Optimal flow in a network with non-constant edges' weights
Your problem is a generalization of the Longest Path problem, which is NP-hard.  If the functions are constant and every conversion increases the amount of money, then there's no reason not to convert all of the money at once.  At that point, you are just looking for the longest path.

Your generalization, allowing partial conversions, non-constant functions, etc. cannot be easier than that.