Consulting Editor. George A. Anastassiou Department of Mathematical Sciences The University of Memphis

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2 Consulting Editor George A. Anastassiou Department of Mathematical Sciences The University of Memphis

3 Alexander Vollert A Stochastic Control Framework for Real Options in Strategic Evaluation Birkhiiuser Boston Basel Berlin

4 Alexander Vollert Universitlit Karlsruhe (TH) Postfach Karlsruhe, D Gennany and McKinsey & Company, Inc. Birkenwaldstr Stuttgart Gennany Library of Congress Cataloging-in-Publication Data Vollert, Alexander. A stochastic control framework for real options in strategic valuation I Alexander Vollert. p.cm. Includes bibliographical references and index. ISBN e-isbn-13: DOl: / Real options (Finance)-Mathematical models. 2. Corporations-Valuation. 3. Business enterprises-valuation. 4. Capital investments-decision making-simulation methods. I. Title. HG4028.V3 V dc AMS Subject Classifications: 60Gxx Printed on acid-free paper Birkhiiuser Boston Birkhiiuser Softcover reprint of the hardcover 1st edition 2003 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhliuser Boston, clo Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN-13: e-isbn-13: SPIN Reformatted from author's files by TEXniques, Inc I Birkhiiuser Boston Basel Berlin A member of BertelsmannSpringer Science+Business Media GmbH

5 Contents Acknowledgment xiii 1 Overview Background and Objectives of the Study 1.2 Organization of the Study Introduction to Real Options Basic Idea Why Flexibility Adds Value Flexibility and Traditional Capital Budgeting Methods Towards a New Investment Paradigm Classification of Real Options Management Perspective Valuation Perspective Discussion of the Real Options Approach When to Use Real Options Advantages of the Real Options Approach Drawbacks of the Real Options Approach Conclusions Real Options and Stochastic Control Real Option Interactions and Stochastic Control Introduction to Impulse Control and Optimal Stopping General Idea Impulse Control Impulse Control Model for Valuing Real Options Problem Formulation Impulse Control Verification Theorem Interpretation and Extensions Combined Impulse Control and Optimal Stopping Problem Formulation Combined Verification Theorem

6 vi Contents 4 Valuing Real Options in a Stochastic Control Framework Equivalence of Stochastic Control and Contingent Claims Analysis Hedging Portfolio and Fundamental Pricing Equation Equivalent Martingale Measure Interpretation and Conclusions Contingency Structure of Option Interactions Example: Timing and Intensity of Investment Extensions: Competition and Time Delay Effects 5.1 Competitive Interaction Exogenous Competition Endogenous Competition. 5.2 Time Delay Effects Case Study: Flexibility in the Manufacturing Industry 6.1 Real Options and Volume Flexibility Model Static Project Value with Fixed Capacity Timing and Intensity Flexible Capacity Timing, Intensity and Flexible Capacity 6.3 Numerical Solution Techniques Finite Difference Methods General Numerical Solution Procedure Numerical Analysis Simulation Results.,. 7 Conclusions and Extensions Bibliography Index

7 List of Figures 2.1 Asymmetric risk profile caused by managerial flexibility 2.2 Time value and intrinsic value of a European call option. 2.3 Sources of firm value Different sources of risk Characteristics of the underlying 2.6 Characteristics of the volatility. 2.7 Characteristics of dividends Characteristics of the rate of return shortfall 2.9 Characteristics concerning the exercise price Characteristics of the risk free interest rate Topics concerning time to maturity 2.12 Different aspects of competition Characteristics of time lags When managerial flexibility is valuable Reversible and irreversible switching. Organization of the study Characteristics of switches Modelling element generalized switching option. Modelling element irreversible switching Modelling element single state..... Modelling element n parallel states... Modelling element infinite time horizon Modelling element abandonment.... Graphical representation of a European option. Graphical representation of an American option Graphical representation of an American compound option Graphical representation of a European compound option. Perpetual option to invest.... Optimal entry and exit decision.... Option interactions Perpetual timing and intensity option

8 viii List of Figures 4.16 Comparison of option values for fixed and variable intensity Percentage of added value by intensity option... " Impact of changing price volatility a on P*(M*), P*(l) and M* Impact of changing marginal production cost c on P*(M*), P*(1) and M* Impact of changing interest rate ron P*(M*), P*(l) and M* Impact of changing rate of return shortfall 8 on P*(M*), P*(1) and M* Impact of changing output efficiency a on P*(M*), P*(I) and M* Impact of unit cost of capacity m on P*(M*), P*(l) and M* Modelling element exogenous competition Graphical representation of the option to invest under competition 5.3 Value of the shared option to invest (model 1) Value of the shared option to invest (model 2) Impact of changing value erosion ratio g on vt (model 1). 5.6 Impact of changing value erosion ratio g on Vi (model 2). 5.7 Impact of changing arrival rate A on vt(g) (model 1) 5.8 Impact of changing arrival rate A on Vi(g) (model 2) 5.9 Impact of changing volatility a on vt(g) (model 1) Impact of changing volatility a on Vi(g) (model 2) Entry and exit decision in monopoly with variable capacity QA Entry and exit decision in duopoly with variable capacities QA and QB Modelling element time lag as generalized switching option Modelling element time lag as generalized timing option Impact of volatility on trigger values with and without time lag, E = Impact of volatility on trigger values with and without time lag, E = Impact of increasing time lag h on trigger values, E = 1 and a 2 = Graphical representation of the product life cycle model Product life cycle model with timing and intensity option Product life cycle model with flexible capacity Product life cycle model with combined timing, intensity and flexible capacity Optimal production rate Q* for capacity levels M = 1.0 and M = Instantaneous annual cash flow f for optimally controlled production rates Output price per unit produced for optimally controlled production rates

9 List of Figures ix 6.8 Static project value and optimal initial capacity Flexible capacity project value and optimal initial capacity 6.10 Timing and intensity option for static project Timing and intensity option for flexible capacity project Demand threshold and optimal capacity for the static project " Demand threshold and optimal capacity for the flexible capacity project Sources of value from the timing and intensity option and the static project Sources of value from the flexible capacity option and the static project Sources of value from the timing and intensity option, the flexible capacity option and the static project Sensitivity analysis Simulated trajectory of the demand parameter level (static project) Optimal production rate trajectory and capacity level (static project) Simulated trajectory of the demand parameter level (timing and intensity) Optimal production rate trajectory and capacity level (timing and intensity) Comparison of cash flow trajectories (static project and timing and intensity) Simulated trajectory of the demand parameter level (flexible capacity project) Optimal production rate trajectory and capacity levels (flexible capacity project) Simulated trajectory of the demand parameter level (flexible capacity project and timing and intensity) Production rate trajectory and capacity levels (flexible capacity 226 project and timing and intensity) Comparison of cash flow trajectories (flexible capacity project and timing and intensity) Simulated pdf of the four models Simulated cdf of the four models Simulated cdf of the four models with capacity restriction Simulated cdf of the four models with capacity restriction and abandonment

10 List of Tables 2.1 Comparison of the option analogy between a call option and the option to invest Comparison of Stochastic Control and Contingent Claim Analysis State description of Trigeorgis' option interaction model Cash flow earned by firm i in each of the four possible states Firm values for each possible state of the option game Qualitative results of strategies and firm values under competition Summary of the variables' names in the growth regime Base case parameter values Project and option values for base case parameters Comparison of simulation results for base case project values Project values for models with capacity restriction Project values for models with capacity restriction and abandonment Project values for the different flexible capacity models Simulation results for restricted capacity project values Simulation results for restricted capacity project values with abandonment

11 Acknowledgment I wish to extend my deep gratitute to those who influenced me a great deal and helped me prepare this text. My first acknowledgement is a deep one. Prof. Dr. Svetlozar T. Rachev (University of Karlsruhe, Germany, and University of California, Santa Barbara) was always generous with his time and counsel; his help and interest are acknowledged with grateful thanks. Along the way, my thinking was influenced by many stimulating discussions with Prof. Dr. Karl-Heinz Waldmann and Prof. Dr. Kuno Egle (both University of Karlsruhe). At various stages their role as a sounding board for various pieces proved extremely fruitful and is deeply acknowledged. My former colleagues at University of Karlsruhe deserve special mention for their intellectual guidance. Dr. Elke Sennewald and Dr. Christian Peter were always inspirational discussion partners; we had a great time together in academia. My current employer, McKinsey & Company, Inc., deserves acknowledgement for providing me with the opportunity to finish this work. I am grateful to Prof. Dr. George Anastassiou (University of Memphis) for recommending this text for publication. Furthermore, I would like to thank the Executive Editor, Ann Kostant of Birkhiiuser, who worked with me on this book. As expected, my greatest gratitude is to my family. Without the constant support of my parents throughout my life this book would not have seen the light of day. For her enduring guidance, encouragement, love and support, I would especially like to thank my wife Christina. Most important is my daughter Johanna Deborah Vollert, who stands by me enduring late nights and weekends, makes it all worthwhile and ensures that my life is full of love and excitement. Finally, I would like to dedicate this book to my late father, Ulrich Vollert.

12 A Stochastic Control Framework for Real Options in Strategic Evaluation