The question of share price volatility has been studied and it is a difficult problem to solve as no economy nor company remains unchanged over statistically relevant lengths of time.

However, the results of this study suggest there is a share price level effect on share price volatility. The author's acknowledge that this is in conflict with efficient markets, but say this is due to investor behavior. Note that reduced volatility only means that day to day percentage price change ranges are reduced. It does not mean that price changes due to idiosyncratic changes in a company's fortune are reduced. Such an event would be explained as a tail event within its expected share price distribution.

https://financebulletin.org/article/view/1853

Approximate factor structures defined by Chamberlain and Rotschild (1983) allow to test whether a given quantitative firm characteristic (the nominal stock price in this paper) is a determinant of the idiosyncratic volatility of stock returns. Our study of 8,000 U.S stocks over the period 1980-2014 shows that small price stocks exhibit a higher idiosyncratic volatility than large price stocks. This relationship is persistent over time and robust to variations in the number of common factors of the approximate factor structure. Moreover, this small price effect does not hide a small-firm effect because it is still valid when we analyze the tercile of large firms. Our result confirms that small price stocks have lottery-type characteristics and, therefore, it is not in line with the efficient market hypothesis.