Short and sweet it's this:
First the grandfather has to actually buy the stock and have it in his name. If he hasn't bought it yet, it doesn't exist as stock. He can buy stock for relatives, but that is now in their name and it doesn't matter whether he dies or not.
The estate of the deceased gets the stock. If there are many descendants and there is no will, it is divided at the behest of the executor of the estate.
It is illegal for the company to reabsorb the stock. It is as good as money.
Probate is required only if the estate is large. Recently, that has been over $200,000 for the federal taxes.
Taxes are paid by the estate (i.e. executor) if it is sold to anyone.
It is best if there is a valid will naming an executor.
The inheritor of the stock can apply to the company and either sell it back to them or put it in their name. There is a form online for most companies that allows the inheritor to state their preferences.
The price of the stock is what Wall Street says the value is at the day of the actual transaction by the company. This could be days after the inheritor notifies the company they want to sell.
Most companies offer the inheritor the opportunity to by more stock, usually to the limit of that for the small investor.
Hope this helps! I'm not a financial pro, but I had recent experience with my mother's stocks.
Artemis