## American and Bermudan Swaptions

 Recall that interest rate swaps can be viewed as an agreement to exchange a fixed-rate bond () for a floating-rate bond (). At the start of the swap the floating bond value must equal the swap principal, which for the purposes of this discussion is normalized to equal \$1. A swaption is therefore an option to exchange a fixed-rate bond for an amount equal to the swap principal (\$1). Consider a payer swaption. If exercised, the holder will have a position with value of reflecting the fact that the holder is long the floating bond and short the fixed bond. The holder will logically only choose to exercise into this position if the payoff given by{71 \* MERGEFOAT (2.8)} the equation above is positive. Because at the option maturity date, we can restate the payoff to the payer swaption as: the payoff to a payer swaption with expiration the time- value of the -maturity fixed-rate bond. Note that the expression in the above equation is just the standard payoff function for a put option. Using similar intuition, the payoff for receiver swaptions can be categorized as that for a standard call option: the payoff to a call option with expiration