Any curve can be represented as a sum of sinusoidal curves, i.e. a Fourier series. Therefore, it becomes mathematically provable that your method, if properly implemented, will cause you to benefit from volatility.

I've never done the proof formally, but the statement to be proved would be something along these lines:
- assume X% allocation bitcoin, and 100-X % allocation fiat at the beginning
- assume price starts at $A and ends at $B, with arbitrary path from A to B
- assume zero spread and zero fees (makes the math simpler, but you'd have to bear in mind this is an oversimplification of the model)
Strategy 1: no buying and no selling at all
Strategy 2: if your percent allocation of wealth in bitcoin rises above or below X% by some fixed amount D (let's say, +/- 5%) due to bitcoin price fluctuations, then buy or sell as needed to keep the % allocation within X +/- D %.

For example: if you set X at 50% at the beginning and D at 5%, then your bot will buy or sell as needed to keep your percent allocation within the range of 45% to 55%

The proof would basically say that Strategy 2 works better than Strategy 1. I think a few more assumptions are needed though. I'm not sure, but I think Strategy 2 is better if we assume that neither bitcoin nor the fiat becomes worthless. Suppose, for example, that the fiat hyperinflates a la Zimbabwe in 2009 or whenever. In that case, Strategy 2 would definitely be a BAD idea, because by the end you would have sold all your bitcoin and you'd have a zillion Zimbabwean dollars worth nothing. But if we assume the fiat currency stays stable, and bitcoin goes to the moon but does so in a very volatile fashion, then I'm pretty sure Strategy 2 can be proven to be superior.

EDIT:
Actually I might also have to assume that you start and end at the same price. Obviously if the price of bitcoin shoots up to $1M, you're better off if you didn't sell any of it at all during the rise, which means Strategy 1 would be better.