I know the factors (like supply and demand) but how the economics know the value of a currency in a certain time exactly? They couldn't possibly take all the factors in the universe instantly into account! E.g. what is the formula that calculates that EUR/USD is equal 1.18792 exactly at this nano-second?
There is no 'authority' that calculates it, it is offer and demand.
Millions of people and organizations buy and sell currencies continuously, and they are willing to buy or sell for a certain price, and that defines what the current price is. Very similar as if you sell your truck to some guy, and you discuss and agree on a price - just more people, more often.