# Given future price probability distribution, what is a strategy that maximizes return?

Say I know the price probability distribution, e.g., lognormal(p,s), of a stock X at a future time T that is perhaps one or two years into the future. p is price and s is a standard deviation.

What should I trade to maximize my expected total return at time T?

Should I for example roll over short-term stock options, long-term stock options, buy the stock using some amount of leverage, etc?

Is there some way or tool that calculates this automatically?

How do I need to change this problem or my thinking so that the solution is not to simply use infinite amounts of leverage?

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Edit:

I am assuming I have some positive amount of capital C to invest.

Available instruments are:

• Long and short stocks
• Long and short call and put options
• Stock futures and swaps
• T-Bills, notes and bonds
• (If it makes things easier we can ignore warrants and convertibles)
• the strategy should be self-financing I assume ? Also you have to make some assumptions on the avaliable instruments you are able to trade. – Probilitator Mar 28 '14 at 7:53
• Are you looking for a dynamic or a static one? – Probilitator Mar 29 '14 at 10:56

## 1 Answer

If you have an accurate 'future price probability distribution' (I will simplify this phrase: prediction), you buy the futures contracts (or any instrument) using leverage in a grid that covers the expected range and trade it aggressively with increased sizes over the range (e.g. martingale). You can also increase the sizes significantly when the 'prediction' is more short term (over a shorter range) as the drawdown is affected by the total range length prices travel from the first trade + all maximum subsequent trades made in that basket.

This is an example spreadsheet which simulates how to setup the grid from the first trade, and calculates the drawdowns and points to breakeven at each trade level.

While you can technically use stocks, you only get 1:2 leverage in most cases. Futures you can have leverage of up to 1:2000 from the exchange (Eurodollar) and day trading can give you 2x or more from your broker (assuming you close your positions out before daily close). Forex derivatives operate similarly. And you really do not have the manipulation or fraud that you do in stock market, assuming you are with a 'proper' broker.