My question is related to this thread, but I'm interested in a special case. Suppose that the price of an asset starts at 100 USD, and changes according to a geometric random walk; one step of 1% either up or down at each second. You start with an equity of 100 USD, and you put 100 USD at each trade.

The strategy is very simple, two orders are set repeatedly: A buy order at $x_{buy}=(100-1)^{n1}$ and a sell order at $x_{sell} = (100+1)^{n_2}$, where $n_1$ and $n_2$ are positive integers.

We know that the price will hit $x_{buy}$ and $x_{sell}$ an infinite number of times, so this strategy is determined to be profitable (however small) in the long term. The problem is, it's an extremely slow strategy.

Suppose that you want to maximize the expected profit over a 1 year period. What values for $n_1$ and $n_2$ are optimal?