1;Pricing of Derivatives on Mean-Reverting Assets;1 1.1;List of Figures;9 1.2;List of Tables;11 1.3;List of Notations and Symbols;12 1.4;1 Introduction;14 1.5;2 Mean Reversion in Commodity Prices;21 1.5.1;2.1 Sources of Mean Reversion;21 1.5.1.1;2.1.1 Convenience Yields;21 1.5.1.2;2.1.2 Kaldor--Working Hypothesis;23 1.5.1.3;2.1.3 Time-Varying Risk Premia;24 1.5.2;2.2 Empirical Evidence of Mean Reversion;25 1.5.3;2.3 Mean Reversion and Volatility: The Samuelson Hypothesis;26 1.6;3 Fundamentals of Derivative Pricing;29 1.6.1;3.1 Derivative Pricing Under the Risk-Neutral Measure;29 1.6.1.1;3.1.1 Introduction;29 1.6.1.2;3.1.2 Change of Measure for Diffusion Processes;31 1.6.1.3;3.1.3 Change of Measure for Jump-Diffusion Processes;34 1.6.1.4;3.1.4 Change of Measure if the Underlyingis not a Traded Asset;37 1.6.2;3.2 Characteristic Functions;38 1.6.3;3.3 Fundamental Partial Differential Equation;40 1.6.4;3.4 European Style Derivatives;43 1.6.4.1;3.4.1 Forwards and Futures;43 1.6.4.2;3.4.2 European Options;44 1.6.4.2.1;Traditional Approach;45 1.6.4.2.2;Carr--Madan Approach;48 1.6.5;3.5 Fast Fourier Algorithms;49 1.6.5.1;3.5.1 Fast Fourier Transformation;49 1.6.5.2;3.5.2 Fractional Fast Fourier Transformation;52 1.6.6;3.6 Recovering Single Option Prices with Gauss-Laguerre Quadrature;54 1.6.6.1;A Question of Computational Efficiency: Explicit or Implicit Schemes?;59 1.6.6.2;The Ode45 Integration Scheme;64 1.7;4 Stochastic Volatility Models;66 1.7.1;4.1 Square-Root Stochastic Volatility;66 1.7.1.1;4.1.1 Comparison with the Tahani Square-Root Model;67 1.7.1.2;4.1.2 Solution for the Characteristic Function;71 1.7.1.2.1;Special Case 1;73 1.7.1.2.2;Special Case 2;74 1.7.1.3;4.1.3 Comparison with the Monte-Carlo Solution;75 1.7.2;4.2 Ornstein--Uhlenbeck Stochastic Volatility;77 1.7.2.1;4.2.1 Comparison with the Tahani OU Model;78 1.7.2.2;4.2.2 Solution for the Characteristic Function;78 1.7.2.2.1;General Case: 1 and (2 / ) N;79 1.7.2.2.2;Special Case 1: 1 and (2 / ) N;80 1.7.2.2.3;Special Case 2: = 1;81 1.7.2.3;4.2.3 Comparison with the Monte-Carlo Solution;82 1.7.2.3.1;Case 1: / is an Arbitrary Noninteger;82 1.7.2.3.2;Case 2: / is a Positive Integer;85 1.8;5 Integration of Jump Components;91 1.8.1;5.1 Simulation of Poisson Processes;92 1.8.2;5.2 Lognormal Jumps of the Underlying;96 1.8.2.1;5.2.1 Non-Mean-Reverting Assets;96 1.8.2.2;5.2.2 Mean-Reverting Assets;97 1.8.2.3;5.2.3 Comparison with the Monte-Carlo Solution;99 1.8.3;5.3 Exponentially and -Distributed Jumps in the Variance Process;100 1.8.3.1;5.3.1 Exponentially Distributed Jumps;100 1.8.3.2;5.3.2 -Distributed Jumps;101 1.8.3.3;5.3.3 Comparison with the Monte-Carlo Solution;102 1.8.4;5.4 Jumps in Both the Underlying and Variance Process;103 1.8.4.1;5.4.1 Independent Jumps;103 1.8.4.1.1;Comparison with the Monte-Carlo Solution;104 1.8.4.2;5.4.2 Correlated Jumps;105 1.8.4.2.1;Exponentially Distributed Variance Jumps;105 1.8.4.2.2;-Distributed Variance Jumps;106 1.8.4.2.3;Comparison with the Monte-Carlo Solution;107 1.9;6 Stochastic Equilibrium Level;110 1.9.1;6.1 Constant Volatility;110 1.9.1.1;6.1.1 Mean-Reverting Equilibrium Level;110 1.9.1.1.1;Special Case: = X;111 1.9.1.2;6.1.2 Brownian Motion with Drift;112 1.9.2;6.2 Integration of Square-Root Stochastic Volatility;114 1.9.2.1;6.2.1 Mean-Reverting Equilibrium Level;114 1.9.2.2;6.2.2 Brownian Motion with Drift;115 1.9.2.2.1;General Case Solution;117 1.9.2.2.2;Special Case 1 Solution;118 1.9.2.2.3;Special Case 2 Solution;119 1.9.2.2.4;Comparison with the Monte-Carlo Solution;120 1.9.3;6.3 Other Model Extensions;121 1.9.3.1;6.3.1 Ornstein--Uhlenbeck Stochastic Volatility;122 1.9.3.2;6.3.2 Model Extensions with Jump Components;123 1.10;7 Deterministic Seasonality Effects;124 1.10.1;7.1 Seasonality in the Log-Price Process;125 1.10.1.1;7.1.1 Constant Volatility;127 1.10.1.2;7.1.2 Square-Root Stochastic Volatility;128 1.10.1.2.1;Comparison with the Monte-Carlo Solution;129 1.10.1.3;7.1.3 Other Model Extensions;130 1.10.2;7.2 Seasona