How do you define "1$ USD peg"? As of right now, the price of DAI is $0.98, 2% off its "peg". Asking as a serious question, since I see this statement a lot.
Dai is not really a traditional peg, as in the sense that there is someone who always trades on the market to maintain the peg (a liquidity provider like in USDT). It's more of an example of a system that has carefully balanced economic incentives. The fun part is that anybody can mint or destroy DAI, and take advantage of arbitrage opportunities.
For example, if I ever see DAI trading at, say, $1.02 then I would quickly mint some new DAI and sell for USD. Likewise, if DAI is at $0.98 then I would buy it back, then burn what I've minted, collecting a profit.
Well, pegs with a defined fluctuation boundary aren't that uncommon, for example, Danish Krona currently is pegged to Euro in such a manner - where it's pegged to within +/- 2.25% of a specific rate.
In essence, if you declare a plausible intent of keeping a rate near 1$ by accepting limitless orders to buy it at 0.98$ and sell it at 1.02$, and can keep that promise, then that's a decent peg.
That's just due to liquidity. The peg is maintained through arbitrage, so of course if you have low liquidity you're going to have fluctuations around the peg.
For example, if I ever see DAI trading at, say, $1.02 then I would quickly mint some new DAI and sell for USD. Likewise, if DAI is at $0.98 then I would buy it back, then burn what I've minted, collecting a profit.
Also note that with traditional USD pegs, they are not always on parity and hover a few cents from time to time. For example the HKD. https://en.m.wikipedia.org/wiki/Hong_Kong_dollar