When stocks rose after last year’s presidential election, DryShips Inc. DRYS +9.70% left the market far behind. The little-known Greek dry bulk carrier’s epic one-week rally pushed its shares up by 1,500% for no apparent reason.

The rally quickly unwound after the shares were briefly suspended by Nasdaq, but the run-up appears to have made possible a flurry of financial maneuvers that may earn the company’s founder a huge windfall, according to calculations by The Wall Street Journal, while small investors suffered hundreds of millions of dollars in losses. Since they peaked, DryShips’s shares are down by 99.9%.

Registered in the Marshall Islands in the central Pacific but based in Athens, the company owns ships that carry bulk cargoes like coal and iron ore. The industry has been battered in recent years by weak commodity prices and an oversupply of ships. Immediately before the share-price surge, the company announced it was suspending principal and interest payments “to preserve cash liquidity.”

George Economou, DryShips’s chairman and chief executive, founded the company in 2004 and listed it on the Nasdaq in 2005.

Mr. Economou and the company didn’t respond to repeated requests for comment on the stock or how Mr. Economou benefited from share sales. They haven’t been accused of any wrongdoing and there is no evidence the company or its CEO engineered the stock rally. Small-company stocks, especially ones in financial distress like DryShips, are often highly volatile as investors try to profit from the big moves.

The sequence of events that could earn Mr. Economou tens of millions in profits began last September. First, through a series of transactions involving the company’s debt, Mr. Economou gained voting control of DryShips without exposure to the common stock, according to securities filings. Filings indicate he owns just 0.01% of the company. Second, the stock price soared, attracting the attention of thousands of fast-trading individual investors. Third, as the rally peaked, the company began issuing stock, which would total more than $500 million, at ever-diminishing prices. Fourth, DryShips used the money to buy ships in deals that benefited Mr. Economou, who earns management fees on its vessels, according to securities filings.

The CEO cemented his control of DryShips two months before the shares took off by converting loans to the company that he owned into a new series of preferred stock that confer 100,000 votes apiece. That stock wasn’t affected by the share price run-up or collapse.

The rally between Nov. 9 and Nov. 16 led the company’s market value to surge from about $5 million to about $80 million.

A day after shares peaked, the company embarked on a series of stock sales totaling more than $500 million so far, according to securities filings. Those documents show the buyer was a British Virgin Islands company called Kalani Investments, but DryShips’s shareholder records don’t list Kalani or any other institution, meaning the firm in turn sold the shares to small investors. DryShips says Kalani is independent of the company. Contact information for Kalani couldn’t be obtained to request comment.

DryShips’s shares would ordinarily be worth pennies following the big decline. The company has avoided that through repeated reverse stock splits that reduce the number of shares outstanding without affecting the company’s value. Despite issuing over $500 million in stock since November, DryShips’s market value is less than $70 million.

The company has used the proceeds from the share sales to buy or acquire options on several hundred million dollars’ worth of ships, likely at attractive prices because of a glutted market. This is one way Mr. Economou has profited. He controls entities that manage the vessels for DryShips for fees that earn him several million dollars a year, according to company filings. He also consolidated over 90% of its debt with a further provision that gives another company controlled by Mr. Economou 30% of any gains earned by the company if certain vessels are later sold.

Individual investors remain obsessed with DryShips. Since the mysterious surge in its price, there have been an average of more than 17,000 mostly bullish mentions a week of DryShips on social investing site StockTwits, a favorite of fast-trading small investors. That is more than 100 times its average weekly mentions before the mid-November frenzy. Message volume is four or five times higher than day-trader darlings Apple Inc.and Tesla Inc.

Lambros Papaeconomou, a longtime observer of DryShips and chief U.S. correspondent at shipping publication Lloyd’s List, calls the size of the share issuance unprecedented in the industry. He says that the first time the company sold stock in November the shares went for $100 each. In its most recent issue, the company has sold Kalani its shares for an average $3.15, according to a securities filing last Tuesday. They closed at $1.23 Wednesday.

The end of the share-selling spree appears to be close. The pace of share issuance to Kalani has slowed and the price keeps falling. But DryShips is still spending the proceeds of the sales. Last week, the company said that it had bought three more ships for $68 million.

Write to Spencer Jakab at spencer.jakab@wsj.com