The no-trade theorem is a fundamental concept in finance based on the assumption that there is no trade between rational investors under certain conditions. This happens when two investors have the same information and the same expectations about the future development of the economy.

This theorem was developed by the economists Milgrom and Stokey. They argued that if two investors have the same information and the same expectation about future economic events, there is no reason for a trade between them. This is because neither would be willing to buy or sell in this scenario because they see the same value for the trade object.

A simple example: Let's say Investor A and Investor B both have shares in a company. They have identical information about the company and the future economy and both expect the value of the shares to remain at the current level or even fall. In this case, investor A would not want to buy shares from investor B because he believes that the shares will not increase in value. At the same time, investor B would not be willing to sell his shares at a lower price because he has the same expectation. Therefore, there is no trading opportunity.

The assumption that all investors have the same information and the same expectations is, of course, only a theoretical simplification. In the real world, there are often differences in information and expectations between investors, which leads to trading opportunities.

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