Valuing a vanilla interest rate swap

Outline how to price a plain fixed-for-floating interest rate swap.

Value the fixed leg and the floating leg off the discount curve and take the difference. The fixed leg is a sum of coupons times discount factors. In single-curve pricing the floating leg plus notional is worth par at each reset, so it equals $N(1-P(0,T_n))$; more generally each floating payment is valued at its forward rate. The par swap rate makes the legs equal: $s=\tfrac{1-P(0,T_n)}{\sum_i\tau_iP(0,T_i)}$.