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University of Warwick Institutional Repository University of Warwick institutional repository: http://go.warwick.ac.uk/wrap A Thesis Submitted for the Degree of PhD at the University of Warwick http://go.warwick.ac.uk/wrap/4063 This thesis is made available online and is protected by original copyright. Please scroll down to view the document itself. Please refer to the repository record for this item for information to help you to cite it. Our policy information is available from the repository home page. 1G 19 1 MENS TAT AI OLEM Models for the Price of a Storable Commodity by Diana Ribeiro Thesis Submitted in partial fulfilment of the requirements to the University of Warwick for the degree of Doctor of Philosophy Financial Options Research Centre, Warwick Business School May 2004 THE UNIVERSITY OF WARWICK Contents List of Tables V List of Figures vii Acknowledgments x Declarations xii Abstract AV Abbreviations xvi Chapter 1 Introduction 1 1.1 Background 1 ......... ................. 1.2 Summary Current Research 5 of ................. 1.3 Objectives 8 .......... ................. 1.4 Overview 9 ........... ................. Chapter 2 Literature Review 13 2.1 Introduction 13 ......... ................ 2.2 The theory of Storage 14 .... ................ 2.2.1 Convenience Yield 14 ................ 2.2.2 Risk-Premium ..................... 18 2.3 Commodity Price Models ................... 19 2.3.1 Structural Models ................... 20 2.3.2 A Partial Equilibrium Real Options Model 41 2.3.3 Reduced Form Models ................. 46 24 Conchicion 65 Chapter 3A Structural Model for the Price Dynamics: Competitive and Monopolistic Markets 70 3.1 Introduction 70 ............................ 3.2 Storage Equilibrium 75 ........................ 3.2.1 Model Formulation 77 ..................... 3.2.2 Linear Inverse Demand Function 85 .............. 3.3 Numerical Implementation 86 ..................... 3.3.1 Space-time Discretization the Explicit Solution 87 and .... 3.4 Results 90 ............................... 3.4.1 Competitive Case 92 ..................... 3.4.2 Monopolistic Case 96 ..................... 3.4.3 Comparison Between the Competitive and the Monopolistic Markets 96 .......................... 3.5 Extension Model to Include Seasonality 100 of the .......... 3.6 Conclusion 101 ............................. Chapter 4A Structural Model for the Price Dynamics: Analysis of the Forward Curve 105 4.1 Introduction 105 ............................ 4.2 Description of the Lattice Model 108 ................. 4.2.1 The Branching Process 108 .................. 4.2.2 The Merging Process 113 ................... 4.3 Calculation of the Forward Curve the Convenience Yield 114 and ... 4.3.1 Results 115 ........................... 4.4 Conclusion 124 ............................. Chapter 5A Contango Constrained Reduced Form Model 126 5.1 Introduction 126 ............................ 5.2 Model Definition 129 .......................... 5.3 Numerical Implementation 134 ..................... 5.3.1 Lattice Description 135 ..................... 5.4 Numerical Results 138 ......................... 5.4.1 Results from Lattice Model 138 the .............. 5.5 Conclusion 146 ............................. Chapter 6A Two-Factor Model for Commodity Prices and Futures Valuation 149 6.1 Introduction 149 ............................ 6.2 Valuation Model 153 .......................... 6.3 Empirical Estimation the Joint Stochastic Process 160 of ....... 6.3.1 State Space Formulation 162 .................. 6.3.2 Empirical Results 165 ...................... 6.4 Conclusion 171 ............................. Chapter 7 Conclusion 173 7.1 Summary Contributions 173 and ... ............... T2 Limitations Further Research 179 and ............... Appendix A Derivation of the Stochastic Dynamic Programming Equa- tion 182 Appendix B Calculation of the Spot Prices Variability 185 Appendix C Lattice Model for the Ornstein- Uhlenbeck Process 187 Appendix D Shepard Local Interpolation 191 Appendix E Derivation Process Followed by In (pt) 193 of the xt - Appendix F Derivation of the Futures Partial Differential Equation 195 Appendix G Derivation of the ODEs and corresponding solution 197 G.1 Derivation ODEs 197 of the .......... G.2 Solution to the system of differential equations (6-14) and (6.15) with initial conditions (6.16) 198 Appendix H Kalman Filter Procedure 201 IV List of Tables 3.1 Value of the Parametersused to obtain the numerical solutions of both competitive and monopolistic markets 91 3.2 zA,, and ým,;, represent the lower and upper values of the grid for "I z-,,, . the exogenous supply rate, z. s,ývj,;, represents the storage capacity. 91 4.1 T represents the maximum time to maturity considered, dt repre- sents the time-step, I represents maximum number of combinations (s, 7.) allowed at each node of the tree, and is the new number (, the takes 116 of 5, z) combinations after merging place........ 4.2 Value of the parameters used to implement the tree for both com- 116 petitive and monopolistic markets .................. lower for 4.3 z,,Ili,. and represent the and upper values of the grid the exogenous supply rate, represents the storage capacity, 116 Values lattice 138 5.1 of the parameters used in the computation ..... 5.2 Sample moments at final T=5 for the log price sample paths c7,. 145 6.1 Light Crude Oil Futures weekly data from 17" of March 1999 to 241h,of December 2003 165 V 6.2 Estimation results and standard errors (in parentheses) for both our model and Schwartz (1997) [82] two-factor model using all the futures contracts FO, Fl, F2, F3, F4, F5 and F6 from 17" of March 1999 to 24'ý` December 2003 167 of ................... 6.3 Summary statistics both our model's and Schwartz (1997) [82] two- factor model's pricing errors in valuing futures contracts during the 17'h March 1999 to 24th December 2003. 168 whole period of of ... 6.4 Market, our model and Schwartz (1997) [82] Two-Factor Model log-returns futures 170 implied volatilities of annualized of prices. ... vi List of Figures 3.1 Competitive Case Storage - rate, u, as a function of the two state variables inventory level, 94 s, and exogenous rate of supply, z. --- 3.2 Competitive Case - Super-position of two graphs for the prices in the absence and in the presence of storage, respectively. The price is represented as a function of the two state variables inventory level, 94 s, and exogenous rate of supply, .-,. ............. Case 3.3 Competitive - Difference between prices in the presence of storage and prices in the absence of storage as a function of supply different fixed levels inventory 95 rate, at of .............. Case function 3.4 Competitive - Price variability as a of supply rate, at different fixed levels 95 of inventory .................. Storage function 3.5 Monopolistic Case - rate, v, as a of the two state variables inventory level, s, and exogenous rate of supply, 98 Super-position for in 3.6 Monopolistic Case - of two graphs the prices the absence and in the presence of storage, respectively. The price is represented as a function of the two state variables inventory level, 98 s, and exogenous rate of supply, ýý.............. Vil 3.7 Monopolistic Case - Difference between prices in the presence of storage and prices in the absence of storage as a function of supply different fixed levels 99 rate, at of inventory.............. Monopolistic Case 3.8 - Price variability as a function of supply rate, different fixed levels 99 at of inventory ................. 4.1 Part the tree for the forward 112 of computing curve .......... 4.2 Competitive case: Evolution of the forward curve when the initial 4.5 the inventory is 120 supply rate is and initial empty......... 4.3 Competitive case: Evolution of the convenience yield curve when initial is 4.5 the initial is 120 the supply rate and inventory empty. .. 4.4 Monopolistic case: Evolution of the forward curve when the initial supply rate is 4.5 and the initial inventory is empty 121 4.5 Monopolistic case: Evolution of the convenience yield curve when initial 4.5 the inventory is 121 the supply rate is and initial empty. .. 4.6 Competitive case: Evolution of the forward curve when the initial supply rate is 4.5 and the initial inventory is equal to 1.125,2.25 4.5, 122 and respectively......................... 4.7 Competitive case: Evolution of the convenience yield curve when the initial supply rate is 4.5 and the initial inventory is equal to 1.125,2.25 4.5, 122 and respectively.................. 4,8 Monopolistic case: Evolution of the forward curve when the initial supply rate is 4.5 and the initial inventory is equal to 1.125,2.25 4.5, 123 and respectively......................... 4.9 Monopolistic case: Evolution of the convenience yield curve when the initial supply rate is 4.5 and the initial inventory is equal to 1.125,2.25 4.5, 123 and respectively ................ Vill 5.1 Forward curves generated by the one-factor mean-reverting model different levels for 5-year 143 at spot price a period ........... 5.2 Forward curves generated by our model at different spot price levels for 5-year 143 a period. ................. 5.3 Convenience yield term structures generated by the one-factor mean- different levels for 5-year 144 reverting model at spot price a period. 5.4 Convenience yield term structures generated by our model at dif- ferent levels for 5-year 144 spot price a period ............. 5.5 Probability density function for the spot prices sample paths at final time T=0 for the unconstrained model. 145 5.6 Probability density function for the spot prices sample paths at final T for 146 time -- 1") our model. .......... 6.1 This figure illustrates the evolution of the forward curve for the market of futures prices and both our model and the Schwartz (1997) [82] 5th November 2002. 169 two-factor model on the of ... 6.2 This figure illustrates the evolution of the forward curve for the market of futures prices and both our model and the Schwartz (1997) [82] 18th July 2001 169 two-factor model on the of ...... 6.3 This figure illustrates the annualized volatility of futures returns implied by our model, Schwartz (1997) [82] Two-Factor Model 170 and the market data ..........
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