The Pricing of Real Options: An Overview Diploma Thesis by Christian von Drathen Universität Fridericiana Karlsruhe (TH) Institut für Unternehmensforschung und Entscheidungstheorie Prof. Dr. Siegfried Trautmann, Supervisor November, 1990. Final Version b Acknowledgements / 1 This diploma thesis is dedicated to my parents Ilse and Gerrit. Acknowledgements / 2 Acknowledgements When I started to research for this diploma thesis last year, I did not know much about option pricing in general, and nearly nothing about the pricing of real options. That this has - hopeful- ly - changed in the meantime is rather the result of successful cooperation and generous sup- port than my personal merit. In particular, I acknowledge the guidance of my supervisor, Prof. Dr. Siegfried Trautmann, on leave to the University of Mainz, and the good preparation in the field of finance by Prof. Dr. Hermann Göppl, University Fridericiana Karlsruhe (TH). Without the generous support of McKinsey & Company, Inc., Germany, this paper could not have been written. In particular, ⁄ would like to thank Dr. Michael Jung, Vienna, and Dr. Thor- sten Grenz, Frankfurt, for a lot of assistance and fruitful advice. Moreover, the McKinsey li- brary and reseach service in Düsseldorf provided me with the opportunity to gain access to va- luable sources of information. The experiment of writing this paper in English significantly increased the number of potential readers. Nevertheless, it contained the hazard of reduced clarity since my native language is German. In this context, I wish to express my sincere thanks to Herbert John Raths, Hamburg, who invested a lot of time and effort to increase the quality of this thesis by helpful advice and stimulating discussions. Last but not least, I must express the gratitude I feel towards my parents who made my studies possible and who supported me over the years. As a little sign of respect this thesis is dedicated to them. Christian von Drathen, London, November, 1990. Contents / 3 The Pricing of Real Options: An Overview Acknowledgements i Contents ii 1. Introduction and Summary 5 PART I: INTRODUCTION TO OPTION PRICING 2. Option Pricing Theory and Contingent Claims Analysis 9 2.1 Introduction - 9 2.2 Option Pricing Theory (OPT) - 11 2.3 Contingent Claims Analysis (CCA) - 24 2.4 Extensions - 31 3. The Theory of Financial and Real Options 39 3.1 Introduction - 39 3.2 Financial Options - 39 3.3 Real Options - 45 4. Numerical Methods to Calculate the Option Value 51 4.1 Introduction - 51 4.2 Methods that Approximate the Value of the Underlying Security - 52 4.3 Methods that Approximate the Value of the PDE - 55 Contents / 4 The Pricing of Real Options: An Overview PART II: MODELS TO EVALUATE REAL OPTIONS 5. Models to Evaluate Real Options 61 5.1 Introduction - 61 5.2 Empirical Relevance and Problems - 64 5.3 Operating Options - 69 5.3.1 Operations with Temporary Shut Down - 69 5.3.2 Abandonment - 80 5.3.3 Operations with Temporary Shut Down and Abandonment - 84 5.4 Investment Opportunities - 92 5.4.1 Single Stage Growth Options - 92 5.4.2 Two Stage Growth Options - 102 5.4.3 Multi Stage Growth Options - 106 5.4.4 Other Applications for Investment Opportunities - 124 5.5 Project Financing - 117 5.6 Conclusion - 132 Appendix: Numerical Procedures to Evaluate Real Options 136 A Abandonment - 136 B Temporary Shut Down and Abandonment - 147 C Construction Time Flexibility - 162 D Project Financing - 175 References 199 1. Introduction and Summary / 5 1. Introduction and Summary A contingent claim is a security whose value depends on those of other, more basic underlying variables. In recent years contingent claims have increased in importantance in the field of fi- nance. Options and Futures are now actively traded on many exchanges. Financial intermedia- tes and their corporate clients regularly issue forward contracts and swaps. Other more specia- lized contingent claims often form part of a bond or stock issue. Even stock itself can be regar- ded as a contingent claim on the value of the underlying firm's assets. Often, the variables underlying the contingent claims are the prices of traded securities. For example, a stock option constitutes a claim contingent upon the the price of the stock. Contin- gent claims, however, can be a function of almost any variable "... from the price of hogs to the amount of snow falling at a certain ski resort." (Hull (1989) p. 1). This paper concentrates on contingent claims, whose underlying variables are values of either real investments or factors that determine their value. If these contingent claims are options, they are called real options. Introduction of different types of real options, and their applications facilitates the presentation of models currently available for quantitative evaluation. It turns out that the applications have a high empirical relevance, where uncertainty about future events is involved. In particular, for capital budgeting decisions concerning operating flexibility or growth opportunities, contin- gent claims analysis (CCA) proves to be an invaluable tool. Two main parts of this paper reflect the relative importance of applications and models. Part I treats the basic concepts necessary for minimal understanding of the pricing of real options. It addresses those unfamiliar with the analysis of contingent claims. Part II, the intellectual bulk of this paper, deals with models to evaluate real options. In part I, as a necessary prerequisite Option Pricing Theory (OPT) and Contingent Claims Analysis (CCA) are introduced with a demonstration of the main concepts to be applied to simple financial options and securities encountered in corporate capital structure. The next step in the assessment of real options, is to introduce options for specific situations. They provide an impression of how to derive more complex real option models. Finally, different real opti- ons themselfes are structured and individually introduced. Readers already familiar with CCA can skip this part. Chapter 2 explains the basic concepts of OPT and CCA. Besides the option terminology and general relations for option prices, three different approaches to derive an exact option pricing formula are presented. The approaches rely solely on arbitrage arguments, that is a portfolio of 1. Introduction and Summary / 6 securities which is continuously immunized against risk must earn the risk-free rate of return. Since OPT covers only the narrow field of traded financial options, the general character of CCA is demonstrated via an example of corporate liabilities. It turns out, that all corporate lia- bilities are contingent claims upon the value of firm's assets, where the risk of bankrupcy is ta- ken into account. An extension in this chapter shows that the value of all contingent claims can be described by a generalized partial differential equation (PDE). The equation is subject to a few specific functions to the claim, but presents a framework to evaluate most contingent claims, nevertheless. Finally, chapter 2 also shows how to relax several standard assumptions in order to arrive at specially tailored models. In addition to different types of stock price movements, special opti- ons - such as compound options, options to exchange one risky asset for another, and options on the minimum and the maximum of two assets - are presented. They prove useful for under- standing the process of deriving models for real options. Chapter 3 demonstrates how to use CCA for the assessment of further corporate liabilities and introduces the different categories of real options. The corporate liabilities addressed are sub- ordinated debt, warrants, convertibles, and callable convertibles. It turns out, that for some se- curities there are representations by simple options, but others can only be evaluated with their own uniquely tailored model. The financial options treated in this section constitute a part of corporate capital structure and typically appear on the liabilities side of corporate balance sheets. On contrary, most real options are part of the assets side of corporate balance sheets. They as- sess management's flexibility to react as uncertainty over future events resolves. Real options fall into three groups: Operating options, investment opportunities, and project financings. Operating options represent management's flexibility to react to changes in profitability by al- tering the operation process. The means to achieve this are the options to temporarily shut down, to abandon, or to switch to a set of more profitable input or output factors. Investment opportunities can either be complete in themselves or entail further investment opportunities. Finally, project financing applies when investments contain both financial and real options. In particular, application of these models allow to consider the effects of operating flexibility, growth opportunities, and bankrupcy risk on all components of project's capital structure. Chapter 4 introduces valuation methods for option pricing models that do not have closed- form, or analytical solutions. For a few simple options, analytical solutions exist for their price. 1. Introduction and Summary / 7 For many realistic financial and real options, however, they do not. In such cases numerical ap- proximation procedures can be employed. One distinguishes between two categories of techni- ques: First, methods that approximate the value of the underlying variable directly, and second, methods that approximate the PDE. The chapter briefly introduces the properties