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I like to ask a practical question regarding the exercise of European Options:

As we know, one may exercise a European option only at maturity $T$.

But for example, if the option can be exercised only today at 21:00:00h, it would be impossible to exercise the option at exactly that time for practical reasons. I know there are cash-based options, but with actual delivery of the underlying, it is not a trivial "in-the-money exercise" decision, because you may want to exercise the option to own the asset for other reasons (for example price impact, transaction costs, market access, time preference) and the exact market price to determine in-the-money may be not available (outdated illiquid) or differ across different trading platforms.

  1. So what exactly happens at maturity of a European option? Is there a timeframe after market closing to decide on the exercise, or is it "in-the-money" automatically tied to some market price?

  2. When do we actually get the underlying asset after exercising?

If the option is not exercised/vested immediately, this might create some arbitrage opportunity because market price may change in between.

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The price of underlying at expiration is determined by "fixing" or "settlement". Different options on different assets have different fixing processes described by exchanges. This information is provided by exchange at which the product is traded. It is often simply the end-of-day settlement price of the underlying. Sometimes, there is a special trading period during which the prices are aggregated to determine the underlying value (i.e., VIX fixing). You don't need to do anything for it, it is automatically exercised into whatever it is supposed to exercise.

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