Table Of ContentBusiness Management / Economics FY
o Systems Evaluation, Prediction, and Decision-Making Series
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Along with the development of economic globalization, many countries have en
begun to relax their controls on their capital accounts. However, the recent financial sg CAPITAL ACCOUNT
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crises in Latin American countries as well as the exchange rate crises in Southeast
Asian countries have shown that there is major risk associated with capital account
liberalization. LIBERATION
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This book details the benefits and risks of capital account liberalization and A
explains how to take an open-door policy at the appropriate time in order to reduce
P Methods and Applications
the risk to the lowest possible level. Supplying a complete mathematical analysis
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framework for the study of the problem of capital account liberalization, it presents
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a few important models that have been developed for the study of capital account
A
liberalization.
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Next, the book examines the influence of capital account liberalization on the
A
stability of financial markets by greatly expanding the scope of ordinary differential
C
equation theory to the analysis of local stabilities. It conveys cutting-edge results
while providing a general yet simple analysis framework, enriched with practical C
experiences from developing countries.
O
This book applies the theory of limit cycles to the study of problems related to U
capital account liberalization and discusses the contagion of financial crises among
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different countries. Many problems related to capital account liberalization are
formulated as optimization models, showing the fact that much broader economic T
issues can be solved by employing optimization methods.
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I
The book concludes by comparing the contagion effect of financial markets
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between nations with a relatively high degree of openness with those characterized
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by a moderate degree of openness. Explaining how to determine optimal capital
inflows and outflows, this book provides you with the understanding required to R
accurately determine the characteristics, backgrounds, causes, and roles of capital
A
account liberalization and relevant capital flows.
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K25108
6000 Broken Sound Parkway, NW Ying Yirong • Jeffrey Yi-Lin Forrest
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K25108 cvr mech.indd 1 4/9/15 11:03 AM
Capital Account
Liberation
Methods and Applications
Systems Evaluation, Prediction, and Decision-Making Series
Series Editor
Yi Lin, PhD
Professor of Systems Science and Economics
School of Economics and Management
Nanjing University of Aeronautics and Astronautics
Capital Account Liberation: Methods and Applications
Ying Yirong and Jeffrey Yi-Lin Forrest
ISBN 978-1-4987-1226-2
Grey Game Theory and Its Applications in Economic Decision-Making
Zhigeng Fang, Sifeng Liu, Hongxing Shi, and Yi Lin
ISBN 978-1-4200-8739-0
Hybrid Rough Sets and Applications in Uncertain Decision-Making
Lirong Jian, Sifeng Liu, and Yi Lin
ISBN 978-1-4200-8748-2
Introduction to Theory of Control in Organizations
Vladimir N. Burkov, Mikhail Goubko, Nikolay Korgin, and Dmitry Novikov
ISBN 978-1-4987-1423-5
Investment and Employment Opportunities in China
Yi Lin and Tao Lixin
ISBN 978-1-4822-5207-1
Irregularities and Prediction of Major Disasters
Yi Lin
ISBN: 978-1-4200-8745-1
Measurement Data Modeling and Parameter Estimation
Zhengming Wang, Dongyun Yi, Xiaojun Duan, Jing Yao, and Defeng Gu
ISBN 978-1-4398-5378-8
Optimization of Regional Industrial Structures and Applications
Yaoguo Dang, Sifeng Liu, and Yuhong Wang
ISBN 978-1-4200-8747-5
Systems Evaluation: Methods, Models, and Applications
Sifeng Liu, Naiming Xie, Chaoqing Yuan, and Zhigeng Fang
ISBN 978-1-4200-8846-5
Systemic Yoyos: Some Impacts of the Second Dimension
Yi Lin
ISBN 978-1-4200-8820-5
Theory and Approaches of Unascertained Group Decision-Making
Jianjun Zhu
ISBN 978-1-4200-8750-5
Theory of Science and Technology Transfer and Applications
Sifeng Liu, Zhigeng Fang, Hongxing Shi, and Benhai Guo
ISBN 978-1-4200-8741-3
Capital Account
Liberation
Methods and Applications
Ying Yirong • Jeffrey Yi-Lin Forrest
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Contents
Preface .........................................................................................................xiii
Acknowledgments ........................................................................................xix
Authors .........................................................................................................xxi
1 From Theory to Visualization: General Analysis
Framework in Finance ............................................................................1
1.1 Mathematical Applications in Finance ..............................................1
1.1.1 Synergetic Approach .............................................................4
1.2 Mathematical Models in Finance ....................................................11
1.3 Mathematical Principles in Finance .................................................17
1.3.1 Financial Balances Equation ...............................................17
1.3.2 A Model in Canonical Form ...............................................18
1.3.3 Rationality of Behavior .......................................................20
1.3.4 Rational Expectations Principle ..........................................21
1.3.5 Model of the Russian Economy in the Crisis Period ...........24
1.3.6 Model of the Japanese Economy in the Crisis Period ..........27
1.4 Mathematical Estimations in Finance .............................................29
1.4.1 Introduction .......................................................................29
1.4.2 Estimation ..........................................................................31
1.4.3 Conclusions ........................................................................33
1.5 Mathematical Approximations in Finance ......................................34
1.5.1 Introduction .......................................................................35
1.5.2 Main Results ......................................................................36
1.5.3 Conclusions .......................................................................40
1.6 Game Theory in Finance ................................................................40
1.6.1 Model Assumptions ............................................................41
1.6.2 Evolutionary Game Model ................................................42
1.6.3 Stability Analysis ...............................................................44
1.6.4 Model Summary .................................................................50
1.7 Visualization Technology in Finance ...............................................51
1.7.1 Introduction .......................................................................51
v
vi ◾ Contents
1.7.2 Self-Organizing Maps .........................................................52
1.7.3 Clustering of the SOM .......................................................54
1.7.4 Identifying Systemic Financial Crises .................................55
2 From Micheal to Heckscher–Ohlin: ODE for Capital
Account Liberation ...............................................................................57
2.1 General Theory of Ordinary Differential Equations ........................57
2.1.1 Basic Concepts of Ordinary Differential Equations ............57
2.1.1.1 Equation with Separated Variables: y′ = f(x)g(y) ......59
2.1.1.2 Homogeneous Differential Equation: y′ = f(y/x) .....60
2.1.1.3 Fractional Homogeneous Differential
⎛ ax+by+c ⎞
Equation: yʹ= f ⎜ ⎟ ..........................60
⎝αx+βy+γ⎠
2.1.1.4 The Logistic Equation .........................................61
2.1.1.5 Homogeneous Equation: Ly := y′ + g(x)y = 0 .........62
2.1.1.6 Nonhomogeneous Equation: Ly = h(x) .................63
2.1.1.7 Bernoulli’s Equation: y′ + g(x)y + h(x)yα = 0, α ≠ 1 .....64
2.1.1.8 Riccati’s Equation: y′ + g(x)y + h(x)y2 = k(x) ..........64
2.1.2 Systems with Constant Coefficients ....................................65
2.1.2.1 Real Normal Forms ............................................67
2.1.2.2 Stability ..............................................................69
2.2 Dynamic Path of Nonperforming Loans: First-Order ODE ............70
2.2.1 Introduction .......................................................................70
2.2.2 Hypothesis ..........................................................................71
2.2.3 The Model ..........................................................................73
2.2.4 Analysis ..............................................................................74
2.2.5 Conclusions ........................................................................75
2.3 Stock Market’s Liquidity Risk: Second-Order ODE .......................77
2.3.1 Introduction ......................................................................77
2.3.2 The Model ..........................................................................78
2.3.3 Exogenous Shocks ..............................................................80
2.3.4 Numerical Example ............................................................82
2.4 Stability of Michael Model under Capital Control:
Two-Dimensional Systems (I) ..........................................................85
2.4.1 Introduction .......................................................................85
2.4.2 The Model ..........................................................................87
2.4.3 Stability Analysis ................................................................89
2.4.4 Conclusions ........................................................................96
2.5 Exchange Rate Fluctuations under Capital Control:
Two-Dimensional Systems (II) ........................................................96
2.5.1 Introduction .......................................................................96
2.5.2 Stability Analysis ................................................................97
2.5.3 Conclusions ......................................................................101
Contents ◾ vii
2.6 Dynamic Optimization of Competitive Agents:
Three-Dimensional Systems ...........................................................102
2.6.1 Introduction .....................................................................102
2.6.2 The Model ........................................................................103
2.6.3 Analysis ............................................................................105
2.6.4 Conclusions ......................................................................108
2.7 Dynamic Heckscher–Ohlin Model: Four-Dimensional Systems ......108
2.7.1 Introduction .....................................................................108
2.7.2 The Model ........................................................................109
2.7.3 Local Stability Analysis ....................................................111
2.7.4 Conclusions ......................................................................112
2.8 Instability: Risk of Capital Flow ....................................................113
2.8.1 Introduction .....................................................................113
2.8.2 Instability σ ......................................................................114
2.8.2.1 σ−g Relations ....................................................114
2.8.2.2 Trade-Off Optimization ...................................116
2.8.2.3 b-Effect and c-Effect .........................................119
2.8.3 Empirical Examples ..........................................................120
2.8.4 Conclusion .......................................................................122
3 From European to Asian Option: PDE
for Capital Account Liberation ..........................................................123
3.1 General Method of Parabolic Partial Differential Equations of
Second Order ................................................................................123
3.2 Pricing of Carbon Emission Cost: Linear Parabolic PDEs (I) ........126
3.2.1 Introduction .....................................................................127
3.2.2 The Model ........................................................................129
3.2.3 The Calculation ................................................................131
3.2.4 Conclusions ......................................................................137
3.3 Pricing of Foreign Currency Option: Linear Parabolic PDEs (II) ......137
3.3.1 Introduction .....................................................................137
3.3.2 The Model ........................................................................138
3.3.3 The Solution .....................................................................139
3.4 Pricing of Credit Default Swaps: Linear Parabolic PDEs (III) .......142
3.4.1 Introduction .....................................................................142
3.4.2 The Model ........................................................................144
3.4.3 The Solution .....................................................................147
3.5 Pricing of Forward Exchange Rate: Linear Parabolic PDEs (IV) .......150
3.6 Pricing of Arithmetic Average Asian Option: Nonlinear
Parabolic PDEs(I) ..........................................................................155
3.6.1 Introduction .....................................................................155
3.6.2 The Lemma ......................................................................156
viii ◾ Contents
3.6.3 Decomposition of the Solution .........................................158
3.6.4 Estimation for Error .........................................................161
3.6.5 Estimation of the Error Term ...........................................162
3.6.6 Conclusions ......................................................................169
3.7 Pricing of European Exchange Options: Nonlinear
Parabolic PDEs (II) .......................................................................170
3.7.1 Introduction .....................................................................170
3.7.2 Foreign Exchange Option with Fractional
Brownian Motion .............................................................172
3.7.3 Conclusions ......................................................................175
4 From Financial Crises to Currency Substitution: Limit Cycle
Theory for Capital Account Liberation ..............................................177
4.1 General Theory of Limit Cycles .....................................................177
4.2 Poincaré Problem: Quadratic Polynomial Differential
Systems (I) ..................................................................................180
4.2.1 Introduction .....................................................................180
4.3 Foreign Assets and Foreign Liabilities: Quadratic
Polynomial Differential Systems (II) ..............................................183
4.3.1 Introduction .....................................................................183
4.3.2 Macroeconomic Model .....................................................183
4.3.3 Dynamics Analysis ...........................................................185
4.3.4 Conclusion .......................................................................187
4.4 Dynamics of Employment: Cubic Polynomial
Differential Systems (I) ..................................................................188
4.4.1 Introduction .....................................................................188
4.4.2 Model ...............................................................................189
4.4.3 Calculation .......................................................................190
4.4.4 Conclusions ......................................................................193
4.5 Contagion of Financial Crisis: Cubic Polynomial
Differential Systems (II) ................................................................194
4.5.1 Introduction .....................................................................194
4.5.2 Model ...............................................................................195
4.5.3 Analysis ............................................................................197
4.5.4 Conclusions ......................................................................201
4.6 Contagion of Currency Crises: Fractional Differential
Systems (I)..................................................................................202
4.6.1 Introduction .....................................................................202
4.6.2 Model and Analysis .........................................................206
4.6.3 Conclusions ......................................................................209
4.7 Contagion of Currency Crises with Extra-Absorption:
Fractional Differential Systems (II) ................................................210
