線性與非線性常微分方程在經濟與財務上之應用---以台灣股價加權指數與德國DAX指數為例

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研究生：	姜林宗叡 TSUNG-JUI,CHIANG LIN
論文名稱：	線性與非線性常微分方程在經濟與財務上之應用---以台灣股價加權指數與德國DAX指數為例 Applications of linear and nonlinear ordinary differential equations in economics and finance --- examples of Taiwan stock index TAIEX and German stock index DAX
指導教授：	繆維中 Wei-Chung Miao 李明融 Meng-Rong Li
口試委員:	劉代洋 Day-Yang Liu 張琬喻 Woan-Yuh Jang 繆維中 Daniel Wei-Chung Miao 李明融 Meng-Rong Li 曾正男 Jeng-Nan Tzeng 謝宗翰 Tzong-Hann Shieh 曾睿彬 Jui-Pin Tseng
學位類別：	博士 Doctor
系所名稱：	管理學院 - 財務金融研究所 Graduate Institute of Finance
論文出版年：	2019
畢業學年度：	107
語文別：	英文
論文頁數：	114
中文關鍵詞：	動態系統與微分方程、二次逼近法、均數回歸、羅吉斯方程、股價指數
外文關鍵詞：	Dynamic system and differential equation, parabola approximation, mean-reversion, logistic equation, stock index
相關次數：	點閱：255 下載：0
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摘要

在總體經濟的研究中，有許多指標被使用來反映一個經濟體的景氣狀況，股價指數即為其中之一。傳統上，金融領域相關研究多以統計模型分析股價指數，時間數列模型以及隨機過程為最常見的分析方法。不同於一般統計模型，本文嘗試應用動態系統與微分方程的數學方法對股價指數建立模型，此研究方法源自於物理學，目前已廣泛應用於各種不同領域，包括生態學、控制學、經濟學等。另外，本文也利用「二次逼近法」與「動態積分」於非線性微分方程求解，以處理研究對象過於動態而不易建模的問題。本文以台灣股價加權指數與德國DAX指數為研究對象，分別建立數種合適的模型，包含單一係數常微分方程動態模型、雙係數非線性常微分方程動態模型、一般化二次微分方程動態模型等，並從中選取最精確的模型來描述及預測指數價格。本文實證結果顯示以此研究途徑分析股價指數，所建立的動態模型有不錯的精確度與預測能力，希冀能提供一個新的觀點來詮釋股價指數的變化趨勢，更寄望未來能應用於相關衍生性商品的評價。

ABSTRACT
Among all kinds of macroeconomic indicators, the stock market prices are an important leading indicator used to reflect investors’ expectations for the future economy. In the past, the stock prices are mostly analyzed by statistical models in finance. Often used models include financial time series models and stochastic process. Different from statistical methods, the approach of the dynamic system and differential equations is applied in this study to model the stock index. This mathematical instrument originates from physics and is widely applied in many research field such as Biology, Control Theory, Economics. However, if the movement of the studied subject is too dynamic, modelling the movement will be difficult. Therefore, “parabola approximation” and “dynamic integration” are specifically used to solve nonlinear differential equations in this study. Several models are built step by step from one-coefficient ordinary differential equations, two-coefficient nonlinear ordinary differential equations to generalized parabola differential equations. Examples of two stock indices, Taiwan TAIEX and German DAX are performed. The best choice of characterizing the stock index prices from the models of the same type is recommended for different series of the stock index prices from the empirical study results. In this way, we can provide a new viewpoint for explaining the trends of the stock indices and in the future we may evaluate the derivatives underlying the stock indices.

TABLE　OF　CONTENTS
Chinese abstract    i
English abstract    ii
Acknowledgements    iii
Table of contents    iv
List of tables    vii
List of figures    viii
Chapter 1    Background    1
Chapter 2    Applications of linear ordinary differential equations
and dynamic system to economics - an example of
Taiwan stock index TAIEX    2
2.1    Introduction    2
2.2    Literature review    3
2.3    Methodology    5
2.4    Empirical study    10
2.5    Discussion    16
Chapter 3    Applications of parameterized nonlinear differential
equations and dynamic systems - an example of the
Taiwan stock index    17
3.1    Introduction    17
3.2    Literature review    17
3.3    Methodology    18
3.4    Empirical study    25
3.5    Discussion    30
Chapter 4    Modelling DAX by applying parabola approximation
method    32
4.1    Introduction    32
4.2    Literature review    33
4.3    Methodology    33
4.4    Empirical study    36
4.5    Discussion    42
Chapter 5    Conclusion    43
Reference    44
Appendix    47
A1    ACF and PACF of daily returns of TAIEX from
2 January 2014 to 20 November 2014    47
A2    p-values of Ljung-Box tests for daily returns of
TAIEX from 2 January 2014 to 20 November
2014    47
A3    3D scatter plot of the first difference of stock index
vs. stock index   by year (1 January 2008
~ 31 December 2014)    48
A4    3D scatter plot of the second difference of stock index
vs. the first difference stock index   by
year (1 January 2008 ~ 31 December 2014)    48
A5    3D scatter plot of the second difference of stock index
vs. stock index   by year (1 January 2008
~ 31 December 2014)    48
A6    The convergence of the simulated theoretical stock
prices by Eq. (9)    49
A7    The convergence of the simulated theoretical stock
prices by Eq. (10)    49
A8    ACF and PACF of TAIEX daily returns from 2 June
2015 to 31 August 2016    50
A9    p-values of Ljung-Box tests for TAIEX daily returns
from 2 June 2015 to 31 August 2016    50
A10    Centering tendency of the dynamic model
coefficients    50
A11    Timeplot of the model A coefficients – α1 and β1    51
A12    Timeplot of the model B coefficients – α2 and β2    51
A13    Timeplot of the model C coefficients – α3 and β3    51
A14    Timeplot of the model D coefficients – α4 and A    52
A15    Timeplot of raw data and the forecasts of model A    52
A16    Timeplot of raw data and the forecasts of model B    52
A17    Timeplot of raw data and the forecasts of model C    53
A18    Timeplot of raw data and the forecasts of model D    53
A19    Timeplot of raw data and the forecasts of model B,
C and the martingale    53
A20    Timeplot of raw data and the forecasts of model A,
D and the martingale    54
A21    Company Constituents of DAX    54
A22    Occurrences of singular integral    55
A23    Alpha of applying RGPM to DAX closing prices
from 1 January 2006 to 12 June 2013    55
A24    Beta of applying RGPM to DAX closing prices from
1 January 2006 to 12 June 2013    55
A25    Gamma of applying RGPM to DAX closing prices
from 1 January 2006 to 12 June 2013    56
A26    Basic statistics of RGPM coefficients    56
A27    Solutions of Ordinary Parabolic Model (OPM)    57
A28    Solutions of Relative Growth-Parabola Model
(RGPM)    63
A29    Proof of solutions to Price Reversion Model
(PRM)    67
A30    Proof of solutions to Velocity Reversion Model
(VRM)    74
A31    Proof of solutions to Price Reversion – Quasi
Logistic Model (PRQLM)    87
A32    Proof of solutions to Dynamic Transformed Logistic
Model    98



LIST OF TABLES
Table 1    Classification of model forecasting ability by MAPE    9
Table 2    MAPEs - different number of days chosen for the means in
the three models    11
Table 3    MAPEs and RMSPEs - forecasts of PRM, PRQLM and the
martingale    15
Table 4    MAPEs and RMSPEs - forecasts of VRM and the
martingale    15
Table 5    MAPEs and RMSPEs - fits of the four dynamic models and
the martingale    28
Table 6    MAPEs and RMSPEs - forecasts of the four dynamic models
and the martingale    30
Table 7    Ratio of coefficient value and the preceding 7-day mean value
for the three model coefficients    41


 
LIST OF FIGURES
Figure 1    PRM fits of TAIEX from 2009-01-07 to 2014-12-31    12
Figure 2    PRQLM fits of TAIEX from 2009-01-07 to 2014-12-31    12
Figure 3    VRM fits of TAIEX from 2009-01-06 to 2014-12-29    12
Figure 4    PRM forecasts of TAIEX from 2014-11-21 to 2014-12-31    13
Figure 5    PRQLM forecasts of TAIEX from 2014-11-21 to 2014-12-31
14
Figure 6    VRM forecasts of TAIEX from 2014-11-20 to 2014-12-30
14
Figure 7    TAIEX forecasts of the three models and the martingale from
2014-11-20 to 2014-12-31    14
Figure 8    Time plot of raw data and the model A fits    27
Figure 9    Time plot of raw data and the model B fits    27
Figure 10    Time plot of raw data and the model C fits    27
Figure 11    Time plot of raw data and the model D fits    28
Figure 12    Time plot of raw data, the forecasts of the four dynamic models
and the martingale    29
Figure 13    Time plot of raw data and the forecasts of model D and the
martingale    30
Figure 14    DAX closing prices from 2006-01-02 to 2013-06-12    36
Figure 15    RGPM fit of DAX closing prices from 2006-01-02 to
2013-06-10    37
Figure 16    Modified RGPM fit of DAX closing prices from 2006-01-02 to
2013-06-10    37
Figure 17    Alpha of RGPM - 4-month interval containing the outbreak of
financial crisis    38
Figure 18    Beta of RGPM - 4-month interval containing the outbreak of
financial crisis    39
Figure 19    Gamma of RGPM - 4-month interval containing the outbreak
of financial crisis    39
Figure 20    Alpha of RGPM - 4-month interval containing the DAX
decrease on 2011/08/01    40
Figure 21    Beta of RGPM - 4-month interval containing the DAX
decrease on 2011/08/01    40
Figure 22    Gamma of RGPM - 4-month interval containing the DAX
decrease on 2011/08/01    41




                                

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全文公開日期 2024/06/30 (校內網路)
全文公開日期 2024/06/30 (校外網路)
全文公開日期 2024/06/30 (國家圖書館：臺灣博碩士論文系統)

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