The following selection covers many important aspects of the options thinking and complements the lessons from the previously analyzed Dixit & Pindyck book. There are comments, for each book, available by clicking their titles (ALL COMMENTS AVAILABLE).

Brealey, R.A. &
S.C. Myers (1991): *Principles of Corporate Finance*

Chapter 21, McGraw-Hill, Inc., fourth ed., 1991

Trigeorgis, L.,
(Ed.) (1995): *Real Options in Capital Investments: Models,
Strategies, and Applications*

Praeger Publisher, Westport, Conn., 1995

Lund, D. and B.
Øksendal, (Eds.) (1991): * Stochastic Models
and Options Values,*

New York: North-Holland, 1991

Shimko, D.C.
(1992): *Finance in Continuous Time. A Primer*

Kolb Publishing Company, 1992

Copeland, T. &
T. Koller & J. Murrin (1990):

*Valuation - Measuring and Managing
the Value of Companies*

Chapter 12, John Wiley & Sons (Ed.), 1990

Lopes, E.P.
(2001): *Opções Reais - A Nova Análise
de Investimentos* ("Real Options - The New Investment
Analysis"), __in Portuguese__.

Ed. Sílabo, Lisboa, 2nd edition, 2001

Forthcoming the analysis of more two books:

Trigeorgis, L.,
(1996): *Real Options - Managerial Flexibility and Strategy in Resource
Allocation*

The MIT Press, 1996

Dixit, A.K.,
(1993): *The Art of Smooth Pasting*

Harwood Academic Publishers, 1993

**Back to the Bibliographical Resources**

**Back to Contents (the main page)**

**Brealey,
R.A. & S.C. Myers (1991):** * Principles of Corporate Finance -
Chapter 21*

A classic corporate finance book, dedicates three chapter for options, one
of then about real options. An interesting example of *growth options*
(or option to make follow-on investments) with the computer "Mark I".
There are others examples and an overview, at an introductory level, of
the Real Options Approach.

One of the authors (Myers) coined the term "Real Options", in
70's (see Myers, S.C.: "Determinants of Corporate Borrowing", in
Journal of Financial Economics, no. 5, November 1977, pp.147-175.

The authors point out that discount cash flow (DCF) does not reflect the
value of management: "DCF implicitly assumes that firms hold real
assets passively. It ignores the options found in real assets". When
uncertainty is present, value calculations "is not a simple matter of
discounting",

The *abandonment option* is presented into * put option*
analogy framework, with two and three periods examples, followed by the
general binomial approach.

The *timing option* is presented, and so the distinction between a
project with timing flexibility and "now or never" project. An
example with three periods is analyzed.

The link between options approach (using binomial method) and decision
tree analysis (DTA) is also presented. The authors (p.525) teach: "there
is no single, constant discount rate for options because the risk of the
option changes as time and the price of the underlying asset change. There
is no single discount rate inside a decision tree".

The "checklist" section shows the various types of financial
options (American, European, calls, puts, with and without dividends) and
the application to real investments, with focus on early exercise role for
real investments.

This is one of best introductory texts on real options.

Back to Selected Bibliography

**Trigeorgis,
L., (Ed.) (1995): ** *Real Options in Capital Investments: Models,
Strategies, and Applications*

Other book exclusively about real options. Although this is not a text
book (is a collection of papers), the papers are grouped into five
thematic parts, preceded by the editor (Trigeorgis) very interesting
paper, which he presents an overview of options literature: see for
example the excellent table 1.1, which is presented the various options
types, and for each type of option, are showed a short description, the
applications, and the main bibliography related. The author learns that
many real options occur naturally (exogenous to the firm) and others may
be planned and built in at some extra cost (many flexibilities of future
expansion, and input or output switching options). He also presents the
expanded (strategic) NPV equation.

The Part I, "Real Options and Alternative Valuation Paradigms"
comprises three papers, one comparing the real options approach with two
others modern and interesting methods (*dynamic DCF analysis* and
*decision analysis*); the second one focus the *decision analysis*
(preference-dependent approach), presenting the concept of *market
utility*; and the last one presents the linkage of capital budgeting
and strategic planning.

The Part II, "General Exchange or Switching Option Interdependence",
comprises also three papers, the first one about the value of flexibility
as a framework for a general model of real options; the second one about
the valuation of American Exchange Options, which the author (Carr)
develop a general formula for pricing these options (with important
special cases); and the last one illustrate the interactions between many
different options types, using both Geometric Brownian and Mean Reverting
motions, concluding that the incremental contribution to project value of
each type of option can be attenuated by *substitute options*
(timing to invest and abandonment) and enhanced by * complementary
options* (temporary stopping and growth options). The part III, "Strategy,
Infrastructure, and Foreign Investment Options", comprises again
three papers, the first one showing the role of real options in mergers,
acquisitions and divestitures; the second one describing the real options
in Japan, where some real options are unique to the Japanese system or at
least more widely seen among Japanese firms; and the last one analyzing
entry & exit of multinational firms, which the effect of the volatile
exchange rates depends on whether the project is of fixed scale or of
variable scale.

The Part IV, "Mean Reversion/Alternative formulations in Natural
Resources, Shipping, and Start-up Ventures", comprises three papers,
one showing the effects of mean-reverting process on projects with
different length, using a model of exponential decay in the term structure
of the volatility; the second one focusing shipping applications using a
more simplified mean-reverting process (arithmetic Ornstein-Uhlenbeck);
and the last one analyzing the start-up venture projects, using an
alternative * jump* formulation for growth option valuation.

The Part V, "Other Applications: Pollution Compliance, Land
Development, Flexible Manufacturing, and Financial Default",
comprises four papers, the first one is about the options faced by a firm
(Georgia Power) due the environmental requirements (Clean Air Act), such
as buy or sell *pollution allowances*, investment in expensive *scrubbers*,
and switch to the more expensive * low-sulfur coal*; the second one
is a application of timing options for land development, which the
assumption of portfolio perfectly correlated with the project is relaxed
(correlation less than 1, but constant), among others features; the third
one is a valuation application for *multiproduct projects*, which
considers both the output yield and the input prices as stochastic in a
lattice contingent claims framework; and the last one is an application
for risky debt valuation, modeled as a put option, extending previous
models of corporate liabilities, by incorporating features of the
bankruptcy law to value default risk.

All the papers' authors are eminent Ph.D. or DBA, and this book can be
used as cases book in MBA schools or as reference for researchers.

Back to Selected Bibliography

** Lund, D.
and B. Øksendal, (Eds.) (1991): ** * Stochastic Models and
Options Values,*

This book is a collection of papers, edited by Norwegian professors from
Economics and Mathematics Departments of the University of Oslo, and is
organized into four parts. This book is a more advance treatment
(interesting for M.Sc and Ph.D. students, not for MBA courses) in
comparison with the others books listed in this page.

The Part I, "Introduction", comprises two papers, the first one
is an overview of real options theory, with considerations about the
growth rate, rate of return shortfall/convenience yield, and risk adjusted
discount rate, and comments about some aspects from the papers of the same
book; the second paper is a summary (by mathematical view) of stochastic
control theory, with emphasis in dynamic programming, introducing some
theorems (like the theorem 5, about the existence of an optimal control,
valid for many problems), and about *singular control*, for problems
which the usual smooth control (and the Bellman equation) not hold.

The Part II, "Financial Option Theory Applied to Real Investment",
in a more practical way, comprises 5 papers. The first one, compares four
models to convenience yield, three of then as function of the spot price
(linear, quadratic, and maximum between a constant value and a linear in
the spot price), and the last, named "autonomous model", which
the term structure of the convenience yield is modeled by a mean-reverting
stochastic process, and empirical analysis is performed for many
commodities (including heating oil). The second paper, again about
convenience yield, is specific for the crude oil, again shows empirical
evidence to model the term structure of the convenience yield as a mean
reverting process, and presents an application to value long-term
oil-linked bonds. The third one, deals with an oil field development in
Norway, which the Government wishes to be initiated not later than a fixed
date (a constraint imposed due the unemployment), analyzing the
possibility of the Government buy back the oil field undeveloped and can
undertake immediate development, or can fix a future date which the
national oil company *must* starts the development. The lost value
due the reduction in the managerial freedom degree (named "cost of a
promise" by the author) is computed for these cases comparing with
the unconstrained case (perpetual opportunity) which reaches, at the
traditional break-even price, 250 millions US$ (the DCF method doesn't see
lost value at this price). The fourth paper is an econometric work about
the predictions of the investment aggregate, showing that neoclassical
models (like *marginal Tobin's q*, see also the comments for Dixit &
Pindyck book) fails, even using sophisticated models incorporating
delivery lags, and also the difficulty of econometric modelling
irreversible investment under uncertainty (equations "extremely"
nonlinear and difficulty to measure key parameters of risk), although the
author shows qualitative evidences (uncertainty delays investment) and
points some ways to perform quantitative empirical tests, to find how
large the uncertainty matters on aggregate investment. The last paper is
an interesting historical overview of the concept of "options",
presenting three different traditions: (a) the *financial tradition*,
with emphasis in assets' value calculation; (b) economic tradition with
emphasis in decision's rule (to make a choice), like early works by Arrow
& Fisher (1974) and Henry (1974) (1), named AFH
model, and has the same sense as financial tradition (with different
emphasis); and (c) the Schmalensee (1972), Bohm (1975) and Graham (1981),
also with economic choice emphasis (2), but relies on
consumers being risk averse, and can be negative. The author points the
similarities between the financial and the AFH traditions, with an example
of hybrid method about the uncertainty in consumer surplus, suggesting new
applications of the financial tradition to problems that has been dealt in
the AFH framework, such as the consumer surplus.

The Part III, "Stochastic Control and Dynamic Programming",
comprises four papers, three of then with mathematical emphasis. The first
one, is an economic paper about *sequential investment* (in contrast
to lumpy investment), where the firm incrementally expands capacity in
response to stochastic demand variations, which is solved using the
advanced tools like *singular stochastic control* mentioned in the
first part (see p.28), and the author says that the Pindyck model (1988)
is a special case of his model, but this is not clear (see comments at
p.14). The second one is about the *high contact* condition (also
known as *smooth pasting* condition) as a sufficient
optimization condition for problems of optimal
stopping, and the author examines the multidimensional case for a
general class of finance/economic problems (including Brennan &
Schwartz, 1985 model, as an example), extending results of early papers.
The third paper considers the invariant control, which
the production is uncertainty (besides the price uncertainty) as is the
case of a firm with several mines, where the rewards from each mine are
determined by the specific metal content of the ore and the metal price
(uncorrelated process), and for others problems which the stochastic
process are not independents. This paper identifies situations in which
this correlation doesn't influence the optimal policies (including an
application of an optimal policy for switching extraction between two oil
fields), and the solution for uncorrelated process is valid also for
correlated process. The last paper examines the shadow
price role in decision making under uncertainty, presenting a
generic optimization problem, and two kinds of constrains (resource and
information, the latter incorporates a process of learning), in a discrete
time approach, using Lagrange multipliers.

The Part IV, "Statistical Models of Natural Resource Exploitation",
comprises two papers. The first one is an application for fishing industry
(mainly for a regulatory agency), which the stock of fish is an unobserved
stochastic variable, and examines how to extract information optimally.
The second and last paper, is a sequential decision problem (continue the
production or abandon an oil field) which examines only the *technical
uncertainty* effect, which the optimal decisions is made considering
the reduction of the technical uncertainty variance, due the informations
collected by the production (see additional comments),
and the authors presents numerical examples, one of then demonstrating the
value of a flexible production technology.

** Shimko,
D.C. (1992): ** *Finance in Continuous Time. A Primer*

This thin book explain modern finance methods in a concise way, and is
useful for both financial and real investment applications. The book is
divided into four chapters, with lots of exercises at the ending of each
chapter. In contrast with most other texts, this one frequently uses *Laplace
transformations* to solve the differential equations (used also in Cox &
Miller, 1965, a book about stochastic processes).

The first chapter is an overview of stochastic processes (arithmetic and
geometric Brownian, mean-reversion, jump process), stochastic tools (Itô
Lemma), and correlations between two stochastic process (see the table at
p.16, for a summary of multiplication for differential variables).

In the second chapter, the frequently occurring differential equations is
presented, which has the same general format in finance problems, and also
important questions such as the homogeneity of these equations, the
discount rate for the assets (showing that the solution is of the same
form in risk-averse and risk-neutral economies), and an introduction of
recursive techniques in asset valuation is also presented.

The third chapter explain the arbitrage principle with applications for
asset valuation. European and American options are analyzed, and the
smooth pasting (or high contact) optimal condition is presented for the
American one.

The last chapter deals with optimal as a maximization of present values,
first with the time-homogeneous problem (a infinitely lived asset or
project), followed by extensions: the multiple state variables case, and
the time non-homogeneous case. Closing the book some aspects of the
classic Merton (1971) optimization problem for consumption and portfolio
rules, is presented.

Back to Selected Bibliography

** Copeland,
T. & T. Koller & J. Murrin (1990): **

*Valuation - Measuring and Managing the Value of Companies*

In this book about valuation of companies, the authors (from a consultant
firm) dedicate one chapter for real options ("Using Option Pricing
Methods to Value Flexibility"). This chapter points that "For
valuations, it is important to remember that options can be found on both
the assets and the liabilities sides of the balance sheet".

Using many simple examples, with event and decision trees analysis (DTA),
is showed that NPV analysis often undervalues assets because it fails to
account for the managerial flexibility (options) embedded in business
decisions. The authors illustrate that "option pricing is a generic
form of decision making that encompasses NPV and DTA as special cases".

There is a little example of oil extraction (pp.359-360) where the option
adds 21% to "OILCO" value. Others examples: mining (option to
shut-down/reopen and abandon, increases in more than 60% the mine value);
Pharmaceutical R&D (option adds more than 80% of the project value);
and so on. On the other side of the balance sheet, for the liability
options, others examples are included.

In short, it is a good introductory explanation for real options, with
focus on valuation of companies.

Back to Selected Bibliography

**Lopes, E.P.
(2001)** *Opções Reais - A Nova Análise de Investimentos ("Real
Options - The New Investment Analysis"), in Portuguese*

The author, Professor Eurico Pereira Lopes, from both Universidade Técnica
de Lisboa and Universidade Lusíada de Lisboa, send me the following
short description of his book:

This book addresses Contingent-Claims Analysis in the valuation of
investment projects, including within the financial analysis, those
components of value linked to the operational and strategic flexibility,
which, due to the failure of traditional DCF capture techniques, are left
to intuition and experience of managers.

It identifies, through Case Study a new type of real option which I
designate "partial guaranteed placed production option," thus
contributing to the extension of Taxonomy of Real Option, and in turn, to
the continued emergence of a Theory of Real Options. It proposes as
criterion of choice of investments opportunities in real assets the
Strategic Adjusted Net Present Value: Strategic Adj. NPV = Adj. NPV +
Value of Options from Active Management. You can consult
www.euricolopes.net to obtain a
complete information about the book in Portuguese.

This expression has been employed by Trigeorgis in several papers.

As Trigeorgis points out, **Expanded (strategic) NPV = static (passive) NPV + value of options
from active management (Option Premium)**

The "Option Premium" can be from the (most) relevant option, or
a "Combined Option Value" (in a multi-options interactions
context). In large companies, with a large portfolio of project
opportunities, the different "synergy effect" in each portfolio
combination, can be added to the above expression, if relevant.

Back to Trigeorgis book's comments

The Arrow & Fisher and Henry early papers, can be seemed as the *qualitative*
foundations of modern irreversibility and real options (flexibility)
concepts, although only after the finance options theory developments, the
modern quantitative approach for real options was established.

The option concept in this context is a kind of risk premium for risk averse consumers, because uncertainty over future valuations of an environmental amenity (see Dixit & Pindyck, 1994, footnote p.426), and is very different of the option concept adopted in this site for economic decisions.

This sufficient optimal condition was first introduced by Samuelson's paper (1965), with both heuristic argument and a rigorous proof in a mathematical McKean's appendix for this paper. Samuelson also introduced the stochastic calculus in finance applications.

Optimal stopping problems means a binary decision (for ex.: wait or
invest), and defines two regions named, respectively, continuation region
(waiting) and stopping region (investment). The line that separates the
two regions is named *free boundary* and defines the optimal points
for economic decisions (or threshold prices line) like investment and
abandonment. See for example, Dixit & Pindyck (1994, pp.103, 108-109,
and 128).

Invariant control means an optimal strategy that remains optimal even if the instantaneous stochastic payoff is multiplied by another (either independent or correlated) stochastic process. See Lund in the same book (p.15).

For the shadow prices that results from a learning by investing process,
see in "Tutorial", the "Technical Uncertainty" item.

For a *continuous time* application of Lagrange multipliers, see
the very interesting paper of G. Chow (1993).

The authors suggest, for investment decisions, that should be considered
the abandon decision with the technical uncertainty and learning effects.
However, as showed by Bjerksund & Ekern (1990), if you have considered
the timing option, the additional abandon option has little effect in
field value.

Other comment is that the several reservoir informations after the
production starting, has little effect for additional variance reductions
(in general, the reduction is significant at the *initial stages*,
with decreasing marginal contribution to variance reduction).

The Stensland & Tjøstheim paper is correct and important to
set the key role of technical uncertainty: the variance reduction with
time by doing (producing or investing), not by waiting (see Dixit &
Pindyck, 1994, chapter 10, section 4), but I believe that a far more
interesting application in petroleum is for ** exploration**
investments decisions, in large oil companies,

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