The question is targeted for symbols trading weekly options. For example, would Friday's
AAPL after-hours trading affect the weekly options contract that are slated for expiration on that day? Or do the contracts expire and cannot be exercised/bought/sold after 4 PM ET?
Typically the settlement price for a financial instrument (such as AAPL stock) underlying a derivative contract is determined from the average price of trading in that instrument during some short time window specified by the exchange offering the derivative. (Read the fine print on your contract to learn the exact date and time of that settlement period.)
Because it's in an exchange's best interest to appear as fair as possible, the exchange will in general pick a high-volume period of time -- such as the close of trading on the expiry date -- in which to determine the settlement price.
Now, the expiry date/time may be different from the last time at which the option can be traded, which may be different from the underlying settlement time. For example, most US equity options currently expire on the Saturday following the third Friday of the month, whereas they can last be traded at end-of-day on the third Friday of the month, and the settlement period may be at a slightly different time on the third Friday of the month. (Again, read the contract to know for sure.) Moreover, your broker may demand to know whether you plan to exercise the option at an even earlier date/time.
So, to answer your question:
After-hours trading can only affect the settlement price of an underlying instrument if the exchange in question decides that the settlement period should happen during after-hours trading. But since no exchange that wants to stay in business would possibly do that, the answer is no.
Contract expiry time, contract exercise time, final contract trading time, and underlying settlement time may all fall at different dates/times. The important one for your question is settlement time.
There is a white paper on "The weekend effect of equity options" it is a good paper and shows that (for the most part) option values do lose money from Friday to Monday. Which makes sense because it is getting closer to expiration. Of course this not something that can be counted on 100%. If there is some bad news and the stock opens down on a Monday the puts would have increased and the calls decreased in value.
Article Summary (from the authors): "We find that returns on options on individual equities display markedly lower returns over weekends (Friday close to Monday close) relative to any other day of the week. These patterns are observed both in unhedged and delta-hedged positions, indicating that the effect is not the result of a weekend effect in the underlying securities. We find even stronger weekend effects in implied volatilities, but only after an adjustment to quote implied volatilities in terms of trading days rather than calendar days."
"Our results hold for puts and calls over a wide range of maturities and strike prices, for both equally weighted portfolios and for portfolios weighted by the market value of open interest, and also for samples that include only the most liquid options in the market. We find no evidence of a weekly seasonal in bid-ask spreads, trading volume, or open interest that could drive the effect. We also find little evidence that weekend returns are driven by higher levels of risk over the weekend.
"The effect is particularly strong over expiration weekends, and it is also present to a lesser degree over mid-week holidays. Finally, the effect is stronger when the TED spread and market volatility are high, which we interpret as providing support for a limits to arbitrage explanation for the persistence of the effect." - Christopher S. Jones & Joshua Shemes
You can read more about this at this link for Memphis.edu