# Most profitable? High % but low probability or Low % but high probability

I have identified a pattern in different assets where a quick spike/flash crash often occurs, dropping the price between -5% and -15% for a few seconds and then going back to previous average.

I am considering setting up buy orders but I do have limited funds. My first instinct was to separate equally my funds in 3 orders: a buy at -5% of price, -10% and -15%.

But then I realized that if most spikes are at -5%, I should have more funds over there. Basically the theory is this: -15% spikes should happen less often, but will yield the most profits. -5% should happen more often but will yield less profits. Technically the -5% spikes should happen three times more often than the -15% to have similar profit if I put all my funds in either case.

Is there a statistical way of separating my buy orders and funds to maximize profits?

• interesting problem: you could pose it as an "asset" allocation problem and therefore use the markowitz formulation but you'll need expected returns and variances of each "asset" which will be difficult to estimate. – mark leeds Dec 15 '18 at 14:37

It depends on the assumptions you are willing to make. If you assume that the 5, 10, 15% events are mutually exclusive, and you can come up with probabilities $$p_1$$, $$p_2$$, $$p_3$$ for each, and you wish to only maximise expected profits, then you have a linear optimization model

$$\max\ p_1 \pi_1 + p_2 \pi_2 + p_3 \pi_3$$

with $$\pi_i$$ the profits under the three scenarios:

$$\pi_1 = 0.05 w_1$$ $$\pi_2 = 0.05 w_1+0.1 w_2$$ $$\pi_3 = 0.05 w_1+0.1 w_2 +0.15w_3$$

The $$w$$ are the proportions of your capital you invest, so you probably want $$w_i \geq 0$$ for all $$i$$ and $$\sum w = 1$$.

There are only three decision variables, so even a simple grid search would do to solve the model. (The solution may not be unique.)

For more-complex assumptions, I would suggest to simulate random paths of your assets, and then find an vector $$w$$ that is optimal for an objective function evaluated for the P/L along those paths. One thing you might particularly be concerned about is risk: after all, when you go to -15% and are invested already, you will also have a drawdown on your P/L.