Why a long a receiver swaption and short a payer swaption with the same exercise rate is equivalent to entering a receive-fixed, pay-floating swap?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A swaption is an option on a swap. A receiver swaption is the right but not the obligation to enter a receive pre-agreed swap rate at a given future date and pay the floating rate.

 

On the exam day, you can be given a question related to swaptions and their equivalencies regarding to swaps.

     

Let's imagine a bank making floating rate loans that wants to hedge the risk of future expected lower interest rates.

 

To do so, the bank could enter a receive fixed, pay floating interest rate swap. Why? Because then it would pay what is low (the relatively lower floating rate compared to the fixed rate and receive what is relatively higher than the floating rate, namely the fixed rate).

 

But the bank, instead, decides to go long a receiver swaption and short a payer swaption with the same exercise rate. 

 

Consider a swaption which expires in 3 months and gives the right but not the obligation to receive a pre-agreed swap fixed rate of 4%, the exercise rate, and pays floating at expiration of the option contract. 

 

We are going to explain why both positions, a long receive fixed,  pay floating swap, on one hand, and a long receiver swaption + a short payer swaption with the same exercise rate and expiration date, on the other hand, have the same payoff.

 

Scenario 1: at expiration, the floating rate is 5% . 

 

The long party to the receive fixed, pay floating rate swap will received 4% and pay 5% at payment date (we suppose that the expiration date of the swaptions and the payment date of the swap match).

 

The long party to the receiver swaption will not exercise its right to enter the swap as the fixed rate receiver and floating rate payer because in that case it would sell low (receiving a relatively lower fixed rate) and buy high (paying a relatively higher floating rate). The only cost incurred will be the premium paid at initiation*.

 

Before moving on to the short party to the payer swaption let’s look at its counterparty, the long payer swaption.

 

The long party to the payer swaption will exercice its right to enter the swap as the fixed rate payer and the floating rate receiver because in that case it would pay low (paying a relatively lower fixed rate) and sell high (selling at a relatively higher floating rate).

 

The short party to the payer swaption, the counterparty to the payer swaption, will then have to received fixed pay floating* (4-5=-1)

 

NB 1*: we consider that the premium paid and received cancel out.

NB 2*: the short payer swaption has the obligation to receive fixed and pay floating (the long party to an option has a right to buy/sell but the short counterparty to an option has an obligation to sell/buy).

 

 

Let’s make a review of both payoffs for scenario 1:

 

Payoff of the receiver fixed payer floating rate swap:

 

Receive 4%, pay floating 5%

 

Payoff= -1% 

 

 

Payoff of the long receiver swaption

+ short payer swaption:

 

0-1

 

Payoff= -1% 

 

Scenario 2: at expiration, the floating rate is 3%.

 

The long party to the receive fixed, pay floating rate swap will receive 4% and pay 3% at the payment date (we suppose that the expiration of the swaptions and the payment of the swap still match).

 

The long party to the receiver swaption will exercise its right to enter the swap as the fixed rate receiver and the floating rate payer because in that case it would sell high (receiving a relatively higher fixed rate) and buy low (paying a relatively lower floating rate). 

 

The long party to the payer swaption will not exercice its right to enter the swap to pay the fixed rate and receive the floating rate because in that case it would pay high (paying a relatively higher fixed rate) and sell low (selling at a relatively lower floating rate).

 

The short party to the payer swaption, the counterparty to the payer swaption, will then keep the premium. 

 

Let’s make a review of both payoffs for  scenario 2:

 

Payoff of the receiver fixed, payer floating rate swap:

 

Receive 4% pay floating 3%: 

 

Payoff= +1% 

 

Payoff of the long receiver swaption + short payer swaption:

 

Received fixed at 4%, pay floating at 3%+ 0 ** :

 

Payoff= +1% + 0%= +1%

 

 

** The 0 here corresponds to the compensation of the premiums associated with the purchase (-p) and sale (+p) of the swaption considered to be of the same amount. The buyer of an option pays the premium and the seller of an option receives the premium.

 

     

As a consequence, being both long a receiver swaption and short a payer swaption with the same exercise price and expiration date is effectively equivalent to being long a receive-fixed, pay floating swap.

 

Note that when rates fall below the exercice rate, the bank is fully hedged and turns its floating rate loan into a fixed rate loan.