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I have sometimes witnessed the price of deep in-the-money option contracts of the same underlying, but different expiration dates, go in different directions by the end of the trading day. This makes no sense to me.

The intrinsic value of both options should be directly correlated to the price of the underlying. For example, in case of put options, if the underlying's price goes up, the intrinsic value of both options should go down in equal amounts.

The extrinsic value should behave similarly. Each day that passes, both options experience time decay, so both option's extrinsic value should go down accordingly (though not necessarily by equal amounts). If the underlying's implied volatility changes, the extrinsic values of both options should move in the same direction as well.

So, how is it possible that one option could end up on any given day with a higher price than the day before, while the other option ends up with a lower price than the day before?

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Illiquid ITM options have wide spreads and can have stale quotes.

Suppose XYZ is $27 and the $25 call is $$2.50 x $3.00. Someone sells at the bid for $2.50 and then with XYZ still at $27, someone buys the call for $3.00. So the option appears to have risen 50 cents when in reality it's just a function of a wide quote.

Meanwhile, another ITM call trades at $2.00 twice in a row and reflects an unchanged status. So one ITM call appears to have risen while the other didn't.

The stale quote scenario would be that when XYZ is $27 (as above), the $25 call trades for $2.50. XYZ rises closes at $29 with no additional trades in the $25 call. The next day, XYZ opens at $29 and the call quote is $4.00 x $4.50. A buy at $4.50 shows a gain of $2.00 yet the stock is unchanged.

One call up and the other one down would just be a variation of the above.

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