Is it permissible for a currency exchange office to invest my transferred amount—which I converted to Syrian currency—and profit from it without my permission?

The money changer is not allowed to dispose of your deposited money without your permission. If he does so, he is liable for the money. Scholars have differed on the recipient of the profit generated from the investment, with several opinions:

First: The profit belongs to the owner of the deposit, because it is the growth of his property. Second: The profit belongs to the public treasury (Bayt al-Mal). Third: It must be given away as charity. Fourth: The profit belongs to the custodian (the money changer), because it is the fruit of his work and effort, and he deserves it due to his liability. Fifth: The profit is to be shared between the custodian and the depositor, similar to a Mudarabah (profit-sharing) arrangement.