And while nothing is impossible, a change in management is not likely, Niles said.
"He took over as CEO when the stock was in the $300 range, the stock is still higher than when he took over," Niles said. "You have to put this in perspective, Apple didn't run the stock up to $700, stupid investors put the stock up at $700. You can't really blame the CEO for that, they are still making more money than they were."
Niles said that he does expect Apple to expand its buyback program sometime during the next several months even though that could help appease investors. But long term, it's going to take more product diversity—like phones in different sizes— to help the company's stock price, he said.
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The fact that Apple is focused on the long term is also a reason that Cook's job is probably not in jeopardy, said Brian White, a senior analyst for Topeka Capital Markets.
"Apple doesn't usually focus on the stock, because they believe if they focus on building great products the stock will take care of itself. Those with investments, though, are pressuring Apple and I could see because of investors that some pressure might be felt internally," White said. "Apple is longer-term oriented and more product focused."
But Colin Gillis, an analyst at BGC, has a tighter timeline for the CEO.
Gillis, who raised his rating to a "buy" from "hold" on Monday and reduced his price target to $500 from $550, said Apple has enough products in the works this year to send the stock back up. But, he added that Cook may face trouble if the stock doesn't doesn't show much improvement by then end of the year.
"I think he is a fantastic COO, let's see what they do on the innovation front to judge what he does as a CEO," Gillis said on CNBC's "Squawk on the Street" on Monday. "He has until the end of the year."