This problem is underspecified unless you tell us something about what the days will be like. Suppose the crystal ball tells you exactly what the day’s closing price will be (ie, a probability distribution function concentrated on one value).
In world A, you have $1k, and on 99% of days, the market will go up by $3, and on 1% of days it will go down by $3. You should always stay in stock, even if this will cause you to lose more than the transaction fee, because you’ll end up paying a second transaction fee on the next day, which the crystal ball didn’t tell you about.
In world B, you have $1k, on 99% of days the market will go down by $3, and on 1% of days it will go up by $3. You should always stay in cash, even if this will cause you to pass up more than the transaction fee, because you’ll end up paying a second transaction fee on the next day, which the crystal ball didn’t tell you about.
This problem is underspecified unless you tell us something about what the days will be like. Suppose the crystal ball tells you exactly what the day’s closing price will be (ie, a probability distribution function concentrated on one value).
In world A, you have $1k, and on 99% of days, the market will go up by $3, and on 1% of days it will go down by $3. You should always stay in stock, even if this will cause you to lose more than the transaction fee, because you’ll end up paying a second transaction fee on the next day, which the crystal ball didn’t tell you about.
In world B, you have $1k, on 99% of days the market will go down by $3, and on 1% of days it will go up by $3. You should always stay in cash, even if this will cause you to pass up more than the transaction fee, because you’ll end up paying a second transaction fee on the next day, which the crystal ball didn’t tell you about.
The problem is underspecified in a more fundamental way: It does not tell you what to optimise!
One needs to specify both the parameter (eg. expected value) and the time (eg. after 1000 days).