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How do I price an Interest rate swap (IRS)?
A forward start interest rate swap is a contract that exchanges payments between two differently indexed legs, starting from a future time instant.
At each time T_i for the set of times T_{alpha+1},..,T_{beta} the fixed leg pays the amount N*tau_i*K where N=nominal value of the IRS K=predetermined fixed rate tau_i : The year fraction between T_{i-1} and T_i
While for the flating side, the leg pays N*tau_i*L(T_{i-1},T_i) where L(t,T): simply compounded spot rate at time t for the maturity T
Let S_{i,j}(t) : Forward swap rate at time t for a swap with first reset date T_i and payment dates T_{i+1},T_{i+2},....,T_j
There are 2 ways to price the IRS:
1.Using discount bond prices
Let P(t,T) : Bond price at time t for the maturity T
Then forward swap rate is: It can also be calculated using forward rates. Let F(t,T,S): simply compunded forward(libor) rate at time t for the expiry-maturity pair T,S set F_j(t)=F(t,T_{j-1},T_j)
2.Using forward rates Swaption formula for forward rates is:
