I am pondering on the existence/impossibility of a trading system (or algorithm) that ALWAYS ends up winning money, no matter how the price of a futures moves. In a context where one can go long or short at will to make a profit.
If the trader has infinite funds, it's possible. He can setup some kind of martingale system and cancel every loss by making an opposite trade.
If the trader has limited funds, there's an intuitive "proof" that no system can win: if it existed, someone would have figured it out by now.
But I'm looking for a formal treatment of the question. Lots of papers deal with trading in a model in which prices movements are associated with probabilities, such as brownian motions and the like. But really, what I'm looking for is a proof that there's no algorithm that can beat ANY price movement.
Meaning, for every trading algorithm that has limited capital, there exists a price movement in which the algorithm incurs a loss - i.e. the trader cannot close all his positions with a positive outcome. This sounds obivous, but I can't find a formal proof. Any references on the topic is appreciated.
I wish to thank everyone for their answers. I helped me think about how to formalize the question better, as some go to overly complicated answers for what I really wanted to ask.
Think of it as a turn-based game played between two independent entities, the trader and the price. On his turn, the trader takes some positions and closes others (positions might be forcefully closed if trader's balance can't cover his positions, like a margin call).
On its turn, the price makes a tick movement, up or down. Then turns alternate in this manner. The game ends when the trader has no open positions.
The trader aims to make a profit, the price aims to make the trader lose money. In other words, assume that the price is kind of sentient, decides on its own movements, plotting against the trader. Who wins this game if both sides play optimally ?