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Would the optimal trading strategy for this stockmarket optimization fantasy be trivial or nearly impossible to compute? -or something in between?
You have an initial amount of money A_o and your goal is to maximize the amount of money you will have at the end of a year by trading stocks. (So you must sell all your stocks by the end of the year.) You have the advantage of knowing a time traveller who gave you data for the entire history of the next year's stock market. Say the market has a fixed number of stocks N and their trading prices are differentiable functions of time. You can buy and sell without paying commission. You can buy in any real number amounts. You can just hold an amount of money that is not invested in stocks, for any time you wish. For simplicity, we''ll say you can't borrow money, can't sell short, or make any money beside that which you make trading.
One idea: At each time t, you put all your money in the stock whose price graph has the steepest slope. If none have a positive slope, you keep your money out of the market. I suppose this would be implemented by a nearly "continuous" trading activity. Perhaps it can only be described as a limit of discrete trades as the time interval between the trades approaches zero.
However, I don't know if that strategy is optimal. The classic stock market strategy is "buy low, sell high". If the stock that currently has the steepest upward slope is expensive, it might be better to take the opportunity to buy a lot of a cheap stock that will eventually go up.
You have an initial amount of money A_o and your goal is to maximize the amount of money you will have at the end of a year by trading stocks. (So you must sell all your stocks by the end of the year.) You have the advantage of knowing a time traveller who gave you data for the entire history of the next year's stock market. Say the market has a fixed number of stocks N and their trading prices are differentiable functions of time. You can buy and sell without paying commission. You can buy in any real number amounts. You can just hold an amount of money that is not invested in stocks, for any time you wish. For simplicity, we''ll say you can't borrow money, can't sell short, or make any money beside that which you make trading.
One idea: At each time t, you put all your money in the stock whose price graph has the steepest slope. If none have a positive slope, you keep your money out of the market. I suppose this would be implemented by a nearly "continuous" trading activity. Perhaps it can only be described as a limit of discrete trades as the time interval between the trades approaches zero.
However, I don't know if that strategy is optimal. The classic stock market strategy is "buy low, sell high". If the stock that currently has the steepest upward slope is expensive, it might be better to take the opportunity to buy a lot of a cheap stock that will eventually go up.
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