A stock exchange office, often disreputable, where people could trade stocks with small amounts of money.
From 'bucket' (possibly meaning discarding or throwing away money) plus 'shop,' a term that emerged in American financial slang by the late 1800s, possibly referencing their small, back-alley locations.
Bucket shops were essentially the illegal day-trading platforms of the 1800s-early 1900s—they operated in legal gray areas and were notorious for cheating customers, leading to their eventual regulation and suppression.
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