Pricing of Derivatives on Mean-Reverting Assets (eBook)

Pricing of Derivatives on Mean-Reverting Assets 1
List of Figures 9
List of Tables 11
List of Notations and Symbols 12
1 Introduction 14
2 Mean Reversion in Commodity Prices 21
2.1 Sources of Mean Reversion 21
2.1.1 Convenience Yields 21
2.1.2 Kaldor--Working Hypothesis 23
2.1.3 Time-Varying Risk Premia 24
2.2 Empirical Evidence of Mean Reversion 25
2.3 Mean Reversion and Volatility: The Samuelson Hypothesis 26
3 Fundamentals of Derivative Pricing 29
3.1 Derivative Pricing Under the Risk-Neutral Measure 29
3.1.1 Introduction 29
3.1.2 Change of Measure for Diffusion Processes 31
3.1.3 Change of Measure for Jump-Diffusion Processes 34
3.1.4 Change of Measure if the Underlyingis not a Traded Asset 37
3.2 Characteristic Functions 38
3.3 Fundamental Partial Differential Equation 40
3.4 European Style Derivatives 43
3.4.1 Forwards and Futures 43
3.4.2 European Options 44
Traditional Approach 45
Carr--Madan Approach 48
3.5 Fast Fourier Algorithms 49
3.5.1 Fast Fourier Transformation 49
3.5.2 Fractional Fast Fourier Transformation 52
3.6 Recovering Single Option Prices with Gauss-Laguerre Quadrature 54
A Question of Computational Efficiency: Explicit or Implicit Schemes? 59
The Ode45 Integration Scheme 64
4 Stochastic Volatility Models 66
4.1 Square-Root Stochastic Volatility 66
4.1.1 Comparison with the Tahani Square-Root Model 67
4.1.2 Solution for the Characteristic Function 71
Special Case 1 73
Special Case 2 74
4.1.3 Comparison with the Monte-Carlo Solution 75
4.2 Ornstein--Uhlenbeck Stochastic Volatility 77
4.2.1 Comparison with the Tahani OU Model 78
4.2.2 Solution for the Characteristic Function 78
General Case: 1 and (2 / ) N 79
Special Case 1: 1 and (2 / ) N 80
Special Case 2: = 1 81
4.2.3 Comparison with the Monte-Carlo Solution 82
Case 1: / is an Arbitrary Noninteger 82
Case 2: / is a Positive Integer 85
5 Integration of Jump Components 91
5.1 Simulation of Poisson Processes 92
5.2 Lognormal Jumps of the Underlying 96
5.2.1 Non-Mean-Reverting Assets 96
5.2.2 Mean-Reverting Assets 97
5.2.3 Comparison with the Monte-Carlo Solution 99
5.3 Exponentially and -Distributed Jumps in the Variance Process 100
5.3.1 Exponentially Distributed Jumps 100
5.3.2 -Distributed Jumps 101
5.3.3 Comparison with the Monte-Carlo Solution 102
5.4 Jumps in Both the Underlying and Variance Process 103
5.4.1 Independent Jumps 103
Comparison with the Monte-Carlo Solution 104
5.4.2 Correlated Jumps 105
Exponentially Distributed Variance Jumps 105
-Distributed Variance Jumps 106
Comparison with the Monte-Carlo Solution 107
6 Stochastic Equilibrium Level 110
6.1 Constant Volatility 110
6.1.1 Mean-Reverting Equilibrium Level 110
Special Case: = X 111
6.1.2 Brownian Motion with Drift 112
6.2 Integration of Square-Root Stochastic Volatility 114
6.2.1 Mean-Reverting Equilibrium Level 114
6.2.2 Brownian Motion with Drift 115
General Case Solution 117
Special Case 1 Solution 118
Special Case 2 Solution 119
Comparison with the Monte-Carlo Solution 120
6.3 Other Model Extensions 121
6.3.1 Ornstein--Uhlenbeck Stochastic Volatility 122
6.3.2 Model Extensions with Jump Components 123
7 Deterministic Seasonality Effects 124
7.1 Seasonality in the Log-Price Process 125
7.1.1 Constant Volatility 127
7.1.2 Square-Root Stochastic Volatility 128
Comparison with the Monte-Carlo Solution 129
7.1.3 Other Model Extensions 130
7.2 Seasonal Impact of Volatility 130
7.2.1 Seasonal Variance According to Richter and Sørensen 130
7.2.2 Modeling of Seasonality in the Variance Process 131
General Case Solution 132
Special Case 1: = 133
Special Case 2: 2 = 133
Model Extensions 134
8 Conclusion 136
References 141