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題名 1996-1999年美國股票群的收益以高頻日移動平均計算之統計與動力性質分析
Statistical and Dynamical Properties of Returns Using High Frequency 1-day Moving Averages For Collections of U.S Stocks Over 1996-1999
作者 王柏淵
Wang, Bo Yuan
貢獻者 馬文忠
Ma, Wen Jong
王柏淵
Wang, Bo Yuan
關鍵詞 朗之萬方程
Lévy穩定分布
自相關函數
隨機行走
布朗運動
擴散係數
Langevin equation
Lévy distribution
autocorrelation function
random walk
Brownian motion
diffusion constant
日期 2012
上傳時間 2-Sep-2013 16:56:32 (UTC+8)
摘要 本研究著重於隨機漫步的理論與應用，並收集S&P500的其中345家交易較為頻繁的公司做為實證的數據。根據高頻率交易一天移動平均 (HF1MA)之下的股票觀測其特徵，發現與多粒子系統的均方位移(MSD)的特徵有相似之處，據此，我們進一步對在不同時間尺度靜態和動態屬性進行了詳細的分析。我們在分析S&P 500其中345家公司在1996 – 1999年各月份的股票數據時，觀察作移動平均的計算對數據的統計分布與動態性質之影響。我們檢驗在有移動平均與沒有移動平均的兩種情況下，市場報酬(log–return)的機率密度函數中心是否符合Lévy分布，分析對單月資料進行統計計算之侷限與技巧，同時我們計算自相關函數並對報酬的機率密度函數如何隨時間尺度的變動進行詳細分析。結果顯示在一天的移動平均下，機率密度函數的中心部份符合Lévy分布，其 α≈1；而在沒有一天移動平均下其 α≈1.6。在新定義的自相關函數中，我們可以分辨在有移動平均與沒有移動平均的情況下其動力性質的特徵。
Based on the observations that the mean square log-return obtained from the high-frequency one-day moving averages(HF1MA) of a collection of stocks share similar features with the mean square displacement of a many particle system described by Langevin equation, we carry out a detailed analysis on the time-scale dependence of static as well as dynamic properties for such averages. We analyze the data of a collection of 345 stocks listed in S&P 500 for each month over the years 1996-1999. We examine if the probability distribution meets Lévy distribution in two cases of moving average & non-moving average, and how the selected interval affect the fitted parameters of the probability distribution. Also we calculate the autocorrelation function and analyze the probability density function of log - return at different time scales in detail. Our results show that the central parts of probability density functions are fitted by Lévy with parameter α≈1 for the averaged data and α≈1.6 for the non-averaged data. With a newly defined autocorrelation function, we can distinguish dynamic features between the averaged data and the non-averaged data.
參考文獻 [1]黃文璋(民國81年), 布朗運動簡介, 數學傳播, 第16卷第4期。
[2] R. N. Mantegna, and H. E. Stanley, Scaling behavior in the dynamics of an
economic index, Nature 376, 46-49 (1995).
[3] M.P. Beccar Varela - M. Ferraro - S. Jaroszewicz – M.C. Mariani, ”Truncated
Levy walks applied to the study of the behavior of Market Indices”
[4] Yanhui Liu, Parameswaran Gopikrishnan, Pierre Cizeau, Martin Meyer,
Chung-Kang Peng, and H. Eugene Stanley, Statistical properties of the volatility
of price fluctuations, Phys. Rev. E. VOL 60 (AUGUST 1999)
[5]陳仁遶(民國91年), “布朗運動: 從物理學到財務學”, 數學傳播, 第26卷第1
期, 17-22
[6]陳宣毅(2005年), “布朗運動：從花粉的無規行走到生物與天文”, 物理雙月刊,
廿七卷三期
[7]龐寧寧(2005年), “布朗運動界面成長與擴散現象”, 物理雙月刊, 廿七卷三期
[8]王子瑜、曹恒光(2005年),”布朗運動、郎之萬方程式、與布朗動力學(Brownian
Motion, Langevin Equation, and Brownian Dynamics)” , 物理雙月刊, 廿七卷三
期
[9] R. N. Mantegna, and H. E. Stanley, “Stochastic Process with Ultra-Slow
Convergence to a Gaussian The Truncated Levy Flight”, Phys. Rev. Lett. VOL73,
2946 (NOVENBER 1994).
[10] W.-J. Ma, C.-K. Hu, and R. E. Amritkar. A stochastic dynamic model for
stock-stock correlations, Phys. Rev. E 70, 026101 (2004)
[11] Wen-Jong Ma, Shih-Chieh Wang, Chi-Ning Chen, and Chin-Kun Hu, Crossover
behavior of stock returns and mean square displacements of particles governed
by the Langevin equation (2013)
[12] Shih-Chieh Wang, Cross-correlations in Taiwan stock market – a computational
statistical physics approach, July 2005
[13] 王碩濱, 東華大學應用物理研究所
[14] Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C. K. Peng, H. E.Stanley, Phys.
Rev. E 60, 1390 (1999).
[15] IAENG International Journal of Applied Mathematics, August 2010
[16] IAENG International Journal of Applied Mathematics, August 2010
描述 碩士
國立政治大學
應用物理研究所
99755010
101
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0099755010
資料類型 thesis
dc.contributor.advisor 馬文忠zh_TW
dc.contributor.advisor Ma, Wen Jongen_US
dc.contributor.author (Authors) 王柏淵zh_TW
dc.contributor.author (Authors) Wang, Bo Yuanen_US
dc.creator (作者) 王柏淵zh_TW
dc.creator (作者) Wang, Bo Yuanen_US
dc.date (日期) 2012en_US
dc.date.accessioned 2-Sep-2013 16:56:32 (UTC+8)-
dc.date.available 2-Sep-2013 16:56:32 (UTC+8)-
dc.date.issued (上傳時間) 2-Sep-2013 16:56:32 (UTC+8)-
dc.identifier (Other Identifiers) G0099755010en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/59446-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用物理研究所zh_TW
dc.description (描述) 99755010zh_TW
dc.description (描述) 101zh_TW
dc.description.abstract (摘要) 本研究著重於隨機漫步的理論與應用，並收集S&P500的其中345家交易較為頻繁的公司做為實證的數據。根據高頻率交易一天移動平均 (HF1MA)之下的股票觀測其特徵，發現與多粒子系統的均方位移(MSD)的特徵有相似之處，據此，我們進一步對在不同時間尺度靜態和動態屬性進行了詳細的分析。我們在分析S&P 500其中345家公司在1996 – 1999年各月份的股票數據時，觀察作移動平均的計算對數據的統計分布與動態性質之影響。我們檢驗在有移動平均與沒有移動平均的兩種情況下，市場報酬(log–return)的機率密度函數中心是否符合Lévy分布，分析對單月資料進行統計計算之侷限與技巧，同時我們計算自相關函數並對報酬的機率密度函數如何隨時間尺度的變動進行詳細分析。結果顯示在一天的移動平均下，機率密度函數的中心部份符合Lévy分布，其 α≈1；而在沒有一天移動平均下其 α≈1.6。在新定義的自相關函數中，我們可以分辨在有移動平均與沒有移動平均的情況下其動力性質的特徵。zh_TW
dc.description.abstract (摘要) Based on the observations that the mean square log-return obtained from the high-frequency one-day moving averages(HF1MA) of a collection of stocks share similar features with the mean square displacement of a many particle system described by Langevin equation, we carry out a detailed analysis on the time-scale dependence of static as well as dynamic properties for such averages. We analyze the data of a collection of 345 stocks listed in S&P 500 for each month over the years 1996-1999. We examine if the probability distribution meets Lévy distribution in two cases of moving average & non-moving average, and how the selected interval affect the fitted parameters of the probability distribution. Also we calculate the autocorrelation function and analyze the probability density function of log - return at different time scales in detail. Our results show that the central parts of probability density functions are fitted by Lévy with parameter α≈1 for the averaged data and α≈1.6 for the non-averaged data. With a newly defined autocorrelation function, we can distinguish dynamic features between the averaged data and the non-averaged data.en_US
dc.description.tableofcontents 誌謝................................................................................................................................I
Abstract..........................................................................................................................II
中文摘要......................................................................................................................III
圖表目錄........................................................................................................................1
第一章 序論................................................................................................................3
第二章 理論背景與方法............................................................................................7
2.1 布朗運動 & 擴散運動 & 朗之萬方程式.......................................................7
2.1.1 隨機行走(random walk)與布朗運動.........................................................7
2.1.2 布朗運動：簡單的一維隨機行走(random walk)問題..............................8
2.1.3 布朗運動：愛因斯坦以機率與擴散的觀點..........................................10
2.1.4 擴散方程式的解(solution of diffusion equation)....................................11
2.2 朗之萬方程(Langevin equation)描述布朗運動..............................................13
2.2.1 朗之萬方程針對速度的描述..................................................................13
2.2.2 朗之萬方程針對MSD的描述................................................................14
2.3 穩定分布(stable distribution)...........................................................................16
2.4 截尾Lévy機率分布(The Truncated Lévy Flight)...........................................18
2.5 自相關函數.......................................................................................................19
第三章 實證分析......................................................................................................21
3.1 高頻一天移動平均線的定義(definition of High Frequency One-day
Moving Average, HF1MA) .............................................................................21
3.2 (對數)報酬 log–return R(t) 的定義與不同的交易 time scale τ...................23
3.3 股價指數報酬對應於朗之萬理論的實證.......................................................24
3.3.1 隨機行走的模擬......................................................................................24
3.3.2 MSLR的意義............................................................................................27
3.3.3 HF1MA與noMA情況下的MSLR比較.................................................28
3.3.4不同時間長度移動平均之下對MSLR的影響........................................31
3.4 股價指數報酬對應於Lévy機率分布的實證..................................................34
3.4.1 標準普爾指數 (S&P500) 報酬分布的尺度特徵........................................36
3.4.2 股價報酬機率密度函數的指數遞減............................................................39
3.4.3 Lévy穩定分布中心附近P(R=0)的 α 值與圖形fitting..............................40
3.4.4 以Lévy穩定分布描述股價報酬P(R=0)........................................................45
3.5 自相關函數.......................................................................................................50
第四章 結論與建議..................................................................................................54
附錄..............................................................................................................................60
參考文獻......................................................................................................................69		zh_TW
dc.format.extent 3594985 bytes-
dc.format.mimetype application/pdf-
dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0099755010en_US
dc.subject (關鍵詞) 朗之萬方程zh_TW
dc.subject (關鍵詞) Lévy穩定分布zh_TW
dc.subject (關鍵詞) 自相關函數zh_TW
dc.subject (關鍵詞) 隨機行走zh_TW
dc.subject (關鍵詞) 布朗運動zh_TW
dc.subject (關鍵詞) 擴散係數zh_TW
dc.subject (關鍵詞) Langevin equationen_US
dc.subject (關鍵詞) Lévy distributionen_US
dc.subject (關鍵詞) autocorrelation functionen_US
dc.subject (關鍵詞) random walken_US
dc.subject (關鍵詞) Brownian motionen_US
dc.subject (關鍵詞) diffusion constanten_US
dc.title (題名) 1996-1999年美國股票群的收益以高頻日移動平均計算之統計與動力性質分析zh_TW
dc.title (題名) Statistical and Dynamical Properties of Returns Using High Frequency 1-day Moving Averages For Collections of U.S Stocks Over 1996-1999en_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1]黃文璋(民國81年), 布朗運動簡介, 數學傳播, 第16卷第4期。
[2] R. N. Mantegna, and H. E. Stanley, Scaling behavior in the dynamics of an
economic index, Nature 376, 46-49 (1995).
[3] M.P. Beccar Varela - M. Ferraro - S. Jaroszewicz – M.C. Mariani, ”Truncated
Levy walks applied to the study of the behavior of Market Indices”
[4] Yanhui Liu, Parameswaran Gopikrishnan, Pierre Cizeau, Martin Meyer,
Chung-Kang Peng, and H. Eugene Stanley, Statistical properties of the volatility
of price fluctuations, Phys. Rev. E. VOL 60 (AUGUST 1999)
[5]陳仁遶(民國91年), “布朗運動: 從物理學到財務學”, 數學傳播, 第26卷第1
期, 17-22
[6]陳宣毅(2005年), “布朗運動：從花粉的無規行走到生物與天文”, 物理雙月刊,
廿七卷三期
[7]龐寧寧(2005年), “布朗運動界面成長與擴散現象”, 物理雙月刊, 廿七卷三期
[8]王子瑜、曹恒光(2005年),”布朗運動、郎之萬方程式、與布朗動力學(Brownian
Motion, Langevin Equation, and Brownian Dynamics)” , 物理雙月刊, 廿七卷三
期
[9] R. N. Mantegna, and H. E. Stanley, “Stochastic Process with Ultra-Slow
Convergence to a Gaussian The Truncated Levy Flight”, Phys. Rev. Lett. VOL73,
2946 (NOVENBER 1994).
[10] W.-J. Ma, C.-K. Hu, and R. E. Amritkar. A stochastic dynamic model for
stock-stock correlations, Phys. Rev. E 70, 026101 (2004)
[11] Wen-Jong Ma, Shih-Chieh Wang, Chi-Ning Chen, and Chin-Kun Hu, Crossover
behavior of stock returns and mean square displacements of particles governed
by the Langevin equation (2013)
[12] Shih-Chieh Wang, Cross-correlations in Taiwan stock market – a computational
statistical physics approach, July 2005
[13] 王碩濱, 東華大學應用物理研究所
[14] Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C. K. Peng, H. E.Stanley, Phys.
Rev. E 60, 1390 (1999).
[15] IAENG International Journal of Applied Mathematics, August 2010
[16] IAENG International Journal of Applied Mathematics, August 2010		zh_TW