“It has been very hard to find a comprehensive option pricing book cover- ing all exchange-traded commodities, until Iain Clark’s book. Commodity Option Pricing is a must-read for option traders, risk managers and quantita- tive analysts. The author combines academic rigor with real-world examples. Practitioners can find an extremely useful toolkit in option pricing andan excellent introduction to various commodities.” Joseph Y. Chen, Chief, Market Analytics, Nexen “Being himself a practitioner with a wealth of experience, Iain knows what is relevant to the daily work of commodities trading desks. He uses his remarkable pedagogical skills to develop the reader’s intuition for the mod- els and products before delving into the practicalities of the commodities markets. Many of the details covered in this book cannot be found elsewhere in the literature. I am pleased to recommend this book to quants and traders, who will soon find themselves relying on it in their daily work.” Paul Bilokon, Director, Deutsche Bank “Option pricing analytics for trading commodity derivatives can be quite different from those of equity and fixed income derivatives. This book fills a gap in current literature by presenting a comprehensive treatise on the risk characteristics associated with pricing and hedging commodity derivatives. The author strikes a fine balance between option pricing theory and financial practice in the markets. The materials are succinctly written, with clear and insightful descriptions of the features of various commodity markets and state-of-the-art pricing models. This book is destined to be a valuable prac- tical guide for practitioners and a useful academic reference for researchers in trading and understanding commodities derivatives.” Yue Kuen Kwok, Professor, Department of Mathematics, Hong Kong University of Science and Technology Commodity Option Pricing For other titles in the Wiley Finance series please see www.wiley.com/finance Commodity Option Pricing A Practitioner’s Guide Iain J. Clark This edition first published 2014 © 2014 Iain J. Clark Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about howto apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with the respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Clark, Iain J. Commodity option pricing : a practitioners guide / Iain J. Clark. pages cm Includes bibliographical references and index. ISBN 978-1-119-94451-5 (cloth) 1. Commodity options. 2. Options (Finance)–Prices. I. Clark, Iain J. II. Title. HG6046.C5736 2014 332.63′28–dc23 2013051276 A catalogue record for this book is available from the British Library. ISBN 978-1-119-94451-5 (hbk) ISBN 978-1-444-36240-4 (ebk) ISBN 978-1-444-36241-1 (ebk) ISBN 978-1-118-87178-2 (ebk) Cover images reproduced by permission of Shutterstock.com Set in 11/13pt Times by Aptara Inc., New Delhi, India Printed in Great Britain by TJ International Ltd, Padstow, Cornwall, UK For Elizabeth Clark (born July 2013). Who made the completion of this book a challenge – in the nicest possible way. Contents Acknowledgements xv Notation xvii List of Figures xix List of Tables xxiii 1 Introduction 1 1.1 Trade, Commerce and Commodities 3 1.2 Adapting to Commodities as an Asset Class 8 1.2.1 Classification of Commodities into Sub-categories 9 1.3 Challenges in Commodity Models 12 1.3.1 Futures 12 1.3.2 Correlation 12 1.3.3 Seasonality 15 1.3.4 American and Asian Features 15 2 Commodity Mathematics and Products 17 2.1 Spot, Forwards and Futures 17 2.1.1 Spot 18 2.1.2 Forwards 19 2.1.3 Futures 20 2.2 The Black–Scholes and Black-76 Models 24 2.2.1 The Black–Scholes Model 24 x Contents 2.2.2 The Black–Scholes Model Without Convenience Yield 25 2.2.3 The Black–Scholes Model With Convenience Yield 26 2.2.4 The Black-76 Model 28 2.2.5 Risk-Neutral Valuation 35 2.2.6 Forwards 36 2.2.7 The Black–Scholes Term Structure Model 38 2.3 Forward and Futures Contracts 39 2.3.1 Forwards 39 2.3.2 Futures 39 2.3.3 Case Study 40 2.4 Commodity Swaps 42 2.5 European Options 44 2.5.1 European Options on Spot 45 2.5.2 European Options on Futures 49 2.5.3 Settlement Adjustments 49 2.6 American Options 50 2.6.1 Barone-Adesi and Whaley (1987) 50 2.6.2 Lattice Methods 53 2.7 Asian Options 54 2.7.1 Geometric Asian Options – Continuous Averaging 54 2.7.2 Arithmetic Asian Options – Continuous Averaging 61 2.7.3 Geometric Average Options – Discrete Fixings – Kemna and Vorst (1990) 62 2.7.4 Arithmetic Average Options – Discrete Fixings – Turnbull and Wakeman (1991) 66 2.8 Commodity Swaptions 70 2.9 Spread Options 73 2.9.1 Margrabe Exchange Options 74 2.9.2 The Kirk Approximation 75 2.9.3 Calendar Spread Options 77 2.9.4 Asian Spread Options 78 2.10 More Advanced Models 78 2.10.1 Mean Reverting Models 79 2.10.2 Multi-Factor Models 88 2.10.3 Convenience Yield Models 94 Contents xi 3 Precious Metals 99 3.1 Gold Forward and Gold Lease Rates 101 3.2 Volatility Surfaces for Precious Metals 103 3.2.1 Pips Spot Delta 104 3.2.2 Pips Forward Delta 104 3.2.3 Notation 105 3.2.4 Market Volatility Surfaces 105 3.2.5 At-the-Money 105 3.2.6 Strangles and Risk Reversals 107 3.2.7 Temporal Interpolation 111 3.3 Survey of the Precious Metals 111 3.3.1 Gold 112 3.3.2 Silver 117 3.3.3 Platinum 119 3.3.4 Palladium 121 3.3.5 Rhodium 124 4 Base Metals 127 4.1 Futures, Options and TAPO Contracts 130 4.1.1 Futures 130 4.1.2 Options 134 4.1.3 Traded Average Price Options 137 4.2 Commonly Traded Base Metals 139 4.2.1 Copper 140 4.2.2 Aluminium 142 4.2.3 Zinc 143 4.2.4 Nickel 145 4.2.5 Lead 146 4.2.6 Tin 148 5 Energy I – Crude Oil, Natural Gas and Coal 151 5.1 Crude Oil 154 5.1.1 WTI 158 5.1.2 Brent 163 5.1.3 Calibration of WTI Volatility Term Structure 171 5.1.4 Calibration of WTI Volatility Skew 174 5.1.5 Brent and Other Crude Markets 177 5.1.6 A Note on Correlation 180 xii Contents 5.2 Natural Gas 180 5.2.1 Deseasonalising Forward Curves 186 5.3 Coal 188 6 Energy II – Refined Products 195 6.1 The Refinery Basket 195 6.2 Gasoline 197 6.3 Heating Oil/Gas Oil 200 6.4 Petroleum Gases and Residual Fuel Oil 203 6.5 Seasonality and Volatility 205 6.6 Crack Spread Options 207 7 Power 213 7.1 Electricity Generation 214 7.2 Nonstorability and Decorrelation 217 7.2.1 Spot Markets 218 7.2.2 Futures and Forward Markets 219 7.2.3 Options Markets 220 7.3 Modelling Spikes in Electricity Markets 220 7.3.1 Reduced Form Models 223 7.3.2 Structural Models 227 7.4 Swing Options 231 7.5 Spark Spread Options 232 8 Agricultural Derivatives 233 8.1 Grains 234 8.1.1 Wheat 236 8.1.2 Corn 239 8.1.3 Rice 240 8.1.4 Oats 241 8.1.5 Barley 241 8.2 Oilseeds 242 8.2.1 Soybeans 242 8.2.2 Canola 244 8.3 Softs 244 8.3.1 Coffee 245 8.3.2 Cotton 247 8.3.3 Cocoa 248 8.3.4 Sugar 249 Contents xiii 8.3.5 Orange Juice 250 8.3.6 Lumber 252 8.4 Pulp and Paper 252 8.5 Livestock 253 8.5.1 Feeder Cattle 253 8.5.2 Live Cattle 254 8.5.3 Lean Hogs 255 8.5.4 Pork Bellies 255 8.5.5 Milk and Dairy 256 9 Alternative Commodities 257 9.1 Carbon Emissions Trading 257 9.2 Weather Derivatives 261 9.2.1 Temperature Derivatives 261 9.2.2 Windspeed Derivatives 263 9.2.3 Precipitation Derivatives 263 9.3 Bandwidth and Telecommunication Trading 264 9.4 Plastics 265 9.5 Freight Derivatives 266 9.5.1 Shipping 266 9.5.2 Pricing and the Baltic Freight Market 268 9.5.3 Forward Freight Agreements and Options 269 Conversion Factors 273 Futures Contract Symbols 275 Glossary 279 References 295 Further Reading 303 Index 307 Acknowledgements Firstly, I would like to thank my wife yet again for her incredible patience and fortitude during the completion of this work, which somehow coin- cided with the happy arrival of our daughter Elizabeth; my efforts to finish the book in nine months were not nearly so successful.