Say I sold a naked put at $30 strike price for an option premium of $1. Would the broker assign 100shares instantly to me when the stock price drops to $30(approximately) or only when price is below $29?
I get to keep the premium if 100shares are assigned but it's pointless if current price is now below $29.
According to the OCC, about 7% of contracts are exercised.
In general, it makes no sense for someone to exercise an option that still has time premium remaining it it because doing so throws away that time premium.
However, there are several reasons for early exercise. If an option trades for less than parity (the bid is less than the intrinsic value), it presents the opportunity for a discount arbitrage should someone sell their option at that price.
There are some less frequent reasons why people exercise early:
They want to own (or be short) the shares
There's a dividend arbitrage available (time premium of an ITM put is less than a pending dividend)
They do a Risk Arbitrage, hoping the stock recovers to the pre-adjusted ex-div close
Neither - If the option is a European-style option, you'd get assigned if the shares are below the strike on the date of option expiry. If the option is an American-style option (options on individual stocks are European, not American), you could get assigned at any time before expiry, but it's very rare - typically it's better for the option holder to sell their option than exercise early.
The broker themselves aren't really assigning anything. It's whoever bought the option that gets to decide whether assign it; the broker simply is a conduit of the assignment. The buyer of the option can do whatever they want, but there isn't much benefit to exercising the option before expiry, or expiry is very close. There's also no point to exercising the option unless it's below the strike price.
What they might do, especially if the stock price is significantly below the strike price, is offer to sell the option back to you at a higher price. For instance, if the stock drops to $28, they might offer to sell the option back to you, but now with a premium of $2.90, allowing both of you to close out your positions and locking in a profit of $1.90 for them. Exercising the option gets them $2.00 for $1.00 net profit, which is less than the $1.90 they could get selling the option, either to you or on the market. (The specific amount of $1.90 isn't necessarily correct, but the general principle of it being worth more than the difference in strike and stock prices should hold).
