在財務時間序列中，如果某些事件導致數據從一種特定分佈轉變為另一種分 佈，則該點稱為變化點。為了估計變化點，我們提出了一種區間型時間序列模型，該模型由每日的最高價、最低價和收盤價組成。基於假設日內對數價格由幾何布朗運動，並使用Girsanov 定理，我們推導了相應的似然函數。最大似然估計（MLEs）使用牛頓-拉弗森法方法獲得。在模擬研究中，我們觀察到所提出的方法在平均方根誤差（RMSE）方面優於僅使用收盤價和開盤價的方法。最後，在實際數據分析中，我們檢測了不同時期標普500 指數跟比特幣的變化點，包括2008 年金融危機、2020 年COVID-19 大流行和2022 年俄烏戰爭，以作為說明。;To identify the change-point for the structure change in financial time series. We propose a symbolic time series model, where the model consists of the daily maximum, minimum, and closing prices. The corresponding likelihood function is derived based on the assumption that the intraday price is driven by geometric Brownian motion. The likelihood function is obtained by using the Girsanov theorem. The maximum likelihood estimates are solved by the Newton-Raphson method. In simulation studies, we observe that the proposed method outperforms the method based on only the closing and opening prices in terms of smaller RMSE. Finally, in real data analysis, we detect change-points for the S&P 500 index and Bitcoin in varied time periods, including the 2008 financial crisis, the 2020 COVID-19 pandemic, and the 2022 Russo-Ukrainian War, for illustration.
