STOCHASTIC MODELLING OF ELECTRICITY AND RELATED MARKETS
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1 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
2 Contents Preface vii 1. A Survey of Electricity and Related Markets The electricity markets ' Electricity contracts with physical delivery Financial electricity contracts The gas market Futures and options on gas The temperature market Other related energy markets Stochastic modelling of energy markets Spot price modelling Forward and swap pricing in electricity and related markets Outline of the book Stochastic Analysis for Independent Increment Processes Definitions Stochastic integration with respect to martingales Random jump measures and stochastic integration The Levy-Kintchine decomposition and semimartingales The Ito Formula for semimartingales Examples of independent increment processes Time-inhomogeneous compound Poisson process Models based on the generalized hyperbolic distributions 51
3 xii Stochastic Modelling of Electricity and Related Markets Models based on the Variance-Gamma and CGMY distributions Stochastic Models for the Energy Spot Price Dynamics Introduction Spot price modelling with Ornstein-Uhlenbeck processes Geometric models Arithmetic models The autocorrelation function of multi-factor Ornstein- Uhlenbeck processes Simulation of stationary Ornstein-Uhlenbeck processes: a case study with the arithmetic spot model Pricing of Forwards and Swaps Based on the Spot Price Risk-neutral forward and swap price modelling Risk-neutral probabilities and the Esscher transform The Esscher transform for some specific models Currency conversion for forward and swap prices Pricing of forwards The geometric case The arithmetic case Pricing of swaps The geometric case The arithmetic case Applications to the Gas Markets, Modelling the gas spot price Empirical analysis of UK gas spot prices Residuals modelled as a mixed jump-diffusion process NIG distributed residuals Pricing of gas futures Inference for multi-factor processes Kalman filtering Inference using forward and swap data Modelling Forwards and Swaps Using the Heath-Jarrow- Morton Approach The HJM modelling idea for forward contracts 156
4 Contents xiii 6.2 HJM modelling of forwards HJM modelling of swaps Swap models based on forwards The market models Modelling with jump processes Constructing Smooth Forward Curves in Electricity Markets Swap and forward prices Basic relationships A continuous seasonal forward curve Maximum smooth forward curve A smooth forward curve constrained by closing prices A smooth forward curve constrained by bid and ask spreads Putting the algorithm to work Nord Pool example I: A smooth curve Nord Pool example II: Preparing a data set and analysing volatility Modelling of the Electricity Futures Market The Nord Pool market and financial contracts Preparing data sets Descriptive statistics A market model for electricity futures Principal component analysis Principal component analysis of the total data set Principal component analysis for individual market segments Estimating a parametric multi-factor market model Seasonal volatility Maturity volatilities Normalised logreturns and heavy tails Final remarks Pricing and Hedging of Energy Options Pricing and hedging options on forwards and swaps The case of no jumps - the Black-76 Formula The case of jumps 247
5 xiv Stochastic Modelling of Electricity and Related Markets 9.2 Exotic Options Spread options Asian options Case Study: Valuation of spark spread options - a direct approach Modelling and analysis of spark spread options Empirical analysis of UK gas and electricity spread Analysis of Temperature Derivatives Some preliminaries on temperature futures Modelling the dynamics of temperature The CAR(p) model with seasonality A link to time series Empirical analysis of Stockholm temperature dynamics Description of the data Estimating the CAR(p) models Fitting an AR(1) model Fitting an AR(3) model Identification of the parameters in the CAR(p) model Temperature derivatives pricing CAT futures HDD/CDD futures Frost Day index futures ' Application to futures on temperatures in Stockholm 314 Appendix A List of abbreviations 319 Bibliography 321 Index 333
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