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Don't they technically expire on Saturday even though the options market closes on Fridays? I feel like the way it stands, this can screw with people with after-hours trading on the underlying.

Or if that premise is not correct, what time do they technically expire?

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Options cease trading at 4 PM EST on the last day of the expiration. Per the NASDAQ, an option owner has until 5:30 PM to exercise. The actual expiration time is not until 11:59 PM EST on Saturday, giving brokers and OCC members time to confirm positions.

These different times can definitely "screw with people with after-hours trading on the underlying." What appears to be about to expire OTM at 4 PM EST on Friday can end up ITM.

If an option is one cent or more ITM at expiration, the Option Clearing Corp (OCC) will automatically exercise your options whether they are long and short (Exercise by Exception). For equity options, you will end up with a long or short position in the underlying (index options are cash settled). On Monday morning you'll be subject to market risk if the underlying reverses in price and it's even possible to receive a margin call.

If you are long the option, you can avoid this risk by designating to the OCC on Friday via your broker that your options are not auto exercised at expiration.

Here's story about this: Options Don’t Expire on Fridays

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The option itself expires midnight on Friday, but practically it is no longer useful after the close of the exercise notification window at your brokerage or clearing firm, because you are not able to do anything with it after that time.

In the United States, in the absence of an instruction (regular or contrary), the OCC will exercise the option for your if the option is at least $0.01 in the money, but this is based on the closing price, not the price in the after hours market at the time. The closing price is not necessarily the price at 4PM, but the official closing print on the tape from the exchange where the underlying stock is listed. Sometimes this might be a few seconds after the close depending on the specialist or market maker and the complexity of the closing auction.

In theory if you held a call option that was in-the money at the close of the market, and then the price dropped in the after-hours market, you might end up with a position you didn't want, but you could see that and instead issue a contrary exercise instruction, provided your broker received it prior to their deadline.

Alternatively you could put a limit order as a hedge in the closing auction, in which case your order would help set the closing price.

It is hard to come up with scenarios where people could be "screwed with". These types of things are usually thoroughly reviewed by the broker dealer's regulators as part of their routine audits, looking for any such impropriety.

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