Lutz Pricing of Derivatives on Mean-Reverting Assets

1. Auflage 2009
ISBN: 978-3-642-02909-7
Verlag: Springer
Format: PDF
Kopierschutz: Wasserzeichen (»Systemvoraussetzungen)

E-Book, Englisch, 137 Seiten

The topic of this book is the development of pricing formulae for European style derivatives on assets with mean-reverting behavior, especially commodity derivatives.

Autoren/Hrsg.

Lutz, Björn

Inhaltsverzeichnis

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 Seasonal Impact of Volatility;130
1.10.2.1;7.2.1 Seasonal Variance According to Richter and Sørensen;130
1.10.2.2;7.2.2 Modeling of Seasonality in the Variance Process;131
1.10.2.2.1;General Case Solution;132
1.10.2.2.2;Special Case 1: = ;133
1.10.2.2.3;Special Case 2: 2 = ;133
1.10.2.2.4;Model Extensions;134
1.11;8 Conclusion;136
1.12;References;141