Mockingjay up in here, with this crystal ball the odds will be ever in our favor.
Buy every day when you've got good odds (more likely to make cash than lose it) to make money tomorrow. Sell every day you start the day owning stock.
Every 2 days you lose 4 dollars in transaction fees. You also gain/lose money equal to the difference in cost between your initial purchase and sail. Since you know the pdf for the outcome, and only play if the odds are in your favor you'll probably make a killing.
1) You neglect to account for the transaction fees - following your model would tell you to spend $4 on fees to capture a $1 gain.
2) Why on earth would you auto-sell the stock without even checking the crystal ball?
Let's suppose you start with $1000 to invest, and the only thing you can invest it in is stock ABC. You are only permitted to occupy two states:
* All assets in cash
* All assets in stock ABC
You incur a $2 transaction fee every time you buy or sell.
Kind of annoying limitations to operate under. But you have a powerful advantage as well. You have a perfect crystal ball that each day gives you the [probability density function](http://en.wikipedia.org/wiki/Probability_density_function) of ABC's closing price for the following day (but no further ahead in time).
What would be an optimal decision rule for when to buy and sell?