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I wonder what would happen if the stock price becomes continuous? I am thinking that if that happens, everyone just places a limit bid order just above the original best bid order or limit sell order below the original sell order. Will there be a trade?

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    $\begingroup$ Continuous with respect to what, time or price? $\endgroup$ – Bob Jansen Feb 22 at 6:38
  • $\begingroup$ All exchanges have a minimum tick size precisely to avoid situations where traders raise/lower their bid/ask by arbitrarily small amounts without ever succeeding in making a trade. This makes the price discontinuous: you have to raise by one tick or stay were you are. $\endgroup$ – Alex C Feb 23 at 9:41
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It is rather philosophical or fantasy question because most of the orderbooks are event-driven, so they are discrete from definition. Honestly, I cannot even imagine how this could be a problem in a continuous time.

However, due to transaction costs, adverse selection and other problem realted to the High Frequency Trading, probably there wouldn't be problems describing by you, because from some point placing new best bid or ask order would be unprofitable. Of course it would also be a situation where new best bid and new best ask order would be matched due to spread equals 1 tick.

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