Yup.
Orders would come in over the phones to the trading floor. Runners would then take those orders to traders in the pit, who would communicate with other pit traders by shouting or using hand signals (which would signify price, amount, and whether they were buying/selling). When both parties signaled to each other that they would trade, each would mark the trade down in their books.
When the days trading was done, the trades would be reconciled and the actual shares/cash transferred between the brokerages.
That is why the plan in Trading Places worked. They sold a bunch of shares they didn’t have at the start of trading (when prices were high), but since they don’t actually settle up until trading is over, they could buy back all of those shares later in the day (when prices were low). They netted out at zero shares owned (they gave the share they bought to the folks to whom they sold shares earlier) but made millions in the process.
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