I am attempting to create a hypothetical risk management system for crypto trading based on the size of my portfolio. I have initial conditions of 200 dollars of initial capital and 200 dollars per trade. I am to create a continuous function that relates the capital to the total amount in a trade. The current ratio is 1:1.
As my capital increases 4 fold, I only want 2 fold increase in the amount that is traded. So if I in turn have 800 dollars in capital, I want 400 dollars to be in each trade; 3200 in capital, 800 dollars to trade and so on. I can map this in a discrete manner, but is there a way to make this a continuous function?
Assuming you meant you want 400 in each trade when the capital goes up to 800, for capital $c$ you should have $200 \sqrt{\frac c{200}}$ in each trade.