從經濟和金融的觀點來看，數據分析和預測一直是關鍵和重要的議題。在實際應用中，大多數模型只考慮股票的收盤價，導致關鍵數據（如最高價和最低價）被排除在外。因此，為了改善基於關鍵數據的參數估計和預測，我們依賴於幾何布朗運動（GBM）框架，利用開盤價、收盤價、最高價和最低價的概似函數來估計參數σ2，並運用反射原理和Girsanov定理。本研究旨在通過馬可夫鏈蒙特卡羅（MCMC）演算法對參數進行估計，並將其與最大概似估計（MLE）方法進行比較，同時使用95-分位數可信區間來評估該演算法在模擬研究中對所提出模型的適用性，並使用相對誤差（RE）指標比較模擬結果。最後，在實證分析中，所提出的方法在將符號數據應用於標準普爾500指數 (S&P 500)的真實數據方面展現了良好的表現。

From an economic and financial perspective, data analysis and prediction are always critical and crucial topics. In real application, most models only consider the closing price of stocks, leading to the exclusion of crucial data, such as the highest and lowest prices. Hence, in general, in order to improve parameter estimation and prediction based on the addition crucial data, relied on the Geometric Brownian Motion (GBM) framework, we obtain the likelihood
function of the opening, closing, highest, and lowest prices to estimate the parameter σ2 by employing the reflection principle and the Girsanov theorem. The purpose of this study is to investigate the performance through the Markov Chain Monte Carlo (MCMC) algorithm for parameter estimation and compare it with the maximum likelihood estimation (MLE)
method. Additionally, we use the 95th percentile credible interval to assess the suitability of the algorithm for the proposed model in the simulation study and to compare the simulation results of each model using the relative error (RE) measure. Finally, in the empirical analysis, the proposed method demonstrates a strong track record in applying symbolic data to real-world data for the S&P 500 index.

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