405-FISwaps.Rmd

# Libor Swaps

```{r, init-LiborSwaps,echo=TRUE,results='hide'}
  library(fBasics)
  library(empfin)
```

An interest rate swap is a contract to exchange the cash flows of a fixed-coupon bond for the cash flows of a floating rate note of identical maturity. The present value of this contract is therefore the difference between the PV of the fixed leg and the PV of the floating leg.

The pricing formulae use the following notations:

$PV_x$

:   PV of the fixed leg

$PV_l$

:   PV of the floating leg

$r_i$

:   zero-coupon rate for maturity $t_i$

$Q$

:   nominal amount

$c$

:   periodic fixed coupon

$k$

:   next floating coupon

$t_i$

:   payment dates, $i=1, \ldots, n$

## IR Swap Pricing

The pricing formulae use the following notations:

$PV_x$

:   PV of the fixed leg

$PV_l$

:   PV of the floating leg

$r_i$

:   zero-coupon rate for maturity $t_i$

$Q$

:   nominal amount

$c$

:   periodic fixed coupon

$k$

:   next floating coupon

$t_i$

:   payment dates, $i=1, \ldots, n$

The PV of the fixed leg is:

$$PV_x = \sum_{i=1}^n c e^{-r_it_i} + Q e^{-r_nt_n}$$

Since the PV of a floating rate note is par at each reset date, the PV of the floating leg is:

$$PV_l = k e^{-r_1t^*_1} + Q e^{-r_1t^*_1}$$

At inception, both legs must have equal value.