On the Valuation of Real Options. Necessary Mathematical Tools and Compelling Approaches in Financial Literature

Bachelor Thesis, 2015

50 Pages, Grade: 1.0


Table of contents

List of Figures

List of Tables

List of Abbreviations


1 What are Real Options?
1.1 Definition
1.2 Comparison to traditional Net Present Value method
1.3 Types of real options and analogy to financial options

2 Real Options Theory
2.1 Literature Review
2.2 Stochastic Processes
2.2.1 The Basic and the Generalized Wiener Process
2.2.2 Itô Process, Geometric Brownian Motion and Itô’s Lemma
2.2.3 Jump-Diffusion Process
2.2.4 Mean-Reverting Process

3 Approaches to Real Option Valuation
3.1 Dynamic Programming
3.1.1 Discrete time optimization
3.1.2 Optimal Stopping
3.1.3 Continuous Time Optimization
3.1.4 Value Matching and Smooth Pasting Condition
3.2 Contingent Claim Analysis
3.2.1 Replicating Portfolio
3.2.2 Spanning Assets
3.2.3 Smooth Pasting
3.3 Simulation Approach
3.4 Comparison of the Approaches

4 Valuing undeveloped petroleum reserves
4.1 Valuation of a developed reserve
4.2 Valuation of an undeveloped reserve
4.3 Numerical examples
4.4 Final Remarks


A Derivation of Itô’s Lemma
B Derivation of expected value and variance for an Ornstein- Uhlenbeck process
C Optimal Stopping Regions and Smooth Pasting References

List of Figures

2.1 Sample paths of the basic and general Wiener process (standard Brownian motion)

2.2 Sample paths of Stocks following a Geometric Brownian Motion . . .

2.3 Sample path of a Jump-Diffusion process vs. pure Brownian motion (fixed jumps)

2.4 Sample path of a Jump-Diffusion process vs. pure Brownian motion (dynamic jumps)

2.5 Sample paths of mean-reverting processes

3.1 Smooth Pasting

3.2 Sample least squared regression for a simple copper mine

4.1 Hitting Boundaries

List of Tables

1.1 Most common real option types

4.1 Comparison of variables for pricing undeveloped oil reserves and fi- nancial call options

4.2 Option values per one dollar of development cost

4.3 Values of undeveloped reserves

List of Abbreviations

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The earliest mention of a real option was found in the writings of Aristotle that leads back to the second century BC. He writes about Thales, a philosopher, who made a fortune by reading tea leaves and forecasted the olive harvest ahead of time. He concluded that the harvest would be large that year and acquired call options by paying money 9 months ahead of time for the right to rent olive presses for the normal renting price during the harvest. Nine months later it turns out that his predictions were right. He profited by paying the normal rental rate and demanding a higher price from olive growers who were excited to have their oil pressed.

In academia, despite its ancient roots, real options evolved from the field of financial options and are viewed as an extension to classical option pricing methods to value real investments under uncertainty. It is superior to traditional project valuation methods as it allows the adaption of later decisions to unexpected future developments. After decades of development, it has become an active field of academic research and is currently a hot topic in academic finance. However, the application of real options by practitioners has been limited.

This thesis aims to provide a comprehensive overview of the valuation method- ologies on real options, their mathematical concepts, and also attempts to give rea- sons to why their application has not yet been fully accepted by practitioners.

In the first chapter, real options are defined and compared to the traditional valuation method, namely, the NPV rule. Then, common real option types are introduced and an analogy to financial options is provided.

The second chapter starts off with a literature review of the major findings in real option pricing theory and continues with exploring a variety of stochastic processes that are important for the valuation models. We first introduce the basic Wiener process and develop its general model and proceed with showing the Itô process and the Geometric Brownian Motion. The chapter finishes with showing some alternative processes such as the jump-diffusion or the mean-reverting process.

In the third chapter, the focus lies on the three major approaches in real option valuation: (1) Dynamic programming, (2) contingent claims, and (3) simulation. The models are developed and a comparison of the approaches is given.

Finally, the fourth chapter provides a real investment case in the face of uncer- tainty where an undeveloped petroleum reserve is priced using one of the approaches.

Chapter 1 What are Real Options?

In this chapter, the basic understanding of real options is discussed. We first define the term ”real option” and then compare real options analysis to the traditional NPV method of project valuation where some major insights on the value of flexibility are highlighted. We then introduce common types of real options and make an analogy to financial options to show similarities and differences.

1.1 Definition

The term real option was first coined by professor Stewart Myers (1976)[32] in his paper Determinants of Corporate Borrowing. He states that ”real options [...] are opportunities to purchase real assets on possibly favorable terms”. Dixit and Pyndick (1995)[13] have a more generalized view on real options as they describe them as opportunities that have ”[...] the right, but not the obligation to take some action in the future”. A more concise definition was introduced by Sick (1995)[39]. He defines real options as “the flexibility a manager has for making decisions about real assets that could involve adoption, abandonment, exchange of one asset for another or modification of the operating characteristics of an existing asset.”

1.2 Comparison to traditional Net Present Value

The traditional approach for valuing investments or projects is commonly known as Net Present Value (N P V ) valuation. To apply this method, we typically make forecasts about future free cash flows and form an expected value E(F CFt). To de- termine the net present value, we simply discount the expected value at the weighted average cost of capital (W ACC) for t periods to obtain the present value and then take out the initial investment (I). A simplified representation of this approach is:1

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The common rule is that if the NPV is greater than zero, shareholders’ wealth is assumed to increase, thus the project should be accepted. However, there are three major issues with the traditional approach that make it inappropriate for valuing projects. The NPV rule makes some implicit assumptions that are often ignored. Firstly, it makes a now or never proposition: If an investment is not made now, it can not be made in the future. Secondly, it makes the assumption that future decisions are set to be fixed once the project has been undertaken. And thirdly, it assumes that expenditures made for an investment can be recovered if it turns out worse than anticipated. Although in reality, (1) most projects can be delayed, and made in the future; (2) the decisions are not fixed and can be changed even when the project has started; and (3) the expenditures associated with the project are irreversible and cannot be recovered. Hence, there is a certain value of flexibility that the conventional NPV analysis does not consider. The real options analysis attempts to capture this value and helps explain why the actual investment behavior of managers differ from the traditional NPV rule.

1.3 Types of real options and analogy to financial

Real options have similar features to financial options as they grant holders the right, but not the obligation, to do certain things in the future that can benefit the owners of the options. However, There are three major differences between real options and financial options:2

1. The underlying asset for the real option is a physical asset whose value is affected by managerial decisions and exogenous risk factors, whereas the un- derlying asset for the financial option is another financial security whose value is affected by exogenous risk factors only.3
2. Real Options are not directly traded in financial markets and therefore have no counter-party involved. In contrast, financial options are sold on financial markets. For every call option purchased, another is sold by a counter-party.
3. The exercise price for real options is the cash outlay upon exercise of any right such as investment costs. For financial options, it is the amount of money that is exchanged for the underlying asset.

Most real options are similar to American type options because they have an earlier exercise feature. They fall roughly into three major groups: (1) operating op- tions, (2) growth options, and (3) compound options. Operating options are based on real investments or projects already in operation and include abandonment, tem- porary suspension, or contraction options. Growth options are based on investments that have not yet been made and thus allow further growth. They include extension, deferral or expansion options. Compound options are options on other options, and there are two types: (a) Simultaneous compound options that are exercisable at the same time, and (b) sequential compound options that occur when an earlier option must be exercised in order to keep later options open. An overview of the real option types described are summarized in Table 1.1 below.

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Chapter 2 Real Options Theory

More than 40 years of academic research on option pricing led to remarkable results in the field of real options theory. This chapter starts off with summarizing the main achievements in that field. Next, the fundamental building blocks for pricing of real options are laid by developing the mathematical tools for real options. We take a brief glimpse of the world of stochastic calculus and introduce some basic stochastic processes such as the Wiener process, the Itô process, and the geometric Brownian motion as well as two alternative processes.

2.1 Literature Review

Black and Scholes’ (1973)[5] seminal paper ”The pricing of options and corporate liabilities” in the field of financial option pricing led to extensive theoretical and applied researches about real options. They laid the intellectual groundwork for pricing options where most recognition were given to the idea of contingent claims. The secrets of option pricing seem to lie in the solution of certain partial differential equations (PDEs) where specific boundary conditions must be satisfied by the value of the contingent claim. However, only few closed-form solutions (e.g. the Black- Scholes formula) exist. Thus, the solution must be approximated.

Schwartz (1977)[35] introduced the finite difference method, and Boyle (1978)[6] the Monte-Carlo simulation method. These conventional opting pricing methods were then extended in order to value investments under uncertainty, namely real options. Myers (1976)[32] was the first to recognize the similarities between finan- cial and real options. Cox, Ross, and Rubinstein (1979)[11] then introduced the binomial approach, a discrete time model, to value real option problems. It is a very intuitive and powerful model that uses risk-neutral probabilities or replicating portfolios to determine the option values. In this model, binomial lattices can be used to solve for American type options as well whereas the Black-Scholes model can only price European type options (and infinite-horizon American type options). In fact, in continuous time the Cox-Ross-Rubinstein model can accurately approxi- mate solutions from the Black-Scholes model e.g. by using a dynamic programming approach. The dynamic programming is a mathematical optimization method that breaks down complex problems into smaller sub-problems. For all approaches, the evolution of uncertainty of an underlying asset is said to follow a stochastic process. With specifying parameters, a drift rate and a variance rate for example, Cox, Ross and Rubinstein (1979)[11] show that the binomial model converges weakly to a log-normal geometric Brownian diffusion.

However, Cox and Ross (1976)[10] point out that this may not hold for real options since many real investments follow a mean-reverting process. Investments that follow a mean-reverting process, e.g. commodity prices, have cash flows that are pulled back to a central value in the long-run. Bhattacharya (1978)[4] shows that in a competitive economy, mean-reverting cash flows are more likely for many investments because cash flow expectations for investments of the same type tend to normalize over time. Laughton and Jacoby (1993)[25] reconfirmed that commodity prices follow mean-reverting processes, and point out that the log-normal geometric Brownian model overestimates the value of the option.

First applications of option pricing methods in the valuation of investments under uncertainty were conducted by, (1) Tourinho (1979)[41] who evaluated a non- renewable natural resources reserve; (2) Brennan and Schwartz (1985)[7] who ana- lyzed a copper mine; (3) McDonald and Siegel (1986)[32] who studied the optimal timing of investment in an irreversible project where the benefits and cost follow continuous time stochastic processes; and (4) Titman (1985)[40] who estimated the value of an undeveloped real estate. Dixit and Pyndick (1994)[14] as well as Tri- georgis (1996)[42] provided a well-rounded summary in the field of real options up until that point.

The diversity of business environments led to the emergence of numerous sub- fields such as real option pricing with elements of microeconomics and business strategy. Notable contributions in that include models with information asymmetry. Grenadier (2002)[18] derived optimal investment strategies in continuous Cournot- Nash equilibria. Grenadier and Wang (2005)[19] analyzed an investment timing problem in an agency setting and show that managers have a more valuable option than the shareholders. Bernardo Chowdhry (2002)[3] showed that an increase in in- formation uncertainty decreases the real option value, whereas an increase in estima- tion of uncertainty increases the real options value. Shibata (2006)[38] examined the real option model in the same setting, but extended it with profit uncertainty that followed a stochastic process which reconfirmed the results obtained by Bernardo and Chowdhry (2002)[3]. Graham (2010)[17] showed in his strategic model with asymmetric information that the outcome is extremely sensitive to the pre-specified information setting. Numerous other researches were conducted, however we regard the above mentioned as the most prominent ones.


1 Cf. Schwartz(2013)[37], page 164.

2 Cf. Copeland, Weston, Shastri (2005)[8], page 307.

3 Under assumption that the financial market is the only source of uncertainty for financial options.

Excerpt out of 50 pages


On the Valuation of Real Options. Necessary Mathematical Tools and Compelling Approaches in Financial Literature
European University Viadrina Frankfurt (Oder)
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corporate finance, realoptions, option, pricing, valuation, derivatives, NPV, Real, Options
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Viet Dung Le (Author), 2015, On the Valuation of Real Options. Necessary Mathematical Tools and Compelling Approaches in Financial Literature, Munich, GRIN Verlag, https://www.grin.com/document/307164


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