When stocks rose after last year’s U.S. presidential election, DryShips Inc. 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...