Real Rainbow Options in Commodity Applications Valuing Multi-Factor Output Options under Uncertainty A Thesis submitted to the University of Manchester for the Degree of Doctor of Business Administration in the Faculty of Humanities 2010 Jörg Dockendorf Manchester Business School Contents LIST OF TABLES ....................................................................................................... 4 LIST OF FIGURES ..................................................................................................... 4 ABSTRACT .................................................................................................................. 5 DECLARATION .......................................................................................................... 6 COPYRIGHT STATEMENT ..................................................................................... 7 ACKNOWLEDGEMENTS ......................................................................................... 8 1 INTRODUCTION ................................................................................................ 9 1.1 Research Objectives and Questions ............................................................. 11 1.2 Contributions to Knowledge ........................................................................ 12 1.3 Thesis Overview........................................................................................... 13 2 REVIEW OF RAINBOW OPTIONS AND THE COMMODITY CONTEXT .................................................................................................................. 18 2.1 Rainbow Options .......................................................................................... 18 2.2 Stochastic Processes for Commodity Prices ................................................ 27 2.3 Overview of Solution Techniques ................................................................ 32 3 RESEARCH PAPER #1 – CONTINUOUS RAINBOW OPTIONS ON COMMODITY OUTPUTS: WHAT IS THE REAL VALUE OF SWITCHING FACILITIES? ............................................................................................................. 39 4 RESEARCH PAPER #2 – CONTINUOUS RAINBOW OPTIONS IN CO- INTEGRATED MARKETS ...................................................................................... 91 5 RESEARCH PAPER #3 – SEQUENTIAL REAL RAINBOW OPTIONS 144 6 CONCLUSION ................................................................................................. 196 6.1 Findings ...................................................................................................... 196 6.2 Critical Discussion ..................................................................................... 199 6.2.1 Problem framing .............................................................................................. 199 6.2.2 Critical Assumptions ........................................................................................ 201 6.2.3 Theory .............................................................................................................. 202 6.2.4 Numerical solution methods ............................................................................. 204 6.3 Directions of Future Research .................................................................... 205 – 2 – REFERENCES ......................................................................................................... 207 APPENDIX A.1. IMPLEMENTATION OF NUMERICAL LATTICE SOLUTION ............................................................................................................... 214 APPENDIX A.2. IMPLEMENTATION OF FINITE DIFFERENCE SOLUTION FOR SEQUENTIAL RAINBOW OPTION ON TWO ASSETS ......................... 219 APPENDIX A.3. IMPLEMENTATION OF FINITE DIFFERENCE SOLUTION FOR SEQUENTIAL RAINBOW OPTION ON MEAN-REVERTING ASSET .................................................................................................................................... 233 APPENDIX A.4. IMPLEMENTATION OF MONTE CARLO SIMULATION FOR SEQUENTIAL RAINBOW OPTION ON TWO ASSETS ......................... 243 APPENDIX A.5. IMPLEMENTATION OF MONTE CARLO SIMULATION FOR SEQUENTIAL RAINBOW OPTION ON MEAN-REVERTING ASSET .................................................................................................................................... 246 Word count: 45,221 – 3 – List of Tables Table 1. Overview of the Rainbow Option Models developed in this Thesis 10 Table 2. Characteristics of the Rainbow Option Models 15 Table 3. Literature Overview on Rainbow Options 23 Table 4. Comparison of Numerical Solution Methods 38 List of Figures Figure 1. The Rainbow Option Models on a Timeline 13 – 4 – Abstract Name of the University: The University of Manchester Candidate: Jörg Dockendorf Degree Title: Doctor of Business Administration Thesis Title: Real Rainbow Options in Commodity Applications: Valuing Multi-Factor Output Options under Uncertainty Date: 2010 This thesis focuses on the valuation of real options when there is flexibility given by the choice between two risky outputs. We develop models to value these rainbow options and to determine optimal operating and investment policies. These models are studied in the context of commodity applications because output flexibility is particularly relevant in volatile commodity markets. We provide insights into the behaviour and sensitivities of option values and operating policies and discuss implications for decision-making. In the early stages of real options theory, research centred on basic options with closed-form solutions, modelling single uncertainty in most cases. The challenge is now to incorporate more complexities in the models in order to further bridge the gap between theoretical models and reality, thereby promoting the widespread application of real options theory in corporate finance. The new option models developed in this thesis are organised in three self- contained research papers to address specific research problems. The first research paper studies an asset with flexibility to continuously choose the best of two risky commodity outputs by switching between them. We develop quasi-analytical and numerical lattice solutions for this real option model, taking into account operating and switching costs. An empirical application to a flexible fertilizer plant shows that the value of flexibility between the two outputs, ammonia and urea, exceeds the required additional investment cost (given the parameter values) despite the high correlation between the commodities. Implications are derived for investors and policy makers. The real asset value is mainly driven by non-stationary commodity prices in combination with constant operating costs. In the second research paper, we study an asset with flexibility to continuously choose the best of two co-integrated commodities. The uncertainty in two commodity prices is reduced to only one source of uncertainty by modelling the spread, which is mean-reverting in the case of co- integration. Our quasi-analytical solution distinguishes between different risk and discount factors which are shown to be particularly relevant in the context of mean- reversion. In an empirical application, a polyethylene plant is valued and it is found that the value of flexibility is reduced by strong mean-reversion in the spread between the commodities. Hence, operating flexibility is higher when the commodities are less co-integrated. In the third research paper, we develop real option models to value European sequential rainbow options, first on the best of two correlated stochastic assets and then on the spread between two stochastic co-integrated assets. We present finite difference and Monte Carlo simulation results for both, and additionally a closed-form solution for the latter. Interestingly, the sequential option value is negatively correlated with the volatility of one of the two assets in the special case when the volatility of that asset is low and the option is in-the-money. Also, the sequential option on the mean-reverting spread does not necessarily increase in value with a longer time to maturity. – 5 – Declaration No portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. – 6 – Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the “Copyright”) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trademarks and other intellectual property (the “Intellectual Property”) and any reproductions of copyright works in the thesis, for example graphs and tables (“Reproductions”), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual