# Why is long a receiver swaption and short a payer swaption equivalent to receive-fixed, pay-floating interest rate swap?

Why on earth, for god’s sake does this equivalence hold?

Why do we need to be long a receiver swaption as well? This doesn’t fit in. Why being short a payer swaption is not enough to emulate the receive-fixed, pay-floating interest rate swap? After all, being short a payer swaption, we are obliged to pay floating interest rate and receive fixed interest rate? Being long a receiver swaption is not necessary as it is just a right (not obligation) to enter a receive-fixed, pay-floating interest rate swap.

I hope somebody can explain this to me or else I will stuarnate!

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Long options have only upside in only one direction.

Short options have only downside in only one direction.

Swaps have upside in one direction and downside in the other direction; you need two options to duplicate that.

This is no different than a forward contract being equivalent to a pair of options on the underlying.

Simplify the complicated side; don't complify the simplicated side.

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I still don’t get it. I’m talking about swaptions. What is the purpose of the long receiver swaption? How does it contribute to the equivalence?

sinep wrote:
I still don’t get it. I’m talking about swaptions.

So am I: swaptions are options; specifically, options to enter into a swap.

sinep wrote:
What is the purpose of the long receiver swaption? How does it contribute to the equivalence?

You’re long the option: you own it, so you can exercise it when it’s favorable.  It’s a receiver option, so it’s an option to enter into a swap to receive a specified fixed rate and pay the prevailing floating rate.  There are many purposes, just as there are many purposes for owning a put option on GOOG.  Two of them are:

• You think that interest rates are going to fall, so you speculate by buying a receiver swaption.  If they fall, you exercise the swaption and receive the fixed rate specified in the swaption while paying a lower floating rate and you make gobs of money.
• You have a short position (pay fixed, receive floating) in a plain vanilla interest rate swap, and you’re worried that interest rates might fall; you hedge by buying a receiver swaption.  If they fall, you exercise the swaption and receive the fixed rate specified in the swaption while paying a lower floating rate and you limit your loss.

sinep wrote:
How does it contribute to the equivalence?

Three things can happen in a receive fixed, pay floating swap:

• The value of the swap is positive (i.e., floating rates have decreased) and you get paid some money
• The value of the swap is zero (i.e., floating rates are (essentially) unchanged) and you get nothing and you pay nothing
• The value of the swap is negative (i.e., floating rates have increased) and you have to pay some money

Three things can happen when you’re long a receiver swaption and short a payer swaption:

• The value of the receiver swaption is positive and the value of the payer swaption is negative (i.e., floating rates have decreased): you exercise your swaption (and the owner of the payer swaption doesn’t exercise it) and you get paid some money
• The value of both swaptions is zero (i.e., floating rates are (essentially) unchanged): neither party exercises its swaption and you get nothing and you pay nothing
• The value of the receiver swaption is negative and the value of the payer swaption is positive (i.e., floating rates have increased): the owner of the payer swaption exercises it (and you don’t exercise yours) and you have to pay some money

Simplify the complicated side; don't complify the simplicated side.

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Side question:

In in swaption, does the short side receive a premium like a regular option and does the pay a premium? Somehow I miss that part in the text.

125mph wrote:
Side question:

In in swaption, does the short side receive a premium like a regular option and does the pay a premium? Somehow I miss that part in the text.

Yes.

It’s an option, and, as with any other option, the buyer pays a premium to the seller.

Simplify the complicated side; don't complify the simplicated side.

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Thank you, S2000magician, for your comprehensive explanation but I still am not able to fully understand that.

The proposition is: receive-fixed, pay-floating interest rate swap = a long receiver swaption + a short payer swaption

In the receive-fixed, pay-floating interest rate swap, do you have an obligation to receive the agreed fixed rate? Yes, you have.

And do you, at the same time, have an obligation to pay the prevailing floating rate? Yes, you have.

With a short payer swaption, do you have an obligation to receive the agreed fixed rate? Yes, you have.

And do you, at the same time, have an obligation to pay the prevailing floating rate? Yes, you have.

Are your obligations equal in both scenarios? Yes, they are.

Why do we need the long receiver swaption then?

sinep wrote:
Thank you, S2000magician, for your comprehensive explanation …

You’re quite welcome.

sinep wrote:
… but I still am not able to fully understand that.

I’ll happily keep going until you do.

sinep wrote:
The proposition is: receive-fixed, pay-floating interest rate swap = a long receiver swaption + a short payer swaption

In the receive-fixed, pay-floating interest rate swap, do you have a obligation to receive the agreed fixed rate? Yes, you have.

Correct.  Whether the net result is favorable or unfavorable to you.

sinep wrote:
And do you, at the same time, have an obligation to pay the prevailing floating rate? Yes, you have.

Correct.  Whether the net result is favorable or unfavorable to you.

sinep wrote:
With a short payer swaption, do you have an obligation to receive the agreed fixed rate? Yes, you have.

Not quite correct.

You have that obligation only when the holder of that option chooses to exercise it.

And that option holder will presumably exercise the option only when it is favorable to him, which is to say only when it is unfavorable to you: when interest rates have increased.

sinep wrote:
And do you, at the same time, have an obligation to pay the prevailing floating rate? Yes, you have.

Not quite correct.

You have that obligation only when the holder of that option chooses to exercise it.

And that option holder will presumably exercise the option only when it is favorable to him, which is to say only when it is unfavorable to you: when interest rates have increased.

sinep wrote:
Are your obligations equal in both scenarios? Yes, they are.

No, they’re not.

With the swap, you have potential upside (when interest rates decrease) and potential downside (when interest rates increase).

With the short payer swaption you have only potential downside (when interest rates increase).

sinep wrote:
Why do we need the long receiver swaption then?

To get the upside when interest rates decrease: you’ll exercise it when it’s favorable to you.

Simplify the complicated side; don't complify the simplicated side.

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Thank you Smagician, I think ich verstehe, Es ist ein schwieriges Konzept

Ja, es ist schwierig.
Bitte.

Simplify the complicated side; don't complify the simplicated side.

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