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PRICING of SWING OPTIONS: a MONTE CARLO SIMULATION APPROACH a Dissertation Submitted to Kent State University in Partial Fulfill

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PRICING of SWING OPTIONS: a MONTE CARLO SIMULATION APPROACH a Dissertation Submitted to Kent State University in Partial Fulfill PRICING OF SWING OPTIONS: A MONTE CARLO SIMULATION APPROACH A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Kai-Siong Leow May, 2013 Dissertation written by Kai-Siong Leow B.S., Iowa State University, 1998 M.S., Oklahoma State University, 2003 Ph.D., Kent State University, 2013 Approved by , Chair, Doctoral Dissertation Committee Dr. Oana Mocioalca , Member, Doctoral Dissertation Committee Dr. Mohammad Kazim Khan , Member, Doctoral Dissertation Committee Dr. Lothar Reichel , Member, Outside Discipline Dr. Maxim Dzero , Member, Graduate Faculty Representative Dr. Eric Johnson Accepted by , Chair, Department of Mathematical Sciences Dr. Andrew Tonge , Associate Dean, College of Arts and Sciences Dr. Raymond A. Craig ii TABLE OF CONTENTS LIST OF FIGURES ......................................... v ACKNOWLEDGEMENTS ..................................... vi 1 Introduction ............................................ 1 1.1 Literature Review ..................................... 1 1.2 Derivative Pricing in Incomplete Market ......................... 3 1.3 Calibration of Energy Spot Price Process ........................ 8 1.4 Organization of the Dissertation ............................. 12 2 Model Description and Formulation ............................... 14 2.1 Swing Optionality ..................................... 14 2.2 Definition of Swing Options ................................ 17 2.3 Stochastic Optimal Control Model ............................ 18 2.4 Dynamic Programming Formulation ........................... 23 2.5 Structural Properties of Optimal Value Functions .................... 26 3 Monte Carlo Pricing Algorithm and Its Convergence ..................... 36 3.1 A Monte Carlo Simulation Method ............................ 36 3.2 Convergence of the Algorithm ............................... 43 3.3 Technical Lemmas ..................................... 48 3.4 Convergence Proof ..................................... 69 4 Numerical Results ........................................ 73 4.1 Swing Call Option ..................................... 73 BIBLIOGRAPHY .......................................... 80 iii A Bootstrapping the Risk-Free Discount Curve .......................... 83 A.1 Cash Deposit on Fed Funds Effective Rate ........................ 85 A.2 OIS Swap .......................................... 85 A.3 Synthetic OIS Swap: Fed Funds/3M LIBOR Basis Swap + 3M LIBOR Swap . 87 A.4 Bootstrapping Algorithm ................................. 88 B The MATLAB code ....................................... 90 C Optimal Exercise Strategy of the Swing Call Option in Section 4.1 ............. 105 iv LIST OF FIGURES 1.1 Monthly NYMEX Natural Gas Futures prices, open interests and trading volumes as of 2:30pm EST, 23 January 2013. Copyright 2013 Bloomberg Finance L.P. Reprinted with permission. All right reserved. Visit www.Bloomberg.com. ............ 9 1.2 Henry Hub natural gas spot price from 5th January 2011 to 23rd January 2013 . 11 1.3 Linear regression on logarithmic spot prices ....................... 12 2.1 System dynamics at t = 0 ................................. 20 2.2 System dynamics at a generic time t ........................... 21 2.3 System dynamics at T ................................... 21 2.4 Optimal value functions at T ............................... 23 2.5 Optimal value functions at a generic time t ....................... 24 2.6 Optimal value functions at t = 0 ............................. 25 4.1 Simulation of Schwartz One-Factor Mean Reverting Process .............. 74 4.2 Option Price Computed by Policy Generated at Each Iterate. ............. 75 4.3 50,000 paths of cumulative quantity exercised through time .............. 77 4.4 86 price scenarios where cumulative quantity at expiry exceed Rmax. ......... 77 4.5 115 price scenarios where cumulative quantity at expiry of the holder is less than Rmin. 78 4.6 Option prices by strikes and penalties .......................... 78 v ACKNOWLEDGEMENTS My most earnest acknowledgement must go to my advisor, Professor Oana Mocioalca, for her guidance, support, encouragement, patience and enthusiasm for this research. Thank you very much! My graduate studies at Kent State has been a wonderful experience. Thanks to the necessary and sufficient financial support that the mathematics department had provided for the first four years of my graduate studies. I would like to extend my thanks to the following individuals at Kent State who have been supportive and/or taught me different areas of mathematics in the past: Dr. Kazim Khan, Dr. Lothar Reichel, Dr. Chuck Gartland, Dr. Hassan Allouba, Dr. Mikhail Chebotar, Dr. Joe Diestel, Dr. Beverly Reed, Ms. Misty Sommers-Tackett, Ms. Virginia Wright, Dr. Volodymyr Andriyevskyy, Dr. Jing Li, Dr. Artem Zvavitch and Dr. Andrew Tonge. Also, I would like to thank my professors at Oklahoma State: Dr. Dale Alspach and Dr. William (Bus) Jaco; and Iowa State: Dr. Stephen Willson. A respectful thanks to Dr. Jaco for showing me the beautiful world of three dimensional manifold topology. I enjoyed your geometric and algebraic topology classes very much. Thanks to all of my colleagues at work. Through collaboration of projects, model validations, maintaining data quality and fixing market risk system issues, you helped improve my risk mea- surement skills. Thanks to Mr. William Kugler, Dr. Vilen Abramov and Dr. Dennis Jarecke for bringing me onboard to the banking industry. Thanks to Mr. Don Schilling, you are a great boss. Thanks to Pastor Huang, Sister Gu and members from the church for your guidance and prayers. With love and gratitude, I give thanks to my father, Peter Leow, and mother, Susan Soo, for all that you gave me. Thanks to my sisters for your emotional support and helps taking care of our mother after the recent untimely departure of our father due to terminal illness. Father, I just wanted to say that I love you and that you are always in my memory. I will continue to work hard, take care of the family and live up to your expectation. My special thanks go to my wife Winnie. You take good care of our adorable daughter, Christy so that I can stay focus on my research and work. Thank you for your love. Thanks to my little Christy for the joy you bring to my life. Also, thanks to my soon-to-be born son Aaron for the late March/early April deadline you have set for me to finish this dissertation. vi Thanks to God for all the blessings you bring to me and my family! vii CHAPTER 1 Introduction We study the problem of pricing swing options, a class of multiple early exercise options that are traded in energy market, particularly in the natural gas and electricity markets. These contracts permit the option holder to periodically exercise the right to trade a variable amount of energy with a counterparty, subject to local volumetric constraints. In addition, the total amount of energy traded from settlement to expiration with the counterparty is restricted by a global volumetric constraint. Violation of this global volumetric constraint is allowed but would lead to penalty settled at expiration. The pricing problem is formulated as a stochastic optimal control problem in discrete time and state space. We present a stochastic dynamic programming algorithm which is based on piecewise linear concave approximation of value functions. This algorithm yields the value of the swing option under the assumption that the optimal exercise policy is applied by the option holder. It is well known that the price of an American option, which is a variant of swing option, has no closed-form formula. Therefore finding a closed-form formula for swing option is unlikely and we have to resort to numerical methods for approximating its price. Numerical techniques employed generally fall into one of the followings: lattice/tree, numerical partial differential equations (PDE), and Monte Carlo simulation. 1.1 Literature Review The first paper on pricing swing option is the paper by Thompson [54], who used a binomial tree algorithm to price take-or-pay contracts, a version of swing options. The paper assumed a simple one-factor geometric Brownian motion in the spot price process. Lari-Lavassani et al. [34] developed another binomial tree whose spot price is a two-factor mean reverting process. Jaillet et al.[27] however employed a trinomial tree for pricing swing options written on natural gas. In PDE- based approach, Kjaer [31] prices electricity swing options for a jump-diffusion spot (day-ahead) price process using finite difference method to solve a partial integro-differential equation. Wilhelm 1 2 [55] uses finite elements. A drawback of lattice-based and PDE-based methods is that an exogenous price process in the form of stochastic differential equation must be specified, similar to models used in financial markets. However, certain energy derivative practitioners are more in favor of fundamental market models for the price process. Unlike spot or forward price models in financial markets that focus on answering the question of how price moves, the fundamental market model looks beyond prices and attempts to answer the question of what causes the price movement. For example, the fundamental market model in Burger et al. [17] is a spot price model comprised of a seasonal autoregressive integrative moving average (SARIMA) forecast of the system load, a deterministic function which specifies the expected relative availability of power plants, an estimate of the price-load
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