Sorry for the rambling, skip to the end for real numbers.
Letās start making stuff up! Suppose the bitcoin price everywhere as a function of time is x(t). We provide sell walls at x(t)+Cx(t) and buy walls at x(t)-Cx(t) where C is the offset. We want to model market volatility in some way and compare balancing to not balancing. In order to achieve this, I will make a pretty bold assumption I think: at a finite balancing time scale, only volatility at that time scale is pertinent when considering whether or not to balance.
This assumption basically says that only market trends that fluctuate on a time period on the order of a week or so matter. That greatly simplifies things so we can look at what happens. Letās give it a go and try x(t) = P + A*sin(wt) where w is a frequency of 1/a week. To model the trend, weāre going to assume a price y(t) corresponding to the local bitcoin price and assume that it outpredicts the bitcoin market by an eighth of a period (45 degrees, pi/4). We also make the assumption that the local market has a velocity of W nbt/period. Fees are labeled with F, but Iām going to have to come back to those later.
So, the price difference:
y-x = A*[sin(wt+pi/4) - sin(wt)]
gives us a profit margin for the local market. We assume A is small compared with P such that second order terms like fees and spread are considered simple functions of P.
The following gives us the profit to be made at any given point in time:
(y-x)/P = W*( AbsoluteValue{ A/P*[sin(wt+pi/4) - sin(wt)] } - C - F )
We wish to take the integral with respect to time. Here, it would be instructive to split the integral into parts and recognize where there is no profit possible. The condition for zeros in this function are:
- or - (C+F)*P/A = sin(wt)-sin(wt+pi/4)
The answer ends up being pretty complicated, so we need to try to simplify further. Letās just use two linear slopes, one with a slope of M and the other with a slope of -M, and a flat line in between at price (P-A); assume local price starts selling continuously until the bottom where it stops making profit. In that case, the normalized profit at any given moment on the down slope is:
1 - A/P - M/Pt - C - F
with zero at t = (1-A/P-C-F)P/M. The bounded integral is then:
3/2P/M(1-A/P-C-F)^2
Letās plug in and simplify. Iām putting in W for t because of bad notation. SAF is spread after fees and = 2(C+F). P/M is a speed at which the price falls and is only instructive when combined with the parameter W. 1-A/P is a volatility ratio that I will label Q. Note that 2Q is strictly greater than SAF or no profit can be made via arbitrage.
3/2*P/M * [ Q^2 - SAF * Q + 0.25 * SAF^2 ]
So, to take some real numbers to it, letās take 10,000 NBT volume over the course of a 10% drop with 1% SAF. That gives us ~90 NBT loss. If we cement that loss by balancing the sides, we can presume to take another one on the way back up. This would be over the course of ~15 days and thus would amount to 3.2% of Nuās operating budget.
Now, lots of assumptions and hand waving, Iāll give you that for sure. However, I do believe that this medium term volatility cost accounts for <10% of our budget. In contrast, a balanced peg on exchange at week long time scales is the very core of our business philosophy.
