Tesla's CEO roiled the stock market on Tuesday with a tweet suggesting that he might seek to take the company private at a price of $420—almost 20 percent above the company's then-current price of around $355:

Am considering taking Tesla private at $420. Funding secured. — Elon Musk (@elonmusk) August 7, 2018

Within minutes, the stock surged another $15 to $370 per share. The value has been swinging wildly ever since.

In a Twitter direct message to Ars Technica, Musk confirmed he was serious about the proposal. "Yes, might take Tesla private," he said.

At a share price of $420, Tesla would be worth more than $70 billion. So it will take some very deep pockets to buy up all of Tesla's publicly held stock.

Most CEOs are careful to avoid making comments that could roil the market for their stocks. But Musk is not a conventional CEO. On April 1 this year, Musk tweeted, "We are sad to report that Tesla has gone completely and totally bankrupt." That turned out to be an April Fools joke.

"I don't have a controlling vote now & wouldn't expect any shareholder to have one if we go private," Musk said in a follow-up tweet. "I won't be selling in either scenario."

Typically, buyouts like this involve a new investor buying out smaller shareholders, since there are regulatory barriers to a privately held company having many small shareholders. But Musk said he'd look for a way to allow current Tesla shareholders to continue holding Tesla stock.

"My hope is all current investors remain with Tesla even if we're private," Musk tweeted. "Would create special-purpose fund enabling anyone to stay with Tesla. Already do this with Fidelity's SpaceX investment."

For years, Musk has enjoyed mocking "shorts"—investors who bet against the value of Tesla's stock. Tesla has long been one of the most-shorted stocks on Wall Street, but so far, the company has not seen the share-price meltdown that shorts expected. Today's Tesla rally could make life more difficult for shorts because brokers often ask shorts to put up more money when a stock they've shorted rises in value.

Even before Tuesday's tweet, Tesla's share price was up nearly 20 percent since last week, when the company reported relatively favorable second-quarter results and reiterated that it wouldn't need to raise additional funds in the second half of 2018.

Musk has a lot riding on Tesla's share price. He personally owns around 20 percent of Tesla's outstanding shares, and his compensation over the next decade is tied to reaching a series of ambitious milestones for Tesla's market value. Tesla's stock will need to rise to around $600 for Musk to reach the first of those milestones: a market capitalization of $100 billion.

The NASDAQ halted trading of Tesla stock at 2:08pm ET on Tuesday.