Rigatos State-Space Approaches for Modelling and Control in Financial Engineering

Systems theory and machine learning methods
1. Auflage 2017
ISBN: 978-3-319-52866-3
Verlag: Springer Nature Switzerland
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 125, 310 Seiten

Reihe: Intelligent Systems Reference Library

The book conclusively solves problems associated with the control and estimation of nonlinear and chaotic dynamics in ?nancial systems when these are described in the form of nonlinear ordinary di?erential equations. It then addresses problems associated with the control and estimation of ?nancial systems governed by partial di?erential equations (e.g. the Black–Scholes partial differential equation (PDE) and its variants). Lastly it an offers optimal solution to the problem of statistical validation of computational models and tools used to support ?nancial engineers in decision making.

The application of state-space models in ?nancial engineering means that the heuristics and empirical methods currently in use in decision-making procedures for ?nance can be eliminated. It also allows methods of fault-free performance and optimality in the management of assets and capitals and methods assuring stability in the functioning of ?nancial systems to be established.

Coveringthe following key areas of ?nancial engineering: (i) control and stabilization of ?nancial systems dynamics, (ii) state estimation and forecasting, and (iii) statistical validation of decision-making tools, the book can be used for teaching undergraduate or postgraduate courses in ?nancial engineering. It is also a useful resource for the engineering and computer science community

Autoren/Hrsg.

Rigatos, Gerasimos G.

Inhaltsverzeichnis

1;Foreword;7
2;Preface;9
3;Acknowledgements;19
4;Contents;20
5;1 Systems Theory and Stability Concepts;28
5.1;1.1 Outline;28
5.2;1.2 Characteristics of the Dynamics of Nonlinear Systems;28
5.3;1.3 Computation of Isoclines;29
5.4;1.4 Stability Features of Dynamical Systems;31
5.4.1;1.4.1 The Phase Diagram;31
5.4.2;1.4.2 Stability Analysis of Nonlinear Systems;32
5.4.3;1.4.3 Local Stability Properties of a Nonlinear Model;35
5.5;1.5 Phase Diagrams and Equilibria;36
5.5.1;1.5.1 Phase Diagrams for Linear Dynamical Systems;36
5.5.2;1.5.2 Multiple Equilibria for Nonlinear Dynamical Systems;37
5.5.3;1.5.3 Limit Cycles;42
5.6;1.6 Bifurcations;44
5.6.1;1.6.1 Bifurcations of Fixed Points;44
5.6.2;1.6.2 Saddle-Node Bifurcations of Fixed Points in a One-Dimensional System;45
5.6.3;1.6.3 Pitchfork Bifurcation of Fixed Points;46
5.6.4;1.6.4 The Hopf Bifurcation;46
5.7;1.7 Chaos in Dynamical Systems;50
5.7.1;1.7.1 Chaotic Dynamics;50
5.7.2;1.7.2 Examples of Chaotic Dynamical Systems;50
6;2 Main Approaches to Nonlinear Control;54
6.1;2.1 Outline;54
6.2;2.2 Overview of Main Approaches to Nonlinear Control;54
6.3;2.3 Control Based on Global Linearization Methods;55
6.3.1;2.3.1 Overview of Differential Flatness Theory;55
6.3.2;2.3.2 Differential Flatness for Finite Dimensional Systems;56
6.4;2.4 Control Based on Approximate Linearization Methods;59
6.4.1;2.4.1 Approximate Linearization Round Temporary Equilibria;59
6.4.2;2.4.2 The Nonlinear H-Infinity Control;60
6.4.3;2.4.3 Approximate Linearization with Local Fuzzy Models;65
6.5;2.5 Control Based on Lyapunov Stability Analysis;67
6.5.1;2.5.1 Transformation of Nonlinear Systems into a Canonical Form;67
6.5.2;2.5.2 Adaptive Control Law for Nonlinear Systems;68
6.5.3;2.5.3 Approximators of System Unknown Dynamics;69
6.5.4;2.5.4 Lyapunov Stability Analysis for Dynamical Systems;70
7;3 Main Approaches to Nonlinear Estimation;74
7.1;3.1 Outline;74
7.2;3.2 Linear State Observers;75
7.3;3.3 The Continuous-Time Kalman Filter for Linear Models;76
7.4;3.4 The Discrete-Time Kalman Filter for Linear Systems;76
7.5;3.5 The Extended Kalman Filter for Nonlinear Systems;78
7.6;3.6 Sigma-Point Kalman Filters;80
7.7;3.7 Particle Filters;83
7.7.1;3.7.1 The Particle Approximation of Probability Distributions;83
7.7.2;3.7.2 The Prediction Stage;84
7.7.3;3.7.3 The Correction Stage;85
7.7.4;3.7.4 The Resampling Stage;85
7.7.5;3.7.5 Approaches to the Implementation of Resampling;87
7.8;3.8 The Derivative-Free Nonlinear Kalman Filter;88
7.8.1;3.8.1 Conditions for solving the estimation problem in single-input nonlinear systems;88
7.8.2;3.8.2 State Estimation with the Derivative-Free Nonlinear Kalman Filter;91
7.8.3;3.8.3 Derivative-Free Kalman Filtering for multivariable Nonlinear Systems;92
7.9;3.9 Distributed Extended Kalman Filtering;93
7.9.1;3.9.1 Calculation of Local Extended Kalman Filter Estimations;93
7.9.2;3.9.2 Extended Information Filtering for State Estimates Fusion;96
7.10;3.10 Distributed Sigma-Point Kalman Filtering;97
7.10.1;3.10.1 Calculation of Local Unscented Kalman Filter Estimations;97
7.10.2;3.10.2 Unscented Information Filtering for State Estimates Fusion;101
7.11;3.11 Distributed Particle Filter;103
7.11.1;3.11.1 Distributed Particle Filtering for State Estimation Fusion;103
7.11.2;3.11.2 Fusion of the Local Probability Density Functions;105
7.12;3.12 The Derivative-Free Distributed Nonlinear Kalman Filter;106
7.12.1;3.12.1 Overview;106
7.12.2;3.12.2 Fusing Estimations from Local Distributed Filters;108
7.12.3;3.12.3 Calculation of the Aggregate State Estimation;110
7.12.4;3.12.4 Derivative-Free Extended Information Filtering;111
8;4 Linearizing Control and Estimation for Nonlinear Dynamics in Financial Systems;112
8.1;4.1 Outline;112
8.2;4.2 Dynamic Model of the Chaotic Finance System;113
8.2.1;4.2.1 State-Space Model of the Chaotic Financial System;114
8.2.2;4.2.2 Chaotic Dynamics of the Finance System;115
8.3;4.3 Overview of Differential Flatness Theory;116
8.3.1;4.3.1 Conditions for Applying the Differential Flatness Theory;116
8.3.2;4.3.2 Transformation of Nonlinear Systems into Canonical Forms;117
8.4;4.4 Flatness-Based Control of the Chaotic Finance Dynamics;118
8.4.1;4.4.1 Differential Flatness of the Chaotic Finance System;118
8.4.2;4.4.2 Design of a Stabilizing Feedback Controller;119
8.5;4.5 Adaptive Fuzzy Control of the Chaotic Finance System Using ƒ;120
8.5.1;4.5.1 Problem Statement;120
8.5.2;4.5.2 Transformation of Tracking into a Regulation Problem;120
8.5.3;4.5.3 Estimation of the State Vector;122
8.5.4;4.5.4 The Additional Control Term uc;123
8.5.5;4.5.5 Dynamics of the Observation Error;123
8.5.6;4.5.6 Approximation of Unknown Nonlinear Dynamics;123
8.6;4.6 Lyapunov Stability Analysis;125
8.6.1;4.6.1 Design of the Lyapunov Function;125
8.6.2;4.6.2 The Role of Riccati Equation Coefficients in Hinfty Control Robustness;130
8.7;4.7 Simulation Tests;131
9;5 Nonlinear Optimal Control and Filtering for Financial Systems;135
9.1;5.1 Outline;135
9.2;5.2 Chaotic Dynamics in a Macroeconomics Model;136
9.2.1;5.2.1 Dynamic Model of the Chaotic Finance System;136
9.2.2;5.2.2 State-Space Model of the Chaotic Financial System;137
9.2.3;5.2.3 Chaotic Dynamics of the Finance System;138
9.3;5.3 Design of an H-Infinity Nonlinear Feedback Controller;139
9.3.1;5.3.1 Approximate Linearization of the Chaotic Finance System;139
9.3.2;5.3.2 Equivalent Linearized Dynamics of the Chaotic Finance System;140
9.3.3;5.3.3 The Nonlinear H-Infinity Control;141
9.3.4;5.3.4 Computation of the Feedback Control Gains;142
9.3.5;5.3.5 The Role of Riccati Equation Coefficients in Hinfty Control Robustness;143
9.4;5.4 Lyapunov Stability Analysis;144
9.4.1;5.4.1 Stability Proof;144
9.4.2;5.4.2 Robust State Estimation with the Use of the Hinfty Kalman Filter;146
9.5;5.5 Simulation Tests;147
10;6 Kalman Filtering Approach for Detection of Option Mispricing in the Black--Scholes PDE;151
10.1;6.1 Outline;151
10.2;6.2 Option Pricing Modeling with the Use of the Black--Scholes PDE;152
10.2.1;6.2.1 Option Pricing Modeling with the Use of Stochastic Differential Equations;152
10.2.2;6.2.2 The Black--Scholes PDE;153
10.2.3;6.2.3 Solution of the Black--Scholes PDE;153
10.2.4;6.2.4 Sensitivities of the European Call Option;154
10.2.5;6.2.5 Nonlinearities in the Black--Scholes PDE;154
10.2.6;6.2.6 Derivative Pricing;155
10.3;6.3 Estimation of Nonlinear Diffusion Dynamics;156
10.3.1;6.3.1 Filtering in Distributed Parameter Systems;156
10.4;6.4 State Estimation for the Black--Scholes PDE;158
10.4.1;6.4.1 Modeling in Canonical Form of the Nonlinear Black--Scholes Equation;158
10.4.2;6.4.2 State Estimation with the Derivative-Free Nonlinear Kalman Filter;161
10.4.3;6.4.3 Consistency Checking of the Option Pricing Model;162
10.5;6.5 Simulation Tests;163
10.5.1;6.5.1 Estimation with the Use of an Accurate Black--Scholes Model;163
10.5.2;6.5.2 Detection of Mispricing in the Black--Scholes Model;163
11;7 Kalman Filtering Approach to the Detection of Option Mispricing in Elaborated PDE Finance Models;166
11.1;7.1 Outline;166
11.2;7.2 Option Pricing in the Energy Market;167
11.2.1;7.2.1 Energy Market and Swing Options;167
11.2.2;7.2.2 Energy Options Pricing Models;168
11.3;7.3 Validation of the Energy Options Pricing Model;170
11.3.1;7.3.1 State Estimation with the Derivative-Free Nonlinear Kalman Filter;170
11.3.2;7.3.2 Consistency Checking of the Option Pricing Model;174
11.4;7.4 Simulation Tests;175
11.4.1;7.4.1 Estimation with the Use of an Accurate Energy Pricing Model;175
11.4.2;7.4.2 Detection of Mispricing in the Energy Pricing Model;176
12;8 Corporations' Default Probability Forecasting Using the Derivative-Free Nonlinear Kalman Filter;178
12.1;8.1 Outline;178
12.2;8.2 Company's Credit Risk Models;179
12.2.1;8.2.1 The Merton-KMV Credit-Risk Model;179
12.2.2;8.2.2 Computation of a Company's Distance to Default;181
12.3;8.3 Estimation of the Market Value of the Company Using ƒ;181
12.3.1;8.3.1 State-Space Description of the Black--Scholes Equation;181
12.4;8.4 Forecasting Default with the Derivative-Free Nonlinear Kalman Filter;183
12.4.1;8.4.1 State Estimation with the Derivative-Free Nonlinear Kalman Filter;183
12.4.2;8.4.2 The Derivative-Free Nonlinear Kalman Filter as Extrapolator;184
12.4.3;8.4.3 Forecasting of the Market Value Using the Derivative-Free Nonlinear Kalman Filter;185
12.4.4;8.4.4 Assessment of the Accuracy of Forecasting with the Use of Statistical Criteria;185
12.5;8.5 Simulation Tests;187
13;9 Validation of Financial Options Models Using Neural Networks with Invariance to Fourier Transform;192
13.1;9.1 Outline;192
13.2;9.2 Option Pricing in the Energy Market;193
13.3;9.3 Neural Networks Using Hermite Activation Functions;195
13.3.1;9.3.1 Generalized Fourier Series;195
13.3.2;9.3.2 The Gauss--Hermite Series Expansion;197
13.3.3;9.3.3 Neural Networks Using 2D Hermite Activation Functions;200
13.4;9.4 Signals Power Spectrum and the Fourier Transform;202
13.4.1;9.4.1 Parseval's Theorem;202
13.4.2;9.4.2 Power Spectrum of the Signal Using the Gauss--Hermite Expansion;203
13.5;9.5 Simulation Tests;204
14;10 Statistical Validation of Financial Forecasting Tools with Generalized Likelihood Ratio Approaches;207
14.1;10.1 Outline;207
14.2;10.2 Neuro-Fuzzy Modelling;208
14.2.1;10.2.1 Problem Statement;208
14.2.2;10.2.2 Determination of the Number and Type of Fuzzy Rules;209
14.2.3;10.2.3 Stages of Fuzzy Modelling;211
14.2.4;10.2.4 Fuzzy Model Validation for the Avoidance of Overtraining;213
14.3;10.3 Fuzzy Model Validation with the Local Statistical Approach;215
14.3.1;10.3.1 The Exact Model;215
14.3.2;10.3.2 The Change Detection Test;216
14.3.3;10.3.3 Isolation of Parametric Changes with the Sensitivity Test;218
14.3.4;10.3.4 Isolation of Parametric Changes with the Min-Max Test;219
14.3.5;10.3.5 Model Validation Reduces the Need for Model Retraining;221
14.4;10.4 Detectability of Changes in Fuzzy Models;221
14.5;10.5 Simulation Results;225
14.5.1;10.5.1 Fuzzy Rule Base in Input Space Partitioning;226
14.5.2;10.5.2 Fuzzy Modelling with the Input Dimension Partition;231
15;11 Distributed Validation of Option Price Forecasting Tools Using a Statistical Fault Diagnosis Approach;234
15.1;11.1 Overview;234
15.2;11.2 State Estimation for the Black--Scholes PDE;236
15.2.1;11.2.1 State-Space Description of the Black--Scholes PDE;236
15.2.2;11.2.2 State Estimation with Kalman Filtering;238
15.3;11.3 Distributed Forecasting Model;239
15.4;11.4 Consistency of the Kalman Filter;242
15.5;11.5 Equivalence Between Kalman Filters and Regressor Models;244
15.6;11.6 Change Detection of the Fuzzy Kalman Filter Using the Local Statistical Approach;246
15.6.1;11.6.1 The Global ?2 Test for Change Detection;246
15.6.2;11.6.2 Isolation of Inconsistent Kalman Filter Parameters with the Sensitivity Test;250
15.6.3;11.6.3 Isolation of Inconsistent Kalman Filter Parameters with the Min--Max Test;250
15.7;11.7 Simulation Tests;252
15.7.1;11.7.1 Distributed State Estimation of the Black--Scholes PDE;252
15.7.2;11.7.2 Simulation Results;253
16;12 Stabilization of Financial Systems Dynamics Through Feedback Control of the Black-Scholes PDE;257
16.1;12.1 Outline;257
16.2;12.2 Transformation of the Black-Scholes PDE into Nonlinear ODEs;258
16.2.1;12.2.1 Decomposition of the PDE Model into Equivalent ODEs;258
16.2.2;12.2.2 Modeling in State-Space Form of the Black-Scholes PDE;261
16.3;12.3 Differential Flatness of the Black-Scholes PDE Model;262
16.4;12.4 Computation of a Boundary Conditions-Based Feedback Control Law;264
16.5;12.5 Closed Loop Dynamics;266
16.6;12.6 Simulation Tests;268
17;13 Stabilization of the Multi-asset Black--Scholes PDE Using Differential Flatness Theory;274
17.1;13.1 Outline;274
17.2;13.2 Boundary Control of the Multi-asset Black--Scholes PDE;275
17.3;13.3 Flatness-Based Control of the Multi-asset Black--Scholes PDE;278
17.4;13.4 Stability Analysis of the Control Loop;280
17.5;13.5 Simulation Tests;282
18;14 Stabilization of Commodities Pricing PDE Using Differential Flatness Theory;285
18.1;14.1 Outline;285
18.2;14.2 Models for Commodities Pricing;286
18.2.1;14.2.1 Elaborated Schemes for Trading Electric Power;286
18.2.2;14.2.2 Commodities Pricing with the Single-Factor PDE Model;287
18.2.3;14.2.3 Commodities Pricing with the Two-Factor PDE Model;288
18.2.4;14.2.4 Commodities Pricing with the Three-Factor PDE Model;290
18.3;14.3 Boundary Control of the Multi-factor Commodities Price PDE;290
18.4;14.4 Flatness-Based Control of the Multi-factor Commodities Price PDE;293
18.5;14.5 Stability Analysis of the Control Loop of the Multi-factor Commodities Price PDE;295
18.6;14.6 Simulation Tests;298
19;15 Stabilization of Mortgage Price Dynamics Using Differential Flatness Theory;300
19.1;15.1 Outline;300
19.2;15.2 Options Theory-Based PDE Model of Mortgage Valuation;301
19.3;15.3 Computation of the Mortgage Price PDE;302
19.4;15.4 Boundary Control of the Multi-factor Mortgage Price PDE;304
19.5;15.5 Flatness-Based Control of the Multi-factor Mortgage Price PDE;307
19.6;15.6 Stability Analysis of the Control Loop of the Multi-factor Mortgage Price PDE;309
19.7;15.7 Simulation Tests;311
20;References;314
21;Index;328