All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
A share is an indefinite cut in the company issuing it, and companies fall into three types: commercial-based, service-based, and a mixture of both.
Anyway, if the company possessed cash and one lunar year elapsed on it then it is liable for Zakah. Therefore, the learned men of the faith should be consulted in order for a person to know whether Zakah is due on the wealth of the company holding his shares or not. If that wealth was liable for Zakah, then he should give the Zakah due on his shares. He should also review the company`s annual report, which reflects how much cash it has, to determine his share in it and pay its Zakah.
Mostly, the price of his shares equals the value of his cut in the company. Therefore, if he gave the Zakah based on the price of the shares then he is free from the liability before Allah.
It is worth pointing out that Zakah is due on wealth once it attains the due Nisab and a lunar year elapses on it. And Allah Knows Best.