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The firm quote is the price a firm or dealer commits to—up to a certain number of shares. Unlike nominal quotes, there isn't much wiggle room or room for negotiation with a firm quote.
Note that firm prices are only for 1 round lot of 100 shares. Larger transactions usually involve some haggling.
A dealer is asked for a price upon which she will transact to trade 100 shares of XYZ, makers of zipper examiners for the masses. The dealer will give a price at which it will buy ($5 per share), and a slightly higher price ($6 per share) at which it will sell. This is the "firm price," so if the other party says "I buy," then the dealer has to sell at the quoted price.