Calibration Of The Schwartz Smith Model For Commodity Prices

Calibration of the Schwartz-Smith Model for Commodity Prices
Ana Luiza Abr˜o Roriz Soares de Carvalho a August 2009

Contents
1 Introduction 2 Commodity Price Models 2.1 2.2 2.3 Kalman Filter Estimation . . . . . . . . . . . . . . . . . . . . . Theory of Storage, Expected Premium and Convenience Yield . One-Factor and Two-Factor Models . . . . . . . . . . . . . . . 2 5 5 7 9 16 16 19 23 2626 30 31 32 35 37 40 40 48 48 51 56 58 63 63 64

3 The Schwartz and Smith Model 3.1 Some basic results . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 3.1.2 Brief Review of Risk Neutral Pricing . . . . . . . . . . . Risk Neutral Model . . . . . . . . . . . . . . . . . . . .

4 Calibration 4.1 4.2 Discretization and Parametrization of the Stochastic Processes Overall Strategy . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.1 4.2.2 4.3 4.4 4.5 Hidden Processes Calibration . . . . . . . . . . . . . . . Likelihood Functions and Parameter Estimation . . . .

Numerical Implementation . . . . . . . . . . . . . . . . . . . . . Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Choosing the best tolerance . . . . . . . . . . . . . . . . Numerical Testsand Validation Exercises . . . . . . . . . . . .

5 Energy Market Data 5.1 5.2 Data Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . .

6 Conclusion A Additional Figures B Computer Codes B.1 Euler-Maruyama Discretization . . . . . . . . . . . . . . . . . . B.2 Calibration Algorithms . . . . . . . . . . . . . . . .. . . . . . . i

List of Figures
2.1 2.2 2.3 Futures prices generated by the Schwartz 1997 one-factor model 14 Futures prices generated by the Schwartz and Smith 2000 twofactor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absolute difference between futures prices generated with one and two factor models . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2 4.3 4.4 4.5 4.64.7 4.8 4.9 5.1 5.2 5.3 5.4 5.5 Discretization using the Euler-Maruyama method . . . . . . . . Estimation error behavior - κ . . . . . . . . . . . . . . . . . . . Estimation error behavior - θ . . . . . . . . . . . . . . . . . . . Estimation error behavior - σ . . . . . . . . . . . . . . . . . . . Estimation error behavior - η . . . . . . . . . . . . . . . . . . . Estimation error behavior - µ . .. . . . . . . . . . . . . . . . . X and Y processes - 10,000 observations . . . . . . . . . . . . . X and Y processes - 30,000 observations . . . . . . . . . . . . . X and Y processes - 65,000 observations . . . . . . . . . . . . . Henry Hub futures prices plot . . . . . . . . . . . . . . . . . . . Henry Hub data plot - 1 month futures . . . . . . . . . . . . . Henry Hub data plot - 12 monthfutures . . . . . . . . . . . . . Henry Hub data plot - 24 month futures . . . . . . . . . . . . . First two eigenvectors from the Henry Hub data . . . . . . . . 15 28 38 38 39 39 39 42 43 43 49 52 52 53 54 15

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List of Tables
4.1 4.2 4.3 4.4 4.5 4.6 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Initial Guess for model parameters . . . . . . . . . . . . . . . . Estimate Results - 10,000, 30,000 and 65,000observations . . . Tolerance test for 10,000 and 30,000 observations . . . . . . . . Iteration errors - Perturbation on initial parameter guesses . . Iteration errors - Random perturbation on initial parameter guesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iteration errors plots - κ . . . . . . . . . . . . . . . . . . . . . 45 47 49 50 50 51 51 51 55 59 60 61 62 36 37 41 44Autocorrelation Functions for 1 month futures . . . . . . . . . Autocorrelation Functions for 12 month futures . . . . . . . . . Autocorrelation Functions for 24 month futures . . . . . . . . . Descriptive Statistics for 1 month futures . . . . . . . . . . . . Descriptive Statistics for 12 month futures . . . . . . . . . . . . Descriptive Statistics for 24 month futures . . . . . . . . . . . ....