Formulating Maximum Profit from Known Multi-Asset Price Paths

Summary

The document poses an idealized optimization problem: given the historical prices of a small set of assets over a fixed interval, find the sequence of trades that maximizes terminal wealth from an initial cash amount. The investor may hold up to a limited number of assets, cannot short, and may trade repeatedly without transaction fees. The central challenge is choosing purchases and sales across time and assets while respecting those constraints.

The text gives no algorithm, derivation, or worked example, so it does not establish whether an efficient solution exists or how to compute one. Its value is in clearly stating a theoretical trading problem and its assumptions. The setup uses future prices as known inputs, making it unsuitable as a directly actionable strategy; it also leaves details such as fractional holdings, cash earning a return, and how the portfolio limit applies unspecified.

Key ideas

• The problem seeks the trading sequence that maximizes terminal wealth when the full price path is known.
• The portfolio can contain at most a small number of assets, and short selling is forbidden.
• Trades may be repeated without transaction fees under the stated setup.
• The document poses the optimization question but provides no solution method or efficiency analysis.

Tags

# Given (past) stock values for N assets, how to find the maximum - theoretical - profit?


# Given (past) stock values for N assets, how to find the maximum - theoretical - profit?












In the past few days I have been thinking about a question which seems trivial, yet I can't think of any efficient way to find the optimal solution...

Here is the problem: imagine you have a portfolio which can be composed of up to N different assets (N being quite small, like 5 or 6). You have X dollars to invest at time t0. You can only sell assets that you have previously bought (no shorting), however you buy and sell (without fees) as many times as you like (within the limits of your portfolio). Now if you know the stock values of these 5 assets between t0 and T, is there an (efficient) way to calculate the trading sequence that would give the maximum profit at time T?

I have tried to look if others have had the same question, but I haven't found any relevant answer: either I have been looking for the wrong keywords or no-one has looked into the problem ... (which is understandable since one would need a time-machine for it to have a practical use!)

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.