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I am analysing the price movement of a U.S. stock in conjunction with its open interest on calls vs puts. If within a month, the call open interest drastically declines (even relative to put open interest), could this lead to a drop in the price of the underlying?

My reasoning goes as follows. Investors buy calls in large quantities with the idea that the stock price will increase in the near term. They were right, the stock price has risen above the exercise price of their calls. When they exercise the calls, they buy the underlying for the strike price and simultaneously sell it for the higher market price. If everyone does this at the same time, won't it drive the price of the underlying down after the exercise? or am I missing something?

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Changes in open interest do not mean that someone exercises the option. On the contrary, it is usually not beneficial to exercise a call option early (definitely not because the price increased above strike).

A thought experiment: if you were to exercise (despite it being unfavourable) and there is physical delivery, someone must deliver that option (hence buy it at market price). If someone subsequently decides to sell it again, it is in theory a zero sum game unless it all were to occur sequentially (everyone decides to exercise at the same time which results in a large need to buy stocks, which drives the price up, before there is a large counter swing from everyone selling).

That does not mean option trading cannot impact the price of the underlying. This paper discusses some channels.

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