... In the past, most studies have ignored the conditions for log approximations. Examples may include, but are not limited to, Duarte-Silva and Kimel (2024), Panagiotidis, Papapanagiotou, and Stengos (2024), Long, Chiah, Zaremba, and Umar (2024), Liu and Kang (2024), Wong (2023), Domenico, Livan, Montagna, and Nicrosini (2023), Blau, Griffith, and Whitby (2023), Chiang and Chen (2023), Simonato and Denault (2023), Ni and Wang (2023), Ausloos, Ficcadenti, Dhesi, and Shakeel (2021), etc. On the other hand, some studies like Tomlinson, Greenwood, and Mucha-Kruczyński (2024), and Herley, Orlowski, and Ritter (2023), used log return and showed that the mean and variance of the log returns are sufficiently close to zero. ...