Mr. Bankman-Fried’s shares are worth around $482 million based on Thursday’s closing price of $8.56. Shares of Robinhood jumped as much as 42% in after-hours trading.

Robinhood is an “attractive investment,” Mr. Bankman-Fried said in the regulatory filing.

“We think it is an attractive investment, too,” Robinhood said in a statement.

Mr. Bankman-Fried disclosed the purchase via a 13D form, which investors use when they plan to take an active role—perhaps by seeking board seats or agitating for a sale. Mr. Bankman-Fried said he has no plans to do those things right now, but he didn’t rule them out.

Investors looking to buy more shares, or perhaps even take control of a company, typically make their presence known using 13D forms. But a Robinhood takeover bid could face formidable challenges. Its founders have more than 60% voting control and, according to a person familiar with the matter, are focused on keeping the company independent as they build it. What’s more, Robinhood stock is down nearly 80% from its July initial public offering price of $38.

The purchase comes at a challenging time for both Robinhood and crypto.

Robinhood added millions of users last year when small-time investors flocked to the stock market. But it has struggled to keep them; monthly active users fell 10% to 15.9 million in March, from 17.7 million in the same month of 2021. Robinhood laid off 9% of its employees late last month.

Cryptocurrencies have plunged recently, a consequence of investors’ reluctance to hold risky assets at a time when the Federal Reserve is moving aggressively to curb inflation. A crash in the price of a key stablecoin pegged to the dollar also roiled the market. Bitcoin fell to its lowest level since December 2020 on Thursday.

Mr. Bankman-Fried’s crypto exchange, FTX, allows traders to make an array of bets, including how stocks will trade after they go public or whether

Donald Trump

will win a second term as president in 2024. An October funding round valued the company at $25 billion.

Robinhood has expanded its crypto offerings in recent months, adding four new tokens to the platform last month. It has offered crypto trading since early 2018.

—Dana Cimilluca contributed to this article.

Write to Orla McCaffrey at orla.mccaffrey@wsj.com