Dr Iain J. Clark

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

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

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

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

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

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