Why are payer swaptions considered puts?

On page 273 of Derivatives, the first paragraph under section 6.1, the curriculum says that payer swaptons are puts and receiver swaptions are calls.

Can anyone explain this? It seems to me that it should be the opposite. Since a payer swaption pays fixed and receives floating, wouldn’t the holder be benefitting if rates rise, just like in a call option? The payout formula for a payer swaption = max[0, FS(o,n,m) - x], with the exercise rate being subtracted out, looks similar to the call option payout formula. I’m bamboozled.

Any help would be appreciated! Thanks.

It’s like a put on the bond is what they might be referring to where the holder of the option benefits from a rate rise. On a lighter note, it looks like a call’s formula but its a swap (so swap the call for a put).

Payer swaption gives you the right but not the obligation to enter into a swap and pay fixed payments. This is analogous to have the option to issue a bond (the bond issuer pays fixed payments). As a holder of a payer swaption, you expect interest rates to rise therefore you like being hedged with this lower-than-market fixed payments.

In a receiver swaption , is like you were buying a bond in the future (when swaption expires) because you receive fixed payments. Therefore it is like a call in a bond. As a holder of a receiver swaption, you expect to interest rates to fall therefore you like to being hedged with this higher-than-market income payments.

I also struggled to comprehend this, the book has its fault, it does not clarify a swaption is a put or call on “what thing” : interest rate or bond. We already know is about the bond value.

Hope this helps.