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Is it correct to say that I risk early assignment only when I am short calls, not short puts? Because the option owner will want to buy stock to capture the dividend, he can only buy it by exercising his call, not his put.

So if I am naked puts, I don't need to worry about early assignment because of dividends?

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Yes.

The put owner isn't likely to put it to you based on the dividend which follows ownership of the shares. He may still exercise the put at his whim, but not for this reason.

  • If the time premium of an in-the-money put is less than the dividend, it offers the possibility of a Dividend Arbitrage (and early exercise). – Bob Baerker Oct 24 '18 at 14:39
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There is no such thing as 'extra assignment risk' when pricing options. The only things that may change daily are stock price, volatility carry cost, and borrow rate if short the underlying.

Carry cost and the effect from its change is so small that it can be ignored.

Increase in volatility increases the amount of time premium in both puts and calls and vice versa for a decrease in IV.

In addition, the price of puts and calls of each series is inextricably linked by arbitrage strategies called conversion and reversal. Ignoring the borrow rate issue, if put premium rises then so does call and vice versa.

Options are exercised early because of the premium's relationship to the stock and strike price. Pending dividends increase put premium and decrease call premiums. If the dividend exceeds the time premium of an ITM put then it sets up a Dividend Arbitrage and increases the likelihood that it will be exercised early.

Any ITM option that trades below intrinsic value is likely to be exercised early (Discount Arbitrage).

For a more in depth answer with examples, see my explanation at:

(When does it make sense to early exercise a deep in the money put option)

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