A swap is a derivative contract entailing the exchange of two different payment streams over the life of the contract. The two parties to a swap are the fixed rate payer (the floating rate receiver) and the floating rate payer (the fixed rate receiver).
On the other hand, a swaption is also an option where the underlying asset is specifically a swap (such as an interest rate swap). That is, swaptions are options on swaps, whereby the holder (buyer) has the right but not the obligation to enter into a swap agreement during the life of the option. In swaptions, the holder (buyer) will be specified either as the fixed rate payer or as the floating rate payer. Consequently, if the option is exercised, the seller of the option (the writer) becomes the counterparty to the swap, i.e., the party to the other leg (either as the floating rate payer or as the fixed rate payer).
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