What is the ruling on currency trading if a person takes money from its owner to exchange it for him through a currency trader, then gives him the money after the exchange and takes a profit from him without the owner's knowledge of the exchange rate, and this takes one or two days?

This transaction is either an exchange (currency exchange) or an agency:

Firstly: If it is an exchange: It is not permissible to exchange currencies except hand-to-hand, as it is authentically reported in the Sahihayn concerning exchange: "If it is hand-to-hand, there is no harm; but if it is deferred, it is not permissible." Therefore, mutual possession is a condition for currency exchange. Based on this, your mentioned transaction is not permissible. However, it is permissible for you to buy pounds with your money and then sell them for riyals hand-to-hand.

Secondly: If it is an agency: Your possession is considered the valid hand-to-hand possession. You must inform the principal of the true price because the agency is based on trustworthiness, and it is not permissible to take the difference. The agent is bound by several rulings, including: executing the agency within the scope of the authorization, providing the principal with necessary information, and returning what belongs to the principal.

An agency is either gratuitous, in which case it is not permissible to profit from the principal, or for a fee, in which case the fee must be determined in advance; otherwise, the hire would be invalid, and you would be entitled to the customary wage (ajr al-mithl).