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Yes but the premium is only received AFTER the trade has settled. The cash requirement is BEFORE the trade happens. Before the trade happened, only the strike price is known so the broker needs to make sure that you have enough cash in the account to support the maximum possible risk on the put which is the entire strike price regardless of the premium. It's not an issue of broker requiring more; it's a matter of the broker covering its a$$.
If you enter a limit order to sell the broker knows with 100%certainty whether you have enough cash in the account. Premium received is cash like any other
cash in your account.
+It has been correctly pointed out by Van.. that the cash must be equal to the strike price in a cash account.
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