What's the correct definition of a call swaption?

financeguy

Active Member

yanyan

Member
I google a lot before I asked question here. BUT there are too many suspicious websites (our risk management professor always says so ). Just listing these links are not convincing. Confirmation from practitioner is more useful.
@Peter, remember Bob said in financial industry, we always use the same term refers to different things. What a terrible thing!
 

pruse

Active Member
I google a lot before I asked question here. BUT there are too many suspicious websites (our risk management professor always says so ). Just listing these links are not convincing. Confirmation from practitioner is more useful.
@Peter, remember Bob said in financial industry, we always use the same term refers to different things. What a terrible thing!
It can be that the same term refers to different things, but that's usually when they take place in different contexts. But I don't think that's the case here. By definition, in a call option you should be long whatever's variable, not fixed.
 

financeguy

Active Member
I google a lot before I asked question here. BUT there are too many suspicious websites (our risk management professor always says so ). Just listing these links are not convincing. Confirmation from practitioner is more useful.
@Peter, remember Bob said in financial industry, we always use the same term refers to different things. What a terrible thing!
it depends what the underlying is that you're talking about
if the underlying is an IRS then call = payer, if the underlying is a bond then put = payer... i assume we're talking about IRS here
 

myampol

Active Member
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.
 

pruse

Active Member
First let me say that we're not debating an intrinsic truth here, but rather which of two definitions is more natural. That said, I again have to disagree in favor of the one that to me makes more sense.

What a call option does in general is, it gives the holder the right to buy something with a variable value at a prescribed (strike) price at a prescribed time in the future. This means that on exercise the holder of the call pays the strike, which is a fixed amount, and in return receives something which until that time was variable. In our case this translates to: the holder of a call swaption, if he chooses to exercise, pays the fixed (prescribed) rate and in return receives floating.

This is the definition that makes more financial sense, is the definition Numerix uses, and is the definition you'll find in @financeguy 's links.
 

yanyan

Member
I should admit that Peter's explanation is logically beautiful if we agree that "What a call option does in general is, it gives the holder the right to buy something with a variable value at a prescribed (strike) price at a prescribed time in the future." It's an interesting aspect overlooked by me before. I usually look at a call option in this way, call option is an option that you can benefit from the rising of the underlying asset. That's why in this case we might have two versions of understanding: underlying could be IR or the bond, and they lead to opposite conclusions. But if we think call in your way, then there will be no ambiguity.
Thanks for pointing out this, although I still don't know whether practitioners all use your definition.:)
 

AlexandreH

Active Member
so Amex got it wrong?? I guess. Someone needs to fix that on the Amex site.
NOW, to end this discussion, for good, (admitting I was wrong because of Amex, but right initially :) ), I found the official Theoretical/Practical definition:
Paul Wilmott book ( Paul Wilmott Introduces Quantitative Finance) P392:
In a call swaption or payer swaption the buyer has the right to become the fixed rate payer; in a put swaption or receiver swaption the buyer has the right to become the payer of the floating leg.


That's it. Problem Solved!!! (If someone wants to argue about this with PW, please, be my guest ) :)
 

bob

Faculty (Undercover)
I actually had a slightly funny miscommunication with a guy I work with over this. I think of a payer as a call on the swap rate, since you would exercise a payer when the rate at expiry is higher than strike, a receiver when it's lower, and it seems very straightforward to me. He, however, thinks of it in cash terms, as if a payer were a put option on the fixed leg of the swap. Similarly, I think of a caplet/floorlet as a call/put on a forward rate, whereas he thinks of it as a put/call on a zero-coupon bond.

This isn't a matter of "right" or "wrong"--all of the interpretations above are valid. This is actually one case where finance gets it right and calls the rate options by different names to distinguish them from their more straightforward counterparts, because evidently call/put isn't really descriptive enough. The cash interpretation (as opposed to the rate interpretation) is evidently quite widespread, since instruments like Eurodollar futures are set up in a way that appeals to that way of thinking but seems completely backwards to me.
 

pruse

Active Member
I'm aware we're arguing between two plausible definitions here. But shouldn't there be a standard definition? Every time someone sells a call swaption, shouldn't everyone know what that means without having to ask every time?
 

bob

Faculty (Undercover)
I'm aware we're arguing between two plausible definitions here. But shouldn't there be a standard definition? Every time someone sells a call swaption, shouldn't everyone know what that means without having to ask every time?
What I'm saying is that payer/receiver (and cap/floor) are the standard conventions. That way it doesn't matter which way the parties involved happen to think of it in put / call terms.

Edit to add:
Since you made me curious, I went and checked some actual swaption trade confirms to see how they handle the issue. The answer, of course, is that they avoid it entirely: Confirms specify the option buyer / seller, and then describe what each of the parties pay in the underlying swap. But in discussions with clients who trade these things, I have always heard (and used) payer/receiver and never put/call.
 

financeguy

Active Member
What I'm saying is that payer/receiver (and cap/floor) are the standard conventions. That way it doesn't matter which way the parties involved happen to think of it in put / call terms.

Edit to add:
Since you made me curious, I went and checked some actual swaption trade confirms to see how they handle the issue. The answer, of course, is that they avoid it entirely: Confirms specify the option buyer / seller, and then describe what each of the parties pay in the underlying swap. But in discussions with clients who trade these things, I have always heard (and used) payer/receiver and never put/call.
Yeah in the broker market the price request will always be for a payer or receiver, never call or put. When a junior shows up on a rate options desk though and gets trained up, you him draw a hockey stick diagram with the rate on the x-axis, and the payer is clearly the call. The cash argument doesn't really make sense just because even though payer/receiver always refers to the fixed leg, you are also doing something with the floating leg as well - so the cash goes both ways in the end! The fixed leg terminology is just something grandfathered in from how people would speak about trading the underlying interest rate swap. If you pay rates or receive rates, you always are talking about what you do with the fixed leg. No rhyme or reason to that, just an easy default way to speak about with things. When they came up with that they weren't thinking about rate options, and so when it came to naming rate options, it just made sense to talk about them in the same way as you talk about the underlying. That doesn't mean we should then think about call and put backwards, it's just terminology. When you exercise a payer, yes you pay the fixed rate but you also receive the floating - so it's not just a one way flow, and so you can't define put or call that way. What is important is that when you buy a payer, you are long the underlying rate and so are long delta, and your hockey stick diagram with the underlying rate on the x-axis is upward sloping from the strike. So it only makes sense that a payer is a call option on the rate and a receiver is a put option on the rate.
 

financeguy

Active Member
so Amex got it wrong?? I guess. Someone needs to fix that on the Amex site.
NOW, to end this discussion, for good, (admitting I was wrong because of Amex, but right initially :) ), I found the official Theoretical/Practical definition:
Paul Wilmott book ( Paul Wilmott Introduces Quantitative Finance) P392:
In a call swaption or payer swaption the buyer has the right to become the fixed rate payer; in a put swaption or receiver swaption the buyer has the right to become the payer of the floating leg.

That's it. Problem Solved!!! (If someone wants to argue about this with PW, please, be my guest ) :)
Also just to add, why should exchanges have any clue what they're talking about? 99.9% of the flow in these derivative markets is OTC; Amex shouldn't be taken as the be all and end all - they're amateurs!
 

yanyan

Member
That's good news :)
What I'm saying is that payer/receiver (and cap/floor) are the standard conventions. That way it doesn't matter which way the parties involved happen to think of it in put / call terms.

Edit to add:
Since you made me curious, I went and checked some actual swaption trade confirms to see how they handle the issue. The answer, of course, is that they avoid it entirely: Confirms specify the option buyer / seller, and then describe what each of the parties pay in the underlying swap. But in discussions with clients who trade these things, I have always heard (and used) payer/receiver and never put/call.
 

vertigo

Member
no such thing as a call swaption. there are only payer swaptions and receiver swaptions. a payer swaption is an option giving the right to enter a payer interest rate swap:

the payer swaption has payoff

nominal * discount(0,t_0) max(0, sum_{i=0...} (t_i-t_{i-1}) * P(t_0,t_i) * [L(t_0;t_{i-1},t_i)-k] )

the receiver swaption has payoff

nominal * discount(0,t_0) max(0, sum_{i=0...} (t_i-t_{i-1}) * P(t_0,t_i) * [k-L(t_0;t_{i-1},t_i)] )

payer swaption is analogous to a call option on the forward LIBOR rate.
receiver swaption is analogous to a put option on the forward LIBOR rate.
 

vertigo

Member
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.
this is garbage, you have no idea what you are talking about. i hope people are not taking this charlatan seriously as he is giving out false information on purpose.

a payer swaption is a call option on forward LIBOR, a receiver swaption is a put option on forward LIBOR. see my previous post.

can you please write down the pay-off of a payer swaption and/or a receiver swaption?

i stopped reading at the second line "fixed (i.e., longer-term)"... no, fixed rate is fixed, it is the forward swap rate, it is not the longer term rate. i dont think you know what a swap is let alone what a swaption is.
 
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