Pricing of Derivatives on Mean-Reverting Assets (Lecture Notes in Economics and Mathematical Systems Book 630)

The Topic Of This Book Is The Development Of Pricing Formulae For European Style Derivatives On Assets With Mean-reverting Behavior, Especially Commodity Derivatives. For This Class Of Assets, Convenience Yield Effects Lead To Mean-reversion Under The Risk-neutral Measure. Mean-reversion In The Log-price Process Is Combined With Other Stochastic Factors Such As Stochastic Volatility, Jumps In The Underlying And The Price Process And A Stochastic Target Level As Well As With Deterministic Seasonality Effects. Another Focus Is On Numerical Algorithms To Calculate The Fourier Integral As Well As To Integrate Systems Of Ordinary Differential Equations. Björn Lutz. Includes Bibliographical References (p. 133-137). front-matter 1 Pricing of Derivatives on Mean-Reverting Assets 1 List of Figures 9 List of Tables 11 List of Notations and Symbols 12 fulltext 14 1 Introduction 0 fulltext_2 21 2 Mean Reversion in Commodity Prices 0 2.1 Sources of Mean Reversion 0 2.1.1 Convenience Yields 0 2.1.2 Kaldor--Working Hypothesis 0 2.1.3 Time-Varying Risk Premia 0 2.2 Empirical Evidence of Mean Reversion 0 2.3 Mean Reversion and Volatility: The Samuelson Hypothesis 0 fulltext_3 29 3 Fundamentals of Derivative Pricing 0 3.1 Derivative Pricing Under the Risk-Neutral Measure 0 3.1.1 Introduction 0 3.1.2 Change of Measure for Diffusion Processes 0 3.1.3 Change of Measure for Jump-Diffusion Processes 0 3.1.4 Change of Measure if the Underlyingis not a Traded Asset 0 3.2 Characteristic Functions 0 3.3 Fundamental Partial Differential Equation 0 3.4 European Style Derivatives 0 3.4.1 Forwards and Futures 0 3.4.2 European Options 0 Traditional Approach 0 Carr--Madan Approach 0 3.5 Fast Fourier Algorithms 0 3.5.1 Fast Fourier Transformation 0 3.5.2 Fractional Fast Fourier Transformation 0 3.6 Recovering Single Option Prices with Gauss-Laguerre Quadrature 0 A Question of Computational Efficiency: Explicit or Implicit Schemes? 0 The Ode45 Integration Scheme 0 fulltext_4 66 4 Stochastic Volatility Models 0 4.1 Square-Root Stochastic Volatility 0 4.1.1 Comparison with the Tahani Square-Root Model 0 4.1.2 Solution for the Characteristic Function 0 Special Case 1 0 Special Case 2 0 4.1.3 Comparison with the Monte-Carlo Solution 0 4.2 Ornstein--Uhlenbeck Stochastic Volatility 0 4.2.1 Comparison with the Tahani OU Model 0 4.2.2 Solution for the Characteristic Function 0 General Case: 1 and (2 / ) N 0 Special Case 1: 1 and (2 / ) N 0 Special Case 2: = 1 0 4.2.3 Comparison with the Monte-Carlo Solution 0 Case 1: / is an Arbitrary Noninteger 0 Case 2: / is a Positive Integer 0 fulltext_5 91 5 Integration of Jump Components 0 5.1 Simulation of Poisson Processes 0 5.2 Lognormal Jumps of the Underlying 0 5.2.1 Non-Mean-Reverting Assets 0 5.2.2 Mean-Reverting Assets 0 5.2.3 Comparison with the Monte-Carlo Solution 0 5.3 Exponentially and -Distributed Jumps in the Variance Process 0 5.3.1 Exponentially Distributed Jumps 0 5.3.2 -Distributed Jumps 0 5.3.3 Comparison with the Monte-Carlo Solution 0 5.4 Jumps in Both the Underlying and Variance Process 0 5.4.1 Independent Jumps 0 Comparison with the Monte-Carlo Solution 0 5.4.2 Correlated Jumps 0 Exponentially Distributed Variance Jumps 0 -Distributed Variance Jumps 0 Comparison with the Monte-Carlo Solution 0 fulltext_6 110 6 Stochastic Equilibrium Level 0 6.1 Constant Volatility 0 6.1.1 Mean-Reverting Equilibrium Level 0 Special Case: = X 0 6.1.2 Brownian Motion with Drift 0 6.2 Integration of Square-Root Stochastic Volatility 0 6.2.1 Mean-Reverting Equilibrium Level 0 6.2.2 Brownian Motion with Drift 0 General Case Solution 0 Special Case 1 Solution 0 Special Case 2 Solution 0 Comparison with the Monte-Carlo Solution 0 6.3 Other Model Extensions 0 6.3.1 Ornstein--Uhlenbeck Stochastic Volatility 0 6.3.2 Model Extensions with Jump Components 0 fulltext_7 124 7 Deterministic Seasonality Effects 0 7.1 Seasonality in the Log-Price Process 0 7.1.1 Constant Volatility 0 7.1.2 Square-Root Stochastic Volatility 0 Comparison with the Monte-Carlo Solution 0 7.1.3 Other Model Extensions 0 7.2 Seasonal Impact of Volatility 0 7.2.1 Seasonal Variance According to Richter and Sørensen 0 7.2.2 Modeling of Seasonality in the Variance Process 0 General Case Solution 0 Special Case 1: = 0 Special Case 2: 2 = 0 Model Extensions 0 fulltext_8 136 8 Conclusion 0 zback-matter 141 References 0 Front Matter....Pages i-xviii Introduction....Pages 1-7 Mean Reversion in Commodity Prices....Pages 9-16 Fundamentals of Derivative Pricing....Pages 17-53 Stochastic Volatility Models....Pages 55-79 Integration of Jump Components....Pages 81-99 Stochastic Equilibrium Level of the Underlying Process....Pages 101-114 Deterministic Seasonality Effects....Pages 115-126 Conclusion....Pages 127-131 Back Matter....Pages 133-137