Numerical Pricing & Risk Management of Energy Commodities Derivatives

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Numerical Pricing & Risk Management of Energy Commodities Derivatives
Konstantinos A. Sofos

Contents
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Numerical Pricing and Risk Management of Energy Commodities Derivatives
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I. Introduction 5
1. Risk Management & Price Modeling 7
1.1. Introduction To Energy Derivatives . . . . . . . . . . . . . . . . . . . 8
1.1.1. Derivatives Contract Types . . . . . . . . . . . . . . . . . . . 9
1.2. Definition of Long and Short Position . . . . . . . . . . . . . . . . . . 10
1.3. Principles Of Hedging And Price Discovery . . . . . . . . . . . . . . . 11
1.3.1. Fair Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2. Standard Energy Options 13
2.1. Energy Options Classes . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1. Financially Settled Options . . . . . . . . . . . . . . . . . . . 13
2.1.2. Physically Settled Options . . . . . . . . . . . . . . . . . . . . 14
2.1.3. Daily and Index or Cash Option . . . . . . . . . . . . . . . . . 15
2.2. Spread Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1. Spark Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3. Strategies With Options . . . . . . . . . . . . . . . . . . . . . . . . . 16
II. Option Pricing Models & Hedging Strategies 17
3. Modeling Tools for Option Pricing 19
3.1. Main Assumption: No Arbitrage . . . . . . . . . . . . . . . . . . . . . 19
3.2. Option Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.1. The Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . 21
3.2.2. Heston’s Stochastic Volatility Model . . . . . . . . . . . . . . 34
III. Numerical Aspects 47
4. Finite-Difference Methods 49
4.1. Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.1. Difference Approximation . . . . . . . . . . . . . . . . . . . . 49
i
Contents Contents
4.1.2. Finite Difference Grid . . . . . . . . . . . . . . . . . . . . . . 50
4.2. European Style Options . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.1. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.2. Explicit (Forward in Time) FDM . . . . . . . . . . . . . . . . 52
4.2.3. Implicit (Backward in Time) FDM . . . . . . . . . . . . . . . 52
4.2.4. The Crank-Nicolson Method . . . . . . . . . . . . . . . . . . . 53
4.3. American Style Options . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.1. Successive Over relation Technique . . . . . . . . . . . . . . . 56
4.3.2. Crank-Nicolson Scheme For American Options . . . . . . . . . 57
4.3.3. American Options as Free Boundary Problems . . . . . . . . . 58
4.4. Option Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5. Heston - Multidimensional PDE . . . . . . . . . . . . . . . . . . . . . 59
4.5.1. Explicit Heston Finite Difference Approach . . . . . . . . . . . 61
4.5.2. Implicit Heston Finite Difference Approach . . . . . . . . . . . 68
IV. Forward and Futures 71
5. Forward Price 73
5.1. Contango and Backwardation . . . . . . . . . . . . . . . . . . . . . . 73
5.2. Forward Price PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3. Convenience Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3.1. Forward Price with Fixed Convenience Yield . . . . . . . . . . 75
5.3.2. Forward Price with Stochastic Convenience Yield . . . . . . . 76
5.4. Schwartz model - Stochastic CY Model . . . . . . . . . . . . . . . . . 77
5.4.1. Joint Distribution of State Variables . . . . . . . . . . . . . . 78
5.4.2. Futures Price . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6. Futures Options 81
6.1. Futures Risk & Neutral Behavior . . . . . . . . . . . . . . . . . . . . 81
6.2. Futures Contract For Constant Interest Rate . . . . . . . . . . . . . . 82
6.2.1. Put-Call Parity: Hedging With Futures . . . . . . . . . . . . . 83
6.2.2. Delta Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.3. Blacks’s Model For Future Options Pricing . . . . . . . . . . . . . . . 85
6.4. American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.4.1. American Option Derivation . . . . . . . . . . . . . . . . . . . 90
6.5. Optimal Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.5.1. The Barone-Adesi and Whaley Model . . . . . . . . . . . . . . 92
6.5.2. Critical Asset Price . . . . . . . . . . . . . . . . . . . . . . . . 96
6.5.3. Futures Option Quadratic Approximation . . . . . . . . . . . 97
7. Appendix 101
Bibliography 105

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