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ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 UMI UNIVERSITY OF OKLAHOMA GRADUATE COLLEGE MULTIFACTOR VALUATION MODELS OF ENERGY FUTURES AND OPTIONS ON FUTURES A Dissertation SUBMITTED TO THE GRADUATE FACULTY in partial fulfillment of the requirements for the degree of Doctor of Philosophy By MARKJBERTUS Norman, Oklahoma 2003 UMI Number: 3082955 UMI UMI Microform 3082955 Copyright 2003 by ProQuest Information and Learning Company. Ali rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 © Copyright by Mark Bertus 2003 All Rights Reserved MULTIFACTOR VALUATION MODELS OF ENERGY FUTURES AND OPTIONS ON FUTURES A Dissertation APPROVED FOR THE MICHAEL F. PRICE COLLEGE OF BUSINESS BY C 3 TABLE OF CONTENTS Title Page ............................................................................................................................... i Signature Page ...................................................................................................................... ii Copyright Page .................................................................................................................... iii Table of Contents ................................................................................................................. iv List of Tables ....................................................................................................................... vi List of Illustrations ..............................................................................................................vii Abstract.............................................................................................................................. viii Chapter 1: Introduction .....................................................................................................1 Endnotes ........................................................................................................5 Chapter 2; World Crude Oil Market ................................................................................ 6 2.1 Crude Oil........................................................................................................ 6 2.2 Supply .............................................................................................................7 2.3 Inventories ................................................................................................... 10 2.4 Demand.........................................................................................................11 2.5 Prices.............................................................................................................14 Endnotes ...................................................................................................... 18 Chapter 3: Commodity Forward and Futures Pricing Models ......................................20 3.1 One-factor ....................................................................................................21 3.2 Two-factor ....................................................................................................39 3.3 Three-factor ..................................................................................................60 3.31 Forward Prices .............................................................................................60 3.32 Futures Prices...............................................................................................70 Endnotes ......................................................................................................91 Chapter 4: Commodity Futures Option Pricing Models ............................................. 115 4.1 Black Scholes ............................................................................................. 116 4.2 One-factor .................................................................................................. 140 4.3 Two-factor ..................................................................................................156 4.4 Three-factor ................................................................................................ 165 4.5 Merton’s model ..........................................................................................176 4.6 Bates’ model ..............................................................................................210 4.7 Hilliard and Reis model ............................................................................ 230 4.8 One-factor jump-difïusion model .............................................................238 4.9 Two-factor jump-diffusion model ............................................................244 4.10 Three-factor jump-difhision model ..........................................................251 Endnotes ...................................................................................................265 Chapter 5: Simulations ............................................................................................... 278 5.1 One-factor futures ..................................................................................... 278 5.2 Two-factor futures .................................................................................... 279 5.3 Three-factor futures .................................................................................. 281 5.4 One-factor Options ................................................................................... 283 5.5 Two-factor Options ..................................................................................285 Chapter 6; Conclusion ................................................................................................ 324 Bibliography ..................................................................................................................326 IV Appendix A: Dynamic Factor Analysis ....................................................................... 330 A.1 Kalman Filter.......................................................................................... 330 Introduction ............................................................................................. 330 Review of Conditional Probabilities ...................................................... 335 Derivation of Kalman Filter ....................................................................342 A.2 The EM Algorithm ..................................................................................352 Appendix B: Arbitrage Pricing Models ........................................................................367 B.l One factor model .....................................................................................367 B.2 Gibson and Schwartz (1990) two factor model ......................................369 Appendix C: Risk Neutral Pricing Methods .................................................................375 C.l Partial Differential Equations ................................................................. 375 C.2 Equivalent Martingale Measures ............................................................ 382 Changing the Means ................................................................................382 Example 1 Operating on the individual realizations .............................. 383 Example 2 Operating on the Probability Distributions .......................... 384 Girsanov Theorem ...................................................................................395 Appendix D: Stochastic Calculus ..................................................................................401 Appendix E: A Discussion of the Feynman-Kac Formula ...........................................411 Example 1 Deterministic discount rate and random cash flow ..............412 Example 2 Stochastic interest rate and a known cash flow ....................416 Example 3 Stochastic discount factor with a stochastic cash flow ........ 421 List of Tables Table 1............................................................................................................................287 Table 2 ............................................................................................................................291 Tables ............................................................................................................................295 Table 4 ............................................................................................................................299