ESTIMATING PROJECT VOLATILITY AND DEVELOPING DECISION SUPPORT
SYSTEM IN REAL OPTIONS ANALYSIS
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. This
dissertation does not include proprietary or classified information.
Hyun Jin Han
Certificate of Approval:
Jorge Valenzuela
Associate Professor
Industrial and Systems Engineering
Steven Swidler
Professor
Finance
Chan S. Park, Chair
Professor
Industrial and Systems Engineering
George T. Flowers
Interim Dean
Graduate School
ESTIMATING PROJECT VOLATILITY AND DEVELOPING DECISION SUPPORT
SYSTEM IN REAL OPTIONS ANALYSIS
Hyun Jin Han
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirement for the
Degree of
Doctor of Philosophy
Auburn, Alabama
December 17, 2007
iii
ESTIMATING PROJECT VOLATILITY AND DEVELOPING DECISION SUPPORT
SYSTEM IN REAL OPTIONS ANALYSIS
Hyun Jin Han
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon the request of individuals or institutions and at their expense. The author
reserves all publication rights.
Signature of Author
Date of Graduation
iv
VITA
Hyun Jin Han, son of BongChul Han and ChunJa Park, was born on August 30,
1969 in Pusan, Korea. He graduated from Korea Military Academy, Seoul, Korea in
March 1992 with a B.A. in Chinese. He was commissioned a second lieutenant in
Republic of Korean Army as a graduate of KMA and started his military service as a
platoon leader. As an officer, his responsibilities included supply control, resource
allocation, and cost analysis & control for acquiring new weapon systems for Korean
Army. He completed his M.S. in Operations Research from Korea National Defense
University in January 2001. In August 2003, he joined Ph.D. program in Auburn
University. He married Yun Kyung Lee, daughter of YoungRea Lee and DongSoon
Lim, on June 12, 1999. On January 22, 2001, their son, Ji Young (David), was born and
on November 8, 2005, their daughter, Grace was born.
v
DISSERTATION ABSTRACT
ESTIMATING PROJECT VOLATILITY AND DEVELOPING DECISION SUPPORT
SYSTEM IN REAL OPTIONS ANALYSIS
HYUN JIN HAN
Doctor of Philosophy, December 17, 2007
(M.S., Korea National Defense University, 2001)
(B.A. Korea Military Academy, 1992)
148 Typed Pages
Directed by Chan S. Park
Today?s uncertain world requires firms to have a system in place that can analyze
the flexibility of their projects. Real options are utilized frequently to quantify the
benefits of taking a particular risk. The real options valuation process provides a
methodology to measure the value of flexibility, and it assists the decision makers in
making the optimal investment decision. The goal of this research is to develop the
methodology for improving the real options application in actual capital investment
decision making.
The Reverse Monte Carlo Simulation model (RMCS), which combines Monte
Carlo simulation and the stochastic processes, is developed as a new volatility estimation
vi
method for risky projects. Compared to previous simulation methods, RMCS results in
more accurate volatility. Then a volatility revision processes based on the previous
volatility estimation processes are proposed. A Bayesian revision process is suggested to
estimate the new volatility when the initial volatility has been estimated by Monte Carlo
simulation. Since specific cases that use typical types of Bayesian conjugate processes
are hard to find in the real world, a Dirichlet conjugate process is applied to estimate
posterior distributions of the future cash flows. After estimating the new distributions of
the cash flows, the revised volatility can be computed using the RMCS approach.
Finally, a new early decision rule is developed in order to make real options more
useful. This rule concentrates on maximizing the expected future project value. Under the
new decision rule, an expected future value of the currently exercised option and the
expected future option value are compared in order to determine the best exercise timing.
An early decision map for ?waiting,? ?early exercise,? and ?early divest? over the entire
option life is developed to automate the decision in case some variables are revised in the
future. The map indicates that increasing volatility enlarges the ?waiting? area while
decreasing volatility shrinks the ?waiting? area. A simulation is applied to validate the
newly developed decision rule by comparing the benefit of the early exercise rule and the
volatility revision during the option life. The new decision rule is found to be useful in
maximizing the expected profit of the delayed investment because the proposed decision
model results in better than or equal to the current decisions model.
vii
ACKNOWLEDGMENTS
First let me praise God who has planned, prepared and accomplished all these
works.
I would like to express my sincere gratitude to my advisor, Dr. Chan S. Park, for
his outstanding insight and generous support throughout my graduate studies. His
guidance and support were vital to the success of this research. I am especially thankful
to all professors and staffs in Industrial and Systems engineering at Auburn University
for their assistance and advices to finish my study.
I would like to thank my wife, Yun Kyung Lee, for her support and help, and I?d
like to express my love to my son David and daughter Grace. My parents and my parents
in law deserve very special appreciations for their enduring love, encouragement, and
support.
Finally I would like to thank Republic of Korean Army for giving me a chance to
study and the financial support.
viii
Style manual or journal used:
Bibliography conforms to those of the transactions of the IEEE
Computer software used:
@Risk Simulation
Microsoft Visual C++ 6.0
Microsoft Office XP
ix
TABLE OF CONTENTS
LIST OF TABLES xi
LIST OF FIGURES xiii
CHAPTER 1 INTRODUCTION 1
1.1 Background 3
1.2 Research Objectives 5
1.3 Study Plan 6
CHAPTER 2 LITERATURE REVIEW 9
2.1 Introduction 9
2.2 Background of Real Options 10
2.3 Volatility Estimation Method in Real Options 14
2.4 Bayesian Revision Process 19
2.5 Decision Timing in Real Options 20
CHAPTER 3 ESTIMATING PROJECT VOLATILITY
USING MONTE CARLO SIMULATION IN REAL OPTIONS 22
3.1 Introduction 22
3.2 Monte Carlo Simulation in Real Options 23
3.3 CA vs. HP Volatility 26
3.4 Developing a Reversed Monte Carlo Simulation 30
3.5 Conclusion Remarks 37
x
CHAPTER 4 BAYESIAN FRAMEWORK TO REVISE THE VOLATILITY
OF REAL OPTIONS 39
4.1 Introduction 39
4.2 Estimating a Continuous Random Distribution from Threepoint Estimates 41
4.3 Bayesian Revision Processes 44
4.4 Volatility Revision Framework in Real Options 46
4.5 Numerical Example 50
4.6 Conclusion Remarks 57
CHAPTER 5 DEVELOPING DECISION SUPPORT SYSTEM IN REAL OPTIONS 59
5.1 Introduction 59
5.2 Developing a New Decision Rule of Real Options 62
5.3 Validation of the New Decision Model through Simulation 74
5.4 Comprehensive Illustrating Example 80
5.5 Conclusion Remarks 88
CHAPTER 6 COMPREHENSIVE APPLICATIONS OF NEW REAL OPTIONS
MODEL 100
6.1 Introduction 100
6.2 Growth Options Framework 101
6.3 Compound Options Framework 110
6.4 Conclusion Remarks 122
CHAPTER 7 CONCLUSIONS 124
BIBLIOGRAPHY 128
xi
LIST OF TABLES
21. Types of real options 12
22. Financial and real options parameters 14
23. Real options volatility estimation methods 18
31. Sample example to simulate the
3
PV distribution 35
32. Cash flows from Copeland and Antikarov (2003) 36
41. Cash flow estimation for XYZ Chemical Company 50
42. Threepoint estimates for the random factors 51
43. Parameters of beta distributions 51
44. New information with the initial threepoint estimates 55
45. Revised discrete approximation for the risky variables 56
51. Condition for the course of action 67
52. Ratios (
V
I
) for early decisions in different volatilities 71
53. Ratios (
V
I
) for early decisions in different riskadjusted rates 72
54. Simulation results for a defer option 77
55. Simulation results for an abandon option 79
56. Decision points of the defer option example 84
57. Decision points of the abandon option example 87
xii
61. Cash flow estimation for the growth opportunity 102
62. Prior belief of the random factors for the 2nd phase investment 102
63. Parameters of beta distribution 104
64. Revised discrete approximation of the growth option 107
65. Comparison of the decision points in case of volatility change 108
66. Cash flow estimation for the compound option 113
67. Prior belief for the random factors 114
68. Parameters of beta distribution 115
69. Compound options parameters 116
610. Revised discrete approximation of the compound option 118
611. Comparison of the decision points in case of volatility change 121
xiii
LIST OF FIGURES
11. Example of defining real options parameters 4
31. Relation between
T
? and ? 30
32. Future value simulation 31
41. Shapes of Beta distribution in a specific parameters 42
42. Bayesian revision processes 45
43. Developing a discrete approximation 48
44. Shape of the demand distribution 52
45. Shape of the unit price distribution 52
46. Shape of the unit variable cost distribution 53
47. Shape of the fixed cost distribution 53
48. Simulated PV distribution 54
51. Binomial tree approach for early exercise decision 63
52. Decision changes in case of different volatilities 73
53. Decision changes in case of different riskadjusted interest rates 74
54. Illustration of simulating a project value 75
55. Illustrations for early decisions: Invest (Left), Divest (Right) 76
56. Decision map for an abandon option 86
57. Decision map for abandon option in case of volatility change 86
xiv
61. Project scenario of a growth option 101
62. Cash flow diagram of the growth opportunity 105
63. Decision framework of the growth option 109
64. Project scenario of a compound option 110
65. Decisions in compound options framework 111
66. Project cash flow of the R&D project 116
67.The decision map for the 2
nd
phase investment 120
68. Decision framework of the growth option 122
1
CHAPTER 1
INTRODUCTION
Recently, real options analysis has been applied to the capital budgeting decisions
under uncertainty. Real options analysis provides an opportunity to improve strategic
investment decisions in an uncertain environment. However, the real options valuation
concept requires some adjustments in order to be useful in management decisions.
Although most managers understand that real options can address investment flexibility,
they do not widely apply the real options valuation method in managing the uncertainty
of their projects.
One of the commonly cited reasons for avoiding real options is that the volatility
of the underlying project is too difficult to estimate. In financial options pricing, it is
possible to estimate the volatility of the financial assets by reviewing the historical price
of the underlying asset. However, in real options volatility is not directly estimated, so
some restrictive assumptions are necessary to value the real options. This research
develops a new method of estimating a risky project?s volatility by comparing the project
return distribution and the future project value distribution generated by Monte Carlo
simulation.
One of the main advantages of real options is that it promotes taking time to
observe the future market movements, thus decreasing the risk of a huge irreversible
2
investment. The ?wait and see? strategy of the traditional options framework means that a
company will wait to proceed until the market is more favorable. The traditional options
framework does not consider the information gathering activities that may take place
during the option?s life. In other words, if a firm purchases a real option, it is assumed
that only the final information for the project is available for their decision. However, the
firm will continue to collect information to help it make the best possible decisions after
they take the investment opportunity. In fact, the volatility of the underlying project,
which is one of the most important variables, can change as time passes. In this study, a
Bayesian revision process is employed to modify initial estimates of volatility.
Deciding the investment timing of the project is another important factor in real
options valuation. However, the lack of research for determining an optimal investment
timing of options has been a barrier to accepting real options as a useful tool in capital
budgeting decisions. Recent research indicates that failing to exercise real options on
time reduces the projects? value much less than predicted, and the question of whether the
real option holders exercise their options optimally has not been researched extensively.
In this research, a new decision rule designed to maximize the future value of a project is
introduced. It incorporates three different decision criteria in the defer options
framework: 1) wait, 2) early exercise, and 3) early divest project; in contrast, the abandon
options framework considers two decisions: 1) early exercise, and 2) wait.
Finally, two real options decision models are demonstrated to explain how to
analyze the investment opportunity with the proposed methods.
3
1.1 Background
Before the real options framework was introduced to resolve the uncertainties of
capital budgeting problems, two different approaches were popularly used: the
Discounted Cash Flow analysis (DCF), which uses the riskadjusted discount rate; and
the MeanVariance approach.
In the DCF approach, the most critical factor is deriving a correct riskadjusted
interest rate which represents the risk a company is taking. The interest rate is then used
either to compute the Net Present Value (NPV) of a project, or it is considered as a
Minimum Attractive Rate of Return (MARR) for decision making through Internal Rate
ofReturn (IRR) criteria. However, accurately selecting the discount rate is one of the
most difficult issues in a DCF analysis. The most commonly used rate in economics is the
Capital Asset Pricing Model (CAPM). With CAPM, a market premium is presumed to be
paid to shareholders for the risk associated with a particular industry. The Weighted
Average Cost of Capital (WACC) is an alternative approach to estimating the risk
adjusted rate of return. With WACC, it is necessary to satisfy all the required returns the
investor anticipated.
The real options valuation model is a different type of decision framework. While
the above frameworks concentrate on the present ?go? or ?nogo? decision, the
fundamental strategy of real options is ?wait and see.? Companies can apply real options
to almost any situation where it is possible to estimate the NPV of a certain project
without flexibility. It is then possible to analyze the project opportunity by considering
the volatility of the project cash flows. Figure 11 shows a defer option framework, which
is a basic type of real options.
4
Theoretically, the traditional decision making processes have difficulties in
addressing the flexibility of investment decisions. In the present volatile market, the ?wait
and see? strategy is of the utmost importance because of the irreversibility of the capital
budgeting decisions. Real options provides two important contributions to the capital
budgeting decision. First, it provides a method for measuring the value of the project
opportunity in several different circumstances: delaying the investment, abandoning the
project, research and development project, or the potential growth of the investment
during a certain time period. Second, real options are wellsuited for deriving the price of
tradable assets such as patents, licenses, or natural resource production projects.
Figure 11. Example of defining real options parameters
5
1.2 Research Objectives
The primary objective of this study is to suggest ways that real options can be
more practical in investment decisions. Although the purpose of applying the real options
model is to address the uncertainty of the investment, it is not widely used to manage the
risk of projects. The most significant problem that needs to be resolved is the
development of a reasonable process to estimate the volatility of a risky project. The
volatility of the underlying asset is one of the most important parameters of the options
theory. In the financial options pricing, it is possible to estimate the volatility of the
financial assets by reviewing the historical price of the underlying asset. However, the
volatility of real options is not directly estimated; therefore some restrictive assumptions
are required in order to value real options. In this research, a new method of Monte Carlo
Simulation is applied to estimate the volatility of a project with an assumption that the
DCF is possible to the project evaluation.
The second objective is the development of a method to revise the volatility of the
real options in case new information is observed during the option?s life. Since real
options are rarely tradable, it is necessary to revise the volatility in order to support the
irreversible investment decision in flexible market conditions. There has been very little
research on this subject until now. This study develops a Bayesian conjugate process in
case of general prior distributions and limited sample sizes. Dirichlet distribution is
applied to revise predictions of the future cash flows, and a new volatility can be
estimated by the simulation with the posterior random factors.
The third objective of this study is to determine the optimal exercise timing of the
real options. Most investments in the project are irreversible, so choosing the right
6
investment timing is critical in real options. Recent research has emphasized that failing
to exercise real options on time reduces the projects? value much less than predicted, but
the question of whether the real option holders exercise their options optimally has not
yet been researched extensively. In this research, a decision map of the real options is
developed by comparing the future value of waiting with the value of exercising. After
simulating the values of the proposed method and the current decision model, a paired t
test is conducted to check the significance of the new decision rules.
Finally, two real options models demonstrate the application of the whole
proposed process of volatility estimation, volatility revision, and decisions. A project?s
growth opportunity and a compound options framework are demonstrated to examine the
changes on decisions.
1.3 Study Plan
Chapter 1 sets forth the background of and the motivation for this research. The
introductory chapter also includes the research?s objectives and the study plan.
Chapter 2 reviews previous studies related to the development of the real options
valuation model and the application of it to various projects. Since the origins of the real
options valuation model are found in financial options pricing, the latter model is briefly
described. Then, some theories as to how to make real options more attractive to
decisionmakers are investigated. Finally, the research on applying real options in the real
world is summarized.
Chapter 3 develops a volatility estimation method by combining Monte Carlo
simulation and stochastic processes. The current methods of estimating volatility through
7
Monte Carlo simulation are examined along with a traditional concept of volatility in
options theory. Then, a new way of estimating the volatility of real options is presented
by considering the option life and the simulated project value followed by a numerical
example of estimating volatility through Monte Carlo Simulation.
Chapter 4 suggests a volatility revision process based on the Bayesian revision
process in case of a general prior distribution with very limited sample sizes. After
demonstrating in a brief introduction how to estimate a cash flow distribution from a
threepoints estimation, Dirichlet conjugate processes are used to generate a posterior
distribution of each parameter of the cash flows. The posterior volatility of the project is
estimated by resimulating the project value distribution.
Chapter 5 develops the concept of decision timing: when to exercise or divest the
real options by comparing the future value of the project. I present a method for making
such a decision by considering the investment opportunity cost concept. A simulation
technique is used to demonstrate the benefit of the new exercise rule. A comparison
between the expected profit of the decision rule set forth in this study and the traditional
rule shows the advantages of the new approach. The fundamental of this simulation is
generating a past dependant project values.
Chapter 6 describes two examples of the research being applied in specific
investment opportunities. This investigation considers a project which has at least two
investment phases. A growth options framework and a compound options framework are
investigated to apply the methods suggested in previous chapters. In the growth options
framework, a revised volatility and project value affect the early exercise decision while
the compound options concentrate on go or nogo decision for the second investment.
8
Different decisions are presented in the two investment scenarios when new information
is collected during the option life.
Chapter 7 presents a brief conclusion along with some suggestions for future
research.
9
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
There are three main objectives in this research. The first objective is to develop a
new method of volatility estimation that uses Monte Carlo Simulation. The second
objective is to propose a Bayesian revision process to enhance decision making by
updating the volatility of real options during the option life. The final objective is to
develop early decision rules for the real options. I include a general decision map with a
ratio between the project value and the investment cost with the goal of automating the
investment decisions at each stage of the option.
In order to further investigate the fundamentals of real options valuation
technique, it is necessary to understand the financial option pricing model because it is
the origin of the real options valuation concept. So, a brief review of the financial option
valuation models?the BlackScholes model and the binomial lattice?will demonstrate
the logic of the real options valuation model. After a brief comparison of real options
valuation and financial options pricing, some conceptual meanings of real options from
previous studies are reviewed in order to expand the use of real options in decision
making. Then the various types of real options and their applications are examined along
with the volatility estimation methods of the projects. After reviewing the research related
10
to revising the volatility of the options, previous studies for improving the strategic
decisions in real options valuation are summarized.
2.2 Background of Real Options
An option is a contract that gives its holder the right but not the obligation to take
action at a predetermined price within a specified time period. There are two basic types
of financial options: call options and put options. A call option gives the holder the right
to buy an asset by a certain date for a predetermined price, while a put option gives the
holder the right to sell an asset. The predetermined price of the asset for the actions is
known as the exercise price, and the date is known as the expiration date. Five parameters
determine the price of the options: 1) the current price of an asset (
0
S ), 2) the exercise
price (K), 3) the expiration date (T), 4) the riskfree interest rate (r), and 5) the volatility
of the asset (? ). There are also two types of options as determined by their exercise
timing: European options, which can be exercised only on the expiration date; and
American options, which can be exercised at any time during the option life.
The valuation of the option premium is of the utmost importance in financial
options because the options are traded in a market such as the Chicago Board Options
Exchange. The two most common option pricing models to appear in the literature are the
BlackScholes model (BS model) and the binomial tree model. In 1973, Fischer Black,
Myron Scholes, and Robert Merton presented a seminal paper in the pricing of stock
options. The BS model had a critical influence on option pricing and hedging options.
Hull (2005) summarizes that they developed a theoretical option pricing model based on
riskfree arbitrage with the following assumptions: 1) the percentage changes in the stock
11
price in a short period of time follows approximate normal distribution, 2) the short
period of time is independent, 3) the underlying stock does not pay any dividend, 4) the
shortterm interest rate is known and constant, 5) the investors can borrow or lend at the
same riskfree interest rate, 6) there are no transaction costs in trading options, and 7) the
options are European.
Cox, et al. (1979) present an algebra technique for pricing an option by
constructing a binomial lattice. Since the binomial lattice model is simple and easy to
understand, it is now a very popular way to explain the invisible logic of options. In the
lattice model, the only assumption is that no arbitrage opportunities exist. Park (2006)
summarizes three different concepts to price an option by binomial lattice: 1) the
replicatingportfolio concept, 2) the riskfree financing concept, and 3) the riskneutral
concept. In spite of their differences, all three approaches yield the same result, so any of
them can be selected to draw the lattice.
2.2.1 Real Options Valuations
Research indicates that managers can choose from seven or eight real option types
to match their investment opportunity. Table 21 is a summary by Amram and Kulatilaka
(1999) and Trigeorgis (2005). A defer option gives the holder the right to wait for a period
of time for the project?s uncertainty to be resolved. In this framework, the company will
have the flexibility to postpone a great portion of its irreversible investment by taking the
option. The defer option is valuable in natural resourcesrelated industries such as
farming, oil extraction, and mining.
12
Table 21. Types of real options
Trigeorgis (2005) Amram & Kulatilaka
(1999)
Defer
Staging (Compound)
Expand
Contract
Temporary Shutdown
Abandon
Switching
Growth
Timing
Growth
Staging (Compound)
Exit
Flexibility
Operating
Learning
A staging option considers a project which has a series of capital outlays over
time. In the staging options point of view, each stage is considered an option for the
investment of the next stage. R&D projects in the biotechnology and pharmaceutical
industries, or the construction of an electric plant are examples of staging projects. The
required outlay for the earlier stage of the project is considered an option premium for the
right to invest in the later stage of the project.
Expand options and the growth options have the similar properties. Both consider
the initial investment and followup investments in light of market conditions. According
to Trigeorgis, the only difference in the two options is that with expand options, the
initially invested project will be expanded if the market condition is good, while in the
growth options framework the initial investment is the foundation for the other projects.
13
A contract option is the opposite of an expand option. In the contract options
framework, if the market is not as favorable as expected, the manager is able to operate
below capacity or reduce the scale of operations. Similarly, a temporary shutdown option
indicates that the plant may cease to operate and then reopen according to market
conditions. In the temporary shutdown options framework, the shifting costs are
considered an option premium. In both the contract options and the temporary shutdown
options framework, the cost of installing the flexible production system is considered an
option premium. These options are useful in natural resource industries such as mining
industry or fashion design and merchandising.
An abandon option gives its holder the right to sell the project for its salvage
value if the project turns out not to be favorable. This option can be valued as an
American put option pricing model. Finally, a switching option allows its holder to install
a flexible system of inputs or outputs for a certain project. An example would be
installing an energy converting system for a production line. The installment cost would
be an option premium in such a case.
The premium involved in taking the investment opportunity is determined by five
parameters similar to financial options pricing: 1) the present value of the project cash
inflows (
0
V ), 2) the investment cost of the project (I), 3) the time to make the investment
(T), 4) the riskfree interest rate (r), and 5) the uncertainty of the project cash flows (? ).
Table 22 summarizes the input parameters of real options and its financial counterpart.
The valuation technique for real options is the same as for the financial options
pricing model. The following example of a defer option explains the real options
valuation.
14
Table 22. Financial and real options parameters
Financial Option Real Option
Stock Price
0
S
PV of the project cash inflows
0
V
Exercise Price X Investment cost of the project I
Time to mature T Time to make investment decision T
Stock Volatility ? Uncertainty of the project value cash flows ?
Risk free rate r Riskfree interest risk r
Assume that a firm has a project with a present cash flow value
0
V which can be
invested in today with a requirement investment cost of I . Under the traditional decision
rule, the positive
0
NPV=V  I suggests that one should invest in the project today, and
the negative NPV indicates a nonattractive project. If it is possible to defer the
investment decision in the nonattractive project up to ?T? years, real options play a role
in valuing the delayed opportunity. The logic for valuing the opportunity appears below.
Assuming the firm takes the investment opportunity, it will invest in the project
until time T if the value of the project is greater than I, otherwise the decision maker will
decide not to invest. So the payoff at time T will be:
T
max(V  I,0) . Then the BS model
or binomial lattice approach will provide the option premium for taking the investment
opportunity.
2.3 Volatility Estimation Methods in Real Options
Financial options derive their price from the value of their underlying financial
assets, such as stocks. Option volatility can be estimated by either historical movements
15
of asset market prices, or by calculating the implied volatility from the BlackScholes
model based on the market price of an option. Estimating the volatility of a real option is
much more difficult because there are no historical returns or current market prices of the
underlying projects. Research indicates that there are essentially six volatility estimation
methods in the previous real options applications including simple assumptions.
2.3.1 Historical Volatility of the Underlying Asset
If the price of a natural resource determines the future cash flow of the project,
then the historical volatility of that resource is considered the volatility of the project.
Kelly (1998) used the binomial lattice approach to determine the investment timing for
the initial public offering of a gold mine. The historical gold price was used to compute
the future project value. Moel and Tufano (2002) also used gold price returns to analyze
the optimal timing in which to close a mine in North America. When Cortazar and
Casassus (1998) analyzed an investment project that expanded the production capacity of
a copper mine, they considered the historical volatility of the price of copper as the
uncertain factor. Cortazar, et al. (1998) also used the volatility of copper prices to
evaluate the environmental investments for the copper production company. Kemna
(1993), Smith and McCardle (1998) and Armstrong, et al. (2004) used the historical
volatility of the price of oil for their decision analysis. Titman (1985) and Quigg (1993)
used land price fluctuations in order to analyze an opportunity to develop the land.
2.3.2 Historical Volatility of the Compatible Asset
When the underlying asset is not tradable in the market, the volatility of the
compatible asset is adapted to compute the project volatility. Benaroch and Kauffman
16
(1999) adapted the historical demand of pointofscale transactions in California when
they were estimating the volatility of a pointofscale transaction project in the New
England area. Insley and Rollins (2005) used the volatility of the price of a sprucepine
fir 2??4? to evaluate the best timing for timber harvesting under the real options model.
2.3.3 Historical Volatility of the Company?s Stock Price
If the project volatility perfectly correlates with the stock price movement of the
company, the volatility of the stock price of a company, this method is useful in the
estimation process. Newton and Pearson (1994) assumed that the stock price of a R&D
company is analogous with the expected NPV of the company?s project. Bollen (1999)
applied a real options framework to value the opportunity of changing a project?s
capacity by setting the demand for the product as a source of risk. He assumed that the
traded asset span changes with demand. Herath and Park (1999) demonstrated that the
stock price of a typical R&D company is perfectly correlated to the company?s R&D
project. Miller and Park (2004 and 2005) did the same in a case involving a maintenance
company.
2.3.4 Historical Volatility of the Industry Index
The historical volatility of the industrial group index has also been used to
estimate the volatility of the project value, particularly in cases where there was a
shortage of past data. Cassimon, et al. (2004) estimated the volatility of a real option
project which values new drug applications and the R&D of pharmaceutical companies
using the pharmaceutical industry?s standard deviation of equity. Jensen and Warren
(2001) valued the research in the ecommerce project by using the average volatility of
six ecommerce companies traded on NASDAQ as the uncertainty factor of their real
17
options model. Teisberg (1994) analyzed a utility power plant construction project using
an option pricing model by considering the six participating firms? historical returns as
the volatility of the project value.
2.3.5 Monte Carlo Simulation
Monte Carlo simulation has been conducted to compute the volatility of the
project itself by using the project?s future cash flow, which is based on the DCF and
plausible scenarios of future uncertainty. The historical data of the project or some
assumptions are used for estimating the distribution of the input parameters in the
simulation approach. Studies by Copeland and Antikarov (2003), Mun (2006), Herath and
Park (2002), and Cobb & Charnes (2004) comprise the body of research that has used
Monte Carlo simulation to estimate volatility.
Table 23 is the summary of volatility estimation methods of recent real options
application papers.
The perspectives of the project will guide the choice of which volatility estimation
method to use. Each method has its limitations and an alternative approach is therefore
needed. Monte Carlo simulation comes the closest of any of the five methods to
accurately estimate project volatility itself, but from a statistical point of view, current
Monte Carlo simulation approaches are inadequate. So, a new volatility estimation
approach uses Monte Carlo simulation to value the investment opportunity correctly is
developed.
18
Table 23. Real options volatility estimation methods
Estimation method Papers Volatility factor
Historical volatility of the
underlying asset
Kelly(1998)
Titman(1985)
Smith & McCardle(1998)
Cortazar & Casassus(1998)
Takizawa & Suzuki(2004)
Cortazar et al.(1998)
Moel & Tufano(2002)
Davis (1998)
Smit (2003)
Kemna (1993)
Quigg(1993)
Armstrong et al.(2004)
Gold price
Land
Oil
Copper
Construction cost
Copper
Gold returns
Precious metal
Flight demand
Oil
Real estate price
Oil
Historical volatility of the
compatible assets
Insley & Rollins(2005)
Benaroch & Kauffman(1999)
Lumber price,
pointofscale transactions
Historical volatility of the
traded asset
Nembhard et al(2005)
Bollen, Nicolas P. B.(1999)
Newton and Pearson(1994)
Hemantha & Park(1999)
Miller & Park(2004)
Miller & Park(2005)
Annualized NPV
Demand of the product.
Expected NPV
A stock price.
A stock price.
A stock price
Historical volatility of the
industrial group index
Cassimon et al.(2004)
Jensen & Warren(2001)
Teisberg(1994)
Pharmaceutical industry
6 ecommerce companies
6 regulated firms
Monte Carlo simulation
Nembhard et al (2003)
Miller, Choi & Park.(2004)
Cobb & Charnes(2004)
Herath & Park(2002)
19
2.4 Bayesian Revision Processes
One of the main advantages of real options is that it can lessen the risk of a huge
irreversible investment by allowing decision makers to take the time to observe future
market movements. Under the traditional options framework, the ?waitandsee? strategy
allows a company to wait to proceed until the market is more favorable. However, in the
real world companies take action to resolve uncertainty once the option is taken in order
to improve their chances of making a profit. The information they obtain will effect the
investment decision.
Bayesian statistics are widely used to revise prior beliefs after observing sample
information. Miller and Park (2005) studied the impact of learning on a multistaged
investment scenario with an assumption of the BS model. In their study, they used the
normal conjugate distribution to compute the value of the acquired information. However,
the real options valuation usually uses the binomial lattice with an assumption that the
future outcome is discrete and not continuous. This is because real options has a kind of
American options perspective. By using the binomial lattice, it is also possible to use any
kind of distribution for option valuations. Therefore some kind of revision process for the
general case is required for the learning real options framework.
In order to develop general processes of volatility revision in case the initial
volatility of the options are estimated by Monte Carlo simulation, it is necessary to
understand the characteristics of a special type of Bayesian revision process, Dirichlet
distribution. Prueitt and Park (1992) presented a method for uncertainty resolution in
generalized cases using Dirichlet revision process. The basis of their approach is to
develop discrete approximations to continuous prior beliefs, record observed samples,
20
and place the observations in discrete categories that correspond to distributions in a
Bayesian framework.
2.5 Decision Timing in Real Options
After deciding to retain an option, the investors must decide the best timing of
exercising or divesting the option in order to maximize their profit. Since the investment
in the project has irreversible characteristics, deciding on the optimal investment timing
is crucial. In financial put option theory, the timing to exercise is defined as the point at
which the value of immediate exercise is higher than that of holding the option until its
expiration date. Financial put option pricing holds that it is best to exercise when the
current value of the project is higher than the expected value of the future flexibility.
However, it is known and proven that early exercise is never optimal in the financial call
option theory, which is applied to the major real options valuation model.
As Copeland and Tufano (2004) mentioned, defining the optimal exercise timing
of the real option is a crucial factor in making real options actually work in the real world.
They suggested that failing to exercise real options on time reduces the value of the
projects much less than predicted. However, in spite of the importance of this issue, it has
not been widely researched.
Brennan and Schwartz (1985) developed an evaluation model for deciding the
optimal investment timing to continue or abandon a mining project by setting stochastic
output prices. McDonald and Siegel (1986) studied a method to determine the optimal
timing of investment in an irreversible project when the benefit and cost of the project
follow Geometric Brownian Motion. By using the simulation technique, they indicated
21
that for riskaverse investors, it is optimal to wait until benefits are twice the investment
costs.
Yaksick (1996) suggested a method for computing the expected exercise timing of
a perpetual American option, and Shackleton and Wojakowski (2002) developed a
numerical expression for computing the expected return and for finding the optimal
timing to exercise real options by using the riskadjusted stopping time method, which is
based on the actual probability distribution of payoff times. Rhys, Song, and
Jindrichovska (2002) summarized recent developments in the topic of ?The timing of real
option exercise.? They reported that only a few studies have been conducted to analyze
this problem, but some progress is being made in the research.
Because capital investments are irreversible, deciding on the best timing of the
investment is one of the most important factors in real options. Some researches have
tried to define the decision timing, but most of their studies have been restricted to
specific cases which can not be widely applied in real world decisions. The previous
studies did not give helpful information which supports the decision of a company during
the option life. A map which guides the rules of action for the real options is one of the
most important information for decision makers, so new investment decision rules of the
real options should be developed.
22
CHAPTER 3
ESTIMATING PROJECT VOLATILITY
USING MONTE CARLO SIMULATION IN REAL OPTIONS
Abstract
Among the five general variables of real options, volatility is one of the most
critical, and it is generally considered to be the only stochastic variable. However,
estimating the volatility of the underlying project is rather problematical, and the
difficulty involved makes some CEOs hesitate to use real options in their analysis. The
five volatility estimation methods that have been used up to this point were discussed in
Chapter 2.
Among those five methods, Monte Carlo simulation is considered fundamental,
and it is used for estimating the volatility of the future project opportunity. This research
suggests a new way of volatility estimation called Reversed Monte Carlo Simulation
(RMCS). A mathematical demonstration indicates that RMCS correctly estimates project
volatility.
3.1 Introduction
There are mainly five volatility estimation methods in previous real options
applications These are 1) considering the historical volatility of the underlying asset, 2)
23
adapting the historical volatility of the compatible asset, 3) using the volatility of the
company?s stock price, 4) applying the volatility of the industrial group, and 5) the
volatility generated by Monte Carlo simulation. Among the five methods, a Monte Carlo
Simulation based volatility estimating method is considered as a fundamental method to
estimate project volatility, and are used for estimating the volatility of the future project
opportunity. However, with a statistical point of view, the current simulated volatility
does not represent the correct volatility of the underlying project; generally it is over
estimated than the correct volatility. Therefore, we have adapted the simulation and the
statistical analysis to develop a new volatility estimation method.
The remainder of this research is organized as follows. Section 2 reviews the
previous applications of Monte Carlo simulation to real options. In section 3, the
statistical meanings of the current Monte Carlo simulationbased volatility estimation
methods are examined by a traditional concept of volatility in option theory. Section 4
presents a new way of estimating project volatility by considering the option life and the
simulated project value. In the section a numerical example of estimating volatility
through Monte Carlo simulation is demonstrated and compared to the current method.
Finally a summary of the new method and some suggestions for future research are
followed in section 5.
3.2 Monte Carlo Simulation in Real Options
3.2.1 Applications of Monte Carlo Simulation in Real Options
Monte Carlo simulation is used with the assumption that the DCF analysis is
possible for the project because the simulation needs the future cash flow distribution of
24
the project in order to estimate the volatility. The historical data of the project or some
assumptions are used for estimating the distribution of the input parameters in this
simulation approach.
Copeland and Antikarov (2003) and Mun (2006) suggested the same concept at
the standard deviation of the rate of return distribution as the volatility of the project.
Before running the simulation, they defined the annual return of the project as the ratio
between the logarithmic value of
1
PV and
0
PV . In this method, it is necessary to set the
denominator
0
PV as a constant variable to simulate the volatility of the project return.
Herath and Park (2002) simulated the project volatility in a different way. They
simulated both the denominator and the numerator with the assumption that the project
cash flows are independent each other. Cobb & Charnes (2004) simulated the real options
volatility in case of correlated cash flow by further developing Hearth and Park?s model.
Each of these methods has its own limitations that prevent it from serving as a general
technique in the many different real options environments.
3.2.2 Volatility Estimation through Monte Carlo Simulation
In most cases, there are multiple uncertainties involved in the underlying project.
Thus it is almost impossible to find the twin securities or tradable assets which would be
necessary if a manager wanted to use the volatility of the financial assets to measure the
volatility of the project. Therefore, a Monte Carlo approach can be useful in estimating
the project?s volatility in real options.
A few studies, such as Herath & Park (2002), Copeland & Antikarov (2003), Mun
(2006), Miller, et al. (2004), Brand?o et al (2005), and Godinho (2006), mentioned the
Monte Carlo Simulation technique as a volatility estimation tool for real options. The
25
fundamental assumption of the Monte Carlo Simulation approach is that it is possible to
estimate future cash flows.
To explain the volatility of the project return, it is first necessary to define some
notations.
t
CF is the cash flow of the project during t
th
year of the total project life T,
and
t
PV is the present market value of the future cash flows from year t+1 to T. Then
the present value of the project
t
PV with a continuous compounding policy can be
defined,
1
exp{ ( )}
T
tk
kt
PV CF r k t
=+
=????
?
(31)
Then the present worth of the project at time t,
t
PW is defined as the sum of
t
PV plus
t
CF .
ttt
PW PV CF=+ (32)
Let z be a random variable that represents the continuous rate of return of project
between time t and t+1. Then,
1
ln
t
t
PW
z
PW
+
??
=
??
??
(33)
The Copeland and Antikarov model (CA model) simulates the standard deviation
of logarithm of the difference between
1
PW and
0
PW by considering the expected
value of
0
PW as a fixed number. So, the volatility of the project is computed by the
standard deviation of the expression
11
00
ln ln
[] []
PW PW
z
EPW EPV
????
==
????
????
(34)
26
While the CA model considers the expected value of
0
PV as a fixed number, the
Herath and Park model (HP model) considers
0
PV as a random variable. That indicates
that both the denominators and the numerators are simulated simultaneously to estimate
the volatility of the project return. Then the simulation processes are the same with the
present project value and that of the first year.
Lately Godinho (2006) pointed out the problems that both of the above methods
have a tendency to overestimate the project?s volatility. He suggested an alternative
simulation technique to estimate volatility: a twolevel simulation model that correctly
represents the amount of project risk. He uses an example to illustrate his claim.
3.3. CA vs. HP Volatility
In this section, the relation between CA and HP volatility is demonstrated. Then, a
new volatility estimation method for the real option is suggested after discussing some
conditions and the assumption for the CA model.
3.3.1 Defining Relations between CA Model and HP Model
In the CA model, the volatility of the project return during the first year is
summarized to the standard deviation of
0
ln( )PV distribution, while the HP model has
2 times higher volatility than the CA model according to the statistical procedures
shown below. Following is the mathematical meaning of the simulated volatility, which is
the standard deviation of the two models. In HP model,
0
ln( )PV and
0
ln( ) 'PV are
independent identical distributions.
27
CA model
()
()()
() ()
()
1
10
0
00
00
0
ln ln ln[ ( )]
[]
ln ln[ ( ]
ln ln( ) ln[ ( ]
ln
r
r
PW
Var Var PW E PV
EPV
Var e PV E PV
Var e PV E PV
Var PV
??
??
??=?
????
??
??
??
=??
? ?
=+?
? ?
??=
??
?
0
[ln( )]Var PV? =
HP model
()
()
()
1
10
0
10
00
000
ln ln ln( ')
'
[ln ] [ln( ')]
[ln( )] [ln( ')]
[ln( )] [ln( ')]
2 [ln ] and ' are i.i.d.
r
PW
Var Var PW PV
PV
Var PW Var PV
Var e PV Var PV
Var PV Var PV
Var PV PV PV
??
??
??=?
????
??
??
??
=+
=?+
=+
= ?
?
0
2[ln()]Var PV? =?
The volatility derived from the HP model is always 2 times higher than the
volatility of the CA model. Godinho (2006) reveals the computation results of the
standard deviation of the HP and CA models using a numerical example. The review
proves that the volatility of the HP model is nevertheless 2 times higher than the
volatility of the CA model.
Because the relationship between the two models is explicit, from this point
forward only the CA model will be considered in order to explain the meaning of the
current volatility.
28
3.3.2 Statistical Interpretation of the Present Models
In real options valuation, we employ techniques which were initially developed
for financial options pricing. So, it is necessary to understand a fundamental concept of
the latter method in order to investigate the volatility of real options. In a financial option,
volatility indicates ?the flexibility of the stock price during a certain time period,? and it
does not represent the stock price distribution at certain point of time.
In order to confirm that the models satisfy the definition of financial options
volatility, we first examined the volatility of the model changes by time changes. The
definition of the volatility of the CA model does not include the value changes. It also
does not reflect any time consideration for the volatility, which is theoretically known to
be t? by the financial options pricing model. In order to demonstrate the problem, a
2year defer option was used. The simulated volatility of the option could be computed
by CA model through,
()
1
0
0
ln ln
[]
PW
Var Var PV
EPV
??
??
? ?=
????
? ?
??
??
(35)
Also, the flexibility during the two years can be computed,
()
()()
() ()
()
2
20
0
2
00
2
00
0
ln ln ln[ ( )]
[]
ln ln[ ( ]
ln ln( ) ln[ ( ]
ln
r
r
PW
Var Var PW E PV
EPV
Var e PV E PV
Var e PV E PV
Var PV
??
??
??=?
????
??
??
??
=??
? ?
=+?
? ?
??=
??
Since the calculated volatilities of above two cases are the same to the volatility of
the one year return, the suggested method has some deficiencies that prevent its
29
calculations from being accurate. The meaning of the volatility according to the CA
model represents the standard deviate on project returns during the whole option life. As
a result, the volatility estimated by the CA model is only correct when the option life is
one year. Thus if the option life is longer than one year the volatility is overestimated, and
if the option life is shorter than one year, it is underestimated.
The second checking point is the statistical meaning of the standard deviation of
the model. The term
0
ln( )PV does not mean the return during a certain time period. It is a
natural log of a present value distribution. Therefore, the meaning of the standard
deviation of CA simulation is the standard deviation of the log distribution of
0
PV , not
the return distribution. The logarithm of the standard deviation of the present value
distribution is computed using the central limit theorem, and one can assume that the
distribution of the project value
0
PV is normal with
2
(, )N ? ? . Then if we take the
log
0
PV distribution, the distribution has parameters of mean m and standard deviation s
which is computed by (36), and (37). (Dixit and Pindyck)
2
2
1
ln( ) ln( 1)
2
m
?
?
?
=? + (36)
2
2
ln( 1)s
?
?
=+ (37)
From the equation (37), it is clear that the volatility estimated by the CA model is
simulating the log of the present value of the project.
In summary, Copeland?s volatility is a transformed standard deviation of the
present project value, and not of the project return. The model does not consider the time
frame, a point which is critical for the option valuation model.
30
3.4 Developing a Reversed Monte Carlo Simulation
This section Dixit and Pindyck (1994) proposed the relationship between the
volatility of the underlying asset (? ) and the volatility of future outcomes of the project
(
T
? ) by studying the perspectives of the normal and the lognormal distribution (see
Figure 31). Moreover, following Ito?s lemma and the general relationship between the
parameters of lognormal and normal, they set the equations (38) and (39) to estimate
0
rT
T
Ve? =? (38)
2
222
0
(1)
TrT
T
Ve e
?
?
?
=?? ? (39)
In order to develop the new method of volatility estimation using Monte Carlo
Simulation, the fundamental of options valuation is applied: the option premium is
calculated based on the value of the project at the end of option life.
Figure 31. Relation between
T
? and ?
0
V
T
u e
?
=
T
d e
??
=
(,)
TTT
V ? ??
T
31
Figure 32. Future value simulation
Using the equation (38) and (39), it is possible to estimate the volatility of the
project if we know the parameters
T
? and
T
? . In order to estimate the future value of
the project in year T, Monte Carlo Simulation technique is used. Figure 32 illustrates the
method to simulate the future worth of project cash flows with a riskfree discount rate.
The future time T represents the end of the option life, not the end of the project life. If
we let the parameters of future cash flow distribution,
T
? , and
T
? , respectively, then
values of those two variables are computed by simulation.
Then the unknown variable ? can be calculated by the equation (310).
2
22
0
ln ( ) 1
T
rT
Ve
T
?
?
??
+
??
??
= (310)
3.4.1 Mathematical Proof for RMCS
The core of real options is taking extra time to make decisions concerning the
irreversible investment. The real option premium is the price of taking the opportunity to
(,)
TT
FV ? ?
Simulation
Option life: T years
32
defer decisions. In the real options valuation, generally the present value of the project is
considered deterministic, and it will either increase or decrease according to its annual
volatility ? . The terminal option premium is calculated by the estimated project value at
the end of the option life. If it is possible to estimate the terminal project value, then it is
also feasible to compute the annual volatility by Reverse Monte Carlo Simulation, as
suggested in this research.
Consider a project which has a present value
0
V . Assume the annual return of the
project (
i
r ) is an identical independent normal distribution, and the known ? and
2
?
are the mean and variance of the return, respectively. Further assume that production will
begin some years after the investment is made. The value of the project (
i
PV ) at the end
of each year i can be expressed by the multiple formation of the project?s present value
and the annual return of the project, as below.
1
212
3312
10
10
20
r
rrr
2
rrrr
3
PV V e
PV PV e V e e
PV PV e V e e e
=
=?=
=?=
null
By using simulation technique, the return of the project during the initial time and
the designated timing is defined by
0
0
ln( ) ln( ) ln(V )
V
i
i
PV
PV=? (311)
If we let '
i
? and '
i
? be the mean and the variance of the return during the
period expressed as equation (311), then
33
At i = 1:
1
'? :
1
1
1
10
0
00
01 0
E[ln( )] E[ln( ) ln(V )]
V
E[ln(V ) ln(V )]
E[ln(V ) ln( ) ln(V )]
E[ln(V )] E[ ] E[ln(V )]
r
r
PV
PV
e
e
r
?
=?
=?
=+?
=
2
1
(')? :
1
1
1
10
0
0
0
2
Var[ln( )] Var[ln( ) ln(V )]
V
Var[ln(V )]
Var[ln(V ) ln( )]
r
r
PV
PV
e
e
?
=?
=
=+
=
At i = 2:
2
'? :
2
0
E[ln( )] 2
V
PV
?=
2
2
(')? :
12
12
2
20
0
0
0
2
Var[ln( )] Var[ln( ) ln(V )]
V
Var[ln(V )]
Var[ln(V)ln()ln( )]
2
rr
rr
PV
PV
e
ee
?
+
=?
=
=++
=
At i = 3:
3
'? :
3
0
E[ln( )] 3
V
PV
?=
2
3
(')? :
123
312
3
30
0
0
0
2
Var[ln( )] Var[ln( ) ln(V )]
V
Var[ln(V )]
Var[ln(V ) ln( ) ln( ) ln( )]
3
rr r
rrr
PV
PV
e
eee
?
++
=?
=
=+++
=
34
At i = t:
'
t
? :
0
E[ln( )]
V
t
PV
t?=
2
(')
t
? :
12
0
0
0
2
Var[ln( )] Var[ln( ) ln(V )]
V
Var[ln(V )]
t
t
t
rr r
PV
PV
e
t?
+++
=?
=
=
null
Thus the annual volatility of a project ? which bears cash inflow at the end of
the t
th
year will be;
'
t
t
?
? = (312)
The variance of the return distribution during the initial and the terminal time
period
2
(')
t
? is computed by (39).
22
22 2
0
'ln( 1)ln( 1)
TT
t rT
T
Ve
??
?
?
=+=+
?
(313)
Since the equation (310) generated from the RMCS and the equation (313) from
the mathematical definition are the same, the suggested RMCS model can be used as the
volatility estimation tool of the project in case of ?t? years option life.
3.4.2 Numerical Proof for RMCS
In order to show that the estimated volatility from RMCS correctly represents the
risk of the project, it is good to present a simple project with a known return volatility. In
our example, the future cash flows are estimated by the current point estimation of the
project value and the random return distribution. All the iterations are recorded to
compute the mean and the variance of future cash flows. The result of this simple project
35
proved that the estimated volatility calculated through RMCS is the same as the pre
assigned volatility. This is an obvious improvement over the CA model, which
consistently computes a higher volatility.
Now consider a project which has a 3year defer option. The expected present
value of the project is $1,000, and its annual volatile return follows a random distribution
2
N(0.1,0.2 ) . It is necessary to simulate the cash inflow at the end of the 3
rd
year because
we deferred the project for 3 years.
The formula to compute the project value at the end of the 3
rd
year is
33 312 12
20
V 1,000
rr rrr rr
3
PV PV e e e e e e e=?= =
By using Monte Carlo Simulation, one finds the mean and variance of
3
PV
which are the parameters for volatility estimation. Table 31 is the result of 1,000
iterations using Microsoft Excel. The simulated mean and standard deviation of
3
PV is
1415.56 and 504.46 respectively
Table 31. Sample example to simulate the
3
PV distribution
Iteration
1
r
2
r
3
r
3
PV
1 0.4348 0.0809 0.4621 2658.95
2 0.0289 0.0469 0.4868 603.56
?
1,000 0.0585 0.3171 0.0406 1397.9
Mean 0.0901 0.1009 0.0961 1415.56
Standard
Deviation
0.2019 0.2015 0.1951 504.46
36
The volatility of the project is computed as 20% which is the same as pre
assigned annual volatility using the mean and variance shown in table 31 with RMCS.
2
2
504.46
ln( 1)
1415.56
20%
3
?
+
==
In contrast, the CA model calculated the simulated volatility as 34.58%.
We then borrowed the data in table 32 from Copeland and Antikarov (2003) and
used a continuous (rather than annual) compound scenario to demonstrate the difference
between the RMCS model and the CA model.
Table 32. Cash flows from Copeland and Antikarov (2003)
0 1 2 3 4 5 6 7
Price/unit 10 10 9.5 9 8 7 6
Quantity 100 120 139 154 173 189 200
Variable cost/unit 6.0 6.0 5.7 5.4 4.8 4.2 3.6
Fixed costs 20 20 20 20 20 20 20
Depreciation 229 229 229 229 229 229 229
EBIT 151 231 280 305 303 284 240
Cash taxes 61 93 112 122 121 114 96
Depreciation 229 229 229 229 229 229 229
Capital expenses 1600 0 0 0 0 0 0 0
Working capital 200 40 24 13 0 13 24
Cash flows 1600 119 327 373 399 411 412 397
37
The option to defer the project for two years is considered the uncertain
investment environment. From the table, we derived the distribution of the present value
of the project with a riskadjusted interest rate. The simulated distribution has parameters
2
00
( , ) (1464.28,88758.15)?? = , and the present value of the project
0
V is 1464.28 when
valuing the option. The CA volatility of the project is calculated as 20.8% by simulation
and 20.2% by the equation (37).
In the RMCS framework, the project value will be recorded at the end of the 2
nd
year because the project will be deferred two years. With risk simulation, the distribution
of the project value is recorded to
2
1,861.46? = and
2
2
143,192.3? = . The volatility of
the project is found to be 14.7% by the equation (313).
3.5 Conclusion Remarks
The real options application presents an opportunity to improve strategic
investment decisions in an uncertain environment. The volatility of the project value is
one of the most critical and the only stochastic parameter in traditional real options;
therefore, accurately estimating it is essential to valuing the future investment opportunity.
However, estimating the volatility of a project is not an easy task because there is no
exact historical data to use in the calculations.
Among the various methods of estimating volatility, Monte Carlo simulation
alone captures the flexibility of the project itself. However, the statistical definition
indicates that the current simulationbased volatility estimating models have some
deficiencies that prevent them from accurately representing the true volatility of a project.
38
This research developed an alternative method of Monte Carlo Simulation, called
Reversed Monte Carlo Simulation (RMCS), in order to enhance the use of real options
valuation in strategic investment decisions. A mathematical demonstration indicated that
RMCS correctly estimates project volatility.
In order to improve the reliability of the volatility derived from RMCS,
generating a correct distribution of the future cash flows is the remaining challenge.
39
CHAPTER 4
BAYESIAN FRAMEWORK TO REVISE THE VOLATILITY
OF REAL OPTIONS
Abstract
This paper studies a Bayesian process in order to apply the learning real options
framework into the strategic investment decisions. In the learning real options framework,
a firm works actively to improve its decision by resolving some uncertainty of the project
during the option life, while the traditional options framework only consider a passive
consequences of waiting. The volatility of the underlying project is estimated by the
Monte Carlo Simulation with the random factors in the cash flow distributions. Re
simulating the revised cash flows with the acquired information of the random factors
regenerates new volatility of the risky project. A Dirichlet conjugate processes are applied
to estimate the posterior distributions of the random factors.
4.1 Introduction
Recently, real options valuation was developed as a method of capital budgeting
decision making. Real options analysis gives a company an opportunity to improve its
strategic investment decisions in an uncertain environment. Under the traditional options
framework, the strategy of option, ?waitandsee,? means that a company will wait to
40
proceed until the market is more favorable. However, a firm works actively to improve its
decision by resolving some of the uncertainty of the project during the option life, while
the traditional options framework only considers the passive consequences of waiting.
The information obtained during the waiting period will effect investment decisions.
Bayesian statistics are widely used in the area of revising prior beliefs after
observing samples. Miller and Park (2005) studied the impact of learning on a multi
staged investment through the BS model. They consider the normal conjugate
distribution in computing the value of the acquired information. However, the real
options valuation usually uses the binomial lattice with an assumption that the outcome
of the future is discrete. This is because the real options has an American options
perspective. If we use the binomial lattice to value the real options, it is also possible to
use any kind of distribution for option valuations. So some kind of revision process for
the general case is required for the learning real options framework.
In this study, Bayesian processes are developed to revise the initial volatility of a
risky project. It is also assumed that only threepoint estimates of the random factors of
the future cash flows are available to analyze the investment decision, because three
point estimates (an optimistic point, a pessimistic point, and a most likely point) are
commonly used for measuring the risk of a project.
After briefly introducing the processes for converting the threepoint estimate into
a beta distribution of the project value, an initial volatility of the project is determined.
Once new sets of information are acquired, a Dirichlet conjugate framework, is utilized to
revise the random factors of the cash flows. Resimulating with the posterior distribution
then generates a new volatility of the risky project.
41
The remainder of this paper is organized as follows. Section 2 reviews the
processes of continuous approximation from the threepoint estimates. Section 3 frames
the volatility revision processes through the Dirichlet conjugate process and Monte Carlo
Simulation. Section 4 suggests a numerical example to show the whole developed
processes. Finally, Section 5 provides concluding remarks and suggestions for future
study.
4.2 Estimating a Continuous Random Distribution from Threepoint Estimates
Decisionmakers begin their assessments by estimating the future aspects of the
project. Cash flow, one of the most important of these aspects, is particularly difficult to
estimate. When there is not a great deal of uncertainty surrounding a project, it is possible
to use probability distributions of future cash flows in the estimation process. However, if
the project is more uncertain, then only three possible outcomes can be estimated without
probability information. If the project is extremely risky, only the outcome intervals can
be estimated. This study assumes that it is possible to estimate at least two points of the
random factors.
Park and SharpBette (1990) stated that it is possible to express most of the
random distributions when the outputs are bounded by beta distributions. Prueitt and Park
(1992) applied the processes for the phasecapacity expansion project.
The standard beta distribution for random variable X is defined as
(2)
() (1 )
(1)(1)
f xxx
? ?
? ?
??
?++
=???
?+??+
, 01x? ? (41)
42
0
0.5
1
1.5
2
2.5
3
0
1
Curve number ? ?
1 3 3
2 1 1
3
22+ 22?
4 0.2 0.2
5 0 1
6 1 0
7 0 0
Figure 41. Shapes of Beta distribution in a specific parameters
1
2
3
4
5
6
7
43
The shape of beta distribution is determined by the parameters ? and ? . The
mean, mode, and variance of beta distribution are also defined by these two parameters,
and figure 41 demonstrates the shapes of the distribution for the different parameters.
1
2
mean
?
??
+
=
++
()
o
mode M
?
? ?
=
+
2
(1)(1)
(2)(3)
variance
? ?
?? ??
++
=
++ ++
Since the standard beta distribution is bounded between 0 and 1, some
adjustments are needed in order to transform a specific distribution which has the three
points H, L, and M to the standard beta distribution. Equation (42) is the transformed
beta distribution with the higher bound H, the lower bound L, and the mode M.
yL
x
HL
?
=
?
()yL HLx=+ ?
1
(2)
() ( ) ( )
(1)(1)( )
f yyLHy
HL
? ?
??
? ?
??
++
?++
=???
?+??+? ?
, LxH?? (42)
The mean, variance, and mode of the transformed distribution are indicated as
follows:
()(1)
[]
2
HL
Ey L
?
??
??+
=+
++
(43)
2
2
()(1)(1)
[]
(2)(3)
HL
Var y
??
?? ??
??+?+
=
++ ++
(44)
44
[]
LH
Mode y
? ?
? ?
+
=
+
(45)
The logic for determining a beta approximation from the threepoint estimates
uses standardized beta distribution. Since the only known value of a transformed
distribution is the Mode (
o
M L
M
HL
?
=
?
), there is still not enough information to compute
? and ? . To solve this problem, an assumption is made that onesixth of the range can
be used as a rough estimate of standard deviation for the distribution. Then the equation
(46) is found.
323223
(7 36 36 ) 20 24 0
oo o o o
MM M M M???+? + ? ? = (46)
Finally, the parameter ? and ? , which determine the shape of the beta
distribution, can be determined by solving the equation (46).
4.3 Bayesian Revision Processes
Bayesian revision is a method for developing a posterior probability distribution
by integrating observations with a prior belief. Because it adds observed information to
the estimation process, it can be very useful in resolving the uncertainty of a project. The
basis of the revision process is the calculation of a posterior probability distribution based
on the prior distribution and the observed information. There are two random variables to
be considered in the revision process. One is related to the event for determining the prior
probability. The other, called a likelihood function, is related to the samplings process.
The simplified concept of the Bayesian theorem is demonstrated in figure 42.
45
Figure 42. Bayesian revision processes
If the prior distribution is discrete, the process can be applied very simply.
However, if the prior distribution is continuous, two cases are possible: discrete
approximation or special types of likelihood function. In a case of continuous prior
distribution, it is necessary that the prior distribution and the datagathering process share
a similar perspective. If a prior random distribution is conjugate with respect to the
sampling process, the posterior distribution is the same as the prior distribution. There are
three well known conjugate processes: 1) a normal prior distribution with a normal
sampling process, 2) a beta prior distribution with a binomial datagathering process, and
3) a gamma prior distribution with a Poisson sampling process.
Prueitt and Park (1992) presented a method for resolving uncertainty in
generalized cases. They argued that one must first develop discrete approximations
compared to continuous prior beliefs, then record the collected sample, and lastly, place
the observations in discrete categories that correspond to distributions in a Bayesian
framework. They stated that even though the nature of the NPV of a project?s cash flow is
a discrete random variable, the circumstances may preclude handling the cash flow as a
continuous variable. Based on their research, three conditions for the application of the
46
discrete approximation are presented below.
1. The uncertainty about the cash flows may not fit a particular probability density
function as;
(11) the prior distribution is based on a histogram that does not fit any
particular distributions,
(12) varying economic or environmental conditions may create multiple modes
condition for the cash flow,
(13) the cash flow may be developed on an incremental basis, and alternative
reactions to specific economic conditions may lead to irregular
distributions.
2. The prior beliefs and the sampling process may not form one of the natural
conjugate distributions.
3. There may be a limited amount of available sample information, and it may not be
appropriate to model them as a specific distribution.
In light of the above conditions, an adjustment is required in order to convert the
continuous range of outcomes to a set of discrete intervals to be made before applying the
Bayesian revision process.
4.4 Volatility Revision Framework in Real Options
The uncertainty of the future is one of the main reasons for purchasing real
options. The investors want to postpone their irreversible investment decision in such
volatile circumstances by paying some premium to take the priority to act. Therefore,
most of the real options applications concentrate on valuing the project opportunity by
47
assuming that the volatility of a risky project is not changed. However, most of the
practitioners are interested in the actions they can take after retaining the options, such as
tracking the uncertainty resolutions and determining the timing of the irreversible
investment. This new information will affect the investment decision. In order to apply
Bayesian statistics for revising volatility, it is assumed that the initial volatility of an
option is estimated through Monte Carlo simulation as demonstrated in chapter 3.
The present value (PV) of a given project is treated as a probability distribution
with unknown parameters because of the uncertainty of the project. While the net present
value analysis uses a riskadjusted interest rate when discounting the future cash flows,
the PV is defined as the sum of the discounted cash flows with a riskfree interest rate.
Although the prior distributions are assumed to be beta by threepoint estimation, the
prior distributions of the random factors do not need to have a specific distribution in this
case.
The distributions of the risk factors of the cash flows will be modeled with a
discrete approximation so that the Dirichlet conjugate processes can be used to define the
probability distribution. The Dirichlet distribution is a family of multinomial distributions
parameterized by the vector ? of nonnegative real numbers. It is a conjugate prior
distribution of the multinomial in Bayesian revision statistics. The probability density
function of the Dirichlet distribution which has k outcomes is:
1
12, 1 2
1
1
(, , ; , , , )
()
k
K
KK
k
fxx x x
L
?
?? ?
?
?
=
=
?
nullnull
48
Figure 43. Developing a discrete approximation
where,
1
1
1
()
0, 0, 1, ( )
K
k
K
k
kk k
K
k
k
k
x x and L
?
??
?
=
=
=
?
?? = =
??
?
??
??
?
?
?
The first step in the Bayesian revision process is to transform the initial
distribution, which is based on threepoint estimation, by dividing the original
distribution into discrete, nonoverlapping intervals. The intervals are not required to be
equal in length, but are presented as such for the sake of clarity. Once the intervals are
defined, corresponding probabilities of the intervals,
k
s? , are determined. Figure 43
illustrates the concept for the development of the discrete approximate distribution.
Once the initial interval probabilities are determined, the
k
s? of the Dirichlet
distribution are calculated by considering the information quality factor (IQF), which
represents the amount of belief for the prior probability. For example, if the prior beliefs
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
12345678
Original distribution
Discrete approximation
49
are considered to be twice the sample amount acquired through the study period, then the
IQF = 2. So the
k
s? are determined by:
IQF
kk
? ?=? for 1, ...,for k K=
The next step is to determine the posterior distribution by conjugating the
observation results,
k
x s , to the prior probability distribution. In some cases, it is
necessary to adjust the weight of the sample as well. Then the posterior probability
"
k
s?
are computed by:
"
11
kk
k KK
kk
kk
x
x
?
?
?
==
+
=
+
??
After the posterior probabilities of all the random factors are determined through
the Bayesian conjugate processes, a new distribution of the present value of the project
can be simulated. Then the posterior volatility of the project is estimated through the
method suggested in chapter 3.
The processes demonstrate how to revise the volatility of real options if a new set
of sample information is obtained. By revising the volatility of real options, the investors
can more accurately determine the optimal timing of their irreversible investment in the
project. The previous Bayesian real options framework just considers the value of
learning by uncertainty resolution from the sampling. However, in the real world it is
very possible that uncertainty will increase after the samples are taken, and the proposed
methods can result in better decision making.
50
4.5 Numerical Example
In this section a numerical example demonstrates the proposed volatility revision
processes. Since the initial volatility estimation is based on Monte Carlo Simulation, the
uncertainty of the future cash flow of a project is assumed as threepoint estimates.
Project descriptions
XYZ Chemical Company, a small manufacturer of a car bumper with
carbohydratereinforced plastic, considers a new project for the next 5 years. Even though
many uncertainties are expected in the future, the most possible cash flows are estimated
in table 41. To develop the tables, the following assumptions are demonstrated.
Table 41. Cash flow estimation for XYZ Chemical Company
EOY 0 1 2 3 4 5
Income Statement
Revenue
Unit price 50 50 50 50 50
Demand 2,000 2,000 2,000 2,000 2000
Total revenue 100,000 100,000 100,000 100,000 100,000
Expenses
Unit variable cost 15 15 15 15 15
Variable cost 30,000 30,000 30,000 30,000 30,000
Fixed cost 10,000 10,000 10,000 10,000 10,000
Depreciation 25,000 25,000 25,000 25,000 25,000
Taxable income 35,000 35,000 35,000 35,000 35,000
Income taxes(40%) 14,000 14,000 14,000 14,000 14,000
Net income 21,000 21,000 21,000 21,000 21,000
Cash Flow statement
Operating Activity
Net Income 21,000 21,000 21,000 21,000 21,000
Depreciation 25,000 25,000 25,000 25,000 25,000
Investment Activity
Investment 160,000
Net cash flow 160,000 46,000 46,000 46,000 46,000 46,000
51
Table 42. Threepoint estimates for the random factors
Random Factor Pessimistic Most Likely Optimistic
Demand (EA) 1000 2000 4000
Unit Price ($) 40 50 55
Unit variable cost ($) 11 15 16
Fixed cost ($) 8,000 10,000 15,000
1. There will be no salvage value for the invested equipment at the end of the project.
2. The equipment will be fully depreciated using straight line deprecation method.
3. The riskfree interest rate 5% is assumed, and it does not change during the project.
4. Only the component described in table 41 is considered for the decision making.
5. The unit price, demand, unit variable cost, and fixed cost are the random variables,
and it is possible to estimate their optimistic, pessimistic, and most likely point.
Table 42 is the summary of their estimates.
Beta distribution models
The distributions of the random factors are defined by the following processes in
order to convert the threepoint estimates to the beta distributions. Table 43 shows the
standardized mode, ? , and ? of the random factors of the project scenario, and
Figures 44 ? 47 illustrate the shapes of the distributions.
Table 43. Parameters of beta distributions
Random Factor ? Mode
?
Demand (EA) 1.819 0.333 3.643
Unit Price ($) 3.643 0.667 1.819
Unit variable cost ($) 3.565 0.8 0.891
Fixed cost ($) 1.469 0.286 3.674
52
Demand distribution Beta(1.819, 3.643, 1000, 4000)
0
1
2
3
4
5
6
7
8
.511.522.533.544.5
Values in Thousands
V
a
lues
x
10^

4
Figure 44. Shape of the demand distribution
Unit price distribution Beta(3.643, 1.819, 40, 55)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
38 40 42 44 46 48 50 52 54 56
Figure 45. Shape of the unit price distribution
53
Variable cost distribution Beta(3.565, 0.891, 11, 16)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
10 11 12 13 14 15 16 17
Figure 46. Shape of the unit variable cost distribution
Fixed cost distribution Beta(1.469, 3.674, 8000, 15000)
0
0.5
1
1.5
2
2.5
3
3.5
7 8 9 10111213141516
Values in Thousands
V
a
lues
x
10^

4
Figure 47. Shape of the fixed cost distribution
54
Volatility estimation
In this section, only the computation of the project volatility is considered. The
mean and the variance of the project value is calculated by the @Risk simulation package.
The shape of the PV distribution using the riskfree interest rate, which is assumed to be
5% in this case, is shown in Figure 48. Then the volatility of the project in case of ?T?
years of option life will be determined by RMCS. In this case, the volatility of the project
is approximately 8% with an assumption that the company wants to defer the project in 2
years, and estimate the cost of taking the option position.
Distribution for PV
Mean =
202368.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
120 164 208 252 296
Values in Thousands
Va
lue
s
in 1
0
^ 5
Figure 48. Simulated PV distribution
55
Table 44. New information with the initial threepoint estimates
Initial belief
Random Factor
Pessimistic Most Likely Optimistic
New
Information
Demand (EA) 1,000 2,000 4,000 2,400
Unit Price ($) 40 50 55 42
Unit variable cost ($) 11 15 16 14.6
Fixed cost ($) 8,000 10,000 15,000 12,000
Posterior volatility estimation
After taking the option for 1 year, the company collects new information for their
project shown in table 44. From the survey of the management group of the company,
the IQF of the initial belief is closer to 2. To make the computation simple, 10 equal
length intervals of the discrete approximation are assumed for all the risky variables. For
the demand distribution example, each interval has 300 events which transform it to the
discrete distribution. Then the median of the intervals are considered as the event of the
variable, and the probability of the intervals are recorded as a corresponding probability,
k
s? , of the event. Then the
k
? of the prior distribution is computed by
2
kk k
IQF? ??=? = .
The next step is to determine the posterior distribution by conjugating the
observation results,
k
x s , to the prior belief. In this case study, it is assumed that the
observation is considered as is. The corresponding probability
"
k
s? of the posterior
distribution is calculated after
"
k
s? is determined. Table 45 is the processes and the
results of the Dirichlet conjugate distribution with the data based on @Risk simulation
56
software. The @Risk simulation package delivered a slightly higher variance for the
project. With the computation through the equation (310), the volatility 9% for the future
option valuations is computed, which is 1% higher than the initial volatility.
Table 45. Revised discrete approximation for the risky variables
* Demand
k
k
?
kk
IQF? ?=?
k
x
"
k
?
"
k
? Events
1 0.0287 0.0573 0.0573 0.0191 1150
2 0.1265 0.2529 0.2529 0.0843 1450
3 0.1974 0.3948 0.3948 0.1316 1750
4 0.2223 0.4447 0.4447 0.1482 2050
5 0.1866 0.3732 1 1.3732 0.4577 2350
6 0.1306 0.2613 0.2613 0.0871 2650
7 0.0736 0.1471 0.1471 0.0490 2950
8 0.0279 0.0558 0.0558 0.0186 3250
9 0.0061 0.0123 0.0123 0.0041 3550
10 0.0003 0.0006 0.0006 0.0002 3850
Sum 1 2 1 3 1
* Price
k
k
?
kk
IQF? ?=?
k
x
"
k
?
"
k
? Events
1 0.0003 0.0006 0.0006 0.0002 40.75
2 0.0060 0.0120 1 1.0120 0.3373 42.25
3 0.0287 0.0573 0.0573 0.0191 43.75
4 0.0714 0.1428 0.1428 0.0476 45.25
5 0.1309 0.2618 0.2618 0.0873 46.75
6 0.1866 0.3733 0.3733 0.1244 48.25
7 0.2217 0.4434 0.4434 0.1478 49.75
8 0.2018 0.4035 0.4035 0.1345 51.25
9 0.1230 0.2460 0.2460 0.0820 52.75
10 0.0297 0.0594 0.0594 0.0198 54.25
Sum 1 2 1 3 1
57
Table 45. Revised discrete approximation for the risky variables (Con?t)
* Variable cost
k
k
?
kk
IQF? ?=?
k
x
"
k
?
"
k
? Events
1 0.0001 0.0003 0.0003 0.0001 11.25
2 0.0028 0.0056 0.0056 0.0019 11.75
3 0.0144 0.0287 0.0287 0.0096 12.25
4 0.0395 0.0789 0.0789 0.0263 12.75
5 0.0846 0.1692 0.1692 0.0564 13.25
6 0.1445 0.2890 0.2890 0.0963 13.75
7 0.2024 0.4048 0.4048 0.1349 14.25
8 0.2554 0.5108 1 1.5108 0.5036 14.75
9 0.2504 0.5009 0.5009 0.1670 15.25
10 0.0060 0.0120 0.0120 0.0040 15.75
Sum 1 2 1 3 1
* Fixed Cost
k
k
?
kk
IQF? ?=?
k
x
"
k
?
"
k
? Events
1 0.0528 0.1055 0.1055 0.0352 8350
2 0.1509 0.3017 0.3017 0.1006 9050
3 0.2173 0.4345 0.4345 0.1448 9750
4 0.2171 0.4342 0.4342 0.1447 10450
5 0.1707 0.3415 0.3415 0.1138 11150
6 0.1101 0.2203 1 1.2203 0.4068 11850
7 0.0563 0.1126 0.1126 0.0375 12550
8 0.0203 0.0406 0.0406 0.0135 13250
9 0.0043 0.0086 0.0086 0.0029 13950
10 0.0002 0.0004 0.0004 0.0001 14650
Sum 1 2 1 3 1
4.6 Conclusion Remarks
Once the investment opportunity is undertaken, tracking the market information
becomes one of the most important actions in determining the best timing for the follow
up strategic investment decisions. A Bayesian revision process is demonstrated in order to
update the volatility of the real options in cases where the initial volatility was estimated
58
through Monte Carlo Simulation. With an assumption of limited sample information, a
probability conjugate process for the general distribution case is applied to estimate the
posterior volatility of the project. The Dirichet conjugate distribution is applied for
estimating the posterior volatility of the project by transforming the continuous prior
distribution to the discrete distribution.
The previous Bayesian learning real options framework considers the value of
learning by uncertainty resolution from the sampling. However, the specific methods to
conjugate the sample information into the prior belief are not studied. This study
demonstrates that Dirichlet revision processes can conjugate the general prior distribution
with a very limited amount of sample information. Correct information on volatility and
the future aspect of the project will make the decisions for irreversible investments more
accurate.
More studies are still necessary to determine the importance of volatility in the
decision making process, because very little research has been conducted regarding
volatility revision after the options are purchased.
59
CHAPTER 5
DEVELOPING DECISION SUPPORT SYSTEM IN REAL OPTIONS
Abstract
One of the most critical issues in real options analysis is determining the optimal
timing of the irreversible investment during the life of the option. Research indicates that
failing to exercise real options on time reduces the projects? value much less than
predicted. However, the question of whether real option holders exercise their options
optimally has not been extensively researched. In this research, a new early decision rule
for real options is developed, and a simulation technique validates this new rule. The
result of the simulation indicates that the new decision rule gives higher profits than the
traditional rule in most cases.
5. 1 Introduction
Recently, managers have addressed the uncertainty in various capital budgeting
decisions by applying an options analysis to their evaluations of the project. This
evaluation technique, real options analysis, provides an opportunity to improve strategic
investment decisions in an uncertain environment, but the real options valuation concept
requires some adjustments in order to be useful in management decisions. One of the
most critical issues in real options is deciding the timing of investment or divestment in
the project during the life of the option. Recent research shows that failing to exercise the
60
real options on time reduces the projects? value much less than predicted. However, the
question of whether real option holders exercise their options optimally has not been
extensively researched. Therefore, the possibility of early action also needs to be
considered in order to make real options more useful.
After deciding to retain the real option, the investors are required to decide the
timing for exercising or divesting the option. Because of the irreversibility of the
investments, deciding on the optimal investment timing is one of the most important
factors in the real options valuation model. In the financial options model, the timing to
exercise is defined as the point at which the value of immediate exercise is higher than
that of holding the option to its expiration date. However, research has indicated that
early exercise is never optimal on a nondividend paying stock in the financial call option
theory, which is applied to most real options valuation models (Hull, 2005).
Copeland and Tufano (2004) stated that defining the optimal exercise timing of
the real option is essential in order for real options to work well in the real world. They
suggested that failing to exercise real options on time reduces the value of the projects
much less than predicted. However, in spite of its importance to decision makers, there
has not been adequate research into selecting the optimal timing for real options. Brennan
and Schwartz (1985) developed an evaluation model wherein they set stochastic output
prices in order to decide the optimal investment timing for continuing or abandoning a
mining project. McDonald and Siegel (1986) studied how to optimally time the
investment in an irreversible project when the benefit and cost of the project follows
Geometric Brownian Motion. They used the simulation technique to show that risk
averse investors should wait until benefits are twice the investment costs.
61
Yaksick (1996) suggested a method to compute an expected exercise timing of a
perpetual American option. Shackleton and Wojakowski (2002) derived a numerical
expression for computing the expected return and for finding the optimal timing for the
exercise of real options by using the riskadjusted stopping time method, which is based
on the actual probability distribution of payoff times. Rhys, Song, and Jindrichovska
(2002) summarized the recent developments in this area, reporting that only a few studies
have been conducted to analyze the problem of timing, but concluding that some progress
is being made in the research.
A new decision rule, based on the binomial lattice valuation model, is presented
here to determine the optimal timing of irreversible investments. The expected profits
under the new decision rule are compared to the traditional rule in order to demonstrate
the advantages of this new approach, and a simulation technique is used to present the
benefit of new early exercise rule in real options. The fundamental of the simulation for
validating the new decision rule is generating pathdependant project values. After
simulating the required amount of the project value path, the expected future values of
the two cases are calculated. Then paired ttests are conducted to observe the differences
between the traditional decision rule and the new decision rule.
Section 2 reviews the general early exercise rule of options and discusses the new
decision model. Section 3 substantiates the new approach by comparing the expected
values of projects using the current options decision rule with those using the new
decision rule. A simulation technique is used to generate the expected future project
values. Section 4 applies the new decision model to a defer option and an abandon option.
Finally, section 5 contains a summary of the research as well as its conclusions
62
5.2 Developing a New Decision Rule of Real Options
Although an early exercise of a financial call option of the nondividend paying
stock is known to be never optimal, the same theory does not apply to the real options
because projects of a firm do not have the same characteristics as financial assets. One of
the most important differences is that in the real options framework, the investment is
irreversible once the project is undertaken, and the invested budget is generally not
tradable. Therefore, deriving the optimal exercise timing of real options requires a unique
approach. The logic of decision analysis is applied to find the optimal decision timing of
the obtained options, because determining the exercise timing and the forfeit timing of
real options is different from financial options. The process of developing the new
decision rule concentrates on the total expected future profit by taking early actions.
5.2.1 Early Exercise in Financial Options
Before relating the details of the new decision model, it is necessary to review the
early exercise decision of the financial option pricing. An American option can be
exercised at any time during its life. The early exercise of the American option is decided
by comparing the value of waiting to the payoff of early exercise. A binomial lattice
valuation model that was originated by Cox, et al. (1979) is applied to calculate the early
decision points.
Figure 51 illustrates the procedures for deciding the early exercise in node ? by
a binomial lattice approach.
63
Figure 51. Binomial tree approach for early exercise decision
The initial stock price
0
S moves to one of two values,
0
Su and
0
Sd, during
the first time interval. The two values also will move to two possible directions, ?up? or
?down,? during the next time interval, and so on. The parameter u represents an ?up?
movement and d a ?down? movement during a time interval t? . The other parameters
in the lattice are p which represents the probability that the stock price takes an ?up?
movement, 1 p? which is the probability that the stock price moves ?down,? and r
the riskfree rate of the model. Then, the processes for deciding on early exercise at node
? are:
1. Compute the option value
2
OVu by waiting.
23 2
[( ) (1 )( )]
rt
OVu p OVu p OVu d e
? ?
=+? ?
1
0
S
0
Su
0
Sd
2
0
Su
0
Sud
2
0
Sd
2
0
Sud
2
0
Sud
3
0
Sd
()OVu
()OVd
3
()OVu
2
()OVu d
2
()OVud
3
()OVd
2
()OVu
()OVud
2
()OVd
0
()OV
3
0
Su
64
2. Compute the immediate payoff.
22
0
max( ,0)OVu S u K=?, for a call option
22
0
max( ,0)OVu K S u=?, for a put option
3. Select the highest value of step 1 and step 2 as the option value of the node ? .
If the value computed by the immediate payoff option from step 2 is higher than
that of waiting, which is defined from step 1, early exercise is preferred in the
node. However, the empirical test demonstrates that early exercise is never
optimal for American call options of the nodividend paying stock, while early
exercise may be possible in an American put option.
5.2.2 New Decision Model
The new idea for determining the early investment points of the real option is
based on the opportunity cost concept, which is defined by comparing the expected future
option value with the expected future profit earned by early action. Two important
assumptions are necessary for the development of the new decision model.
? The first assumption is that once the option is exercised, the projected profit is
immediately realized and is available for other investment purposes.
? The second assumption is that the investment in the other projects will earn the
riskadjusted rate of return of the company, compounded continuously. These two
assumptions are widely used in the engineering economics analysis of the capital
investment decision.
Before explaining the method of making decision in detail, it is necessary to
define some notations.
65
Notations
? T : Option life, t? : Length of the option period
?
T
N
t
=
?
: Number of time period during the option life
? n : Time node, 0,1,2,...,nN=
? m : Value states in each time node. 0,1,2,...,mn=
?
nm
V : Estimated project value at
th
m highest value of time n .
?
nm
OV : Option value at node nm . For the call style option model, since we initially
considered a defer option model, the option values at the end nodes are defined as
Nm Nm
OV V I=? and the values at the other time nodes are defined as,
(1) (1)( 1)
[(]
rt
nm n m n m
OV OV p OV p e
? ?
+++
=?+ ???, 0,1,2,..., 1nN= ? , 0,1,2,...,mn=
where, p : riskneutral probability
For a putstyle real option such as an abandonment option, the option values at the
very last time nodes are determined by
Nm Nm
OV I V= ? and the value of the other
00
V
0N
V
NN
V
0n
V
nm
V
nn
V
null
null
null
null
null
Nm
V
66
time nodes are determined by,
(1) (1)( 1)
[(]
rt
nm n m n m
OV OV p OV p e
? ?
+++
=?+ ???, 0,1,2,..., 1nN= ? , 0,1,2,...,mn=
?
nm
PV : Estimated early exercise profit at node nm .
For the Call style
nm nm
PV V I= ? , 0,1,2,..., 1nN= ? , 0,1,2,...,mn=
For the Put style
Nm Nm
PV I V= ? , 0,1,2,..., 1nN= ? , 0,1,2,...,mn=
Decision rule
The decision of early exercise is based on the option valuation model and the
capital budgeting decision rule. In the new decision rule, the profit from the early
investment will earn the riskadjusted interest rate (R) which is generally higher than the
riskfree interest rate. Three actions are determined by the new decision rule.
? Invest if the future value of the profit realized by the direct investment is higher
than the expected option value.
? Divest when the expected option value reaches zero.
? Wait to observe the movements of the underlying project value.
Table 51 demonstrates the conditions for early action at time n after retaining
the real options.
5.2.3 Early Exercise Points for Lower Volatility Projects
For the purposes of this study, three assumptions are made in the development of
the early exercise conditions for real options: 1) the riskadjusted interest rate is higher or
equal to the riskfree interest rate, RRiskfrerate? , 2) the investment decision will be
based on the expected value criteria, and 3) the decision on early investment is
investigated once a year, 1t?=.
67
The first step is to determine the condition of the early exercise points for a
specific case. If all the terminal payoffs which are initiated from the current project value
through the lattice approach are nonnegative, which is common in projects of relatively
lower volatility, then the conditions of the early exercise decision are derived as below
with the remaining option life n . If the project value at time ( )Nn? is higher than the
right hand side of equation (51), the decision is for early exercise at that time. The value
*
()Nnm
V
?
represents the minimum project value for early exercise when n years are
remaining. The process to derive the equation (51) is in Appendix 51.
(1)
*
()
()
knr
Nnm kr
Ie e
V
ee
??
?
?
=
?
, 2,3,nN= null (51)
Equation (51) implies that the project value for deciding the minimum
investment conditions of the real option has no correlation with the risk factors of the
project. Rather, the condition is a function of the remaining time, the riskfree rate, and
the riskadjusted interest rate of the company. The equation (51) is resulted in the low
flexibility of the project; therefore, further studies are necessary for more risky projects.
Table 51. Three decision options
Course of Action Conditions
Exercise the option
(1) (1)( 1)
0[ (1 )]
MARR t
nm n m nm
OV p OV p PV e
??
+++
# include
# include
# include
# include
const int NumNode = 100;
double Max(double,double);
void main()
{
double value;
double invest;
double sigma;
double risk_free;
double marr;
double t;
cout << " Enter Initial Project Value: "<< endl;
cin >> value;
cout << " Enter Investment cost: "<< endl;
cin >> invest;
cout << " Enter the volatility: "<< endl;
cin >> sigma;
cout << " Enter the risk free rate: "<< endl;
cin >> risk_free;
cout << " Enter the MARR of the company: "<< endl;
cin >> marr;
cout << " Enter the remaining option life: "<< endl;
cin >> t;
cout << " Enter a delta of the binomial tree: "<< endl;
cin >> delta;
double u = exp( sigma * sqrt(delta));
double d = exp( 1 * sigma * sqrt(delta));
double p = ( exp(risk_free * delta)  d) / ( u  d );
double q = 1  p ;
double Project_V[NumNode][NumNode];
double Immediate_Profit[NumNode][NumNode];
double Option_V[NumNode][NumNode];
double Exp_Option[NumNode][NumNode];
double Future_Value[NumNode][NumNode];
98
// Developing a monetary tree //
int N = int((t/delta)+1);
Project_V[0][0] = value;
for ( int k = 0; k < 10000; k++){
double function = 10;
Project_V[0][0] = Project_V[0][0] + 0.01;
for ( int i = 1; i < N; i++){
Project_V[0][i] = 0;
Project_V[i][0] = Project_V[i1][0] * u;
}
for ( i = 1; i < N; i++ ){
for ( int j = 1; j < N ; j++){
if (i < j)
Project_V[i][j] = 0.0F;
else
Project_V[i][j] = Project_V[i1][j1] * d;
}
}
// Compute the profit by immediate exercise//
for ( i = 0; i < N; i++ ){
for ( int j = 0; j < N ; j++){
if (i < j)
Immediate_Profit[i][j] = 0.0F;
else
Immediate_Profit[i][j] = Project_V[i][j]  invest;
}
}
// Compute the option premium of the end node//
for ( i = 0; i < N; i++ ){
for ( int j = 0; j < N ; j++){
Option_V[i][j] = 0;
}
}
for ( int j = 0; j < N; j++ ){
Option_V[N1][j] = Max(0.0F,Immediate_Profit[N1][j]);
}
// Compute the option premium of each node//
for ( i = N2; i >= 0; i ){
for (int j = 0; j < N; j++ ){
if( i < j)
continue;
else
Option_V[i][j] = exp( risk_free * delta)
*((Option_V[i+1][j] * p) + (Option_V[i+1][j+1] * q));
99
}
}
// Compare two numbers in each node//
// Compute the expect option value at the next period //
for ( i = 0; i < N; i++ ){
for ( int j = 0; j < N ; j++){
Exp_Option[i][j] = 0;
}
}
for ( i = N2; i >= 0; i ){
for (int j = 0; j < N; j++ ){
if( i < j)
continue;
else
Exp_Option[i][j] = ( OptionV[i+1][j] * p +
Option_V[i+1][j+1] * q );
}
}
// Compute the future value of current period with MARR//
for ( i = 0; i < N; i++ ){
for (int j = 0; j < N; j++ ){
Future_Value[i][j] = Immediate_Profit[i][j]*exp(marr * delta);
}
}
// minimize the difference between two values//
function = Exp_Option[0][0]  Future_Value[0][0];
if (function <= 0.00001)
break;
}
cout << "Minimum project value is:" << Project_V[0][0] << endl;
}
double Max(double x , double y)
{
double temp;
if ( x <= y){
temp = x;
x = y;
y = temp;
}
return x;
}
100
CHAPTER 6
COMPREHENSIVE APPLICATIONS OF NEW REAL OPTIONS MODEL
6.1. Introduction
In order to examine the impact of the last three chapters? proposals on decision
making, two real options decision models are presented. The application of the new
volatility estimation method; the volatility revision process; and the new decision rules
for two specific growth project opportunities, a growth option and a compound option,
are demonstrated. It is assumed in the examples that a growth option occurs when a
company needs to make an initial investment in order to support followon investment.
Projects that require phasedexpansions or a webbased technology investment are
examples of projects that have growth options perspectives. In the growth options
framework, investing in the initial small scale investment provides the option for the
followon large scale project opportunities, and the amount of loss from the initial
investments represents the option premium.
Compound options are similar to growth options, but while growth options
concentrate on phaseexpansions of the already developed project, compound options
apply to projects that have more than a twophase investment. A good example of a
compound option is a R&D project for a new medicine, because the investment decision
in a pharmaceutical project depends on its future revenue at the end of multiple stages.
101
Figure 61. Project scenario of a growth option
6.2. Growth Options Framework
Figure 61 demonstrates a brief of a growth option scenario. In the growth options
framework, the loss from the initial investment ( )
II
NPV I V? is considered as an option
premium to take the opportunity to expand the project into the next phase of production.
It is assumed that the followon investment is available at any time during the option life.
The option parameters of the growth option are defined as follows: the present value of
the estimated project value
II
V represents
0
V , the required largescale investment cost
II
I is I, time to make the investment (T), riskfree interest rate (r), and the uncertainty of
the project cash flows (? ). The valuation process for growth options is the same as for
the defer options valuation model. Since real options permits the early exercise option,
the general binomial lattice model can be applied for the option valuation. In the growth
options framework, when the computed option premium (OP) through the valuation
I
I
II
I
I
V
II
V
T
102
model is higher than ( )
II
NPV I V? , the required cost to take the largescale investment
opportunity, the investment should be initiated.
The following example is provided in order to apply the processes demonstrated
in the previous three chapters into the growth options framework. Consider a project of
XYZ Chemical with an opportunity to expand the production level at 3 years from the
initial investment. The expected cash flows are summarized in table 61. All the values
and costs in the table are the expected values.
Table 61. Cash flow estimation for the growth opportunity
EOY 0 1 2 3 4 5 6
Income Statement
Revenue
Unit price 20 20 20 24 24 24
Demand 1,000 1,000 1,000 2,500 2,500 2,500
Total revenue 20,000 20,000 20,000 60,833 60,833 60,833
Expenses
Unit variable cost 10 10 10 15 15 15
Variable cost 10,000 10,000 10,000 36,339 36,339 36,339
Fixed cost 6,000 6,000 6,000 11,728 11,728 11,728
Depreciation 3,333 3,333 3,333 7,500 7,500 7,500
Taxable income 667 667 667 5,266 5,266 5,266
Income taxes(40%) 267 267 267 2,107 2,107 2,107
Net income 400 400 400 3,160 3,160 3,160
Cash Flow statement
Operating Activity
Net Income 400 400 400 3,160 3,160 3,160
Depreciation 3,333 3,333 3,333 7,500 7,500 7,500
Investment Activity
Investment 10,000 30,000
Net cash flow 10,000 3,733 3,733 26,267 10,660 10,660 10,660
103
Table 62. Prior belief of the random factors for the 2
nd
phase investment
Random Factor Pessimistic Most Likely Optimistic
Demand (EA) 1000 Unknown 4000
Unit Price ($) 15 25 30
Unit variable cost ($) 11 15 16
Fixed cost ($) 10,000 11,430 15,000
It is also estimated that there are four uncertainty factors among the elements of
the cash flow table: 1) the unit price, 2) demand, 3) unit variable cost, and 4) fixed cost.
Because of the high degree of uncertainty involved in estimating the random distribution,
only threepoint estimates are demonstrated. Typically, it is very difficult to estimate the
demand of the 2
nd
phase, so only the possible outcome ranges are suggested. The prior
beliefs of the random variables are shown in table 62.
6.2.1 Initial Option Valuation
6.2.1.1 Initial Volatility Estimation
The distributions of the random factors are defined by the processes in chapter 4,
which explains how to convert the threepoint estimates to beta distributions. The shapes
of the risky variables are defined by three parameters: 1) the standardized mode, 2) ? ,
and 3) ? . Note that the demand distribution is converted to Uniform random distribution
since only the upper and the lower bounds are known. Table 63 shows the standardized
mode, ? , and, ? of the scenario?s random factors.
104
Table 63. Parameters of beta distribution
Random Factor ? Mode
?
Demand (EA) 0  0
Unit Price ($) 3.643 0.667 1.819
Unit variable cost ($) 3.565 0.8 0.891
Fixed cost ($) 1.469 0.286 3.674
Once the distributions of the risky elements in the cash flow table are determined,
the volatility estimation processes becomes simpler through the use of the model
suggested in Chapter 3. The @Risk simulation package calculates the mean and the
variance of the 2
nd
investment at the end of the 3
rd
year as $28,521 and $17,621
respectively. Since the expected present value with the riskfree interest rate of 6% is
$23,823, the volatility of the project is derived to 33% by the following formula:
2 2
0
22 20.36
0
17621
ln ( ) 1 ln ( ) 1
23823
33%
3
rT
e e
T
?
?
?
????
+ +
????
?
??? ?
== =
6.2.1.2 Option Premium and Investment Decision
Once the volatility of the real option is defined, the next step is to evaluate the
project opportunity. From the NPV approach, the project is not favorable because of the
negative NPV 4.17K.
( ) 10 3.7 ( / ,12%,3) 1.11NPV initial K K P A K=? + =?
(2 ) [ 30 10.7 ( / ,12%,3)]( / ,12%,3) 3.06NPV nd K K P A P F K=? + =?
( ) 1.11 3.06 4.17NPV Total K K K=? ? =?
105
However, the options framework gives a different result than the NPV approach.
Since the calculated growth option premium is 2.11K through the binomial lattice model
with the parameters
0
V = 18.2K , I = 30K , T = 3 years , 0.33? = , 0.06r = , and
1t?=, the strategic net present value by taking the investment opportunity is computed
to 1K.
SNPV = 1.11K + 2.11K = 1K
Finally, the decision will be to take the 2
nd
phase investment opportunity by
investing in the initial stage of the project even if the NPV of the initial investment is
negative. Figure 62 is an overall cash flow of the XYZ project for a growth options
framework with exercise time T.
Figure 62. Cash flow diagram of the growth opportunity
10K
30K
3
3.7K 3.7K 3.7K
10.7K 10.7K 10.7K
106
6.2.2 Volatility Revision
One year after initiating the project, a new set of information is collected so the
company needs to reevaluate the project opportunity by combining the observed
information to their initial belief. The observation indicates a new demand of 2,400EA, a
unit price of $24, a unit variable cost of $14.6, and a fixed cost of 12,000. From the
survey to the company?s management group, the IQF of the initial belief is closer to 3. To
make the computation simple, 10 equal length intervals of the discrete approximation are
assumed for the risky variables. For the demand distribution example, each interval has
300 events when it is transformed to the discrete distribution. The median of the intervals
is considered as the event of the variable, and the probability of the intervals is recorded
as a corresponding probability,
k
s? , of the event. Then the
k
? of the prior distribution
is computed by 3
kk k
IQF? ??=? = .
The next step is to determine the posterior distribution by conjugating the
observation results,
k
x s , to the prior belief. In this case study, the observation is assumed
as is. The
"
k
s? of the posterior probability are calculated after
"
k
s? are determined.
Table 64 is the result of the Bayesian conjugate distribution with the data based on the
@Risk simulation software. The simulation shows that the standard deviation of the cash
flow at the end of the 3
rd
year is $15,082, and the expected value of the project at the end
of the 1
st
year is $25,029. The new volatility is computed to 35% by equation (310), and
the expected value of the project at the end of year 1, $20.1K, is still less than its
investment cost. Therefore, it is advisable to wait until the end of the option life. The
logic behind the decision to wait will follow in section 6.4.
107
Table 64. Revised discrete approximation of the growth option
Variables k
k
?
kk
IQF? ?= ?
k
x
"
k
?
"
k
? Events
1 0.10 0.30 0.30 0.075 1150
2 0.10 0.30 0.30 0.075 1450
3 0.10 0.30 0.30 0.075 1750
4 0.10 0.30 0.30 0.075 2050
5 0.10 0.30 1 1.30 0.325 2350
6 0.10 0.30 0.30 0.075 2650
7 0.10 0.30 0.30 0.075 2950
8 0.10 0.30 0.30 0.075 3250
9 0.10 0.30 0.30 0.075 3550
10 0.10 0.30 0.30 0.075 3850
Demand
Sum 1 3 1 4 1
1 0.0003 0.0009 0.0009 0.0002 15.75
2 0.0060 0.0180 0.0180 0.0005 17.25
3 0.0287 0.0861 0.0861 0.0215 18.75
4 0.0714 0.2142 0.2142 0.0536 20.25
5 0.1309 0.3927 0.3927 0.0982 21.75
6 0.1866 0.5598 0.5598 0.1340 23.25
7 0.2217 0.6651 1 1.6651 0.4163 24.75
8 0.2018 0.6054 0.6054 0.1514 26.25
9 0.1230 0.3690 0.3690 0.0923 27.75
10 0.0297 0.0891 0.0891 0.0223 29.25
Price
Sum 1 3 1 4 1
1 0.0001 0.0003 0.0003 0.0001 11.25
2 0.0028 0.0084 0.0084 0.0021 11.75
3 0.0144 0.0432 0.0432 0.0108 12.25
4 0.0395 0.1185 0.1185 0.0296 12.75
5 0.0846 0.2538 0.2538 0.0635 13.25
6 0.1445 0.4335 0.4335 0.1084 13.75
7 0.2024 0.6072 0.6072 0.1518 14.25
8 0.2554 0.7662 1 1.7662 0.4416 14.75
9 0.2504 0.7512 0.7512 0.1878 15.25
10 0.0060 0.0180 0.0180 0.0045 15.75
Variable
Cost
Sum 1 3 1 4 1
1 0.0528 0.1584 0.1584 0.0396 10250
2 0.1509 0.4527 0.4527 0.1132 10750
3 0.2173 0.6519 0.6519 0.1630 11250
4 0.2171 0.6513 0.6513 0.1628 11750
5 0.1707 0.5121 0.5121 0.1280 12250
6 0.1101 0.3303 1 1.3303 0.3326 12750
7 0.0563 0.1689 0.1689 0.0422 13250
8 0.0203 0.0609 0.0609 0.0152 13750
9 0.0043 0.0129 0.0129 0.0032 14250
10 0.0002 0.0006 0.0006 0.0002 14750
Fixed
Cost
Sum 1 3 1 4 1
108
6.2.3 Decisions on Growth Options
It is necessary to summarize the option parameters before automating the
processes for calculating the points of the early exercise at the end of each year. In this
case study, the initial option parameters are shown below.
? Project value (
00
V ) = $18.2K, Investment cost = $30K
? Riskadjusted interest rate (R) = 12%, Riskfree interest rate = 6%
? Volatility of the project (? ) = 33%, T = 3 years , 1tyear? =
Table 65 is the summary of the suggested decision criteria at the end of years 1, 2,
and 3. The table indicates that the decision at the end of the 1
st
year is ?wait? if the
expected value of the 2
nd
phase project is more than $15.51K and less than $84.86K. The
decision will be ?invest? if the estimated project value is higher than $84.84K, and
?divest? if the value is lower than $15.51K.
In the case of volatility revision in the growth options framework, no further
investment decision remains because the investment was already made. This means that
only the investment decision map will be revised through the volatility changes, since the
initial investment to select the 2
nd
project opportunity is considered as a sunk cost during
the option life because of its irreversibility.
Table 65. Comparison of the decision points in case of volatility change
Exercise Forfeit
EOY
Initial ? Revised ? Initial ? Revised ?
1 84.86? 84.86? 15.51? 14.90?
2 58.26? 58.26? 21.57? 21.14?
3 30? 30?
109
A. Initial volatility
B. Revised volatility
Figure 63. Decision framework of the growth option
The only change to take place through collecting more information is the early
exercise decision, as demonstrated in chapter 5. Figure 63 demonstrates how decisions
change according to volatility.
In this example, the volatility increases approximately 2% by RMCS with the
revised cash flows. With the new parameters, it is calculated that the early exercise points
: Exercise
: Wait
: Forfeit
1 2 EOY
15.51
21.57
84.86
58.26
Value
: Exercise
: Wait
: Forfeit
1 2 EOY
14.90
21.14
84.86
58.26
Value
? 20.1
110
have not changed because the value of
I
d
has not increased over the initial exercise
point. However, the early forfeit points are moved down. As the revised investment
decision map show, if the expected present value of the project is estimated as $20.1K at
the end of the first year, the decision should be to wait a little longer to see the future
characteristics of the project.
6.3 Compound Options Framework
A sequential investment is considered through the compound options framework.
Since there are multiple investment stages in the project, early investment for the latter
phase is not permitted before the completion of the previous stages. This is true even if
the future is estimated to be favorable. Figure 64 illustrates a simplified compound
options framework.
Figure 64. Project scenario of a compound option
1
T
1
T
0
I
1
I
2
I
T
V
Fixed Flexible
Research
Development
Production
111
From the figure, the initial investment cost,
0
I , is viewed as an option premium
for acquiring the right to invest in the proceeding projects. With the options framework,
the present value of the future cash inflows,
T
V , is considered as the current project value,
while the investment cost of each stage is the exercise price. Therefore a binomial lattice
approach, which is a slightly different process than the simple option valuation model,
should be applied to value the stagebased investment opportunity.
Figure 65. Decisions in compound options framework
2
T
1
T
0
T
22
max( ,0)
T
OV V I=?
121
max( ,0)OV OV I= ?
??
0
V
0
uV
0
dV
1
0
T
uV
1
0
T
dV
12
()
0
TT
uV
+
12
()
0
TT
dV
+
12
(1)
0
TT
udV
+?
12
(1)
0
TT
ud V
+?
?
Option value:
Flexible investmentNo early exercise
112
Figure 65 represents the binomial lattice used in order to value the compound
option premium. Compared to the general lattice model, there is one more decision node.
In the figure, the additional decision immediately follows the first stage of the project.
Since the binomial lattice approach uses the dynamic programming skill, the decision is
based on the option premium of the initial computation.
Below are the logical processes for handling a project evaluation that has a
compound option perspective, as suggested by Park (2006):
Step 1: Calculate the initial underlying asset value by calculating the up and down
factors and determining the present value of the future cash flow
0
V over the planning
horizon.
Step 2: Calculate the longerterm option value and the shorterterm option value,
because the value of a sequential option is based on the earlier option.
Step 3: Calculate the combinedoption premium.
6.3.1. Compound Options Scenario
A pharmaceutical firm is considering developing a new drug. It is known that the
development process has 6 stages excepting the additional postmarketing test: 1)
discovery, 2) preclinical testing, 3) clinical phase I, 4) clinical phase II, 5) clinical phase
III, and 6) FDA approval. However, the process can be shortened to 3 stages: 1) research,
2) development, and 3) production. In this example, the three stages are considered in
order to collect the information for the project.
It will take 2 years to complete the research stage, which requires a $0.5M
investment cost including labor and initial laboratory preparations. Upon completion of
the research, $3M in preclinical and clinical test costs must be expended in order to
113
proceed to the next stage. The estimated testing times are 3 years after the success of the
research. Including the government approval fee and the cost of building the mass
production system, $7M is necessary for the investment. The expected cash flows of the
project are summarized in the Table 66, and the firms? riskadjusted interest rate is 15%.
Consider the riskfree interest rate as constant at 5% during the project.
Table 66. Cash flow estimation for the compound option
EOY 0 1 2 3 4 5 6 7 8 9 10
Income Statement
Revenue
Unit price 0.08 0.08 0.08 0.08 0.08
Demand 200 200 200 200 200
Total revenue 15.5 15.5 15.5 15.5 15.5
Expenses
Unit variable cost 0.04 0.04 0.04 0.04 0.04
Variable cost 7.5 7.5 7.5 7.5 7.5
Fixed cost 2 2 2 2 2
Taxable income 5.98 5.98 5.98 5.98 5.98
Income taxes(40%) 2.39 2.39 2.39 2.39 2.39
Net income 3.59 3.59 3.59 3.59 3.59
Cash Flow statement
Operating Activity
Net Income 3.59 3.59 3.59 3.59 3.59
Investment Activity
Investment 0.5 3 7
Net cash flow 0.5 3 7 3.59 3.59 3.59 3.59 3.59
114
Table 67. Prior belief for the random factors
Random Factor Low bound Most Likely High bound
Demand (EA) 50 Unknown 350
Unit Price (M$) 0.02 0.08 0.12
Unit variable cost (M$) 0.01 0.03 0.10
Fixed cost (M$) 0 2 4
Since there are so many uncertainties in the future market, the survey of the
project management team suggests the threepoint scenario estimates for the four
uncertain variables: 1) demand, 2) unit price, 3) unit variable cost, and 4) fixed cost.
Table 67 expresses the current beliefs as to the random variables with a pessimistic,
optimistic, and most likely point.
6.3.2 Initial Option Valuation
6.3.2.1 Initial Volatility Estimation
Before deciding to take the project opportunity, it is necessary to determine the
project?s flexibility. The distributions of the random factors are converted to the beta
distributions through the beta approximation processes shown in chapter 4. Note that the
demand distribution is converted to Uniform because only the upper and the lower
bounds are known information. Table 68 represents the standardized mode, ? , and ?
of the project scenario?s random factors.
115
Table 68. Parameters of beta distribution
Random Factor ? Mode
?
Demand (EA) 0  0
Unit Price (M$) 3.4805 0.6 2.3203
Unit variable cost (M$) 1.0303 0.778 3.6108
Fixed cost (M$) 2.2982 0.5 2.2982
Once the distributions of the uncertain factors are determined, the volatility
estimation processes are simplified by using the model suggested in Chapter 3. The
@Risk simulation package calculates the mean and the variance of the cash inflows at the
end of the 5
th
year with a riskfree interest rate of 5% as $15.55M and $15.59M
respectively. Since the expected present value of the cash flows is $12.18M, the volatility
of the project is computed to 37%.
6.3.2.2 Option Premium and Investment Decision
With the NPV approach, the project is not acceptable because of the negative
NPV 0.26M.
()0.5NPV research M=?
( ) 3 ( / ,15%, 2) 2.27NPV Development M P F M=?=?
( ) [ 7 3.59 ( / ,15%,5)]( / ,15%,5) 2.51NPV Production M M P A P F M=? + =
( ) 0.5 2.27 2.51 0.26NPV Total M M M M=? ? + =?
116
Figure 66. Project cash flow of the R&D project
To apply the compound options framework, it is necessary to define the
parameters of the compound options. The cash flow diagram of the pharmaceutical
project (Figure 66) determines the option parameters found in Table 69.
In the compound options framework, the initial investment of $0.5M is
considered a premium for taking the opportunity to invest in the followon projects. The
option premium for taking the following investment opportunities is computed through
the binomial lattice valuation method as $0.52M, with 1tyear? = . Since $0.5M is
cheaper than the option premium, investment in the research phase of the project is
recommended.
Table 69. Compound options parameters
Parameters
0
V
1
T
2
T
1
I
2
I
?
r
Value 5.98M 2 3 3M 7M 37% 5%
2 3
1
4
10
5
12.04V =
117
6.3.3 Volatility Revision
One year after initiating the project, a new set of information is collected so the
company needs to reevaluate the project opportunity by combining the observed
information to their initial belief. The sample indicates a new demand of 205EA, a unit
price of $0.1M, a unit variable cost of $0.05M, and a fixed cost of $2.75M. According to
the company?s management group survey, the IQF of the initial belief is closer to 3. Ten
equallength intervals of the discrete approximation are assumed for the risky variables in
order to simplify the computation. The median values of the intervals are considered as
the events of the variables, and the probabilities of the intervals are recorded as a
corresponding probability,
k
s? , of the events. Then the
k
? of the prior distribution is
computed by 3
kk k
IQF? ??=? = .
The next step is to determine the posterior distribution by conjugating the
observation results,
k
x s , to the prior belief. In this case study, it is assumed that the
observation is considered as is. The
"
k
s? of the posterior probability are calculated after
the
"
k
s? are determined. Table 610 is the result of the Bayesian conjugate distribution
with the data based on the @Risk simulation software. The simulation result shows that
the standard deviation of the cash flow at the end of the 5
th
year is $16.09M and the
expected value of the project at the end of the 1
st
year is $14.11M. Therefore, the new
volatility is computed to 40% by equation (310). The expected value of the project at the
end of year 1 is estimated to $7.59M. In this particular case, the volatility of the project is
slightly increased by conjugating new information during the waiting period.
118
Table 610. Revised discrete approximation of the compound option
Variables k
k
?
kk
IQF? ?= ?
k
x
"
k
?
"
k
? Events
1 0.10 0.30 0.30 0.075 65
2 0.10 0.30 0.30 0.075 95
3 0.10 0.30 0.30 0.075 125
4 0.10 0.30 0.30 0.075 155
5 0.10 0.30 0.30 0.075 185
6 0.10 0.30 1 1.30 0.325 215
7 0.10 0.30 0.30 0.075 245
8 0.10 0.30 0.30 0.075 275
9 0.10 0.30 0.30 0.075 305
10 0.10 0.30 0.30 0.075 335
Demand
Sum 1 3 1 4 1
1 0.0006 0.0018 0.0018 0.0004 0.025
2 0.0112 0.0336 0.0336 0.0084 0.035
3 0.0466 0.1398 0.1398 0.0350 0.045
4 0.1022 0.3066 0.3066 0.0767 0.055
5 0.1696 0.5088 0.5088 0.1272 0.065
6 0.2137 0.6411 0.6411 0.1603 0.075
7 0.2070 0.6210 0.6210 0.1552 0.085
8 0.1609 0.4827 0.4827 0.1207 0.095
9 0.0767 0.2301 1 1.2301 0.3075 0.105
10 0.0115 0.0345 0.0345 0.0086 0.115
Price
Sum 1 3 1 4 1
1 0.0981 0.2942 0.2942 0.0735 0.0145
2 0.2100 0.6299 0.6299 0.1575 0.0235
3 0.2222 0.6665 0.6665 0.1666 0.0325
4 0.1934 0.5801 0.5801 0.1450 0.0415
5 0.1376 0.4128 1 1.4128 0.3532 0.0505
6 0.0823 0.2470 0.2470 0.0617 0.0595
7 0.0398 0.1193 0.1193 0.0298 0.0685
8 0.0138 0.0413 0.0413 0.0103 0.0775
9 0.0029 0.0086 0.0086 0.0021 0.0865
10 0.0001 0.0004 0.0004 0.0001 0.0955
Variable
Cost
Sum 1 3 1 4 1
1 0.0027 0.0082 0.0082 0.0021 0.2
2 0.0308 0.0925 0.0925 0.0231 0.6
3 0.0929 0.2788 0.2788 0.0697 1
4 0.1637 0.4911 0.4911 0.1228 1.4
5 0.2098 0.6294 0.6294 0.1573 1.8
6 0.2098 0.6294 0.6294 0.1573 2.2
7 0.1637 0.4911 1 1.4911 0.3728 2.6
8 0.0929 0.2788 0.2788 0.0697 3
9 0.0308 0.0925 0.0925 0.0231 3.4
10 0.0027 0.0082 0.0082 0.0021 3.8
Fixed
Cost
Sum 1 3 1 4 1
119
However, the acquired information in the compound options framework plays a
different role in the decision compared to simple options. Since there are two phases for
valuing the option premium, two decision scenarios should be considered for selecting
the investment timings. The first scenario is that the volatility is updated before the
development phase investment, and the second scenario is that the volatility is changed
after the development phase investment. In other words, if the new information is
collected before the 2
nd
investment, then it affects the decision on the next investment.
However, when the information is collected after the last investment decision is made, it
is necessary to find the early exercise points.
6.3.4 Effect of Revising Volatility on Decision Making
In the compound options framework, two decisions remain following the
purchase of the research option. Since it is assumed that the early exercise option is not
available during the research phase, the collected information will affect the decision
whether or not to invest in the development phase. In this case, simply follow the option
valuation model with the posterior volatility and the project value. The other cases
consider the change in volatility during the development phase of the project. In this case,
it is assumed that early exercise is possible.
6.3.4.1 Volatility Change before the 2
nd
Investment
Assume that a new set of information is observed to make possible a revision of
the original decision one year after the research phase investment. The new set of
information is assumed to be the same as the scenario which was introduced in section
6.3.3. In this case, since early actions are not available, the decision is delayed until the
end of the 2
nd
year, and the decision then would be whether or not to take another option.
120
The required cost to invest in the development phase is viewed an option premium for
obtaining the right to invest in the production phase, while the previous investment cost is
considered a sunk cost.
In order to support decision making at this point, the relative volatility and the
project value graph, which justify further investment, are derived in Figure 67. The bold
line in the figure represents the border of two investment decisions: 1) invest, or 2) do not
invest. By reducing the volatility because of the new information, the expected project
value should be increased to stay in the investment decision region. Figure 68
demonstrates what would happen if the volatility changes from 37% to 40%. In this case,
if the expected value of the future project is less than 7.98M and higher than 7.80M, ?do
not invest? was the better option without the volatility revision. However, it is a better
decision to invest in the project if the real volatility is 40%.
Figure 67. The decision map for the 2
nd
phase investment
V
8
7.8
7.6
7.4
7.2
?36% 38% 40% 42% 44% 46% 48% 50%
Invest
Do not invest
7.98
7.80
121
6.3.4.2 Volatility Change after the 2
nd
Investment
In case of volatility revision after the 2
nd
phase investment, only the final decision
remains. The two previous investments to obtain the 3
rd
phase investment opportunity are
considered a sunk cost because of its irreversibility. The only changes in this stage will
affect the early exercise decisions under the rule shown in chapter 5.
Assume that the development phase investment was already undertaken with the
initial volatility, and that a new set of information is collected one year from the
development investment. The new volatility is assumed to be 20% using the processes
demonstrated in section 6.3.3. To automate the processes of defining the points of early
actions at the end of each year, it is necessary to apply the processes demonstrated in
section 5.2. Here, we use the option parameters for this case study: the riskadjusted
interest rate (R) of 15%, the riskfree interest rate (r) of 5%, and the option life (T) of 3
years. Table 611 is the summary of the suggested decision criteria at the end of years 3, 4,
and 5. This table shows that if the expected present value of the project one year after the
2nd investment is made is $14M, then the decision should be ?Exercise? in light of the
revised volatility, while it was ?Wait? under the initial volatility. Figure 68 demonstrates
how decisions change according to volatility.
Table 611. Comparison of the decision points in case of volatility change
Exercise Forfeit
EOY
Initial ? Revised ? Initial ? Revised ?
3 14.04? 13.33? 3.34? 4.69?
4 10.25? 10.25? 4.84? 5.73?
5 7? 7?
122
A. Initial volatility
B. Revised volatility
Figure 68. Decision framework of the compound option
6.4 Conclusion Remarks
This chapter presents all of the sequences, from estimating the project cash flows
to the early action rule of a real option, needed to apply real options to capital budgeting
decisions. A Dirichlet conjugate distribution is applied to revise the volatility of the
project when new information is acquired. A new decision map is developed in case the
volatility changes. Since one of the most important purposes of the real options valuation
: Exercise
: Wait
: Forfeit
1 2 EOY
4.69
5.73
13.33
10.25
Value
? 14
: Exercise
: Wait
: Forfeit
1 2 EOY
3.34
4.84
14.04
10.25
Value
123
model is deriving a value of the investment opportunity, examples of intuitive growth
options and compound options are presented.
Since the only investment decision involved in buying the option is already made
even in a growth option, the remaining decision is related to the irreversible large
investment, which is the same as the pure investment in a defer option. In the growth
options framework, only information collected during the option life will affect the early
exercise decision. The decisionmaking process for compound options is different
because of the future decisions that remain. There can be two possible schedules for
volatility revisions in such cases: before or after the 2
nd
investment. The new volatility
will impact the decision on the 2
nd
investment when such new information is collected
before the 2
nd
investment, while the early exercise rule is applied when the information
arrives after the investment.
124
CHAPTER 7
CONCLUSIONS
Today?s uncertain world requires firms to have a system in place that can analyze
the flexibility of their projects. Real options are utilized frequently to quantify the
benefits of taking a particular risk. The real options valuation process provides a
methodology to measure the value of flexibility, and it assists the decision makers in
making the optimal investment decision. The goal of this research is to develop the
methodology for improving the real options application in actual capital investment
decision making. In order to accomplish the goal, three proposals are suggested and
studied. However, as demonstrated in the text, the three objectives are quite
complementary.
The initial proposal to enhance the use of real options is developing a DCFbased
volatility estimation. Most researchers argue that determination of the correct project
volatility is the most important parameter of real options valuation, but it is very difficult
to estimate the true volatility of a project. Since the DCF analysis is popular in today?s
capital budgeting decisions, it is reasonable to estimate the flexibility of the project by
using the already existing information. So, in this study, the Reverse Monte Carlo
Simulation model (RMCS), which combines Monte Carlo simulation and the stochastic
processes, is developed as a new volatility estimation method for risky projects.
125
Secondly, volatility revision processes based on the previous volatility estimation
processes are proposed. Even though real options gives a firm an opportunity to improve
its strategic investment decisions in an uncertain environment, the traditional options
framework did not consider the data gathering activity that usually takes place in the real
world. In this study, a Bayesian revision process is suggested to estimate the new
volatility when the initial volatility has been estimated by Monte Carlo simulation. Since
specific cases that use typical types of Bayesian conjugate processes are hard to find in
the real world, a Dirichlet conjugate process is applied to estimate posterior distributions
of the future cash flows. After estimating the new distributions of the cash flows, the
revised volatility can be computed using the RMCS approach.
Finally, this study proposes a new early decision rule in order to make real options
more useful. This rule concentrates on maximizing the expected future project value.
Under the new decision rule, an expected future value of the currently exercised option
and the expected future option value are compared in order to determine the best exercise
timing. An early decision map for ?waiting,? ?early exercise,? and ?early divest? over the
entire option life is developed to automate the decision in case some variables are revised
in the future. The map indicates that increasing volatility enlarges the ?waiting? area
while decreasing volatility shrinks the ?waiting? area.
A simulation is applied to validate the newly developed decision rule by
comparing the benefit of the early exercise rule and the volatility revision during the
option life. Then the expected profits under the new decision rule and the traditional rule
are compared in order to demonstrate the advantages of the new approach. The new
decision rule is found to be useful in maximizing the expected profit of the delayed
126
investment because the proposed decision model results are better than or equal to the
current decisions model.
Two real options decision models are demonstrated in order to apply the proposed
processes to specific investment opportunities. In the growth options framework, the
revised volatility and the project value affect the early exercise decision, while the
compound options framework concentrates on go or nogo decisions for the next phase
investments. The information obtained during the option life has an effect on the early
exercise decision in the growth options framework. The new volatility will impact the
decision on the 2
nd
investment when such new information is gathered before the 2
nd
investment, while the early exercise rule is applied when the information arrives after the
2
nd
investment.
Our research on volatility estimation through RMCS provides a practical volatility
of real options, especially when an estimate of the project?s future cash flows is available.
We also researched revising the volatility through a Bayesian revision process in case the
initial volatility is based on RMCS. Then an early decision map of the real options is
developed to automate the decisions. Below are several future research opportunities in
real options volatility and decision making.
? Estimating a correct distribution of the cash flows in order to improve the
reliability of the volatility obtained through RMCS
? Valuing an investment opportunity in case the estimate of the future project value
is dependent on the past value.
? Developing an algorithm for implementing the Dirichlet conjugate process in
volatility estimation so that the volatility revision process will be easier
127
? Application of the new decision rule to various investment opportunities in order
to verify that the rule works in real world projects
128
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