State-Space Approaches for Modelling and Control in Financial Engineering (eBook)

Foreword 7
Preface 9
Acknowledgements 19
Contents 20
1 Systems Theory and Stability Concepts 28
1.1 Outline 28
1.2 Characteristics of the Dynamics of Nonlinear Systems 28
1.3 Computation of Isoclines 29
1.4 Stability Features of Dynamical Systems 31
1.4.1 The Phase Diagram 31
1.4.2 Stability Analysis of Nonlinear Systems 32
1.4.3 Local Stability Properties of a Nonlinear Model 35
1.5 Phase Diagrams and Equilibria 36
1.5.1 Phase Diagrams for Linear Dynamical Systems 36
1.5.2 Multiple Equilibria for Nonlinear Dynamical Systems 37
1.5.3 Limit Cycles 42
1.6 Bifurcations 44
1.6.1 Bifurcations of Fixed Points 44
1.6.2 Saddle-Node Bifurcations of Fixed Points in a One-Dimensional System 45
1.6.3 Pitchfork Bifurcation of Fixed Points 46
1.6.4 The Hopf Bifurcation 46
1.7 Chaos in Dynamical Systems 50
1.7.1 Chaotic Dynamics 50
1.7.2 Examples of Chaotic Dynamical Systems 50
2 Main Approaches to Nonlinear Control 54
2.1 Outline 54
2.2 Overview of Main Approaches to Nonlinear Control 54
2.3 Control Based on Global Linearization Methods 55
2.3.1 Overview of Differential Flatness Theory 55
2.3.2 Differential Flatness for Finite Dimensional Systems 56
2.4 Control Based on Approximate Linearization Methods 59
2.4.1 Approximate Linearization Round Temporary Equilibria 59
2.4.2 The Nonlinear H-Infinity Control 60
2.4.3 Approximate Linearization with Local Fuzzy Models 65
2.5 Control Based on Lyapunov Stability Analysis 67
2.5.1 Transformation of Nonlinear Systems into a Canonical Form 67
2.5.2 Adaptive Control Law for Nonlinear Systems 68
2.5.3 Approximators of System Unknown Dynamics 69
2.5.4 Lyapunov Stability Analysis for Dynamical Systems 70
3 Main Approaches to Nonlinear Estimation 74
3.1 Outline 74
3.2 Linear State Observers 75
3.3 The Continuous-Time Kalman Filter for Linear Models 76
3.4 The Discrete-Time Kalman Filter for Linear Systems 76
3.5 The Extended Kalman Filter for Nonlinear Systems 78
3.6 Sigma-Point Kalman Filters 80
3.7 Particle Filters 83
3.7.1 The Particle Approximation of Probability Distributions 83
3.7.2 The Prediction Stage 84
3.7.3 The Correction Stage 85
3.7.4 The Resampling Stage 85
3.7.5 Approaches to the Implementation of Resampling 87
3.8 The Derivative-Free Nonlinear Kalman Filter 88
3.8.1 Conditions for solving the estimation problem in single-input nonlinear systems 88
3.8.2 State Estimation with the Derivative-Free Nonlinear Kalman Filter 91
3.8.3 Derivative-Free Kalman Filtering for multivariable Nonlinear Systems 92
3.9 Distributed Extended Kalman Filtering 93
3.9.1 Calculation of Local Extended Kalman Filter Estimations 93
3.9.2 Extended Information Filtering for State Estimates Fusion 96
3.10 Distributed Sigma-Point Kalman Filtering 97
3.10.1 Calculation of Local Unscented Kalman Filter Estimations 97
3.10.2 Unscented Information Filtering for State Estimates Fusion 101
3.11 Distributed Particle Filter 103
3.11.1 Distributed Particle Filtering for State Estimation Fusion 103
3.11.2 Fusion of the Local Probability Density Functions 105
3.12 The Derivative-Free Distributed Nonlinear Kalman Filter 106
3.12.1 Overview 106
3.12.2 Fusing Estimations from Local Distributed Filters 108
3.12.3 Calculation of the Aggregate State Estimation 110
3.12.4 Derivative-Free Extended Information Filtering 111
4 Linearizing Control and Estimation for Nonlinear Dynamics in Financial Systems 112
4.1 Outline 112
4.2 Dynamic Model of the Chaotic Finance System 113
4.2.1 State-Space Model of the Chaotic Financial System 114
4.2.2 Chaotic Dynamics of the Finance System 115
4.3 Overview of Differential Flatness Theory 116
4.3.1 Conditions for Applying the Differential Flatness Theory 116
4.3.2 Transformation of Nonlinear Systems into Canonical Forms 117
4.4 Flatness-Based Control of the Chaotic Finance Dynamics 118
4.4.1 Differential Flatness of the Chaotic Finance System 118
4.4.2 Design of a Stabilizing Feedback Controller 119
4.5 Adaptive Fuzzy Control of the Chaotic Finance System Using ƒ 120
4.5.1 Problem Statement 120
4.5.2 Transformation of Tracking into a Regulation Problem 120
4.5.3 Estimation of the State Vector 122
4.5.4 The Additional Control Term uc 123
4.5.5 Dynamics of the Observation Error 123
4.5.6 Approximation of Unknown Nonlinear Dynamics 123
4.6 Lyapunov Stability Analysis 125
4.6.1 Design of the Lyapunov Function 125
4.6.2 The Role of Riccati Equation Coefficients in Hinfty Control Robustness 130
4.7 Simulation Tests 131
5 Nonlinear Optimal Control and Filtering for Financial Systems 135
5.1 Outline 135
5.2 Chaotic Dynamics in a Macroeconomics Model 136
5.2.1 Dynamic Model of the Chaotic Finance System 136
5.2.2 State-Space Model of the Chaotic Financial System 137
5.2.3 Chaotic Dynamics of the Finance System 138
5.3 Design of an H-Infinity Nonlinear Feedback Controller 139
5.3.1 Approximate Linearization of the Chaotic Finance System 139
5.3.2 Equivalent Linearized Dynamics of the Chaotic Finance System 140
5.3.3 The Nonlinear H-Infinity Control 141
5.3.4 Computation of the Feedback Control Gains 142
5.3.5 The Role of Riccati Equation Coefficients in Hinfty Control Robustness 143
5.4 Lyapunov Stability Analysis 144
5.4.1 Stability Proof 144
5.4.2 Robust State Estimation with the Use of the Hinfty Kalman Filter 146
5.5 Simulation Tests 147
6 Kalman Filtering Approach for Detection of Option Mispricing in the Black--Scholes PDE 151
6.1 Outline 151
6.2 Option Pricing Modeling with the Use of the Black--Scholes PDE 152
6.2.1 Option Pricing Modeling with the Use of Stochastic Differential Equations 152
6.2.2 The Black--Scholes PDE 153
6.2.3 Solution of the Black--Scholes PDE 153
6.2.4 Sensitivities of the European Call Option 154
6.2.5 Nonlinearities in the Black--Scholes PDE 154
6.2.6 Derivative Pricing 155
6.3 Estimation of Nonlinear Diffusion Dynamics 156
6.3.1 Filtering in Distributed Parameter Systems 156
6.4 State Estimation for the Black--Scholes PDE 158
6.4.1 Modeling in Canonical Form of the Nonlinear Black--Scholes Equation 158
6.4.2 State Estimation with the Derivative-Free Nonlinear Kalman Filter 161
6.4.3 Consistency Checking of the Option Pricing Model 162
6.5 Simulation Tests 163
6.5.1 Estimation with the Use of an Accurate Black--Scholes Model 163
6.5.2 Detection of Mispricing in the Black--Scholes Model 163
7 Kalman Filtering Approach to the Detection of Option Mispricing in Elaborated PDE Finance Models 166
7.1 Outline 166
7.2 Option Pricing in the Energy Market 167
7.2.1 Energy Market and Swing Options 167
7.2.2 Energy Options Pricing Models 168
7.3 Validation of the Energy Options Pricing Model 170
7.3.1 State Estimation with the Derivative-Free Nonlinear Kalman Filter 170
7.3.2 Consistency Checking of the Option Pricing Model 174
7.4 Simulation Tests 175
7.4.1 Estimation with the Use of an Accurate Energy Pricing Model 175
7.4.2 Detection of Mispricing in the Energy Pricing Model 176
8 Corporations' Default Probability Forecasting Using the Derivative-Free Nonlinear Kalman Filter 178
8.1 Outline 178
8.2 Company's Credit Risk Models 179
8.2.1 The Merton-KMV Credit-Risk Model 179
8.2.2 Computation of a Company's Distance to Default 181
8.3 Estimation of the Market Value of the Company Using ƒ 181
8.3.1 State-Space Description of the Black--Scholes Equation 181
8.4 Forecasting Default with the Derivative-Free Nonlinear Kalman Filter 183
8.4.1 State Estimation with the Derivative-Free Nonlinear Kalman Filter 183
8.4.2 The Derivative-Free Nonlinear Kalman Filter as Extrapolator 184
8.4.3 Forecasting of the Market Value Using the Derivative-Free Nonlinear Kalman Filter 185
8.4.4 Assessment of the Accuracy of Forecasting with the Use of Statistical Criteria 185
8.5 Simulation Tests 187
9 Validation of Financial Options Models Using Neural Networks with Invariance to Fourier Transform 192
9.1 Outline 192
9.2 Option Pricing in the Energy Market 193
9.3 Neural Networks Using Hermite Activation Functions 195
9.3.1 Generalized Fourier Series 195
9.3.2 The Gauss--Hermite Series Expansion 197
9.3.3 Neural Networks Using 2D Hermite Activation Functions 200
9.4 Signals Power Spectrum and the Fourier Transform 202
9.4.1 Parseval's Theorem 202
9.4.2 Power Spectrum of the Signal Using the Gauss--Hermite Expansion 203
9.5 Simulation Tests 204
10 Statistical Validation of Financial Forecasting Tools with Generalized Likelihood Ratio Approaches 207
10.1 Outline 207
10.2 Neuro-Fuzzy Modelling 208
10.2.1 Problem Statement 208
10.2.2 Determination of the Number and Type of Fuzzy Rules 209
10.2.3 Stages of Fuzzy Modelling 211
10.2.4 Fuzzy Model Validation for the Avoidance of Overtraining 213
10.3 Fuzzy Model Validation with the Local Statistical Approach 215
10.3.1 The Exact Model 215
10.3.2 The Change Detection Test 216
10.3.3 Isolation of Parametric Changes with the Sensitivity Test 218
10.3.4 Isolation of Parametric Changes with the Min-Max Test 219
10.3.5 Model Validation Reduces the Need for Model Retraining 221
10.4 Detectability of Changes in Fuzzy Models 221
10.5 Simulation Results 225
10.5.1 Fuzzy Rule Base in Input Space Partitioning 226
10.5.2 Fuzzy Modelling with the Input Dimension Partition 231
11 Distributed Validation of Option Price Forecasting Tools Using a Statistical Fault Diagnosis Approach 234
11.1 Overview 234
11.2 State Estimation for the Black--Scholes PDE 236
11.2.1 State-Space Description of the Black--Scholes PDE 236
11.2.2 State Estimation with Kalman Filtering 238
11.3 Distributed Forecasting Model 239
11.4 Consistency of the Kalman Filter 242
11.5 Equivalence Between Kalman Filters and Regressor Models 244
11.6 Change Detection of the Fuzzy Kalman Filter Using the Local Statistical Approach 246
11.6.1 The Global ?2 Test for Change Detection 246
11.6.2 Isolation of Inconsistent Kalman Filter Parameters with the Sensitivity Test 250
11.6.3 Isolation of Inconsistent Kalman Filter Parameters with the Min--Max Test 250
11.7 Simulation Tests 252
11.7.1 Distributed State Estimation of the Black--Scholes PDE 252
11.7.2 Simulation Results 253
12 Stabilization of Financial Systems Dynamics Through Feedback Control of the Black-Scholes PDE 257
12.1 Outline 257
12.2 Transformation of the Black-Scholes PDE into Nonlinear ODEs 258
12.2.1 Decomposition of the PDE Model into Equivalent ODEs 258
12.2.2 Modeling in State-Space Form of the Black-Scholes PDE 261
12.3 Differential Flatness of the Black-Scholes PDE Model 262
12.4 Computation of a Boundary Conditions-Based Feedback Control Law 264
12.5 Closed Loop Dynamics 266
12.6 Simulation Tests 268
13 Stabilization of the Multi-asset Black--Scholes PDE Using Differential Flatness Theory 274
13.1 Outline 274
13.2 Boundary Control of the Multi-asset Black--Scholes PDE 275
13.3 Flatness-Based Control of the Multi-asset Black--Scholes PDE 278
13.4 Stability Analysis of the Control Loop 280
13.5 Simulation Tests 282
14 Stabilization of Commodities Pricing PDE Using Differential Flatness Theory 285
14.1 Outline 285
14.2 Models for Commodities Pricing 286
14.2.1 Elaborated Schemes for Trading Electric Power 286
14.2.2 Commodities Pricing with the Single-Factor PDE Model 287
14.2.3 Commodities Pricing with the Two-Factor PDE Model 288
14.2.4 Commodities Pricing with the Three-Factor PDE Model 290
14.3 Boundary Control of the Multi-factor Commodities Price PDE 290
14.4 Flatness-Based Control of the Multi-factor Commodities Price PDE 293
14.5 Stability Analysis of the Control Loop of the Multi-factor Commodities Price PDE 295
14.6 Simulation Tests 298
15 Stabilization of Mortgage Price Dynamics Using Differential Flatness Theory 300
15.1 Outline 300
15.2 Options Theory-Based PDE Model of Mortgage Valuation 301
15.3 Computation of the Mortgage Price PDE 302
15.4 Boundary Control of the Multi-factor Mortgage Price PDE 304
15.5 Flatness-Based Control of the Multi-factor Mortgage Price PDE 307
15.6 Stability Analysis of the Control Loop of the Multi-factor Mortgage Price PDE 309
15.7 Simulation Tests 311
References 314
Index 328