlying stock are both based on the stock price movements (Hull, 1997). The formula is derived by assuming the movements of the stock price follow Geometric Brownian Motion (Hull, 1997): dS = Sdt + Sdz By considering the portfolio over a short period of time the Black Scholes differential equation is formed (Hull, 1997): The solution to this differential equation is the option pricing tool discovered by Black and Scholes for a call option (Hull, 1997):
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,

Where

√

,

√

=

√

and

√

= ф

The input parameters are: 1) 2) S: Stock Price. This is the value of the stock at time zero. X: Strike price – This is the price the owner of the call option agrees to pay for

the stock at the time the option expires. 3) : Volatility, Volatility is a measure of how spread the stock is. As a result of

this it also represents the uncertainty of the future stock price. (Ways to estimate volatility, 2006) Implied volatility is the volatility that can be found by iterated possible volatilities into the Black Scholes analysis if all other variables are known (Hull, 1997) Volatility can be calculated as follows:
√

Where: n + 1 = number of observations Si = Stock price : Length of time interval ln ∑ 4) 5)
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T: Time until the option expires. Risk free rate – this is the current risk free rate.

The value of the call option at time T can be summarised as follows: =max {0, A put option is summarised as follows: As =max {0, .

0

0

As such the pricing formula for a put option is: p S, t e
T

XN

d

SN

d e

T

with d1, d2 and N(x) defined as above.

5.5. Financial Option Characteristics It is useful to note the important characteristics of financial options in order to better understand the use of financial option analysis in determining the value of a real option: 1) 2) The higher the volatility the higher the value of the option. (Hull, 1997) The larger | | is, the larger the value of the option. That is, the further

away the money (above or below) the less value the option has. (Howell et al., 2001) 3) Financial options are determined under the assumption of risk neutrality.

(Hull, 1997) 4) 1997) 5.6. Black Scholes Analysis of Real Options As suggested by Myers (1977) the value of a real option can be determined by inserting the appropriate real values into the Black Scholes pricing formula. Each of the different types of real options can be seen to contain the characteristics of either call or put options (Kodukula and Papudesu, 2006): 1) Option to abandon: In this case, is the expected cash flows and X is the this real option has The longer the time until expiration the larger the value of the option. (Hull,

salvage value. As the option will be exercised only when the characteristics of a put option.
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2)

Option to expand: In this case,

is the expected cash flows if the option is

exercised and X is the price of exercise. As the option will be exercised only when this real option has the characteristics of a call option. 3) Option to wait: In this case, is the expected cash flows if the option is

exercised and X is the price of exercise. As the option will be exercised only when this real option has the characteristics of a call option. 4) Option to choose: This option comprises of a combination of the three main

types of real options, the options to wait, expand or abandon an investment. The owner of the option to choose either has the right to abandon, expand or wait and as such the option corresponds to the appropriate call or put financial option. The Black Scholes Pricing formula only represents the true price of a real option if the assumptions mentioned in Section 5.4 are true for the real option. Using Black Scholes analysis to determine the value of a real option also requires the investor to be able to determine all of the input parameters for the pricing formula. The cash flows represented in figure 3 based on a real option to expand are used as an example to illustrate the input parameters for Black Scholes pricing formula.

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4 2 0 0 Cost or Benefit ($M) ‐2 ‐4 ‐6 ‐8 ‐10 ‐12 Costs Benefits 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Time (Yrs)

Figure 3 – Example Set of Cash Flows Figure 4 represents the various input parameters for Black Scholes analysis, which are explained below.

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15

10

Cost or Benefit ($M)

5 Costs Benefits 0 0 ‐5 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Strike Price S(t=0) S(t=T)

‐10

‐15

Time (Yrs)

Figure 4 – Representation of Various Parameters for Black Scholes Analysis The input parameters of a real option for the Black Scholes pricing formula are as follows: S (t=0): The Value at the Underlying stock. This is calculated by a risk free discounted cash flow valuation of the future cash flows that will occur after exercise. These cash flows are black and grey in the figure 4 and the value of S (t=0) is represented by the dotted bar at time zero. T: The time until expiration, measured in years. This is the time until the owner of the real option has to decide whether or not to exercise the option. σ: Volatility. It can be determined a number of ways as discussed in Section 5.9; it is a representation of the uncertainty of the future cash flows, or the volatility of the underlying asset. r: Risk free rate. This is the return rate at which money can be saved without any risk; it is usually the bond rate or London Interbank Offered Rate (LIBOR) which is the highest rate at which an investor can save money without risk. In Australia the Bank Bill Swap Reference Rate (BBSW) is used.
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X: Strike price: This is the price of expansion and is represented by the hollow bar in figure 4. The cost of the real call option at time t, denoted c for underlying asset S is then calculated using the Black Scholes Pricing Formula: c S, t
S X

e

T

SN d e
T

T

XN d

Where d
S X

√T

,

d

T √T

= d e

σ√T

t and

N x

√

dx = ф x

The cost of the real put option at time t, denoted c for underlying asset S is then calculated using the Black Scholes Pricing Formula: p S, t e
T

XN

d

SN

d e

T

with d1, d2 and N(x) defined as above.

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5.7. Criticisms of Real Options Analysis The use of Black Scholes analysis to determine the value of a real option is questioned in the literature for a number of reasons. The following difficulties are raised in regard to using financial option pricing tools when determining the value of a real option: 1) The arbitrage principle is difficult to accept for all real options since real

assets are not freely traded. The arbitrage principle is a key assumption of financial option pricing. Damodaran (1999) suggests being cautious when using values from option pricing models when the assets cannot be freely traded. Borison (2003) and Howell et al. (2001) also claim that this is an issue that needs resolving before using real options analysis. 2) Howell et al. (2001) and Damodaran (1999) question the assumption that a

real option follows geometric Brownian motion. Their main concern is that the asset would not follow a continuous process but rather have price jumps. This would cause an underestimate for the price of the option when it is out of the money and an overestimate when it is in or close to the money. 3) The difficulty of precisely determining the volatility of a real option is

discussed by Amram and Kulatilaka (1999) and Kodukula and Papudesa (2006). Damodaran (1999) is also concerned that the volatility of the real asset will change during the long life of real options, which contradicts the assumption of real options analysis that the volatility for the life of the option can be determined and is constant. While volatility has relevant meaning for stock options it has no implicit meaning for real options. 4) To use financial pricing to determine the value of a real option the investor

must have the ability to exercise their option instantaneously. Damodaran (1999) believes that this is not the case for many real options. 5) The decision relating to a financial option cannot change the value of the

underlying asset, while the decision relating to the real option can. (Howell et al., 2001) 6) Real options analysis ignores the risk involved in delaying an engineering

project (Eschenbach, Lewis and Hartman, 2009)

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5.8. Real Options Difficulties in the Industry While Copeland and Antikarov (2003) suggest that by 2010 real options analysis will be the most prominent form of analysing capital investments, Block (2007) and Putten and MacMillian (2004) suggest otherwise. Real options are not popular among industry professionals; they have suggested that the real options overestimate the uncertainty of projects. This suggestion is based on the fact that options analysis is defined for financial options that have their own bases and assumptions (Putten and MacMillian, 2004). Block (2007) surveyed the Fortune 1000 on their use of real options analysis. Out of the 279 usable responses only 40, accounting for 14.3%, indicated that they use real options. Out of those 40 companies only 1 used Black Scholes analysis to determine the value of the real option. This represents only 0.36% of all usable responses. The reason for the small representation of Black Scholes analysis in real options analysis is due to the difficulty in determining the value of the five input variables for the Black Scholes Pricing Formula (Block, 2007). Out of the remaining 85.7%, of the responses, the main four reasons for companies not using real options were lack of support from top management, acceptance of discounted cash flow, lack of understanding of real options analysis and risk of overestimation in the value of a real option. The reasons are listed in order of popularity with ‘lack of support from top management’ being chosen by 42.7% of the companies. (Block, 2007)

25

5.9. Suggested Solutions to Real Options Analysis Difficulties The challenges to real options are discussed at length in the literature and whilst some solutions have been suggested, there is no common agreement on how to resolve the issues listed in Section 5.7 and Section 5.8. In order to resolve the issue of ‘no arbitrage’ in real options, Borison (2003) suggests using the real options analysis approach of Amram and Kulatika (1999), who suggest that a portfolio of stocks can be found that would replicate the returns of the real option. The volatility of the real option could also be taken as the volatility of the portfolio. This could be done for example by an oil company that uses a portfolio consisting of the price of oil, or an oil company on the stock exchange. This suggestion, however, does not solve the issues if no stock portfolio can be found to replicate the real option. An example of a real option without a replicating stock portfolio would be the option to expand a local library by building stronger foundations. Borison (2003) also suggests that investors with experience and understanding of an industry could estimate a subjective volatility for a real option. To resolve the issue of price jumps having an effect on the volatility of the real option Damodaran (1999) recommends using higher volatility estimates for real options out of the money and lower estimates for real options in or close to the money. Koduka and Papudesu (2006) list the following five different methods for determining the volatility of a real option: 1) Logarithmic Cash flow returns method: This method takes the natural logarithms of the relative returns of the cash flows that are used to determine the present worth of the underlying asset. The standard deviation of these natural logarithms is the volatility factor. This method is mathematically valid and uses the actual variability of the assets cash flows; however it cannot be used when negative cash flows exist. 2) Monte Carlo Simulation: This method takes the natural logarithms of many simulations to calculate a distribution of the volatility factor.

26

3) Project Proxy Approach: This method uses the historical cash flows data of an existing project to calculate the volatility factor. The project must have similar cash flows and risks. 4) Market Proxy Approach: Similar to the project proxy approach in the sense that a similar existing project is used, however, in this case the closing stock price of a publicly traded company is used. The company must have similar cash flows and risks. 5) Management Assumption Approach: The Management Assumption Approach uses management estimates of optimistic (S cash flows over a specific period (t). The volatility of a real investment can then be calculated using one of the following three equations: ln S S 2√t S S 2√t ln σ S S 4√t ), pessimistic (S ) and average (S )

σ

ln σ

The large quantity and diversity of solutions could be viewed by industry as making real options analysis a high risk tool when defining an assets value. Putten and MacMillan (2004) offer a way in which investors may feel more comfortable using real options. They suggest that investors should integrate real options analysis and net present value analysis when determining whether or not to invest in a project. They put forward different times in which projects should be analysed using either one or both of these tools.

27

5.10. A New Approach to Valuing Real Options Carmichael and Balatbat (2009) criticise Black Scholes analysis for the high level of sophistication it requires as well as the difficulty in defining volatility for valuing a real option. In order to solve these issues as well as a number of the issues discussed above, Carmichael and Balatbat (2009) suggest using a second order moment approach to determining the value of real options. The second order moment approach to valuing a real option is similar to the net present value approach as it also takes the present worth of each cash flow. However, instead of taking the deterministic present worth by discounting each cash flow for risk, this approach takes the probabilistic present worth of the cash flows, including the uncertainty of each cash flow, reflected through incorporating the variance of the cash flows. The sum of the present worth of each cash flow is considered the mean of the investment and the sum of the variances of each cash flow is considered the variance of the investment. Assuming the probabilistic distribution of the investment is normally distributed, the value of the option is the upside value of the present worth of the investment, which equals the truncated mean of the positive side of the present worth distribution. By modifying the discounted cash flow analysis to include the probabilistic present worth of each cash flow the second order moment approach can calculate the present worth of each cash flow with a risk free rate of return. This is because the uncertainty of the cash flows is accounted for in the variance of each cash flow. This adapted approach allows the real option, which is the upside potential of the investment, to be valued. This approach can be simplified to the following four steps: 1) The estimated value of each cash flow and variance can be determined by

utilising a suggestion by Carmichael and Balatbat (2008) to use a PERT (Program Evaluation and Review Technique) style of thinking to determine the mean and variance. This technique requires the investor or an expert in the industry to state the pessimistic (TP), optimistic (TO) and most likely (TL) value of each cash flow. The expected value or mean (TE) then equals (TP + 4TL + TO)/6 and

28

The standard deviation (SD) = (TP‐TO)/6 The variance (Var) = SD^2 This formula is based on the assumption that the range of Optimistic to Pessimistic is approximately equal to 6 standard deviations, which is approximately 99% of the probability density function. (Carmichael, PERT and PNET, 2010) The expected values for either the mean or the variance can be substituted by subjective values estimated by the investor. 2) The mean and variance of the present worth of the investment is determined

as follows: Where Y is defined as a number of cash flow components made up of k = 1,2..,m and Z is defined as Z E[Z ] = ∑ Var [Z ] = ∑ Y EY Var Y 2∑ ∑ Cov Y , Y Y …Y :

The mean and variance of the present worth of the investment is calculated as follows: E[PW] = ∑ Var [PW] = ∑
EZ

Z

V

2∑

∑

C

Z ,Z

Var PW 3) The value of the present worth is said to have a normal distribution with a .

mean of E[PW] and a variance of Var [PW] and a standard deviation of SD Under the assumption that each of the cash flows are mutually independent random

variables, with finite means and therefore all moments are finite, then the Lindeberg Central Limit Theorem states that the sum of these random variables for cash flows at any time, t, will approximate towards a normal distribution when the number of random variables is large.

29

If W equals the value of the present worth over time, it can be said that: W ~ N E PW , Var PW The probability function of which equals f(x) can be stated:

Where µ = E[PW] and σ = SD[PW] = Var PW This distribution is represented in figure 5:
0.09 0.08 Frequency of x's Occurence 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 ‐20 ‐10 0 x 10 20 30

Figure 5 – Normal Distribution of the Present Worth of the Investment 4) The second order moment approach values the option price as the mean of the upside of the present worth. This is termed the upside value. The upside of the present worth is the portion of the distribution that is above zero which is represented in figure 6:

30

0.09 0.08 Frequency of x's Occurence 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 ‐20 ‐10 0 X 10 20 30

Figure 6 – Upside of the Present Worth of the Investment As such: c = Upside Value [PW] = E [PW upside] c = E [ PW | PW > 0 ] This is the mean of a truncated normal distribution which can be expressed in the following way: µ E [ PW | PW > 0 ] = µ

ф

µ

The right hand side of this equation can be solved using standard normal look up tables.

31

5.11.

Input Parameters for the Second Order Moment Approach

The second order moment approach has four input parameters, the mean of each of the cash flows, the variance of each of the cash flows, the risk free rate of return and the time until expiration. The cash flows in figure 7, which are based on a real option to expand, are used as an example to illustrate the input parameters for second order moment approach.
4 2 0 0 Cost or Benefit ($M) ‐2 ‐4 ‐6 ‐8 ‐10 ‐12 Costs Benefits 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Time (Yrs)

Figure 7 – Example Set of Cash Flows Figure 8 represents the input parameters for the Second Order Moment Approach.

32

6 4 2 Cost or Benefit ($M) Variance 0 0 ‐2 ‐4 ‐6 ‐8 ‐10 ‐12 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Costs Benefits S D ∑Var[PW] ∑E[PW]

Time (Yrs)

Figure 8 – Representation of Parameters of the Second Order Moment Approach E: Mean value of each cash flow. This is the estimated mean of each cash flow. The mean of each cash flow is calculated without adjusting for risk. The uncertainty of the investment is incorporated in the second order moment of the cash flows, which is defined as the variance. The strike price in the option to expand is considered a negative cash flow in the year that the option expires. So that: Value of the Asset = Cash Flows – Strike Price The strike price in the option to abandon, or the option to sell, could be considered a positive cash flow in the year that the option expires and all other cash flows are considered opposite to what they were; that is, costs become benefits and benefits become costs. However, a simpler way is to consider the strike price as a negative cash flow and then take the negative of the sum of all cash flows such that: Value of the Asset = Strike Price – Cash Flows = – (Cash Flows – Strike Price)
33

In this way the option will be exercised when the strike price is larger than the existing cash flows, which is the case in real options to abandon or sell. SD or Var: Variance of Each cash flow. These are determined using the PERT (Program Evaluation and Review Technique) based method as discussed in Section 5.10 such that SD
M M

. Where Max and Min are the most optimistic and pessimistic value for

each cash flow as determined by the owner of the asset. The uncertainty caused by time is implicitly included in the evaluation of the maximum and minimum for each cash flow, as such Max Min will increase as the time until expiration increases.

r: Risk free rate. This is the return rate at which money can be saved without any risk; it is usually the bond rate or London Interbank Offered Rate (LIBOR) which is the highest rate at which an investor can save money without risk. In Australia the Bank Bill Swap Reference Rate (BBSW) is used. T: The time until expiration, measured in years. This is the time until the owner of the real option has to decide whether or not to exercise the option 5.12. Equivalence between Second Order Moment Approach and Black

Scholes Analysis Carmichael and Balatbat (2009) state that the Black Scholes formula would calculate “something similar” to the value found by the second order moment approach. It is the aim of this thesis to demonstrate equivalence between the second order moment approach and Black Scholes analysis in determining the value of a real option. A measure of equivalence between different methods of determining real options already exists in literature in regard to the equivalence between Black Scholes analysis, Monte Carlo simulation and binomial trees. Mun (2006) suggests that Black Scholes analysis, Monte Carlo simulation and binomial trees all calculate an equal value for a particular real option even though there is a difference of 12% from the highest value to the lowest.
34

5.13. Summary Real options analysis can be used by investors to incorporate the value of potential or existing real options in an investment. While Black Scholes analysis is one technique that is very popular in determining the value of financial options investors are hesitant to utilise it as a tool for real options due the difficulty in the determining the input parameters. The second order moment approach of Carmichael and Balatbat (2009) offers a more intuitive tool for investors. This thesis demonstrates the equivalence of the second order moment approach with Black Scholes analysis for determining the value of a real option.

35

6. METHODOLOGY
6.1. Introduction The purpose of this thesis is to show that the second order moment approach of Carmichael and Balatbat’s (2009) offers investors an equivalent method to valuing real options as Black Scholes analysis. Equivalence between Black Scholes analysis and the second order moment approach is determined in three parts in the methodology. Firstly, a mathematical explanation of each approach is presented and a comparison of each approach’s input variables are outlined. In the second part of the methodology, a spreadsheet model is used to analyse four example cases of real options to determine similarity of the resulting real option values. The third Section analyses a case based on an actual industry real option to show that each approach determines a similar value of real options in industry. For the four basic cases and the industry based real option sensitivity analyses are carried out on the input parameters for each approach; specifically, cash flows, rate of return, strike price, standard deviation, volatility and time until expiration.

36

6.2. Mathematical Equivalence The mathematical comparison explains in detail how each approach values real options. A mathematical explanation of each approach is outlined to illustrate the similarities and differences of each approach. First the Black Scholes method is explained step by step, detailing the way in which the method captures the upside value of a real option. Black Scholes’ input parameters are explained in detail, as well as the effect that changes to each of the parameters has on the value of an option. Black Scholes analysis’ use of probability distributions is also explained and discussed. The second order moment approach is also explained step by step detailing the way in which the method captures the upside value of a real option. The determination of each input parameter and the effect that each parameter has on the value of the option is discussed. The method’s use of probability distributions is also illustrated. The comparison of the two approaches is based on a real option to expand, and then extrapolated to all other types of real options. The detailed mathematical analysis of each method demonstrates that each approach calculates the value of the option in the same way. Specifically that each method determines the probability that exercise is profitable and calculates the value of the investment when this occurs. After each method is explained and the similarities between the approaches are discussed, a direct comparison of the parameters of each method is carried out. The input parameters which are compared are the value of the underlying asset, the likelihood of exercise, the risk free rate, the uncertainty, the time until expiration and the present worth. Finally any relevant issues not resolved in the direct comparison are discussed. These issues include the value of the asset at expiration, the comparison of volatility and variance, the uncertainty of the strike price, call and put options, likelihood of occurrence, risk neutrality and the time until expiration.

37

6.3. Example Real Option Values Determined by Both Approaches 6.3.1. Introduction Four real option examples are valued by each method in order to demonstrate equivalence between Black Scholes analysis and the second order moment approach. The real options are valued by each method in a spreadsheet model. A set of cash flows and data is input into the model and the value of the option, as calculated by each method, is displayed. Four basic cases are modelled; one option to expand which is close to the money, one option to expand which is far from the money, one option to expand that has both positive and negative cash flows and one option to abandon. A call option is considered ‘out of the money’ if the strike price is below the current value of the asset and ‘in the money’ if the strike price is above the current value of the asset. The opposite is true for a put option. An option is considered ‘close to the money’ if it is either just ‘in the money’ or just ‘out of the money’ and considered ‘far from the money’ if it is either ‘deep in the money’ or ‘deep out of the money’. When an option is close to the money the uncertainty of the asset has a large influence on the price of the option. The four cases are chosen because they represent a wide range of types of real options. All types of real options can be constructed by using either a real option to expand, a real option to sell or a combination of both. The real option to expand, which is far from the money, and the real option to expand, which is close to the money, are modelled to determine the effect that uncertainty has on the equivalence of the two approaches. Finally the real option to expand that has both positive and negative cash flows is tested to ensure that equivalence between the two approaches exists even when there are negative cash flows. The cash flows for each year are assumed annual and that the expected means of the benefits and costs are already calculated. It is also assumed that the expiration year and strike price are known. For each basic case sensitivity analyses are carried out to ensure that the equivalence between each method exists over a large number of real options. The standard deviation, volatility, cash flows, risk free rate of return and strike price are each changed.
38

6.3.2. Calculating the Value of a Real Option with Black Scholes Analysis In order to calculate the value of a real option using Black Scholes analysis the five input parameters for the Black Scholes pricing formula are required. S (t=0): The value of the asset at time zero. This is calculated by a risk free discounted cash flow valuation of the future cash flows that occur if the option is exercised. Each cash flow is multiplied by 1/ (1+r)^n where r is the risk free rate and n is time from now measured in years. T: The time until expiration, measured in years. This is the time until the owner of the real option has to decide whether or not to exercise the option. σ: Volatility. It can be determined a number of ways as discussed in Section 5.9. It is a representation of the uncertainty of the future cash flows, or the volatility of the underlying asset. For this model the volatility is determined by ln 2√

) and average (S ) estimates for the

Where values for the optimistic (S

), pessimistic (S

value of the asset over a specific period (t) are determined based on arbitrary uncertainty values. The optimistic, pessimistic and average estimates are approximately equal across both methods to ensure a similar value of uncertainty is used. r: Risk free rate: Risk free rate. This is the return rate at which money can be saved without any risk; it is usually the bond rate or London Interbank Offered Rate (LIBOR) which is the highest rate at which an investor can save money without risk. In Australia the Bank Bill Swap Reference Rate (BBSW) is used. For modelling the values of the real options the risk free rate of 5%, which is close to the current BBSW rate, is used as the default value. X: Strike price: This is the price of exercising the option. Different values for the strike price are used in the four cases.

39

The value of the option using Black Scholes analysis is then determined in the spreadsheet model by substituting the input parameters into the Black Scholes pricing formula for either a call or put option. c S, t p S, t
S X

e e

T

SN d e XN d

T

XN d SN d e

T

T

Where d
S X

T √T T

,

d N x

√T

= d

σ√T

t and

√

e

dx = ф x

6.3.3. Calculating the Value of a Real Option with the Second Order Moment Approach The mean and variance of each cash flow, the risk free rate of return and the time until expiration are required in order to calculate the value of a real option using the second order moment approach. These input parameters are input into the spreadsheet model. The strike price is considered a negative cash flow and included with the other known cash flow estimates, such that: Value of the Asset = Cash Flows – Strike Price The mean of the present worth is then calculated by summing the present worth of each cost and benefit, such that: E[PW] = ∑
EZ

For the option to abandon or the option to sell, the option is exercised when the strike price is greater than the cash flows so:
40

Value of the Asset = Strike Price – Cash Flows = – (Cash Flows – Strike Price) Therefore the strike price is still considered a negative cash flow and the mean of the present worth is equal to the negative of the sum of the present worth of each cost and benefit, such that: E[PW] = ‐ ∑
EZ

The variance of each cash flow is determined by the PERT based method with the standard deviation (SD) = (TP‐TO)/6 and the variance (V) = SD^2. Where pessimistic (TP), optimistic (TO) and most likely (TL) value of each cash flow is approximated with the same level of uncertainty used with Black Scholes analysis of the real option. Since the effect that time has on the uncertainty of the cash flows is not explicitly accounted for in the PERT based method, investors will implicitly change their optimistic, pessimistic and most likely estimates as time increases. In the model used in this section the uncertainty for each cash flow is multiplied by a factor of time to account for the higher uncertainty as time increases. This factor ranges from time/3 to time/7 depending on the size of the cash flows. The factor range from time/3 to time/7 is used because it has been found by the author of this thesis, through extensive modelling, to accurately account for the factor of time in real options. For each individual case the factor remains constant as the individual input parameters are changed in order to show that the sensitivity analyses accurately show equivalence between the two methods. For the case where there are negative cash flows the factor for the positive and negative cash flows are equal. After the variance for each cash flow is established a general standard deviation for all the cash flows is chosen so that a sensitivity analysis can be done on the effects that volatility and variance have on the values of the real option using each method. The overall variance for all cash flows are equal using both the general standard deviation and the individual standard deviations determined with the PERT based method. The present worth of the standard deviation is taken as follows: Var [PW] = ∑
V Z

2∑

∑

C

Z ,Z

41

Var PW In each of the cases analysed the covariance of cash flows is assumed to be zero. This is for the sake of simplicity and also because it is consistent with the assumption of the central limit theory that each of the cash flows are independent. The value of the real option is then determined in the spreadsheet model by using the second order moment approach’s formula: c,p = Upside Value [PW] = E [PW upside] = E [ PW | PW > 0 ] µ c,p = = µ

ф

µ

where µ = E[PW] and σ = SD[PW] = Var PW

The distribution of the present worth is drawn for specific cases with the upside shaded in to illustrate the upside of the present worth. 6.3.4. Four Cases Analysed Four basic scenarios are modelled in order to compare the values that each method determines for each scenario. Case 1: Option to expand, close to the money. Both the second order moment approach and the Black Scholes analysis are used to determine the value of the real option in this case. An option to expand is modelled, which is the same as a call option. In order to allow the uncertainty of the option to have a relatively large effect on the value of the option an option that is close to the money is chosen. That means the value of the underlying stock is close to the value of the strike price. Case 2: Option to expand, far from the money. Both the second order moment approach and the Black Scholes analysis are used to determine the value of the real option in this case. An option to expand is modelled, which is the same as a call option. In order to illustrate the similarity of the two methods when the uncertainty of the option has a relatively small effect on the value of the option, an option that is far from the money is chosen. That means the value of the underlying stock is far from the value of the strike price. An option that is deep in the money is analysed.
42

Case 3: Option to expand, with some negative cash flows. Both the second order moment approach and the Black Scholes analysis are used to determine the value of the real option in this case. An option to expand is modelled, which is the same as a call option. In this case negative cash flows are used in addition to positive cash flows. Case 4: Option to Abandon. Both the second order moment approach and the Black Scholes analysis are used to determine the value of the real option in this case. An option to abandon is modelled, which is the same as a put option. For each model different input variables are changed to determine the effects that each input has on the results from each method. By changing different input parameters individually, the thesis attempts to determine if the two methods remain similar over a large range of real option cases, illustrating that each basic case are not the only cases where the two methods calculate the same value. Keeping all other parameters equal the following items are changed to see how they affect the resulting real option values from each method: 1) Standard deviation and volatility 2) Estimated benefits and standard deviation 3) Risk free rate of return 4) Strike Price 5) Negative Cash flows for case 3 For each case the expiration date of 10 years and cash flows from the 10th through to 30th year are used.

43

6.3.5. Analysis of Cases In order to determine equivalence between the two methods the values calculated for each real option, by Black Scholes analysis and the second order moment approach, are compared. A percentage difference less than 10% between the two approaches is considered equivalent. This is because real options analysis is used in combination with other financial tools to determine the value of real options. In addition to this, strategic decisions are based on factors other than financial, for example political and long term investment choices. Defining equivalence as a difference of less than 10% is slightly more conservative than Mun (2006) who suggests that Monte Carlo simulation, binomial trees and Black Scholes analysis are equivalent when they calculate values within 12% of each other.

44

6.4. Case Based on Actual Industry Real Option In order to demonstrate the equivalence of the second order moment approach with Black Scholes analysis in determining the value of a real option a case that is based on an actual industry real option is modelled and the value of the option determined using each approach. For confidentiality reasons the actual case could not be used, however the data and information is similar to the actual case. In the actual case, Black Scholes analysis was used to determine the value of the real option. The case based on the actual real option is of an investor who has an apparent option to sell a prison facility at a particular time at a particular price. The investor has been contracted by the State to operate the prison until 2020. In the year that the contract expires the State may or may not renew the contract with the investor. If the contract is not renewed the State will need to build a new prison or buy the existing prison from the investor. Therefore, if the State does not renew the contract they will be willing to pay for the existing prison up to the cost of building a new prison. As such the investor can be thought of as owning an apparent option to sell the prison back to the state in the year that the contract ends for the price of building a new prison. A more detailed explanation of the case can be found in Section 7.4.2. The case is modelled in a spreadsheet with all input parameters taken from the example case. The same spreadsheet and approaches are used as explained in Section 6.3. Due to the difficulty of equating volatility and variance, a sensitivity study of the effect that each parameter has on the option value calculated by each method is carried out.

45

6.5. Conclusion The equivalence of the second order moment approach with Black Scholes analysis is determined through a mathematical comparison as well as through the analysis of various example real option cases. The mathematical comparison of each approach attempts to show that in every case the two methods value real options in the same way. The four examples of real option cases attempt to show that the use of both methods result in the same value for any given real option. In particular, a real option based on an actual case is analysed to test the ability of the second order moment approach to value an actual real option with the result being compared to the value calculated by Black Scholes analysis.

46

7. COMPARISON
7.1. Introduction By comparing the second order moment approach with Black Scholes analysis this thesis attempts to show that the second order moment approach can be used as a more intuitive substitute tool for valuing real options. The two approaches are first compared mathematically to determine the similarities and differences between the approaches. Secondly, a model is used to determine how the two methods value different example real option cases. Finally a case based on an actual industry real option is valued using each method.

47

7.2. Mathematical Equivalence 7.2.1. Introduction Both Black Scholes analysis and the second order moment approach, as discussed in Section 5.4 and Section 5.10 respectively, can be used to determine the value of a real option. The mathematical comparison of each method demonstrates the equivalence between them. 7.2.2. Generic Real Option Case In order to demonstrate the equivalence between Black Scholes analysis and the second order moment approach the cash flows in figure 9, which are based on a real option to expand, is used as an example:
4 2 0 0 Cost or Benifit ($M) ‐2 Costs ‐4 ‐6 ‐8 ‐10 ‐12 Benefits Strike Price 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Time (Yrs)

Figure 9 – Example Set of Cash Flows Figure 9 represents the cash flows from a real option; at a predetermined time T (10 years in the example) the owner of the real option has the right but not the obligation to spend a predetermined amount ($10M in the example) to expand their investment and receive additional future cash flows.

48

Both Black Scholes analysis and the second order moment approach can be used to determine the value of a real option to expand, which is equal to the amount that an investor is willing to pay for the option. 7.2.3. Black Scholes Analysis Approach to Real Options Figure 10 represents various values used by Black Scholes analysis to value real options:
15

10

Cost or Benefit ($M)

5 Costs Benefits 0 0 ‐5 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Strike Price S(t=0) S(t=T)

‐10

‐15

Time (Yrs)

Figure 10 – Representation of Parameters from Black Scholes Analysis Black‐Scholes equation is based on the following assumptions (Hull, 1997): 1. Short selling is permitted. 2. There is no cost of transactions and the stocks are perfectly divisible 3. The derivatives have no dividends 4. Riskless arbitrage opportunities do not exist 5. Stock trading is continuous 6. The risk free rate stays constant 7. The log of stock prices follows Geometric Brownian motion with constant drift and volatility.
49

Black Scholes analysis requires the following input parameters to determine the value of a real option (Copeland and Antikarov, 2003): S (t=0): The value at the underlying asset. This is calculated by a risk free discounted cash flow valuation of the future cash flows that are received if exercise occurs. These cash flows are grey and black in figure 10 and the value of S (t=0) is represented by the dotted bar at time zero. T: The time until expiration, measured in years. This is the time that the owner of the real option has to decide whether or not to exercise the option. σ: Volatility. It can be determined a number of ways as discussed in Section 5.9; it is a representation of the uncertainty of the future cash flows, or the volatility of the underlying asset. r: Risk free rate. This is the rate at which money can be saved without any risk; it is usually the bond rate or London Interbank Offered Rate (LIBOR) which is the highest rate at which an investor can save money without risk. In Australia the Bank Bill Swap Reference Rate (BBSW) is used. X: Strike price: This is the price of expansion and is represented by the hollow bar in figure 10. The cost of the real option at time t, denoted c, for underlying asset S, is then calculated using the Black Scholes Pricing Formula (Hull, 1997): ,

Where

√

,

√

=

√

and

√

= ф

50

Black Scholes analysis assumes that the value of S follows geometric Brownian motion. That is . is represented by the horizontally shaded bar in figure 10. The expressions d1 and d2 are also determined by assuming that the value of S follows geometric Brownian motion, specifically that the underlying asset is log‐normally distributed and follows a generalized Wiener process. (Hull, 1997) is the cumulative probability distribution function of a standard normal variable which is a normally distributed variable with a mean of 0 and a standard deviation of 1. is represented in figure 11 by the thick black line and as an example N(5) is the area shaded in grey.
0.45 0.4 Frequency of x's Occurence 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 ‐5 ‐4 ‐3 ‐2 ‐1 0 x 1 2 3 4 5

. Using this assumption the value of

is projected forward to

Figure 11 – Probability Distribution of Standard Normal Variable

51

The Black Scholes Pricing Formula can be deconstructed to understand how it represents the value of the option (Hull, 1997): The pricing formula suggests that the value of the option at time T is equal to: =max {0,

0

The present worth of this is then calculated: c = *

The pricing formula can be deconstructed into two components. Component 1: The first component of if Therefore As discussed m = log is the contingent receipt of the underlying investment or 0 if Prob (Hull, 1997) having a mean of: :

is log normally distributed, with log

And a standard deviation of: s = √ As such the expected value of 1 truncated at X is equal to

That is to say that value of the contingent receipt of the underlying investment is equal to the mean of the expected value of truncated at X. (Nielsen, 1993)

52

Component 2: is the probability that the option will be exercised which equals Prob( Therefore By combining the two components the value of the option at time t = T can be stated: = = ( Prob = or 0 if Prob

is represented in figure 10 as the horizontally shaded bar minus the hollow bar, or the value of the overall option investment at time t = T. The value of the asset at time t = 0 is determined by taking the present worth of the value of the option at time t = T. discounts the value of the option at time t = T to its present worth at time t = 0. represents the continuous present value. In summary c = the present worth of Or c = PW [ | 0] does not equal . If it did then it could be said times by the probability that .

It should be noted that that: Prob =

includes the random movement of the investment S, which

However this is not true as

is based on its geometric Brownian motion. If it was true, then a negative option value would be possible which can never exist. In fact: Prob >

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7.2.4. Second Order Moment Approach to Real Options Figure 12 represents various values used in the second order moment approach to valuing a Real Option:
6 4 2 Cost or Benefit ($M) Variance 0 0 ‐2 ‐4 ‐6 ‐8 ‐10 ‐12 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Costs Benefits S D ∑Var[PW] ∑E[PW]

Time (Yrs)

Figure 12 – Second Order Moment Representation of the Cash Flows The first step of the second order moment method is to determine the input parameters. E: Mean value of each cash flow. This is the estimated mean of each cash flow. The mean of each cash flow is calculated without adjusting for risk. The uncertainty of the investment is incorporated in the second order moment of the cash flows which is defined as the variance. The strike price in the option to expand is considered a negative cash flow in the year the option expires. So that: Value of the Asset = Cash Flows – Strike Price or Var: Variance of Each cash flow. These are determined using the PERT (Program Evaluation and Review Technique) based method as discussed in Section 5.10 such that
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. Where

and

are the most optimistic and pessimistic value for

each cash flow as determined by the owner of the asset. The uncertainty caused by time is implicitly included in the evaluation of the maximum and minimum for each cash flow and as such is larger as the time until expiration increases.

r: Risk free rate. This is the rate at which money can be saved without any risk; it is usually the bond rate or London Interbank Offered Rate (LIBOR) which is the highest rate at which an investor can save money without risk. In Australia the Bank Bill Swap Reference Rate (BBSW) is used. T: The time until expiration, measured in years. This is the time the owner of the real option has to decide whether or not to exercise the option After the input parameters are determined the second order moment approach takes the sum of the discrete cash flows and variances. Where is defined as a number of cash flow components made up of k = 1,2..,m and is … 2∑ ∑ , :

defined as E[ ] = ∑ Var [ ] = ∑

The present worth of the means and variances respectively can be determined as follows: E[PW] = ∑ Var [PW] = ∑ 2∑ ∑
,

These values represent the mean and variance of the investment at time = 0 and are represented in figure 12 by the hollow and dotted bars respectively. Also the standard deviation of the present worth can be stated: Var PW The value of Present Worth has a probability distribution with a mean of E[PW] and a variance of Var [PW] and a standard deviation of
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.

Under the assumption that each of the cash flows are mutually independent random variables, with finite means and therefore all moments are finite, the Lindeberg Central Limit Theorem states that the sum of these random variables for cash flows at time t will approximate towards a normal distribution when the number of random variables is large. It should be mentioned that this method does not rely on the cash flows approximating towards normality. The mean and variance of the cash flows offer an insight into the probability distribution of the value of the option whether or not the cash flows approximate towards a normal distribution. The only benefit of cash flows, which can be said to approximate towards a normally distribution, is that a more accurate probability distribution can be determined. Defining ~ The probability function of which equals f(x) can be stated: to equal the present worth of the asset over time it can be said that: ,

Where µ = E[PW] and σ = SD[PW] = Var PW This distribution is represented in figure 13:

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0.09 0.08 Frequency of x's Occurence 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 ‐20 ‐10 0 x 10 20 30

Figure 13 – Probability Distribution of the Present Worth of the Investment The second order moment approach values the option price as the mean of the upside of the present worth. This is termed the Upside Value. The upside of the present worth is the portion of the distribution that is above zero which is represented in figure 14:
0.09 0.08 Frequency of x's Occurence 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 ‐20 ‐10 0 x 10 20 30

Figure 14 – Upside of the Present Worth of the Investment As such: c = Upside Value [PW] = E [PW upside]

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c = E [ PW | PW > 0 ] This is the mean of a truncated normal distribution which can be expressed a number of ways: E [ PW | PW > 0 ] = where µ = E[PW] and σ = SD[PW] = Var PW

ф

This can be solved using standard normal look up tables. The following is also true: E [ PW | PW > 0 ] = E [ PW | PW > 0 ] = | 0

To compare this probability distribution to N(x) in the Black Scholes pricing formula it should be noted that the shaded area in figure 14 equals: P(PW>0) = 1 – P (PW