Advanced Series on Statistical Science & Applied Probability Vol. I I STOCHASTIC MODELLING OF ELECTRICITY AND RELATED MARKETS Fred Espen Benth JGrate Saltyte Benth University of Oslo, Norway Steen Koekebakker University ofagder, Norway World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI
Contents Preface vii 1. A Survey of Electricity and Related Markets 1 1.1 The electricity markets ' 3 1.1.1 Electricity contracts with physical delivery 3 1.1.2 Financial electricity contracts 5 1.2 The gas market 8 1.2.1 Futures and options on gas 10 1.3 The temperature market 11 1.4 Other related energy markets 14 1.5 Stochastic modelling of energy markets 18 1.5.1 Spot price modelling 19 1.5.2 Forward and swap pricing in electricity and related markets 24 1.6 Outline of the book 32 2. Stochastic Analysis for Independent Increment Processes 37 2.1 Definitions 37 2.2 Stochastic integration with respect to martingales 41 2.3 Random jump measures and stochastic integration 43 2.4 The Levy-Kintchine decomposition and semimartingales.. 45 2.5 The Ito Formula for semimartingales 48 2.6 Examples of independent increment processes 49 2.6.1 Time-inhomogeneous compound Poisson process... 49 2.6.2 Models based on the generalized hyperbolic distributions 51
xii Stochastic Modelling of Electricity and Related Markets 2.6.3 Models based on the Variance-Gamma and CGMY distributions 55 3. Stochastic Models for the Energy Spot Price Dynamics 59 3.1 Introduction 59 3.2 Spot price modelling with Ornstein-Uhlenbeck processes.. 60 3.2.1 Geometric models 66 3.2.2 Arithmetic models 74 3.3 The autocorrelation function of multi-factor Ornstein- Uhlenbeck processes 78 3.4 Simulation of stationary Ornstein-Uhlenbeck processes: a case study with the arithmetic spot model 82 4. Pricing of Forwards and Swaps Based on the Spot Price 89 4.1 Risk-neutral forward and swap price modelling 89 4.1.1 Risk-neutral probabilities and the Esscher transform 95 4.1.2 The Esscher transform for some specific models... 99 4.2 Currency conversion for forward and swap prices 100 4.3 Pricing of forwards 104 4.3.1 The geometric case 104 4.3.2 The arithmetic case 114 4.4 Pricing of swaps 118 4.4.1 The geometric case 119 4.4.2 The arithmetic case 122 5. Applications to the Gas Markets, 129 5.1 Modelling the gas spot price 129 5.1.1 Empirical analysis of UK gas spot prices 130 5.1.2 Residuals modelled as a mixed jump-diffusion process 136 5.1.3 NIG distributed residuals 139 5.2 Pricing of gas futures 142 5.3 Inference for multi-factor processes 146 5.3.1 Kalman filtering 147 5.3.2 Inference using forward and swap data 150 6. Modelling Forwards and Swaps Using the Heath-Jarrow- Morton Approach 155 6.1 The HJM modelling idea for forward contracts 156
Contents xiii 6.2 HJM modelling of forwards 160 6.3 HJM modelling of swaps 164 6.3.1 Swap models based on forwards 168 6.4 The market models 172 6.4.1 Modelling with jump processes 176 7. Constructing Smooth Forward Curves in Electricity Markets 181 7.1 Swap and forward prices 183 7.1.1 Basic relationships - 183 7.1.2 A continuous seasonal forward curve 184 7.2 Maximum smooth forward curve 187 7.2.1 A smooth forward curve constrained by closing prices 187 7.2.2 A smooth forward curve constrained by bid and ask spreads 190 7.3 Putting the algorithm to work 191 7.3.1 Nord Pool example I: A smooth curve 191 7.3.2 Nord Pool example II: Preparing a data set and analysing volatility - 195 8. Modelling of the Electricity Futures Market 203 8.1 The Nord Pool market and financial contracts 205 8.2 Preparing data sets 206 8.3 Descriptive statistics 208 8.4 A market model for electricity futures 214 8.5 Principal component analysis 215 8.5.1 Principal component analysis of the total data set.. 217 8.5.2 Principal component analysis for individual market segments 220 8.6 Estimating a parametric multi-factor market model 224 8.6.1 Seasonal volatility 226 8.6.2 Maturity volatilities 227 8.7 Normalised logreturns and heavy tails 231 8.8 Final remarks 235 9. Pricing and Hedging of Energy Options 237 9.1 Pricing and hedging options on forwards and swaps 238 9.1.1 The case of no jumps - the Black-76 Formula... 238 9.1.2 The case of jumps 247
xiv Stochastic Modelling of Electricity and Related Markets 9.2 Exotic Options 254 9.2.1 Spread options 254 9.2.2 Asian options 260 9.3 Case Study: Valuation of spark spread options - a direct approach 262 9.3.1 Modelling and analysis of spark spread options... 264 9.3.2 Empirical analysis of UK gas and electricity spread. 268 10. Analysis of Temperature Derivatives 277 10.1 Some preliminaries on temperature futures 277 10.2 Modelling the dynamics of temperature 280 10.2.1 The CAR(p) model with seasonality 281 10.2.2 A link to time series 283 10.3 Empirical analysis of Stockholm temperature dynamics... 285 10.3.1 Description of the data 285 10.3.2 Estimating the CAR(p) models 287 10.3.2.1 Fitting an AR(1) model 289 10.3.2.2 Fitting an AR(3) model 296 10.3.2.3 Identification of the parameters in the CAR(p) model 300 10.4 Temperature derivatives pricing 301 10.4.1 CAT futures 302 10.4.2 HDD/CDD futures 305 10.4.3 Frost Day index futures ' 312 10.4.4 Application to futures on temperatures in Stockholm 314 Appendix A List of abbreviations 319 Bibliography 321 Index 333