二位元序列中連續1的記憶效應初探

在複雜的金融系統中，許多變動因素影響著市場，人們在揭開這些因素的努力中觀察到許多典型化的實況(stylized facts)；其中一個重要的結果為:各種計量的分布呈現著冪次定律(power-law)。本論文回歸到一個基本的問題:在簡單的布朗運動中是否能夠、或者如何產生冪次定律的特性；藉由將股價的下跌與上漲視為二位元的0與1，將一支股票的時間演變，視為一個二位元序列，所要分析的就變成序列中連續1的分布性質。我們的模型中，在某一時間點的下一個位元(股價變動)，可以是任意產生(交易不依靠記憶)或是根據交易歷史來決定(根據記憶進行交易)；這樣的時間序列最終會收斂到某種穩定狀態。我們發現，有記憶的序列，其抵達收斂狀態的平均時間與記憶長度呈現冪次定律的關係，而在不同記憶長度下得到的抵達時間之機率密度函數，在經過尺度轉換後，其曲線彼此間有很好的重疊性。此結果顯示金融數據所呈現的冪次定律分布，或可連結到系統內在的尺度不變性。

關鍵字

二位元字串；連續列；持續時間；冪次定律

並列摘要

In a complex financial system, there are many changing factors that are responsible for the time evolution of the market. There have been efforts to reveal those factors, which result in the observations of many “stylized facts”. One important observation is the presence of power laws in the distributions of various quantities. In this thesis, the author chose to explore such properties on a fundamental level to find whether, or how power laws can be generated in simple Brownian motion. The author uses a one-dimensional model to explore the fall and rise of stock prices, treating them as 0 and 1 in binary, respectively. The time evolution of the price changes of a stock is then realized as a binary sequence. The analysis goes to find the distributions of runs of ones (sections of consecutive ones) in binary sequences. In our model, the next bit (price change) at each time step is determined, either at random (trading without memory) or in accord with the history (trading with memory). The time sequence eventually converges to some steady state. It is found in this study that, for the sequences with memory, the mean arrival time of convergence is a power law function of the memory length. After scale transformation, the curves of the probability density distributions of arrival times for different memory lengths overlap with each other nicely. The result suggests the power-law properties in the distributions of financial data may be related to some underlying scaling behavior of the system.