Explain the pricing of interest rate swaps, and work out their no-arbitrage value and say what it means

A swap's fixed leg is priced like a par bond: the swap rate is (1 − final discount factor) ÷ sum of discount factors for the payment dates. It is a blend of FRA rates, not an average of spot rates, and gives the swap zero value at initiation. Later, the fixed-rate payer's value is V = (current swap rate for the remaining term − original fixed rate) × sum of the current discount factors × notional. The payer gains when rates rise; the receiver's value is the negative. The trap is multiplying the rate difference by the number of payments instead of the sum of discount factors.