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

Abstract

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.