Nope. It should be the opposite. Think about it... when you're long a call, it makes sense to have a long position in the floating rate. In other words, you would exercise if the floating rate is above the fixed rate, because you would be receiving the floating rate and paying the fixed rate.
Sorry, but this isn't correct either.
By your argument, you would exercise a swaption to pay fixed / receive floating only when the floating (i.e., short-term) rate exceeds the fixed (i.e., longer-term) rate.
Perhaps what you wrote differs from what you meant, but it is not correct. At the moment of expiry of the swaption, the immediate value of the short-term rate is
not directly relevant to the exercise decision.
Right now the "floating rate" -- i.e., LIBOR -- is very low, while the forward curve is upward-sloping.
The fixed rate on a par swap of any maturity is well above the current 3-month libor rate.
The current rate on a forward-starting swap (at the expiry of the swaption) is going to be even higher.
However, this "upward sloping yield curve" does not always need to be the case. There are periods when the yield curve is inverted, and there are periods when it is nearly "flat."
The decision to exercise or not exercise a swaption is based upon the difference between the market rate for a swap
at the time the swaption expires, vs. the
strike of the swaption. Although it is theoretically possible to trade a swaption at any strike, typically the strike of most swaptions that are actually traded will reflect the ATMF (at-the-money-forward) rate which prevailed at the time when that swaption was originally traded.
Note that the quoted rate for a swap reflects the fixed-rate leg; for a vanilla swap, the floating rate side is usually 3-month LIBOR.
Suppose that today the market rate for the fixed leg of a five-year swap is
5 percent, and suppose that today the market rate for a
1-year-into-five-year "forward starting" swap is
6 percent.
Most swaptions are executed "at-the-money-forward", so if you were to seek either a payer or a receiver swaption now, for a five-year swap which would commence in one year's time and terminate 6 years from now, the easiest strike to obtain would be 6 percent, as that is ATMF. Also, by put-call parity, the price (whether in "basis points", or in dollars, or however you care to express it) for the
payer and for the
receiver would be equal.
Jump to one year from now.
If the then-market rate for a five-year swap happens to be 6 percent, then both the payer and the receiver swaption expire at-the-money, and hence worthless.
If, however, rates have risen further, so the market rate for a five-year swap has increased to
7 percent, then the "right to
pay" only 6 percent is "in the money", while the "right to
receive" 6 percent is worthless. Therefore, if you were long the 6%
payer swaption you would exercise it, while if you were long the 6%
receiver swaption you would let it expire worthless.
By exercising the 6%
payer swaption when the market rate for a five-year swap is 7%, you are establishing a swap where you are
paying 6% fixed and receiving floating. The value of this transaction is positive to you, and negative to your counterparty, who would rather receive 7% fixed instead of only 6% fixed. Thus, the PV of the fixed leg
is less than the PV of the floating leg, and you are "
putting" the (undesirable) low-coupon fixed leg to the counterparty who was unfortunate enough to have sold you the
payer swaption.
This is why a payer swaption is a put.
Note that the value of 3-month LIBOR at that moment does NOT enter into the calculation.
Three-month could be one percent, but there would have to be a very steep increase in the forward libor rates over the next 5 years in order for the market swap rate to be 7 percent.
Or, three-month LIBOR -- and all projected future LIBOR rates -- could be flat, at 7 percent -- which would give a flat curve for all swap rates: everything would be 7 percent (well, up to the adjustments for the different day-count and compounding conventions between the fixed leg and the floating leg, but let's assume that away for now.)
Or, the three-month LIBOR rate could be 10 percent -- but there would have to be a significant decline in the forward LIBOR rates in order for the then 5-year swap rate to be equal to seven percent.
In any of the above cases, the holder of a 6%
payer swaption would exercise it and receive immediate value -- regardless of the value of the short-term (floating) rate -- because the market value of a 6% swap in a 7% environment would make it desirable to "
put" this swap to the counterparty who had sold you the option.
On the other hand, suppose that swap rates have not risen. Indeed, suppose that they have remained unchanged -- so the 5-year swap rate, one year from now, happens to be 5 percent (the same value as it is today.) In this case the 6%
payer swaption is worthless -- why would anyone enter into a swap
paying 6% fixed when a market-rate swap could be had where one would only pay 5%? However, the 6%
receiver swaption is "in-the-money": you would exercise this option to enter into a 5-year swap where you are
receiving 6% fixed, which you couldn't get in the market, as the current rate for a 5-year swap is still just 5 percent.
By exercising the 6%
receiver swaption when the market rate for a five-year swap is 5%, you are establishing a swap where you are
receiving 6% fixed and paying floating. The value of this transaction is positive to you, and negative to your counterparty, who would rather pay the market rate of 5% fixed instead of the strike rate of 6% fixed. Thus, the PV of the fixed leg
is more than the PV of the floating leg, and you are "
calling" the (desirable) high-coupon fixed leg away from the counterparty who was unfortunate enough to have sold you the
receiver swaption.
This is why a receiver swaption is a call.
Again I could repeat the above interest rate curve argument (replacing each "7" with a "5") to explain why the value of the floating rate upon expiry of the swaption does not directly enter into the exercise decision -- the value of the floating rate could be higher than, equal to , or less than the strike rate, but
the exercise decision is solely based upon the difference between the market rate for a new par swap vs. the strike rate which was established at the time when the swaption was written.
I hope this clarifies things, and we can (hopefully) put this topic to rest.