The quoted margin is a contract term. The required margin is what the market currently demands for that issuer's credit and liquidity at that tenor, and it moves every day.
Everything about a floater's price follows from the gap between those two numbers.
When the required margin equals the quoted margin, the note prices at par on a reset date. When the market requires more than the note pays, the note prices below par, and the discount is what makes up the shortfall. When the market requires less, it prices above par.
How is the discount margin found?
The discount margin is the required margin backed out of an observed price. It is the spread that, added to the assumed future reference rate, discounts the note's expected cash flows to the market price.
Assuming the reference stays at its current level is a simplification everyone makes and nobody should forget. It is a convention for making the number computable, and a steep forward curve makes it a poor one.
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Take a three-year quarterly floater with twelve periods remaining, a quoted margin of 80 basis points, a current reference rate of 4.8000% and a market price of 99.35. Discounting the assumed cash flows at the reference plus a solved spread gives a discount margin of 103.7766 basis points. The gap of 23.7766 basis points over the quoted margin is why the price sits 0.65 points below par. Feed 103.7766 basis points back through the pricer and it returns 99.3500000000, which is the verification that matters.
Set the discount margin equal to the quoted margin on the same note and the price is 100.0000000000 exactly. Set it to 40 basis points and the price is 101.1044637403; set it to 120 basis points and the price is 98.9092494793. Near par the note moves about 0.0274 per 100 for each basis point of margin, which is the sense in which a floater has almost no duration to the reference rate and a real duration to the spread.
Does a floating rate note always trade at par?
Now the qualifier, because the identity is routinely stated more strongly than it deserves. A floating-rate note prices at par at a reset date, and only when the required margin equals the quoted margin. Neither condition is automatic. Between reset dates the coupon for the current period is already fixed while the discount rate for the remaining stub is not, so the note behaves for those few weeks like a very short fixed-rate instrument, and its price moves.
The arithmetic is small and worth doing once. On the note above, the period-one coupon is fixed at 5.6500% for 91 days, which is 1.4281944444 per 100. Thirty days into the period the reference for the remaining 61-day stub has risen 60 basis points to 5.4500%, and the required margin is still 80 basis points, so the note will reprice to par at the next reset. Discounting the 100 plus the fixed coupon over the stub at 6.2500% gives a full price of 100.3652980588; accrued interest on an actual/360 basis for 30 days is 0.4708333333; the flat price is 99.8944647254, which is 0.1055352746 below par, or $10,553.53 on $10,000,000 of face. Had the index not moved, the flat price would have been 99.9955351693.
The gap is small because the stub is short, and it grows with the length of the reset period. The honest form of the statement runs like this. A floater is pinned to par at each reset, sags or rises between resets in proportion to the remaining stub, and parts from par permanently when the required margin parts company with the quoted one.
A note whose issuer has been downgraded three notches will not print at par on any reset date. The quoted margin is a term of a contract signed when the credit was better.