Investment in Energy Assets Under Uncertainty

Numerical methods in theory and practice

Contents

Part I Investment Under Certainty

1 Valuation Made Simple: No Uncertainties, Just Time 3

1.1 Some Preliminaries 3

1.1.1 Simple and Compound Interest 3

1.1.2 Discounting 5

1.2 Cash Flow Streams: Annuities, and Perpetuities 5

1.2.1 Annuities 6

1.2.2 Perpetual Annuities 7

1.2.3 Annuities and Perpetuities Under Continuous

Compounding 8

1.2.4 Increasing Annuities 8

1.3 Management and Value 9

1.4 Dynamic Programming 10

1.4.1 A Friendly Introduction:

Charting the Shortest Route 10

1.4.2 Maximizing Profit from Mineral Extraction 12

1.4.3 A Rigorous Exposition 17

1.5 Where Next? 18

References 19

Part II Investment Under Uncertainty

2 Theoretical Foundations 23

2.1 Mean-Variance Analysis in a Single Period 23 2.1.1 Characteristics of Asset Returns 23 2.1.2 Characteristics of Portfolio Returns 26 2.1.3 Riskless Borrowing and Lending 29 2.2 The Standard Capital Asset Pricing Model 31

2.3 Single-Period Risk-Neutral Pricing 35

2.3.1 State Prices 35

2.3.2 Risk-Neutral Valuation 36

2.4 Forward and Futures Markets 37

2.4.1 A Primer 37

2.4.2 Futures Prices, Spot Prices, and Storage Costs 42

References 43

3 Analytical Solutions 45

3.1 Stochastic Price Models 45

3.1.1 The Geometrie Brownian Motion 46

3.1.2 The Inhomogenous Geometrie Brownian Motion 51 3.2 Annuities and Futures Contracts Under

the Above Processes 55

3.2.1 Annuities Under the GBM 55

3.2.2 Annuities Under the IGBM 56

3.2.3 Futures Contracts Under the GBM 57 3.2.4 Futures Contracts Under the IGBM 58 3.3 Fundamental Pricing Equation: The Perpetual Option 60

3.3.1 The GBM 60

3.3.2 Example 1: Optimal Timing Under Certainty

(Finite-Lived Option) 62

3.3.3 Example 2: Optimal Time to Invest Under a GBM ... 64 3.3.4 Example 3: Two correlated GBMs 69

3.3.5 The IGBM 71

3.3.6 Example 4: Optimal Time to Invest

Under an IGBM 73

3.4 Pricing Formulas for European Options 75

References 75

4 Binomial Lattices 77

4.1 Introduction 77

4.2 The Basic Setting: Binomial Lattice Under a GBM 78 4.2.1 Determining the Parameters of the Lattice 79 4.2.2 The Finite-Lived Option to Invest 82

4.2.3 Extensions 83

4.2.4 Example 1: One Time Step Per Year 86 4.2.5 Example 2: One Hundred Time Steps Per Year 87 4.2.6 Example 3: Convergence to the Perpetual Option 88 4.2.7 Example 4: Decreasing Investment Cost

(One Step Per Year) 89

4.2.8 Example 5: Decreasing Investment Cost

(One Hundred Steps Per Year) 89

4.2.9 Example 6: Convergence to Perpetual

Option (Decreasing Investment Cost) 90 4.3 The Finite-Lived Option to Invest Under the IGBM 90 4.3.1 Example 7: One Time Step Per Year 92 4.3.2 Example 8: One Hundred Time Steps Per Year 93

Contents xiii 4.3.3 Example 9: Convergence to the Perpetual Option 93

4.4 Bi-dimensional Binomial Lattice s 93

4.4.1 Example 10: Two GBMs 94

4.4.2 Example 11: Two GBMs; Approximation

to the Perpetual Option 95

4.4.3 Two IGBMs 95

4.4.4 Example 12: Two IGBMs, One Step Per Year 97 4.4.5 Example 13: Two IGBMs with One

Thousand Steps 98

4.4.6 One GBM and One IGBM 98

4.5 Trinomial Lattice with Mean Reversion 99

References 102

5 Finita Difference Methods 103

5.1 Introduction 103

5.2 The Implicit Finite Difference Method 104 5.3 The Explicit Finite Difference Method 106

5.4 Relationship with Lattice Models 107

5.5 Example 1: Valuation of a European Real Option 108

5.6 The Crank-Nicolson Method 110

5.7 Example 2: Valuation of an American Put Option 111 5.8 Example 3: Valuation of a Long-Term

American Put Option 112

References 112

6 Monte Carlo Simulation 113

6.1 Introduction 113

6.2 The Basic Setup: Only One GBM Underlying Variable 113

6.2.1 Use of Random Numbers 115

6.2.2 Example 1: Comparison with a GBM Annuity 115 6.2.3 Example 2: A GBM Annuity with Jump

(Convergence to Peipetual Annuity) 117 6.2.4 Example 3: A GBM Annuity with Jump (<j> = 0.50). . . 118 6.2.5 Example 4: Valuation of a European Option

by Simulation 118

6.2.6 Variance Reduction Techniques 119

6.2.7 Example 5: Valuation of a European Option by Simulation with Sobol Low-discrepancy

Sequences 119

6.3 Monte Carlo Simulation and American Options Valuation. . . . 120 6.3.1 Example 6: Valuation of an American Option

by Simulation 120

6.3.2 Example 7: Valuation of an American Option

6.3.3 Example 8: The American Put Option by LSMC,

Binomial Lattice, and Finite Differences 121 6.3.4 Example 9: Long-Term American Put

(Three Approaches) 122

6.3.5 Example 10: An IGBM Underlying Variable 122 6.4 The Gase of Several Underlying Variables 123 6.4.1 Two GBMs: The Cholesky Factorization 123 6.4.2 Example 11: One Hundred Steps Per Year,

Two GBMs 124

6.4.3 Example 12: European Option with a GBM

and an IGBM (with Stochastic Interest Rate) 125

Appendix 126

References 133

Part III Investments in the Energy Sector

7 Economic and Technical Background 137

7.1 Introduction 137

7.2 Coal-Fired Power Plants 140

7.3 Natural Gas-Fired Stations 140

7.4 Gasification Plants 144

7.5 Wind Parks 144

7.6 Futures Markets 148

References 150

8 Valuation of Energy Assets: A Single Risk Factor 151

8.1 Introduction 151

8.2 Gase 1: An Advanced Gas/Oil Combined Cycle 151 8.3 Gase 2: A New Scrubbed Coal-Fired Station 153

8.4 Case 3: An Oil Well 156

9 Valuation of Energy Assets: Two Risk Factors 159

9.1 Introduction 159

9.2 Case 1: An Advanced Gas/Oil Combined Cycle 159 9.3 Case 2: A New Scrubbed Coal-Fired Station 162 10 Valuation of Energy Assets: Three Risk Factors 167

10.1 Introduction 167

10.2 Case 1: An Advanced Gas/Oil Combined Cycle 167 10.3 Case 2: A New Scrubbed Coal-Fired Station 172

Contents xv 11 Value Maximization and Optimal Management

of Energy Assets 177

11.1 Introduction 177

11.2 Case 1: A Natural Gas-Fired Power Plant

("On" or "Off"; no Switching Costs) 178 11.3 Case 2: A Coal-Fired Power Plant

("On" or "Off"; no Switching Costs) 180

References 183