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how market makers set the time factor to calculate option greeks on the expiration day? does they set time equal 1/24or 2/24 when only 1hour or 2hour left? what frequency market makers update new time to calculate greeks so that they hedge better?

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That's usually taken care of in software, and expiration tenors are always set fractionally, because it is both (a) easier and (b) more reliable not to have two different code paths. Allow me to note as well that "days to expiration" is often only for display to humans, with systems working internally using years, seconds, or voltime.

Back when we were market-making options, we didn't use an intraday volatility profile but I always thought we should have and I'm sure quality options market-makers today use one.

Since you are asking questions at this rather elementary level (no offense intended), then aside from "fractional days" you should be at least as motivated to learn about

  • those intraday volatility profiles
  • overnight versus live trading volatility
  • holiday management
  • pin risk
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